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Question 565853
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You are kidding, right? Which one of several hundred math books in use today, each of which costing in excess of $75, shall I pull off of my shelf of all possible math books so that I can see the diagram? This is Algebra.com. It is NOT the Psychic Hot Line |
Here is a new site that has some nice dynamic calculus tutorials. and here is a link to another good tutorial site called Visual Calculus TI Calculator Guide.. Here is a great link that lets you look up functions on your calculator in an alphabetical list, and then shows you how to do it... simply Great.
HEY, WAY COOL, FREE SOFTWARE
This link will download the WINPLOT program. The file is a self extracting compressed file, just double click to expand. It will create a new file called winplots that is the execute file you want to run when you run the program. And if that didn't convince you, you can also download a Discrete Math software program that is also a great tool.
Another graphing software program which is FREE is called GraphCalc.
You will find it at
Here is another great interactive Algebra and Geometry software, and it is also free. They call it GEOGEBRA.
Lots of Middle Grade teachers have asked about interesting math games for their students which are both educational and entertaining. This link will download a set of Arcade games including John Conway's Game of Life, the 15 puzzle, ghost mazes, and several others. Here are the links to documents I have written about assorted topics.
And for the stats TEACHER a FREE demo of FATHOM my very favorite software for statistics and probability simulations. Students can order a student version for less than $40.
Here is an index of DISCOVERY UNITS using the GEOMETER'S SKETCHPAD that I have written. Some are about GEOMETRY, and some are about ALGEBRA. I hope to have more added soon, so keep checking back. If you do not have Geometer's Sketchpad you can get a FREE DEMO |
Essential Mathematics 9 (ICSE Board264 Our Price:224 You Save: 40
(15%)
This item is presently Out of Stock.
Essential Mathematics 9 (ICSE Board) Book Description
About the Book :
Essential Mathematics is a series of two books for Classes 9 and 10. This series is based on the latest syllabus prescribed by the Council for the Indian School Certificate Examinations, New Delhi. Salient features of the books:Each chapter has a large number of solved problems to illustrate the concepts and methods. Stress has been laid on concept building. The text is lucid and to the point. In the exercises, problems are graded from simple to complex A list of important definitions, formulae and results are provided at the end of each chapter in the form of Points to Remember. Test your knowledge at the end of each chapter tests the childs learning.
Popular Searches
The book Essential Mathematics 9 (ICSE Board) by Paj Lewis
(author) is published or distributed by Ratna Sagar P. Ltd. [8183323677, 9788183323673].
This particular edition was published on or around 2007-1-1 date.
Essential Mathematics 9 (ICSE Board |
Cliffs Quick Review For Geometry - 01 edition
Summary: When it comes to pinpointing the stuff you really need to know, nobody does it better than CliffsNotes. This fast, effective tutorial helps you master core geometry concepts -- from perimeter, area, and similarity to parallel lines, geometric solids, and coordinate geometry -- and get the best possible grade.
At CliffsNotes, we're dedicated to helping you do your best, no matter how challenging the subject. Our authors are veteran teachers and talented wri...show moreters who know how to cut to the chase -- and zero in on the essential information you need to succeed. ...show less
Ed Kohn, MS is an outstanding educator and author with over 33 years experience teaching mathematics. Currently, he is the testing coordinator and math department chairman at Sherman Oaks Center for Enriched StudiesEXTENSIVE MARKINGS. Intact & readable. PLEASE NOTE~ we rated this book USED~ACCEPTABLE due to likely defects such as highlighting, writing/markings, folds, creases, ETC. We ship from Dallas within 1 d...show moreay & we LOVE our customers |
Is All About Math - Julio Cesar de la Yncera
Main site for a mathematical Video PodCast that will cover a big range of topics in college-level mathematics accessible to the majority of the students with some mathematical inclinations. The Video PodCast will also try to be inspirational for the non-mathematicallyKarismath - CLSO Learning Systems, Inc.
Math topics, including Algebra, are presented visually, through lessons (Flash animations that explain a target concept), demos (Flash animations that explain every exercise template to be used) and worksheets (PDF format) that can be downloaded. Each
...more>>
Khan Academy - Salman Khan
Salman Khan has recorded over 1400 YouTube videos on a variety of topics, largely math and finance. The Khan Academy, which Khan founded, is a not-for-profit organization with the mission of providing a high quality education to anyone, anywhere. To keep
...more>>
Learning Math is Fun - Kevin Cornell
Learn the order of operations by watching a video of the PEMDAS song. Before becoming an elementary school principal, singer "Mister C" taught 5th grade science and served as a math coach. See also Cornell's science site at
...more>>
Learning Wave Communications - Human Relations Media
Learning Wave is a producer of videos, CD-ROMs and other supplementary materials for math education including interactive games such as Absurd Math. Under the name HRM Video it has created programs such as middle school math mysteries, applications-basedMAA Bookstore - Mathematical Association of America
A searchable list of books, with descriptions, in the following categories: Algebra; Analysis; Applied Mathematics; Calculus; Career Information; Computing and Computers; Elementary Models; Games, Puzzles, and Popular Exposition; Geometry and Topology;
...more>>
Maisonet Math - Shane Maisonet
This website provides math worksheets in pdf form for public download. Online quizzes also provide instant feedback to indicate level of mastery. Online videos of various math concepts are also available.
...more>>
Math and Soccer Video - Sharp Tack Productions
A video revealing the role played by numbers, graphs, geometry, algebra, angles, area, surveys, and chance data in the game of soccer; for example, how a goalie relies on the properties of angles to know where to stand when defending the goals, and on
...more>>
Math Dork Interactive Math
Animated algebra tutorials, on-line quizzes, and games. Animations require Flash 3. Full access to this site involves a monthly fee; free lessons include graphing points in the coordinate plane and "The Golden Rule of Algebra."heatre - Sadie Bowman and Marc Gutman
Blog of the performers starring in "Calculus: The Musical!" This comic "review" blend of sketch comedy, musical theatre, and lecture about the concepts and history of calculus emerged as a teaching tool from the classroom of Gutman, who "... found that
...more>>
Mathematics - Annenberg Media
The video, software, and print guides in the math collection show concrete examples of good teaching and active learning in all sorts of settings: public school classes, multi-age classes in rural areas, bilingual classes, magnet and charter schools,Math Learning Site
This site offers a library of over 2000 lessons in video format, as well as pretests, exams, and a performance report accessible by parents and/or teachers. Lessons are accessible by subscription, with a money-back guarantee and an online demo.
...more>>
Math Magic by the Human Calculator Scott Flansburg
Scott Flansburg, "The Human Calculator," created Math Magic to be a fun math activity that makes learning basic math skills easy. For kids, Math Magic offers an entertaining video with an easy-to-use workbook that features math worksheets which cover
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The Mathman - Don Cohen
Materials for sale for K-12 students, teachers and parents; pre-calculus. Materials include Get Ready for Calculus (Calculus By and For Young People book, CD-ROM, worksheet book, videotapes and map) and Changing Shapes with Matrices. Patterns, visualization,
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MathMol - NYU/ACF Scientific Visualization Laboratory
A starting point for those interested in molecular modeling, one of the fastest growing fields in science, from building and visualizing molecules to performing complex calculations on molecular systems. Using molecular modeling scientists will be better
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Mathnasium Math Learning Centers - Mathnasium LLC
After school learning centers where students, grades pre-K to 12, "go to boost their math skills." Browse a directory of Mathnasium tutoring centers in the US and abroad. Read about the Mathnasium Method, created by former Los Angeles teacher Larry Martinek.
...more>> |
This pre-algebra course provides basic principles, concepts and techniques that are necessary for students' success in higher level mathematics courses. Content material includes using the four operations with integers and rational numbers, algebra basics such as the order of operations, using variables and combining like terms. 1-step and 2-step equations, and inequalities, will be explored as well as multi-step equations. Designated for students with little or no algebra background. |
This book explains, as clearly as possible, tensors and such related topics as tensor products of vector spaces, tensor algebras, and exterior algebras. You will appreciate Yokonuma's lucid and methodical treatment of the subject. This book is useful in undergraduate and graduate courses in multilinear algebra.
Tensor Spaces and Exterior Algebra begins with basic notions associated with tensors. To facilitate understanding of the definitions, Yokonuma often presents two or more different ways of describing one object. Next, the properties and applications of tensors are developed, including the classical definition of tensors and the description of relative tensors. Also discussed are the algebraic foundations of tensor calculus and applications of exterior algebra to determinants and to geometry. This book closes with an examination of algebraic systems with bilinear multiplication. In particular, Yokonuma discusses the theory of replicas of Chevalley and several properties of Lie algebras deduced from them.
Readership
Graduate students as well as experts in theoretical and mathematical physics, differential and integral equations and mathematical analysis.
Reviews
"This book provides a well-organized introduction to tensors and related topics and could be useful for students of different levels (including Ph.D. level). The author includes a number of exercises at the end of each chapter." |
GCSE Maths Foundation Tier (L00042-121304)
It may be possible to join some courses after the start date.
Please email
[email protected]
to enquire.
Weekdays/Time:
Fri (09:30 to 12:30)
Number of Weeks:
17 course follows the AQA Foundation Tier syllabus 4360. It is a modular syllabus suitable for the post-16 student. The tier of Assessment is C - G. There is no coursework. This is a busy and comprehensive course with a final written examination.
Entry Requirements:
In order to enrol onto this course you need to have established maths skills at Level 1 as a minimum. If you have recently (within the last three academic years) achieved a Grade D in GCSE Maths then this will be accepted as an appropriate standard, otherwise we will give you an online initial assessment when you come for an interview.
You will also need to have good English speaking, listening, reading and writing skills (ideally at Level 2).
Course Content:
This GCSE Maths Syllabus is comprised of three separate units as follows:
Unit 1 – Statistics and Number assesses skills in statistics and probability. Some appropriate aspects of Number Theory are also incorporated. Where appropriate these skills are assessed within contexts relevant to everyday life.
This Module accounts for 26.7% of your overall final mark and Grade.
Unit 2 – Number and Algebra concerns all of the number topics such as the 4 operations, Ratio, Fractions, Percentages and Standard Form of a number amongst others. This unit concentrates on aspects of Number which can be tested without a calculator. It also focuses on aspects of Algebra, such as manipulating and solving equations, wherein a calculator is not required. This Module accounts for 33.3% of your overall final mark and Grade.
Unit 3 – Geometry and Algebra assesses the more traditional areas of Mathematics such as Algebra, Shape and Measure. This includes Area and Volume, the Pythagoras Theorem, Trigonometry, Simultaneous Equations, Lines and many other topics which will undoubtedly be familiar to you. The focus is on all of the geometry and measures content as well as areas of Algebra where a calculator is required and graphical methods in algebra. This is the larger Module and accounts for 40% of your final mark and Grade. Your tier of entry on this Module determines your final tier of entry. Please note your tutor will determine your level of entry for this Module.
Assessment Methods:
For Unit 1, the assessment is managed through 1 x 1-hour written paper; you are expected to use a calculator for this paper. You will sit this exam in Spring 2013 on one weekday. The date and time will be confirmed for you on course. Unit 2 is assessed through one written paper which is for 1 hour and 15 minutes. This is a non-calculator paper and will be scheduled during June 2013. Unit 3 is assessed through a 1 hour and 30 minute paper. Each unit exam is scheduled on a separate day in June 2013. You will have the option of re-sitting Unit 1 in June 2013. Your tutor will discuss this option with you as appropriate.
Additional Information:
You should be aware that although this is a Foundation Tier course, we only deliver material up to Grade C. Because of the limited time available to deliver a wide ranging and complex syllabus, we will spend time in the class concentrating on the material which will enable you to achieve a High Grade up to C. Please be aware that there is a significant level of commitment required of you in order to successfully complete this course.
If you would like more information on course fees and
how to get additional help to pay for them, please click here
Bring To First Class:
You will need the following equipment:
• Writing pad and selection of pens
• Mathematical tool kit
• New GCSE Maths AQA Modular Student Book Foundation 1 by Brian Speed et al, published by Collins ISBN 9780007340126
• New GCSE Maths AQA Modular Student Book Foundation 2 by Brian Speed et al, published by Collins ISBN 9780007340164.
• Scientific calculator – preferably one of the Casio FX series which will cost in the region of c£9.00 – though if you already have a Scientific calculator, it should suffice for the course. Please discuss this with your tutor.
You need to bring these resources with you to each session unless directed by the tutor.
Attendance:
Regular attendance is important. We also ask that you are punctual for each session. You should also be aware that the examinations linked to this course are a compulsory aspect of the course. When you sign the learning agreement with the college you are agreeing to attend each week, be punctual and sit the examinations. Please see FAQs.
Further Study and Career Opportunities:
GCSE Maths at Grade C is an entry requirement for a significant number of HE courses and in particular Teacher Training courses. It is also a qualification required and sought by a significant number of employers.
How to Enrol:
Please ensure you have fully read this information sheet first.
This course needs tutor approval before you can enrol.
You will need to complete an Application Form and email it to [email protected]
The Information, Advice and Guidance team (IAG team) will contact you to arrange an interview.
If you are aged between 16-18 please complete the Young Person's Application Form and return it to ([email protected])
Frequently Asked Questions:
"How much homework will I need to do outside of class?"
You should allow at least 6 hours per week outside of class time to concentrate on homework. This will include research and preparing assignments. This course is continuous, and each week the tutor will progress through the material which comprises the syllabus. Because of the demands of the course, there is limited time for the tutor to spend on each topic. This is why it is important for you as a learner to keep up with the homework and to work as a pro-active learner in the classroom; contributing in a positive and constructive manner and asking questions when you are unsure. Your tutor will collect, mark and return your homework to you with comments to help and guide you with your study.
"What will happen if I miss a class?
You should endeavour to attend all sessions. However, if you are unable to attend the tutor will make the materials available through MOODLE. We will also encourage you at the start of the course to work with another learner and ask him/her to be a study buddy; someone in particular whose notes you can copy on your return and who will collect materials from the tutor for you when you are absent.
"When will the exams take place?"
The examinations are scheduled to take place in June 2013. The Modules are examined through non-calculator based and calculator based written papers. These examinations are run by the college on prearranged days and times. These dates and times are set by a range of external bodies and as such are non-negotiable. The examinations are normally scheduled on a week day and as such you may need to make arrangements with work or family for these days. Your tutor will inform you of these dates as early as possible on the course to enable you to make any necessary arrangements. The examinations will take place in the college. It is your responsibility to ensure that you arrive on time with the necessary Mathematical tools you need as well as current and valid identification.
"Is there any tutorial support on the course?"
No. However, the college does offer a Revision course which is scheduled in Term 3. There are also Mock Examination Days scheduled in Term 3. These additional courses will further support you in your preparation for the final examinations in June 2013.
If you have declared a specific learning difficulty/ disability, you will be offered an interview with our additional learning support co-ordinator, who may arrange 1:1 tutorials with a learning support tutor, or other support.
Outside of your class time, contact time with the course tutor is minimal. Material will be uploaded onto our online learning environment, MOODLE, and you should use this platform to communicate with the tutor and other learners during the course. This will be discussed with you at the start of your course. Access to a computer outside of class time would be an advantage.
"But I am no good at Maths!?"
If you meet the entry criteria, then that is a good start! However, you need to have the interest and drive (motivation) to succeed. A significant number of people can succeed, and indeed have succeeded in previous years, on this course even though they thought the same. This is largely because they needed the qualification (a High Grade C or better in GCSE Maths) to gain entry onto another qualification. In other words they had a reason to work and succeed on the course. If you have a particular goal in mind, then although you may not like or be confident with Mathematics, you will succeed if you keep this goal in mind and work through the year as directed by your tutor.
"What if I want to do the course but not take the exams?"
Please do not enrol. This course is available to you free of charge and in this format because of the support from the SFA who are the funding body for the college. They expect us to ensure that all candidates who enrol onto the course do so knowing that the summative examinations are integral to the course. In other words, they expect you to take the examinations as part of the course. Further to this, if you enrol but do not take the examinations you are regarded in the eyes of the SFA and therefore the college as a failure and this can have an adverse effect on our results data (statistics) for the year which in turn affects our funding and the provision we can plan to offer. So it is important that you realise, before you enrol, that there is a significant level of commitment expected of you on this course. If you are interested in developing your Mathematics skills and knowledge without having to face the pressure of these GCSE examinations, you may wish to consider a SfL Entry Level Numeracy course. Please read the course descriptors for these courses on the college website.
"What if I have achieved a Grade C before and I now need a Grade B or higher?"
This course is not suitable for you. You should consider a GCSE Maths Higher Tier course. |
This course is deigned for students who need additional time to understand and develop mathematical concepts. In addition to their regular Math class, students will use an online computer program called ALEKS to support and strengthen their math skills. Topics covered include number sense, solving equations, linear functions, quadratics, and exponential functions. This is a course designed to help students pass their regular math course. In addition to working with ALEKS, students will prepare and study for the Oregon State Assessment Test in Mathematics.
State Standards:
H.1A Numeracy
H.2A Linear Functions and Systems
H.3A Quadratic and Exponential Functions
H.1S Analyze and interpret empirical data
H.2S Probability
Guidelines:
1. Students will be courteous and respectful to others.
2. Do not disrupt the learning process.
3. Students will be prepared for class!!!
4. Show P.R.I.D.E
Important:
Cell phones may not be used during class time. Cell phones may not be used as calculators.
Music Playing devices will be left to teacher discretion.
No FOOD or DRINK will be allowed in the computer lab.
Materials: Needed everyday for class!
Calculator – Scientific Calculator is required for this course!
Pencils w/erasers
Planner
Grading Procedures;
Assignments: No homework!
10 points weekly for ALEKS
Students are required to work on ALEKS 90 minutes a week to gain full credit. Points will be rewarded as follows:
90 + minutes 10 pts
80– 89 minutes 9pts
70 – 79 minutes 8pts
60 – 69 minutes 7pts
50 – 59 minutes 6pts
40 – 49 minutes 5pts
30 – 39 minutes 4pts
20 – 29 minutes 3pts
10 – 19 minutes 2pts
Less than 10 minutes 0pts
5 points daily for participation on math with the classroom activity of the day such as large group activity, mini-assignment, homework time, math game, or ALEKS.
Students must work on Math Homework, when not working on the ALEKS program. Some days all students will be required to participate in a review activity together. Occasionally, math practice web sites including math games will be an option for the day's participation.
Tests:
Once every two weeks, students will be assessed in the Aleks program. Each assessment will be worth a total of 10 points. Students will be awarded points for showing improvement from the previous assessment. At the end of the course, a Final assessment will be given in the Aleks program.
Attendance Policy:
Students are encouraged to attend class everyday. If a student is absent, they will still be accountable for completing 90 minutes in ALEKS per week and must see Miss Kandle for a make-up assignments for the daily participation points. |
MATH32012 Commutative Algebra - 2012/13, Semester 2
Online Test
The Online Test is currently accessible via the MATH32012 course content page in Blackboard.
You may retake the test for revision purposes (e.g., to practise
the computation of Gröbner bases). It will not affect your coursework mark.
(The coursework marks have been finalised and are available via the Grade
Centre in Blackboard.)
Module description and prerequisites
You should have general facility for dealing with algebraic structures: complex numbers, sets, groups, rings, fields. For this reason, MATH20212 Algebraic Structures 2 is a prerequisite.
About the course
Many find MATH32012 Commutative Algebra the most advanced abstract algebra course they take as part of their degree.
Nevertheless, the content of the course is not just a sequence of theorems and proofs.
You are expected to learn methods of algebraic computation relating to polynomials in several variables.
Solving equations has been a driving force of algebra at least since the Babylonians learned to solve quadratic equations some 3700 years ago. The subject matter of this course is, however, informed by more recent developments.
The work of Hilbert in late 19th - 20th century was key to the modern treatment of multivariate polynomials and provided a basis for commutative algebra and algebraic geometry. His result that every (consistent) system of polynomial equations over an algebraically closed field has at least one solution is known as the Nullstellensatz.
But an efficient method of finding such solutions by elimination was not found until 1965, when Buchberger invented Gröbner bases.
In the course, key theorems about the ring of polynomials in several variables will be rigorously proved.
Algorithms relating to polynomials will be explained and supported by examples. This includes factorising polynomials into irreducible factors and computing a Gröbner basis of an ideal.
Results and methods of Commutative Algebra have applications in various branches of mathematics and computer science. Here are some puzzles which we may use in the course as an illustration for the main content. You are welcome to have a go at solving them!
Question 1 (Fermat, 17th century).
Find all integers a "sandwiched" between a square and a cube.
Question 2.
How many ways are there of placing 8 queens on a chessboard so that no two queens attack each other? What about n queens on an n×n chessboard?
Question 3.
How many distinct Sudoku boards are there? (A Sudoku board is a 9×9 square
with a number from 1 to 9 in each cell, satisfying the Sudoku constraints.)
images from Wikimedia commons
Coursework
There will be 2 pieces of assessed coursework:
Assessed homework 1 (see a link above): a take-home problem sheet
set on Wednesday 27 February (week 5),
due on Tuesday 12 March (week 7) at 4pm.
Blackboard-based online test: a timed, open-book test which the students complete online; multiple attempts are allowed
Previous years' exams
Commutative algebra exam papers from years 2008-2012 are available here. |
Costs
Course Cost:
$299.00
Materials Cost:
None
Total Cost:
$299Other Materials
Description
Students will become familiar with the vocabulary, method, and meaning in the statistics which exist in the world around them. This is an applied course in which students actively construct their own understanding of the methods, interpretation, communication, and application of statistics. Each unit is framed by enduring understandings and essential questions designed to allow students a deep understanding of the concepts at hand rather than memorization and emulation. Students will also complete several performance tasks throughout the year consisting of relevant, open-ended tasks requiring students to connect multiple statistical topics together. The TI-83+/84 OR 89 calculator and computers will be used to explore the world of data and the patterns which can be found by analyzing this information as well as statistical relationships.
Standards Alignments
All SVL curriculum is aligned to Spokane Public School district curriculum, which is explicitly designed to support rigorous curriculum in order to stretch each learner to achieve at their highest ability through our district�s Written, Taught and Tested Initiative . Spokane�s curriculum engages students with higher order thinking questioning and assessments assist in determining how teachers can differentiate to meet student�s needs. Because SVL works with students individually, teachers assist each student to reach maximum learning by using SVL curriculum and other teacher developed resources to help each student reach standard. |
The Basic Math DVD Series helps students build confidence in their mathematical knowledge, skills, and ability.
In this episode, the graphing calculator in introduced in the context of statistics. Students will learn how statistics can be used to analyze sets of data in order to measure tendency and variation. The concept of outlier is introduced, as well as the box plot, graphic displays of data, and the qualitative analysis of data. Grades 3-7. 30 minutes on DVD. |
Course Offerings in Mathematics
Below is a list of available courses offered by the Mathematics Department. Consult the Registrar's Office and the College Catalog for registration information.
MATH 109 - INTRO TO QUANTITATIVE REASONING
This course presents mathematical ideas in a real world context. Topics covered include critical thinking and problem solving, the mathematics of finance, basic statistical principles, mathematics and the arts, and the theory of voting.
Hours credit: 3. Students considering Curricular Studies should not register for this course, as students may not receive credit for both MATH 109 and MATH 208
MATH 113 - ELEMENTARY MATHEMATICAL MODELING
This course explores mathematical models of natural phenomena such as population growth and radioactive decay. Analysis of data using computer technology. Linear, quadratic, general polynomial, exponential, and logarithmic models will be discussed.
Hours credit: 3. Not open to students who have completed Mathematics 119R or above, except by departmental recommendation.
MATH 119 - PRECALCULUS
A study of the properties of various functions, including polynomial, trigonometric, exponential, and logarithmic. Analytic geometry of conic sections.
Hours credit: 3. Prerequisite: Mathematics 113 or the equivalent. Not open to students who have been placed into Mathematics 149 or above, except by permission of the Department.
MATH 149 - CALCULUS I
Limits, continuity, and differentiation of algebraic functions of one variable. Applications to curve sketching, optimization, and rates of change. The definite integral applied to finding the area under a curve.
Hours credit: 3. Prerequisite: Mathematics 119R or the equivalent.
MATH 150 - CALCULUS II
A continuation of Mathematics 149R. Volumes and surface area of solids of revolution. Lengths of curves. The logarithm and exponential functions. Techniques of integration. Areas in polar coordinates. Improper integrals, infinite series, and power series.
Hours credit: 3. Prerequisite: Mathematics 149R or permission of the Department.
MATH 227 - ELEMENTARY APPLIED STATISTICS
An introduction to statistics, including probability, binomial distributions, normal distributions, sampling theory, testing hypotheses, chi-square tests, and linear regression.
Hours credit: 3. Not open to students who have satisfactorily completed Mathematics 343. A student may receive credit for only one of these courses: MATH 227, POL 231, or PSYC 227. Offered second semester.
MATH 229 - ADVANCED MATHEMATICAL PROBLEM SOLVING
In this course, students will be expected to solve and present solutions to a collection of problems gathered from various mathematics competitions. Problem solutions may involve the techniques of classical algebra, geometry, calculus, and combinatorics.
Hours credit: 1.0. Prerequisite: Permission of the Instructor. May be repeated for credit up to a maximum of 4 hours.
MATH 250 - CALCULUS III
An introduction to vector calculus. Differential and integral calculus of more than one variable. Vector fields, including Green's, Stokes', and the Divergence Theorems.
Hours credit: 3. Prerequisite: Mathematics 150R and either MATH 241 or PHYS 115, or permission of the Department.
MATH 318 - FOUNDATIONS OF GEOMETRY
A study of modern geometries, including finite projective and Non-Euclidean geometries. Geometric transformations and synthetic geometry.
Hours credit: 3. Prerequisite: Mathematics 150R or permission of the Department. Offered alternate years.
MATH 331 - DIFFERENTIAL EQUATIONS
First order linear and non-linear equations, second and higher order linear equations, series solutions, Laplace transforms, and systems of linear differential equations. Applications, primarily to mechanics and population dynamics.
Hours credit: 3. Prerequisite: Mathematics 150R and 241, or the course may be taken concurrently with MATH 241 by permission of the Department.
MATH 353 - MATHEMATICAL MODELING
The construction and analysis of mathematical models to solve problems in the physical and social sciences. Dynamical systems are emphasized with a particular concentration on linear and non-linear discrete dynamical systems. Topics may include dimensional analysis, stability, chaos, and fractals.
Hours credit: 3. Prerequisites: MATH 150R and 241. Offered alternate years.
MATH 420 - NUMERICAL ANALYSIS
A study of algorithms for solving mathematical problems using computers. These problems include finding the roots of functions, solving systems of linear equations, interpolation, approximate integration, and solving differential equations. Implementation of these algorithms on the computer will be an important part of the course.
Hours credit: 3. Prerequisite: MATH 150R and 241. Prerequisite or corequisite: CSCI 156. Offered alternate years.
MATH 443 - INTRODUCTION TO ANALYSIS
A rigorous study of limits, continuity, differentiation, and integration of functions of a real variable. Hours credit: 3. Prerequisite: Mathematics 250 or permission of the Department.
MATH 494 - SENIOR SEMINAR
Readings on the history of mathematics from the seventeenth century through modern times. Oral reports by students and faculty on topics of interest in mathematics. Written reports are also required.
Hours credit: 3. Prerequisite: Mathematics 360 and 443. |
Mathematical Induction
In this lesson our instructor talks about mathematical induction. He answers the question what is induction and does examples of mathematical induction. He talks about the historical background of induction where he discusses the French mathematician Pierre de Fermat. He also talks about Leonhard Euler. Lastly, he talks about the principle of mathematical induction. Four extra example videos round up this lesson.
This content requires Javascript to be available and enabled in your browser.
Mathematical Induction
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. |
This is a transition course between lower divison mathematics and upper division mathematics. It involves critical thinking, creativity, and analytical reasoning. Lower divison mathematics consists mainly of repletion and memorization. Upper division mathematics is more abstract and involves proving theorems. This class serves as an introduction to various advanced topics in mathematics such as Geometery, Trigonometry, and Statistical Analysis.
Course:
Number:
Grade Level:
Prerequisites:
Credit:
Algebra 1, Part 1
3131
8, 9, 10, 11, 12
Math 8
1
Algebra 1, Part 1 provides students with the basic algebra skills necessary to move to a higher level mathematics course. This course was designed to eliminate or reduce math anxiety by teaching Algebra at a slower pace. Thus, it makes mathematics understandable and applicable to everyday life. The student will learn computation with rational numbers such as intergers, fractions, and decimals and solve application problems. The student will use applications with polynomials, equations, and inequalities.
Course:
Number:
Grade Level:
Prerequisites:
Credit:
Algebra I
3130
8, 9, 10, 11, 12
None
1
This course includes types of numbers, algebraic vocabulary, properties and operations of numbers, simplifying expressions, solving equations and inequalities, and graphing. Finding and using prime factors, square roots, repeating decimals, as well as using polynomials, rational expressions, and radicals are also part of this course. Mastery of graphing, solving equations with two variables, and solving quadratics is required.
Course:
Number:
Grade Level:
Prerequisites:
Credit:
Geometry
3143
9, 10, 11, 12
Algebra I
1
The geometry course is a one year mathematics course that includes both plan geometry and three-dimensional geometry. The course is considered necessary to demonstrate a reasonable knowledge of mathematics for students who plan to pursue a college education. Simple algebraic equations are integrated into the course and presented as a means of solving some geometry problems. Geometric proofs and problem solving develop analytical reasoning skills and improve the ability to apply logic to analysis of problems.
Course:
Number:
Grade Level:
Prerequisites:
Credit:
Algebra II
3135
10, 11, 12
Algebra I
1
Algebra II is mandatory for students seeking the Advanced Studies Diploma and for those students planning a higher education in math or science. Concepts of Algebra I are reviewed and strengthened. Emphasis will be placed on the study of complex numbers, coordinate geometry, linear systems, functions, conic sections, logarithms, and an indirection to progressions and series.
Course:
Number:
Grade Level:
Prerequisites:
Credit:
Advanced Algebra and Trigonometry
3161
11, 12
Algebra II
1
Advanced Algebra and Trigonometry is a course that includes an extensive and comprehensive treatment of trigonometry. The course includes algebra topics not covered in previous courses, such as analytical geometry; exponential and logarithmic functions; sequences and series; matrix algebra and determinants. The course is designed as preparation for math analysis or for freshman mathematics in college. |
Complex Multiplication and de Moivre's Formula
Polynomial Arithmetic
Numerical Integration
RSA Signatures
Links
UTeach UMass Lowell - UTeach prepares science and engineering undergraduate students to become middle school or high school teachers.
Focus on Mathematics - a partnership funded by the National Science Foundation to improve student achievement by providing mathematics teachers with the content knowledge and skills valuable in their profession. |
Chapter 1. Matrices and Systems of EquationsMatrix AlgebraA is called a square matrix of order n if m = n.An 1 n matrix is called a row vector.An m 1 matrix is called a column vector.Math1111VectorsChapter 1. Matrices and Systems of EquationsMath1
Homework II1. Your company has compiled the following data on the small set of products that comprise the specialtyrepair parts division. Perform ABC analysis on the data. Which products do you suggest the firm keep thetightest control over? Explain.S |
hands to explore and build proficiency and eventually to replicate... I've previously taken regular calculus classes with engineers and won... This is not the same at all. We were solving real problems every day...Students work through problems using an online interactive textbook D...When teams become stuck on a problem Chiel or a teaching assistant m...
hands, to explore and build proficiency, and, eventually, to replicate and build on recent math models used in the biological sciences. The course is cross-listed as both a biology and biomedical engineering class.
"I've previously taken regular calculus classes with engineers and wondered what would the classes ever be useful for," said Kate Coyle, a biology major who completed the Dynamics class and graduated this semester. "Labs I've had in biology and physics show you the protocol and the expected result.
"This is not the same, at all. We were solving real problems every day."
Students work through problems using an online interactive textbook, Dynamics of Biological Systems: A Modeling Manual Chiel wrote and the computer programming language Mathematica, which scientists worldwide rely on to build mathematical models of complex systems. Chiel's book is available free to students as well as teachers who may want to use it as is or as a model for their own classes.
When teams become stuck on a problem, ,Chiel or a teaching assistant makes suggestions, gives clues and tries to coax out the answer. After success, teachers quiz individuals about how they found the solution and what they'd learned.
The class of 30 is spread out among hexagonal tables. Teams power up their laptops and go to work. Each day the teachers rotate to a different group of students, and after each class they compare notes on who has mastered the skills and who needs extra help, Gill said.
When the second half of the semester begins, teams choose a mathematical model that was recently published in a scientific journal, begin reconstructing and analyzing it and then writing in detail what they learn. The students then extend the model to answer new questions that they ask themselves, and write up results as if they were writing for a scientific journal.
Coyle and her teammates Valencia Williams and Joshua DeRivera focused on a pa |
Contemporary Mathematics for Business and Consumers - With CD - 5th edition
Summary: With a unique step-by-step approach and real-life business-based examples throughout, CONTEMPORARY MATHEMATICS FOR BUSINESS AND CONSUMERS, is designed to help students overcome math anxiety and confidently master key mathematical concepts and their practical business applications. The text is designed to let students progress one topic at a time, without being intimidated or overwhelmed. Each chapter features numerous exercises, including Excel spreadsheet problems, ...show morereal-life business scenarios, and detailed calculator sequences, to provide immediate practice to reinforce learning and hone essential skills. The MathCue CD-ROM, included with each new text, is one-of-a-kind tutorial-practice software that gives students an extra edge by providing an endless supply of practice problems keyed to each performance objective in the text. ...show less
0324568495SellBackYourBook Aurora, IL
0324568495VeryGood
Better World Books Mishawaka, IN
Sorry, CD missing. GreatCampus_Bookstore Fayetteville, AR
Used - Good TEXTBOOK WITH CD!!!! 5th Edition8032456849380324568493-5-1
$2.70 +$3.99 s/h
VeryGood
Bookbyte-OR Salem, OR
Has minor wear and/or markings. SKU:9780324568493-3-0 |
Cresskill CalculusA good approach is to first identify all the quantities which can be counted or measured and the relationships between them. Students then find it easier to set up tables and relevant equations. I help students learn this approach |
Mathematics
SCC offers a wide variety of mathematics courses to prepare students for work in the fields of mathematics, science, health, education, business and more. Courses offered range from basic developmental mathematics through calculus, differential equations, linear algebra and transition to theoretical mathematics. SCC offers traditional, hybrid and online courses.
Mission Statement
To make mathematics accessible, to provide quality mathematical content with concerned instruction and to provide a broad range of courses so that students may acquire the necessary mathematical skills to meet their career and personal goals. |
AS Maths Unit 1 & 2: Core 1 and 2 – these cover the essential pure mathematical methods that can be applied to real world scenarios. Topics include solving equations, graphs and transformations, coordinate geometry, logarithms and exponentials, sequences and series, trigonometry, and differentiation and integration. Unit 3: Mechanics 1 – this unit introduces mathematical modelling in physical situations, and studies motion in one or two dimensions (including the constant acceleration equations), forces on static objects, Newton's Laws of Motion, momentum and projectiles.
AS Further Maths Unit 1: Further Pure 1 – this unit extends your knowledge on quadratic equations, trigonometry, inequalities and calculus. It also introduces imaginary numbers/ complex numbers, Matrices, Series and Numerical methods. Unit 3: Statistics 1 – in this unit students work with real data sets extending the work they have covered in GCSE Maths, such as the calculation of the numerical measures mean, median and mode, and the practical applications of correlation and regression. Elementary probability theory is also studied, and the Binomial and Normal distributions are introduced. Unit 3: Decision 1 – this module covers a number of techniques which are important in the solution of large-scale organizational and business problems. It also introduces networks, algorithms, pseudo-code, sorting and linear programming.
Year 13 Students study a further 6 units to gain A Level Mathematics and an A Level Further Mathematics. A level Maths
Units 4 & 5: Core 3 and 4 – these units develop the pure mathematical techniques from AS Level and introduce practical applications such as differential equations. New topics include numerical methods and vectors. Unit 6: Mechanics 2 – students study more advanced physical situations including circular motion, moments and centers of mass, work and energy, further kinematics, and applications of different equations.
A level Further Maths
Units 4 & 5: FP2 and FP3 – these modules take some of the FP1 topics further and introduces some new topics like summation of finite series, De Moivre's theorem, Hyperbolic functions, Polar coordinates, and second order differential equations. Unit 6: Statistics 2 – this unit introduces such practical applications as hypothesis testing and contingency tables. New probability distributions include the Poisson distribution and the Rectangular distribution.
Assessment
No Coursework
Examination (100%)
Examining Board – AQA.
Special Entry Requirements Grade A or A* in GSCE Mathematics. Please note that students take Further Maths as part of a five AS package, for which you need at least 8 GCSE passes at grade A*-B across 7 different subjects. In exceptional circumstances, if you do not fulfill the above criteria, but are a Mathematics specialist with grade A*, please consult with a senior manager who may allow you to take this as part of a 4 AS package.
Prohibited Options
Statistics and Use of Mathematics.
Career and Progression Opportunities
Students with Further Mathematics commonly take up top professional careers in a wide variety of areas. These include Insurance and Actuarial Work, Finance, Management, Operational Research, Civil, Mechanical and Electrical Engineering, Architecture, Government, Medical or Pharmaceutical Statisticians, Business Analysts and others. This is a very valuable option for students considering Mathematics, Physics, Engineering or Economics degrees.
Other Information
Students will be required to purchase a Casio graphical calculator from College.
The Maths department has a Twitter account (@solmaths) - follow them to learn more about what to expect from studying a Maths-related A level. |
for complete understanding of the logic behind the collaborating calculators, you wil |
Book summary
The text presents a circular function approach to trigonometry by demonstrating connections between the familiar algebra and the new language of trigonometry. This method-along with foreshadowing-is used throughout the text to provide students with a comfortable base for learning something new from something old or familiar. With just a few connections to algebra, students have the tools to understand the circular functions, their domains and ranges, and the relationship between the circular functions and the functional values. The approach immediately launches the student into the concept of periodic functions, their applications, graphs, and use in modeling many periodic phenomena. Beginning with this approach provides the student with a common thread that can be used to discover, connect and understand the remaining concepts of trigonometry. [via] |
Relations: Find/Domain/Range The learner will be able to
find the domain of the independent variables and the range of the dependent variables represented by a graph, a set of ordered pairs, or a symbolic expression.
Strand
Scope
Source
Relations
Master
NM: Content Standards, 2002, Grade 9-12, pg 45: AFG.6
Functions: Study/Explain The learner will be able to
study and explain middle and end (asymptotic) behavior of linear, quadratic, and exponential functions, and draw the graphs of functions.
Strand
Scope
Source
Functions
Master
NM: Content Standards, 2002, Grade 9-12, pg 45: AFG.10
Functions: Notation The learner will be able to
describe or apply function notation.
Strand
Scope
Source
Functions
Master
NM: Content Standards, 2002, Grade 9-12, pg 45: AFG.5
Functions: Domain/Range The learner will be able to
apply the composition of functions, and determine the domain, range, intercepts, zeros, and local maxima or minima of the final function.
Strand
Scope
Source
Functions
Master
NM: Content Standards, 2002, Grade 9-12, pg 45: AFG.11
Functions: Variables The learner will be able to
recognize the independent and dependent variables from a real life scenario.
Strand
Scope
Source
Functions/Relations
Master
NM: Content Standards, 2002, Grade 9-12, pg 45: AFG.7
Functions/Relations: Equation The learner will be able to
use an equation to convey the relationship between two variables.
Strand
Scope
Source
Functions/Relations
Master
NM: Content Standards, 2002, Grade 9-12, pg 46: AFG.4
Functions/Relations: Graph The learner will be able to
use a graph to convey the relationship between two variables.
Coordinate Plane: Midpoint/Distance The learner will be able to
find the midpoint and distance between two points with a coordinate system and associate these concepts with geometric figures.
Strand
Scope
Source
Coordinate Geometry
Master
NM: Content Standards, 2002, Grade 9-12, pg 49: GT.2
Coordinate Plane: Comprehend The learner will be able to
illustrate a comprehension of the construction of the coordinate plane, understand the names of the origin, coordinate axes and four quadrants, draw and label them accurately, determine the coordinates of a given point, and plot a point with given coordinates.
Strand
Scope
Source
Coordinate Geometry
Master
NM: Content Standards, 2002, Grade 9-12, pg 49: GT.1
Real Numbers: Powers/Roots The learner will be able to
compute powers and roots of real numbers, both rational and irrational.
Strand
Scope
Source
Real Numbers
Master
NM: Content Standards, 2002, Grade 9-12, pg 44: AFG.12
Real Numbers: Understand/Describe/Use The learner will be able to
understand, describe, and use equivalent representations for the same real number including integers, decimals, percents, ratios, scientific notation, numbers with integer exponents, inverses (reciprocal), and prime factoring.
Strand
Scope
Source
Real Numbers
Master
NM: Content Standards, 2002, Grade 9-12, pg 43: AFG.7
Linear Equations: Solve/One Variable The learner will be able to
find the solution to one-variable linear equations.
Strand
Scope
Source
Linear Equations
Master
NM: Content Standards, 2002, Grade 9-12, pg 46: AFG.4
Linear Equations: Write The learner will be able to
write the equation of a line that passes through two coordinate points.
Strand
Scope
Source
Linear Equations
Master
NM: Content Standards, 2002, Grade 9-12, pg 46: AFG.10
Linear Equations: Formulate/Point-Slope The learner will be able to
formulate the equation of a line through the use of the point-slope formula.
Strand
Scope
Source
Linear Equations
Master
NM: Content Standards, 2002, Grade 9-12, pg 46: AFG.12
Linear Equations: Determine The learner will be able to
determine if lines are parallel, perpendicular, or coincide when given two linear equations.
Strand
Scope
Source
Linear Equations
Master
NM: Content Standards, 2002, Grade 9-12, pg 49: GT.3
Linear Equations: Verify Points The learner will be able to
check that a point lies on a line when the equation of the line is known.
Strand
Scope
Source
Linear Equations
Master
NM: Content Standards, 2002, Grade 9-12, pg 46: AFG.12
Linear Equations: Graph/Table of Values The learner will be able to
apply a table of values to graph linear equations on a coordinate plane.
Strand
Scope
Source
Linear Equations
Master
NM: Content Standards, 2002, Grade 9-12, pg 46: AFG.7
Linear Equations: Real World Models The learner will be able to
represent real world situations with linear equations.
Strand
Scope
Source
Linear Equations
Master
NM: Content Standards, 2002, Grade 9-12, pg 46: AFG.1
Linear Equations: Fit/Data The learner will be able to
fit a linear model to a data set by applying technological tools.
Strand
Scope
Source
Linear Equations
Master
NM: Content Standards, 2002, Grade 9-12, pg 52: DAP.4
Linear Inequalities: Solve/One Variable The learner will be able to
find the solutions to one-variable linear inequalities.
Strand
Scope
Source
Linear Inequalities
Master
NM: Content Standards, 2002, Grade 9-12, pg 46: AFG.4
Linear Inequalities: Illustrate/Real The learner will be able to
illustrate real world situations using linear inequalities.
Strand
Scope
Source
Linear Inequalities
Master
NM: Content Standards, 2002, Grade 9-12, pg 46: AFG.1
Graphs: Comprehend/Symmetry The learner will be able to
comprehend symmetry of graphs. |
MS Pre-Algebra B is the second half of a series of two courses that are designed to prepare the student for more advanced work in Algebra I. The course emphasizes concepts in solving linear equations, graphing linear equations, angles, two- and three-dimensional geometry, integrating algebra with geometry, and data, statistics, and probabilities. The course offers graphics, explanations, and practice exercises before formative assessments |
Author:
Hao Wang, Wenlong Wang
This is a free textbook from BookBoon.'Algebra is one of the main branches in mathematics. The book...
Type: Open Textbook
Date Added: Jan 29, 2013
Date Modified: Jan 29, 2013 |
Mathematics PhD
The general aim of the program leading to the Ph.D. in mathematics is to prepare students to become productive research scholars capable of communicating their knowledge to students and to the mathematical community. The program is planned to develop in the student a fundamental understanding of certain basic fields of mathematics, a deep understanding of the major field of interest, the ability to formulate and recognize significant research problems, and the ability to analyze problems and reach solutions and to transmit ideas to others |
Students will need to pass course competencies in addition to the course in its entirety to earn credit. It is considered best practice. For background information on Competency and Competency Assessments, please visit the high school website. The link is located on the right of the home page under "School Info".
CP Algebra 2A Course Competencies:
The student will solve and graph simple and compound linear inequalities.
The student will graph singular and systems of linear equations. The students will solve systems of linear equations by elimination and substitution methods.
The student will solve quadratic equations using a variety of methods.
The student will apply synthetic division and the factor theorem to factor and solve expressions and equations.
The student will use basic properties of logarithms.
The student will simplify rational expressions and complex fractions. The student will solve rational and fractional equations.
The student will perform basic arithmetic and geometric series and sequences operations.
CP Pre-Calculus Course Competencies:
The student will demonstrate and apply knowledge of commonly used algebraic functions and transformations.
The student will demonstrate and apply knowledge of exponential and logarithmic equations, their graphs, and properties.
The student will demonstrate and apply knowledge of trigonometric functions, identities and equations.
The student will demonstrate and apply knowledge of graphs and systems of linear inequalities as well as linear programming.
The student will demonstrate and apply knowledge of limits, graphically and algebraically.
Fundamentals of College Algebra Course Competencies:
The student will be able to add, subtract, multiply, divide and simply real numbers.
The student will be able to solve linear equations and inequalities.
The student will be able to translate and solve word problems.
The student will be able to add, subtract, multiply, divide and simply polynomials.
The student will be able to solve quadratic equations by factoring.
The student will be able to add, subtract, multiply, divide and simply rational expressions.
The student will be able to graph linear equations.
The student will be able to solve systems of equations by the elimination method.
X2 grading codes - describes why an assignment did not earn a point value.
AB
Absent and did not turn in by designated time.
Calculates as a zero
BG
Being Grade
No score
DD
Didn't Do (and said so)
Calculates as a zero
DH
Collected – Did Not Hand In
Calculates as a zero
EXC
Excused from assignment
No score
FAIL
Failed
Competency Assessment
INC
VERY Incomplete
Calculates as a zero
MI
Missing
Calculates as a zero
NC
No Credit -- no work shown, answers only, illegeble, more than half wrong, did not check answers
Calculates as a zero
OPT
Optional assignment
Scored as designated
OWE
Owes (b/c of absences or other circumstances)
Will change to point value when handed in of changes to DH after designated time |
This is the second course in a two semester sequence
designed for elementary, middle school, and special education majors. The purpose
is the continuation of the study of mathematical topics from Math 215 with an emphasis
on algebraic notation, sets and functions, basic geometric concepts of measurement,
length, area, perimeter, surface area, volume, and the Pythagorean Theorem. Some
elementary probability and statistics, including some educational statistics, will
be included. Every semester. Prerequisites: UTC Math Placement Level 30 or Mathematics
214 or 215 with a grade of C or better or Math ACT 26 or above or approval of the
Mathematics Department Head. This course will meet General Education Mathematics
requirement, but not General Education Statistics requirement.
245 Introduction to Differential and
Difference Equations (3)
First order and second order linear
differential and difference equations, systems of equations and transform methods.
Every semester. Prerequisite: Mathematics 161 with a minimum grade of C. Pre- or
Corequisite: Mathematics 212 with a minimum grade of C.
Introductory concepts of sets, functions,
equivalence relations, ordering relations, logic, methods of proof, and axiomatic
theories with topics from combinatorics, graph theory, or abstract algebra. Fall
and spring semesters. Prerequisite: Mathematics 161/162 with minimum grades of C.
This course is a prerequisite for Mathematics 321, 350, 403, 410, 412, 422, 430
and 452. Mathematics majors should enroll in it at the end of the sophomore year
or beginning of the junior year.
303 Discrete Structures (3)
Concepts and techniques of several
areas of discrete mathematics with emphasis on areas often applied to computer science.
Topics will include formal logic, induction, recursion, algorithms, counting methods,
languages and grammars, and finite state machines. Fall and spring semesters. Prerequisites:
Mathematics 161/162 and Computer Science 150 with minimum grades of C.
307 Applied Statistics (3)
Introduction to probability and statistical
methods with applications to various disciplines. A study of some basic statistical
distributions, sampling, testing of hypotheses, and estimation problems. Fall and
spring semesters. Prerequisite: Mathematics 161 with a minimum grade of C. Credit
not allowed in both Mathematics 307 and 407-408.
321 Introduction to Modern Algebra
(3)
Integral domains, rings, fields, groups,
elementary number theory, and other selected topics. Prerequisites: Mathematics
212 and 300 with minimum grades of C or approval of the Mathematics Department Head.
350 Fundamental Concepts in Analysis
(3)
Classical treatment of the basic concepts
of calculus: limits, continuity, differentiation, Riemann integration, sequences
and series of numbers and functions. Prerequisites: Mathematics 255 and 300 with
minimum grades of C.
401 Mathematics of Interest (3)
Mathematical theory of interest with
applications, including accumulated and present value factors, annuities, yield
rates, amortization schedules and sinking funds, depreciation, bonds and related
securities. Recommended for students planning to take actuarial exams. Prerequisites:
Mathematics 161 with a minimum grade of C, or approval of the Mathematics Department
Head.
403 Introduction to Graph Theory (3)
An overview of graph theory and its
applications including trees, planar graphs and graphical invariants such as domination,
coloring, and matchings. Prerequisites: Mathematics 300 with a minimum grade of
C.
A continuation of Math 407 with an
introduction to the theories of point and interval estimation, hypothesis testing,
regression and correlation analysis, goodness of fit, chi-square, t and F distributions.
Prerequisite: Mathematics 407 with a minimum grade of C.
Intermediate applied statistical analysis
and model building. Covers One and Two Factor Analysis of Variance, Simple and Multiple
Regression and Correlation, and Time Series Analysis. This course is recommended
for students planning to take actuarial exams. Prerequisite: Mathematics 307 or
407 or Engineering 322 with a minimum grade of C, or approval of the Mathematics
Department Head.
The nature of libraries of procedures
for solving mathematical problems, including the design parameters, selection of
algorithms, problems with reliability and transportation. Several major mathematical
packages will be examined in detail, including IMSL, LINPACK, EISPACK, MINPACK.
On demand. Prerequisites: Computer Science 118 or approval of the department head;
Mathematics 245 and Engineering 225 with minimum grades of C.
430 The Historical Development of Mathematics
(3)
Examination of central ideas, major
developments, and important issues in mathematics from ancient times to the present.
Historical overview of the evolution of the discipline through comparative examination
of specific theories and results. Prerequisite: Mathematics 300 with a minimum grade
of C.
440 Applied Analysis (3)
Vector analysis through Stokes'
Theorem and the Divergence Theorem. Topics in advanced calculus including implicit
functions, Jacobians, interchange of limit processes, and uniform convergence of
series of functions; introduction to power series solutions of differential equations.
Prerequisites: Mathematics 245, 255 with minimum grades of C. |
Ok you people out there that want the easy way out in your geometry, algabra, and pre-calc class
i got a math program for you that does most of what you need for these classes.
ne thing from: Area, surface area, volume, cramers rule, Conics, distance formula, end behavior, midpoint, pascals triangle, quadratic formula, reducing radicals, and slope.
It runs right out of the program menu, or if you really want to be secretive, it works under mirage (which can hide it from teachers)
I was hoping to sell it for like 2 bucks....but don't know if it will work that way... but if you want it..email me and i'll give it to you for free
Because I like to help people learn programming and stuff, I have setup a few yahoo groups where people can learn from me/eachother, share files/programs, post messages, etc. - Trust me, I am among the best, so I think it would be very beneficial.
I am very willing to teach people everything I know whenever I have free time (I like to work on a one-to-one basis). Check out my groups if you are at all interested or curious.
Below are the web addresses for my groups (CAREFUL, I had to put a space after each slash, so if you copy and paste, delete the spaces):
You could try searching for one first. For example, you could probably find one or more in 83plus/basic/math (such as baseic.zip or bases.zip) or in 83/basic/math.
I don't want to sound rude or condescending, but you really should look first for what you want before you ask someone to do something for you. This site especially has most math programs a beginner like you or I would need.
Once you've learned enough and need a program that doesn't exist (at least on ticalc), one should hope that you would have the ability and knowledge to write what you need by yourself or at least to search the many resources available to you on the Internet and other places. This goes for most every beginner, not just you.
I'm not trying to be high and mighty by calling you and others a beginner; I still consider myself to be a beginner in many ways.
hi i am taking AP Calculus. please, i am begging you. someone has to create a super program for this class. it isn't a hard class, but it takes time to solve some stuff. also, a calculus program would save me time in contests. well see you and thanks
A lot of programs calculate everything for you in the background, giving you a final answer. What about a program that gives you the formula? I can't remember formula, and I don't trust values of programs, I'd prefer to just see the formula.
Option 1: If it is Mirage compatible (add a ":" to the beginning of the first line), you simply highlight it in Mirage and press "tan(". To the right, the properties should read either "LOCKED:Y" or "L:Y".
Option 2: Send it to your computer (PC or Mac) and open it with 83+ Graph Link. There should be a box next to "Protected". Check the box.
If you don't have either of those two programs, e-mail me at [email protected]
Ok, this is kind of similar.
I'm the only person at my school who knows any bit how to program calculators, so, like anyone would do, I sell them. However, lately, people have been giving each other the programs instead of buying them from me. Is there a way to stop people from being able to send programs????
thats pretty underhanded ... but, back to the point if you aren't sending then mirage its easy to send them a shell that can detect hidden progrsms and run them but not unhide them. You can send them the program from mirage so you dont have to unhide it. they will be able to run the program from the shell but if you dont give anyone mirage then they cant send them. |
. 60+ examplesMath software for students studying precalculus. Can be interesting for teachers teaching precalculus. Math Center Level 1 consists of Graphing calculator 2D, Advanced Calculator, and Simple Calculator called from the Control Panel. Simple calculator is a general purpose calculator.Advanced Calculator is a step farther in complexity comparing to the Simple Calculator.Graphing Calculator 2D has two panels |
The content. Analysis is about the concepts of function, derivative, and integral (quoted from the preface for the Shilov's book). As a starting point in this study we will look into the structure of real line and explore some of the axioms of the real number system.
The primary focus in the initial part of the course will be on the Completeness Axiom and various equivalent forms of it. We will study in detail the concepts of limit, derivative, integral , and the series. Although you have already study all those concepts in your Calculus courses, I am sure that you will realize that there is a lot of room to improve our basic understanding of those concepts. Real Analysis is one of the key courses in the foundations of mathematics, and most of the time in this course will be spent into looking back into the foundations of Calculus. However, we would also like to use our improved understanding of
Calculus to move toward the greater level of abstractions, and to learn something about some of the more "modern" developments of Analysis. In particular, we will study the concept of Metric Spaces a little bit.
1. Program Assessment - Course Objectives Here is a concise list of course goals following the format of Math Department
Assessment Plan. In Section
1
2 MATH 420 - REAL ANALYSIS SPRING 2005
Course-level Outcomes. Students shall study foundations of the real number system, limits, continuity, derivative and integral.
Mathematical Reasoning: Students will try to achieve an in-depth understanding of the ideas presented. This includes making sure statements and claims made are checked for accuracy, and an appropriate justification is given.
Problem Solving: A number of calculus problems will be revisited, and many new solved. This course will challenge and develop students' problems solving skills to the limit.
Communication: The main mode of communication in this course will be written (homework, exams). However, in-class participation (oral communication) is essential, as well.
Technology: Use of a Computer Algebra System (Derive, Maxima) and
LATEX.
2. Course Philosophy and Procedure
In order to succeed in this course, you should really immerse yourself totally in doing mathematics. The key strategy in solving problems is "not to give up". This course is truly a problem solving course.
Most of the course time will be spent on limits, derivatives, integrals, ... . You will see again some of the stuff that you are familiar with, but sooner or later you will stumble over some basic stuff that you actually do not know very well.
You should certainly build up on your strengths (the stuff you more or less know), but make sure that you do not neglect working on the weaknesses, as well. Use homework and exams as directions, but you should really guide yourself in the work of filling in the gaps in your knowledge in order to be able to meet the goals set up by the homework and exam problems.
Mathematics is not a spectator sport. It is learned by doing. Viterbo University is striving to be a Learner-centered institution. That entails an expectation of maturity and taking responsibility for their learning on the part of students. I see my job as one helping you succeed in this learning process.
In spite of my best efforts, I may not always manage to say things the way which best leads to your full comprehension. You can also help me by providing as much of a feedback as you can. I will try to do a formal evaluation survey around the middle of the semester. Other than that, I find the questions in class, and especially when someone comes to my office for assistance, very helpful.
As a further assistance to you:
• About a week prior to any exam, you will receive a practice exam which will be, in terms of format and type of problems, very much like the actual exam.
• I am asking you to keep a The Learner's Journal. This is to be a separate notebook that should contain a record of your study/practice on daily basis.
I would also like you to keep a time log - date, hour from-to - for each study session. I would prefer that you use a pen for writing in that journal. If you are going to use pencil, then please do not use erasers, and in any case, do not tear pages out. For a learning to take place, you have to try to do
MATH 420 - REAL ANALYSIS SPRING 2005 3 something. In trying, you are likely to make mistakes. The real learning will start taking place once you start understanding and correcting your mistakes.
You turn that journal in together with your exam, and then you will be graded for the portion of that journal that covers the period preceding that current exam. Up to 30% of the exam score is possible to earn this way.
The elements that will play the key role in grading the journal are
– Organization - readability: In order to evaluate, I have to able to read it first. I should not have a difficult time navigating through those notes.
– Mathematical correctness.
– The quality of the work and the amount of time spent on studying.
• Take-home problems: These assignments should test/help a better integration of material. Some will include more difficult problems. One of those assignments will be a group HW. In general, I encourage you to find some time to study together, but unless stated otherwise, the HW is to be written up on your own.
I will try to space those assignments so that you could have some time to catch up. This should leave significant room for exploring the book on your own, and I encourage you to find your own balance between solving some problems in full, and just sketching solutions to some. You should try to read, meaning to the point where you really understand the question, most of the book problems.
The work in class, your book, HW, and practice exams should give you a pretty clear idea what is that you are expected to learn. It is your job to, perhaps through trial and error, find learning strategies that work best for you. Remember, learning is something you do, rather than something I do to you.
Help. I am used to you asking questions in class, coming to my office, working in groups, asking questions by e-mail. Hope all of that will continue.
There is a growing mount of Internet resources, too. Just go to Yahoo, or Google and search for "real analysis". In particular, you may want to look at
Some of the HW will involve some use of technology (CAS and LATEX). The details will be given later.
One of the writing assignments will be graded in two parts - the second part will require you to come to my office and explain your reasoning, answer some questions.
In all your work, written and oral, it is essential to provide explanations, justify your reasoning.
My grading scale is
A=80%, B=60% , C=40%, D > 30% .
The following exceptions to that scale are possible:
• An A on the final exam (more than 180/200 points) will mean an A for the final grade as well.
4 MATH 420 - REAL ANALYSIS SPRING 2005
• If one is passing the course by the time of the final exam, but earns less than 30% points (a score less than 60/200), that will result in an F for the final grade.
3. Some details and examples
Specific Course Goals. To study rigorous foundations of calculus; extend basic knowledge of functions, limits, derivatives and integrals. Students are expected to learn to state definitions and theorems precisely, and be able to prove theorems stated. In particular, the proofs involving the concept of limit are going to be of central importance.
The process of working toward those goals will involve looking back into everything you have learned in mathematics so far, and to subject those concepts to the following key questions:
• What do I really know about ...?
• What does ...mean?
• Is what I just said about ...true?
• How do I know if (why) it is true?
Let me try to clarify this a little bit by looking into an example.
Example: For some Math 155 students, conquering a problem such as
p8 + p32 −
p18 = 3p2
represents a major undertaking; something they consider worth of including into their portfolio for the semester.
Now, when you look at what is involved into justifying the above result, a number of problems present themselves. For example, we have to use the rule
pAB = pApB , A,B 0.
Can you prove that rule?
Another key rule here is
AC + BC = (A + B)C .
How about proving this one It is much more difficult question than one before.
In fact, it is so basic that there is nothing more basic to use as a help in proving it. So, we have to accept that rule as an axiom. Note: I would like to encourage you to read [4, Chapter 4] here!
Moreover, the rules are used by applying them to "existing" mathematical objects (in this case, real numbers). But, what do we mean by p2. Can you prove that such a number exists? What do we even mean by asking a question like that?
The goal of this course is not only to learn how to answer the questions as those (that is how to do certain rigorous proofs), but we would also like to develop the corresponding mathematical awareness, so that we do not overlook those "simple" questions when solving problems.
Ultimately, this kind of training leads to deepening our understanding of what mathematics is about. We would like to use that improved understanding to bring about other two basic goals of this course, which are:
• Move beyond the concepts we are familiar with. We will study the concept of metric spaces.
Fall 1999 - Final. To illustrate the goal of "improving Calculus skills", let's consider the final exam I gave to the Real Analysis class in Fall 1999.
Problem 1. • State the definitions of a lower/upper bound of a set of real numbers.
• State the definition of infimum.
• State the theorem about existence of an infimum of a set bounded below.
• Extra credit: Should I say "the infimum"?
• Prove that the set
S = 3,
5
2,
7
3,
9
4, . . . has an infimum, and find that infimum.
• State the Completeness Axiom.
• Prove that the Completeness Axiom is equivalent to the Infimum Theorem
above.
Problem 2. Define
f(x) =
1Xn=1
n2 + 1
n! xn .
(a) Use the ratio test to show that f is defined for all real x.
(b) Prove that f is continuous at x = 0.
Problem 3. State the definition of the derivative f0(a) of a function f at the point a of its domain.
Use that definition to find the derivative of f(x) = 3x at an arbitrary point of
Df .
Problem 4. Find,
lim
x!0
sin (ln (1 + x))
ln (1 + sin (x)) .
State and prove all the rules used in the process. In the case of L'Hospital's rule, just state it. The proof would be an extra credit.
Problem 5. Probably most of the functions we have encountered so far would be continuously differentiable. That is, if f is differentiable at x = a, then f0 is a continuous function at
x = a.
Is this statement a theorem?
Hint: Show that the function
f(x) = (x2 sin 1
x , if x 6= 0
0, if x = 0
is a counterexample.
6 MATH 420 - REAL ANALYSIS SPRING 2005
A student that has passed a Calculus sequence should have no difficulty in understanding almost all of the questions on that final. Being able to answer them completely is a different matter.
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DVD-based GCSE Maths Learning Tool for UK Students Available at TopMathsDVD.co.uk
London, United Kingdom (I-Newswire) September 12, 2012 - The General Certificate of Secondary Education or GCSE – an academic qualification awarded in a specified subject – is a necessity for secondary students in England, Wales and Northern Ireland, with equivalent levels in key skills. When applying to universities in the United Kingdom, most admissions have, among other things, GCSEs for qualification purposes. In this regard, it is essential for students to ensure that they understand GCSE subjects and achieve the desired grades.
Catering to Mathematics as specific subject, the GCSE Maths Master DVD provides the complete DVD study kit. The tool is published and presented by a senior tutor at private Maths tuition bureau in the UK, Top Grade Tutoring, which has successfully helped over one hundred students, with 85% achieving grade A*-C.
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The GCSE Maths Master DVD works on a computer or DVD player, presenting bite sized topics to make learning easier to digest. Practical worked examples to support understanding, as well as top tips to handle the toughest exam questions are all offered through the study kit – all with engaging Hollywood style CGI effects.
Detailed at TopMathsDVD.co.uk, the GCSE Maths Master DVD prevents people from going round in circles and wasting time trying different ways to revise. The tried and tested method has helped thousands of students get the grade that they want in GCSE Maths revision.
The GCSE Maths Master available in higher level and foundation level, which is a refreshing new approach to revising for GCSE Maths, designed to make revising simpler, quicker and more fun. The tool is particularly suitable for all exam boards and syllabi, and compatible for linear and modular exams. The comprehensive content includes top exam hints and tips |
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By Gwendolyn Lloyd, Beth Herbel Eisenmann, Jon Star, Rose Mary Zbiek
Why do some equations have one solution, other two or even more solutions, and some no solutions? Why do we sometimes need to "switch" the direction of an inequality symbol in solving an inequality? What could you say if a student described a function as an equation?
How much do you know...and how much do you need to know?
Helping your students develop a robust understanding of expressions, equations, and functions requires that you understand this mathematics deeply. But what does that mean?Focus on the ideas that you need to understand thoroughly to teach confidently.
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This book focuses on the essential knowledge for mathematics teachers about statistics. It is organized around four big ideas, supported by multiple smaller, interconnected ideas--essential understandings
This book is a collection of the best of NCTM's Addenda series, grades 5-8 and includes problems and examples that represent critical content for today's middle school curriculum. The problems focus on the four key practices:
• Roles of representation • Generalization • Problem solving • Connections in mathematics learning and teaching
This book has More4U, which includes additional resources online. Download activities, classroom materials, and blackline masters. Look inside book for access code.
The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research. |
Perfect for families that already own the sold-separately Teaching Textbooks Teaching Textbooks Math 7 Extra Workbook & Answer Key, this set includes four CD-ROMs that contain step-by-step audiovisual solutions to each homework and test problem. Topics covered include basic arithmetic, including fractions, decimals, and percents; geometry (e.g. how to find the area of a circle); statistics and probability; simple graphing concept; equations and inequalities; and math in the real world. A digital gradebook grades answers as soon as they are entered and calculates percentages for each assignment. Though this CD-ROM set may technically be used without the workbook, students will then have to write out each problem; won't be able to work away from the computer; and won't receive the written summaries available in the textbook. Teaching Textbooks Grade 7.
The new hybrid CD-ROMs feature multiple improvements, including:
Easy multiple user setup (built into program)
An area where parents can access all of their students' gradebooks.
An editable gradebook where you can reset a particular lesson(s) without having to uninstall and reinstall. |
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Study quickly and more effectively; Get the big picture without spending hours poring over dull texts Schaum's Outlines give you the information teachers expect you to know in a handy and succinct format—without overwhelming you with unnecessary details. You get a complete overview of the subject—and no distracting minutiae. Plus,you get plenty of practice exercises to test your skill. Compatible with any classroom text,Schaum's lets you study at your own pace and reminds you of all the important facts you need to remember—fast! And Schaum's is so complete it's the perfect tool for preparing for graduate or professional exams! Students of mathematical economics apply complex formulas—a challenging task that even the best students find daunting. But this Schaum's guide demystifies tough problems and gives you plenty of fully worked examples! Chapters include: Review. Economic Applications of Graphs and Equations. The Derivative and the Rules of Differentiation. Uses of the Derivative in Mathematics and Economics. Calculus of Multivariable Functions. Calculus of Multivariable Functions in Economics. Exponential and LogarithmicFunctions. Exponential and Logarithmic Functions in Economics. Differentiation of Exponential and Logarithmic Functions. The Fundamentals of Linear (or Matrix) Algebra. Matrix Inversion. Special Determinants and Matrices and Their Use in Economics. Linear Programming: A Graphic Approach. Linear Programming: The Simplex Algorithm. Linear Programming: The Dual. Integral Calculus: The Indefinite Integral. Integral Calculus: The Definite Integral. Differential Equations. Difference Equations. Second-Order Differential Equations and Difference Equations. The Calculus of Variations
Schaum's Outlines contains hundreds of worked-out solutions to problems covered in any college |
Intermediate Algebra, 1e, authored by Sherri Messersmith presents content in bite-size pieces, focusing not only on learning mathematical concepts, but also explaining the why behind those concepts. For students, learning mathematics is not just about the memorization of concepts and formulas, but it is also about the journey of learning how to problem solve. By breaking the sections down into manageable chunks, the author has identified the core places where students traditionally struggle, and then assists them in understanding that material to be successful moving forward. Proven pedagogical features, such as You Try problems after each example, reinforce a student's mastery of a concept. While teaching in the classroom, Messersmith has created worksheets for each section that fall into three categories: review worksheets/basic skills, worksheets to teach new content, and worksheets to reinforce/pull together different concepts. These worksheets are a great way to both enhance instruction and to give the students more tools to be successful in studying a given topic. The author is also an extremely popular lecturer, and finds it important to be in the video series that accompany her texts. Finally, the author finds it important to not only provide quality, but also an abundant quantity of exercises and applications. The book is accompanied by numerous useful supplements, including McGraw-Hill's online homework management system, MathZone as well as ALEKS.
MESSERSMITH is rigorous enough to prepare students for the next level yet easy to read and understand. The exposition is written as if a professor is teaching in a lecture to be more accessible to students. The language is mathematically sound yet easy enough for students to understand |
93.3- "Cracking theCode of Algebra" or "Cracking One's Head on Algebra"
Thursday, April 14, 2011: 10:00 AM-11:00 AM
116 (Convention Center)
Lead Speaker:
ExWksp: Borenson and Associates
How does Hands-On Equations® enable 80 percent of inner-city fourth graders to succeed with such basic equations as 4x + 3 = 3x + 10? If algebra is a foreign language to your students, this session is for you! |
Edie M. Brown / Academic Therapist
People often say that mathematics is a universal language. It is the essence of cognition - thinking with numbers, imagery, and language. For the people who understand mathematics, the language of numbers turns into imagery, letting them calculate and verify mathematics while seeing its logic. Through multisensory instruction, I integrate all of the senses to help create a mathematical picture of concepts and theories, forming the foundation of understanding.
Mathematics
Math literacy means having the mechanics and skills to perform calculations without looking them up and knowing what situations these skills may be appropriate. The true "math literate" is not someone who can successfully complete a test on recently acquired processes and techniques, but one who can apply those abilities to situations that occur in a variety of situations in the long term.
In secondary and postsecondary math, there should be a broad focus encompassing a wide variety of career choices. This includes common foundations of math ideas and applications. Students should be adept at the integration of mathematical concepts. These areas include algebra, geometry, probability, statistics, calculus, and discrete mathematics. |
Ramsey theory is a fast-growing area of combinatorics with deep connections to other fields of mathematics such as topological dynamics, ergodic theory, mathematical logic, and algebra. The area of Ramsey theory dealing with Ramsey-type phenomena in higher dimensions is particularly useful. Introduction to Ramsey Spaces presents in a systematic... more...
Dyslexia, Dyscalculia and Mathematics will be an essential resource for teachers, classroom assistants, and SENCOs who help dyslexic and dyscalculic children with their understanding of mathematics. Written in an accessible style with helpful illustrations, this practical book reveals helpful ways in which to tackle both simple and complex concepts... |
Algebra Unplugged is unlike any math book you have ever read. It explains why algebra problems have to be solved in a particular way. Most textbooks give just a brief introduction to the next set of problems, and then students are expected to complete the practice sets. But how many times has the student (or parent!) not understood what exactly was being asked of them? Understanding the "why" is essential to some children.
The author uses funny, and sometimes odd, comparisons to try to help the student understand the concepts and vocabulary of algebra. I know the author is writing to teens, but some of his comparisons were a little "out there" for me. As the parent of a child with Asperger's who struggles with math, I was glad to have a chance to review this item. I was wondering if it would be able to help him understand algebra a bit better when the time comes in another year or two. I'm going to have to let him be the judge of that. Because he sees things very literally, I'm not sure that he'll understand some of the comparisons.
With more than 250 pages, this book covers any topic you can think of--negative numbers, working with fractions, solving difficult equations, graphing, and much more. If your child needs a little extra help understanding the basic facts of algebra, you may find it helpful to use Algebra Unplugged side-by-side with your chosen textbook. It's often very easy to just "go through the motions" with math and make good grades without understanding what is truly going on, but I don't think that is the best way to learn math. Algebra Unplugged is likely to fill in the gaps for many confused students . . . and probably some parents as well!
Product review by Kris Price, Assistant to the Publishers, The Old Schoolhouse Magazine, LLC, May 2007 |
This lecture introduces the idea of a path integral (scalar line integral). Dr Chris Tisdell defines the integral of a function over a curve in space and discusses the need and applications of the idea. Plenty of examples are supplied and special attention is given to the applications of path integrals to engineering and physics, such as calculating the centre of mass of thin springs.
A basic introduction on how to integrate over curves (line integrals). Several examples are discussed involving scalar functions and vector fields. Such ideas find important applications in engineering and physics.
Vector Addition and Scalar Multiplication, Example 1. In this video, we look at vector addition and scalar multiplication algebraically using the component form of the vector. I do not graph the vectors in this video (but do in others). |
Algebra - Wikipedia, the free encyclopedia Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. For historical reasons, the word "algebra" has several related ... |
MERLOT Search - materialType=Simulation&category=2526&sort.property=overallRating
A search of MERLOT materialsCopyright 1997-2013 MERLOT. All rights reserved.Sun, 26 May 2013 02:03:48 PDTSun, 26 May 2013 02:03:48 PDTMERLOT Search - materialType=Simulation&category=2526&sort.property=overallRating
4434Population Modeling Applet
In this applet, the user applies Euler's Method to modeling population growth using the Malthus exponential model and the Verhulst constrained growth model. After finding the Euler solution, the user can check the solution with the Adaptive Euler Approximation or with a slope field. Also, the user can enter an exact solution obtained from separating variables (or whatever) and again check the Euler solution graphically.Math Warehouse
This site has has interactive explanations and simulations of math from alegrbra to trigonometry. Just click the "interactive" tab on the top left menu and you can choose different simulations. It includes, the complete definition of parabolas, reaching beyond the ability to graph into the realm of why the graph appears as it does. It also has vivid descriptions of angles including circle angles for geometry. It also has calculators for principal nth roots, gdc, matrices, and prime factorization. It's definitely worth checking out. Quote from site: "A parabola is actually a locus of a point and a line. The point is called the focus and the line the directrix. That means that all points on a parabola are equidistant from the focus and the directrix. To change the equation and the graph of the interactive parabola below just click and drag either the point A, which is the focus, or point B, which controls the directrix." This is an interactive site that allows people to change the graph to understand why directrix and focus dictate parabolic graphs. Adding apples and oranges
To calculate the value of an apple and an orange from 2 purchases.About mental arithmetic, with a pre-algebra tool introducing the Gaussian elimination.In the mirror site, there's the Android 2.2 (and up) version of this program.Adding apples oranges and pears
To calculate the value of apple, orange, pear from 3 purchases.About mental arithmetic, with a pre-algebra tool introducing the Gaussian elimination.Adding apples oranges pears and lemons
To calculate the value o apple, orange, pear and lemon from 4 purchases.About mental arithmetic, with a pre-algebra tool introducing the Gaussian elimination.GAUSS - A Graphic Calculator
A free, interactive tool to graph any mathematical function. Developed with Macromedia Flash and Action Script programming language.How much pizza to order?
Application on addition of fractions.Working with italian pizzas, improper fractions and mixed fractions.The program raises the problem from a hypothetical party where the guests ask for a specific pizza fraction. Guests can only choose two types of pizza for each party.The program calls for the total amount you end up ordering pizza.In order to solve the most difficult problems is a tool to help solve, finding the common denominator.There is a video as support material: soundtrack: 'O surdato nnammurato' (written by: Aniello Califano Composer: Enrico Cannio) Voice by: Beniamino Gigli) (In Neapolitan language)This program has a castilian (spanish) version and catalan version in the nummolt site:״Cuanta pizza encargamos?״ & "Quanta pizza encarreguem?״There's a free Android (V2.2 and up) version in the mirror link: (multilingual version: en, ca, es, fr, it) Virtuales para Matemáticas
Los Manipuladores Virtuales son los objetos visuales que ayudan a ilustrar las relaciones matemáticas y sus aplicaciones. estos manipuladores permiten a los estudiantes para examinar visualmente, explorar y desarrollar conceptos.La Biblioteca Nacional de Manipuladores Virtuales (NLVM)de la Universidad Estatal de Utah posee una colección NLVM de más de 100 programas de software interactivo, llamado "applets״, son un medio eficaz para acelerar y profundizar la comprensión de los estudiantes de matemáticas.Razones trigonometricas en un triángulo rectángulo
Compara las razones trigonométricas de un ángulo agudo en el triángulo rectángulo. De una manera didáctica, luego de indicar como se determinan las razones trigonométricas, queda la idea que las razones trigonométricas dependen de los lados del triángulo, siendo que dependen del ángulo. Con esta presentación interactiva queda en evidencia que las razones trigonométricas efectivamente dependen de los ángulos agudos del triángulo rectángulo. Sólo basta seguir las actividades que se encuentran en la parte inferior. Toda la presentación interactiva está realizada en geogebra.Sumando peras y manzanas
To calculate the value of pear and apple from 2 purchases.About mental arithmetic, with a pre-algebra tool introducing the Gaussian elimination.Version in Spanish. In spanish is necessary to change oranges for pears.In spanish the quote is "You can't add pears and apples" (No se pueden sumar peras y manzanas)Para calcular el valor de una pera y una manzana a partir de dos compras.Sobre cálculo mental, con una herramienta de preálgebra que inicia en la eliminación de Gauss-Jordan. (Y la suma de polinomios) |
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Description
Calculus B introduces integration of functions, differential equations, and applications of integration. The student will�calculate antiderivatives using a variety of methods including substitution. The student will evaluate integrals using a variety of methods including numerical integration. Then the student will understand and apply Riemann sums, definite integrals, and the Fundamental Theorem of Calculus. In particular, the student will differentiate and integrate logarithmic, exponential, and inverse trigonometric functions. The student will solve simple differential equations, which can be solved by separation of variables, and�use the calculations�to solve applied problems. The student will use integration to determine the area between two curves, volume, and surface area. Finally, the student will apply integration to determine work, center of mass, and fluid force. |
Exponentials Logarithms
Dawn Leigh Anderson University of Georgia Athens, Georgia Ann Willis Sebrell Godwin High School Richmond, Virginia Rebecca Berg Bowie State University Bowie, Maryland Donald W. Smith Albuquerque Academy Albuquerque, New Mexico
Historical Modules Project
Sponsored by the Mathematical Association of America Funded by the National Science Foundation
Table of Contents
Introduction Suggested courses for using module activities HISTORICAL AND CULTURAL OVERVIEW Timeline Visual timeline From the Top of the Mountain Transparencies John Napier Lesez Euler Portrait of Euler Frontispiece from Euler's Introductio in Analysin Infinitorum Contents of Euler's Introductio in Analysin Infinitorum Euler's thoughts on studying mathematics EXPONENTIAL FUNCTIONS Exponent Notation According to Newton Properties of Exponential Functions Properties of Exponents Compound Interest Activity: The Manhattan Purchase What Will Happen If ... Euler and Population Growth Logistic Growth Newton's Law of Cooling Euler and Exponential Functions: An Investigation Calculating ex Equations Using e 23 26 32 38 45 49 54 62 67 73 78 7 9 10 17 18 19 20 21 22 4 6
2
LOGARITHMIC FUNCTIONS REFERENCES Bibliography Websites 135 137 82 92 102 109 113 118 121 127
3
Introduction
The Principles and Standards for School Mathematics released in April 2000 by The National Council of Teachers of Mathematics, advise the reader that "the secondary school mathematics program must be both broad and deep" (p. 287). In light of this guidance, the study of Logarithmic and Exponential functions plays a very important role in secondary education, particularly in providing a way to model some of the real world problems that students meet outside of the classroom. Such problems are found in diverse places as medicine and finance, as well as, advanced mathematics and politics. More likely than not, if one asked those who studied mathematics in secondary school 25 years ago why they studied exponentials and logarithms, other than to learn about slide rule computation, many would say it was because the topic was a chapter in their textbook that had to be covered. One wonders why something that made a major impact on mathematics four centuries ago became so meaningless to the secondary students of the latter half of the 20th century, the "new math" era. Fortunately, textbooks which embody the Principles and Standards for School Mathematics today reward the student with the theory of exponents and logarithms along with the study of contemporary applications giving meaning to the study. In 1989, the Curriculum and Evaluation Standards for School Mathematics, which The Principles and Standards for School Mathematics were designed to build upon, were released. The document encouraged that attention be focused "on the need for student awareness of the interaction between mathematics and the historical situations from which it has developed and the impact that interaction has on or culture and our lives" (p.6). There is no doubt that the study of the history of exponents and logarithms is a prime example of how "theoretical mathematics (which) has burgeoned in its diversity and deepened in its complexity and abstraction...has become more ... vital to our technologically oriented society" (p.6). Goals of the Module: 1. To understand the historical background leading to the development of exponents, exponentials, and logarithms and to see how mathematical concepts evolve over periods of time. 2. To provide students with opportunities to apply their knowledge of exponents and logarithms to various concrete situations and problems in a historical context. 3. To show students the connection between exponential functions and logarithmic functions, that is the inverse relationship they share with one another. 4. To develop in students an appreciation of the history connected with the development of exponents and logarithms.
4
How to use this module: The authors realize this module contains more material than most teachers can squeeze into an already overcrowded syllabus. It is recommended that the teacher review the material and pick and choose those topics that best fit teacher and students' needs. The material can be made to fit many different types of objectives, and it can be made to fit many different lesson plans. It may even be of interest to teachers of subjects other than mathematics. General historical material may be used for group or whole class discussion. Some quotations and biographical sketches may be more appropriate for introductory material while some are best used within activities. Students should be able to locate the places mentioned on a world map. The written assignments that are required in some of the activities could also include connections with units in social studies, science, economics, etc. If more comprehensive studies are required or desired, the authors have included a bibliography for students and teachers interested in the further study of content in the module. Time Schedule Time needed will depend on the activities and materials the individual teacher chooses to use and the level of the class. The activities may be used to introduce the topic and then further work may be accomplished using exercises found in student textbooks. The teacher is in no way obliged to ask students to perform all these activities; one is not constrained to follow the suggested sequence in which the activities are listed nor to follow the teacher hints. The authors believe that the history will enable students to understand the mathematics better. It is further suggested that the mathematics teacher team teach parts of this module with a history or social science teacher, if such an arrangement can be made. Materials Needed It is highly recommended that the teacher post a world map. Other materials which may be needed in these activities are: machine tape, straightedge, compass, ruler, lined paper, graphing paper, graphing calculator or graphing utility, computers. .Classroom Organization: When using the exploratory activities and projects of this module appears to be the appropriate time to have the class organized into cooperative groups. Using a constructivist approach, the lessons have been created with questions or hints that are designed to guide the student toward "discovering" the relevant results. The quotations of mathematicians may be best presented to the class as a whole on overhead transparencies; copies may be made for personal study by the students.
5
Suggested Courses for Using Module Activities
Activity Exponent Notation According to Newton Properties of Exponential Functions Properties of Exponents Compound Interest Activity: The Manhattan Purchase What Will Happen If....? Euler and Population Growth Logistic Growth Newton's Law of Cooling Euler and Exponential Functions: An Investigation Calculating ex Equations using e Algebra 1 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Algebra 2 Precalculus/ Trigonometry
6
Historical and Cultural Overview
A Time Line for Exponents and Logarithms
This is a brief listing of the contributions of various individuals to the development of exponents and logarithms. Diophantus (Greek living in Alexandria (?), c. 250) gave rules for multiplying and dividing exponents in Arithmetica Al-Karaji (Islamic, d. 1019) Formulated rules for exponents in al-Fakhri Al-Samaw'al (Islamic, c. 1125 1174) used negative exponents in al-Bahir (1144) Nicole Oresme (French, c. 1320 1382) discussed fractional and irrational exponents in Algorismus proportionum (c. 1360) Nicolas Chuquet (French, c. 1445 c. 1500) introduced exponential notation in Triparty (1484) Michael Stifel (German, 1487 1567) used the word `exponent' in Arithmetica integra (1544); correlated the terms of an arithmetic sequence with the terms of a geometric sequence; used negative exponents John Napier (Scottish, 1550 1617) began to work on the concept of logarithms about 1590; published the first description and table of logarithms in Mirifici logarithmorum canonis descriptio (1614); invented the word `logarithm' by combining the Greek words logos (ratio) and arithmos (number) Joost Brgi (Swiss, 1552 1630) independently of Napier also developed the equivalent concept of logarithms; published his table in 1620 in Progress tabulen Henry Briggs (British, 1561 1630) collaborated with Napier; wrote Arithmetica logarithmica (1624) giving values of logarithms base 10 for the integers from 1 to 20,000 and 90,000 to 100,000 to 14 decimal places William Oughtred (British, 1575 1660); using logarithms, designed the rectilinear slide rule around 1621, an early mechanical device used for computation Adriaan Vlacq (Dutch, 1600 c. 1666) wrote Het tweede deel van de Nieuwe telkonst (1627), giving the values of the logarithms of integers from 1 to 100,000 to 10 decimal places Ren Descartes (French, 1596 1650) gave the modern symbolization for exponents in La geometrie (1637)
7
Alfonso Antonio de Sarasa (Belgian, 1618 1667) in Solutio problematis a Mersenno propositi (1649) gave the first identification of the logarithmic properties of the area bounded by the x-axis under the curve y = 1/x Nicolaus Mercator (Danish, c. 1619 1687) in Logarithmotechnia (1668) gave the first published method of computing logarithms using infinite series, based on the work of de Sarasa and Gregory of St. Vincent Isaac Newton (British, 1642 1727) in c. 1650 wrote the inherent calculus relationship between the natural logarithm and the area bounded by y = 1/x Gottfried Wilhelm Leibniz (German, 1646 1716) published Nova methodis (1684) stating the calculus relationship between the natural logarithm and the area bounded by y = 1/x Leonhard Euler (Swiss, 1707 1783) in Introductio in analysin infinitorum (1748) defined exponential functions and then logarithmic functions as their inverses
8
Visual Timeline
0 200 400 600 800 1000 1200 1400 1500 1600
Diophantus (c. 250) rules for multiplication and division of numbers with exponents
Al-Karaji (c. 1000) rules for exponents Al-Samaw'al (c. 1144) negative exponents Oresme (c. 1350) fractional and irrational exponents Chuquet (1484) exponential notation Stifel (1544) negative exponents
1700
Napier (1614) table of logarithms Burgi (1620) table equivalent to logarithms Oughtred (1621) rectilinear slide rule Briggs (1624) Arithmetica logarithmica with Napier Descartes (1637) exponential symbolism de Sarasa (1649) logarithmic properties of the area bounded by the x-axis and the area under the curve y = 1/x Mercator (1668) computing logarithms using infinite series Newton (1670) calculus relationship between natural logarithms and de Sarasa's work Leibniz (1684) calculus relationship between natural logarithms and de Sarasa's work
Euler (1748) defined exponential functions and defined logarithmic functions as their inverses
9
From the Top of the Mountain
It is an example of the truth that from the top of the mountain one can often see how the climb might have been made easier by deviations which to the climbers might well seem to be courting unnecessary difficulties. Lord Moulton, The Invention of Logarithms, Its Genesis and Growth (NTV, 21) The discovery of logarithms is attributed to the Scottish mathematician John Napier; in this section, you will be able to learn about Napier's life, how he formulated the first view of logarithms, and how they were quickly transformed during his own lifetime into something different. The temptation is severe to recast all of Napier's work through the lens of our present interpretation of logarithms, but the actual story is every bit as impressive. Moulton's quote above is useful to keep in mind when we consider the path which Napier took. John Napier was born into an influential and well-to-do family in 1550 in the family estate at Murchiston Castle near Edinburgh, Scotland as the eldest son of Archibald and Janet Napier. Before his death (also at Murchiston) in 1617, he was to become a theologian and an ardent supporter of the Protestant faith, an inventor of weaponry, and an exceptional mathematician. In mathematics, he was known for his work on spherical trigonometry, his invention of an early form of a mechanical calculator, and in particular, his discovery of and work with logarithms. It is also the case that Napier's name was spelled a variety of different ways, including by himself, among them being: Napeir, Nepeir, Neper, Nepper, Naper, Napare, and Naipper. We will adopt the spelling in ordinary usage at the present time. We really know quite little about Napier's early life. It seems he was a student at University of St. Andrews, but he never earned a degree there. He may have traveled abroad, but by 1571 he had returned to the family estates in Scotland where he remained for the rest of his life. He married Elizabeth Stirling in 1572, and after she died in 1579, he married Agnes Chisholm. He had a son and a daughter by his first wife and five sons and five daughters by his second. At this time in Scotland, religious strive was rampant. King James VI of Scotland had designs on the English throne, and there were rumors of the involvement of King Philip II of Spain in those plans. The issue of whether Scotland would become Catholic or Protestant became a widespread concern, and Napier came down very firmly on the side of the Protestants. Scotland suffered through a civil war in 1570-72, and in 1588 the English defeated the Spanish Armada. In 1593 Napier wrote the Plaine Discovery of the Whole Revelation of St. John; to gain a sense of this book, consider the following quote from a letter addressed to King James which formed the dedication: "...let it be your Majesty's continuall study...to reforme the universall enormities of your country, and first...to begin at your Majesty's owne house, familie and court, and purge the same of all suspicion of Papists and Atheists and Newtralls..." (NTV, 42) The book was widely read; between 1593 and 1645 there were 5 English editions, 2 Dutch editions, 5 French editions, and 3 in German. In it, among other things, Napier declared the Pope to be the Antichrist and predicted "that the day of God's judgement appears to fall betwixt the years of Christ 1688 and 1700."
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There is strong evidence that probably as early as 1594, Napier had begun to investigate certain properties of calculations which would make working with the trigonometric values of sines considerably easier to do in order to facilitate the computations involved in astronomy. The German mathematician Michael Stifel wrote the Arithmetica integra in 1544, and it contained a clear statement relating the arithmetic sequence of integers (1, 2, 3...) with the corresponding geometric sequence of 2 raised to those powers ( 2 1 , 2 2 , 2 3 , ...), and this may have suggested the possibility to Napier of using sums to compute products. Stifel even called the powers `exponents' and discussed what we would describe as "when you multiply the numbers, you add the exponents." Napier was also visited by Dr. John Craig, a member of the Scottish royal court, who met had the Danish astronomer Tycho Brahe in 1590 and seen some of Tycho's work with prosthaphaeresis. This was a long-known method of using "product to sum" formulas: namely, such trigonometric identities as 2 cos x cos y = cos(x + y) + cos(x y). Whatever the impetus, in 1614 Napier published the Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Canon of Logarithms) and then posthumously in 1619 the Mirifici Logarithmorum Canonis Constructio (Construction of the Wonderful Canon of Logarithms) was published. It appears as if Napier wished his work to be as widely accessible as possible, and to this end he aided in Edward Wright's translation of the Descriptio into English in 1616. In this translation he explained that his intent with logarithms was to aid in the calculations involved in multiplication, division, and the taking of roots: Seeing Napier coined the word `logarithm' from the Greek words logos (ratio) and arithmos (number), and his original definition takes some getting used to, compared to our present interpretation of a logarithm. Keep in mind as well that he originally was working strictly with the sines of angles. In proposition 26 in the Constructio he states: "The logarithm of a given sine is that number which has increased arithmetically with the same velocity throughout as that with which radius began to decrease geometrically, and in the same time as radius has decreased to the given sine." (Napier, p. 19) T b g a S i
In the figure above, imagine that TS is a line segment of 10 7 units, and that bi is a ray from b. (Napier chose 10 7 for his radius so that he could use values of sines accurate to 7 decimal places but treat them as integers. It is also the case that Napier was one of the first mathematicians to widely use the decimal point as we now write it to separate the integer from the decimal fractional part of a number.) A point g starts at T and moves towards S at a rate
11
equal to its distance from S; at the same time that g starts at T, point a starts at b and moves towards i at the same rate as g's initial velocity. The distance from b to a is then the logarithm of the length from g to S. There are some immediate conclusions that can be reached, stressing the difference from this original view of logarithms and our present perspective. Let Nap. log stand for Napier's definition. First, if g is still at T, then the point a is still at b; therefore, Nap. log 10 7 = 0 and Nap. log is a decreasing function. Second, it can be shown that Nap. log (xy) = Nap. log x + Nap. log y Nap. log 1. Recasting Napier's description in the language of differential equations and x solving, it is possible to demonstrate that Nap. log x = 10 7 log 1 7 ; however, it is critical to 10
e
stress that Napier did not interpret his logarithms in anything approaching the modern sense of an exponent for some given base. It may be reassuring to us to solve a differential equation to obtain something more familiar, but Napier's vision clearly did not start there. From the top of the mountain, we can see that Napier's path and our more recent path of regarding logarithms as exponents clearly both get us to the top, but the paths themselves are very, very different. Napier regarded one of the essential properties of logarithms as "the logarithms of similarly proportioned sines are equidifferent." (Napier, p. 23) In other words, for a given k, log (kx) log x = log (ky) log y. This can clearly be considered, in light of the laws of logarithms ky kx which Napier himself knew, as equivalent to log = log . In another interesting side x y note, Napier's proposition 51 states: "All sines in the proportion of two to one have 6931469.22 for the difference of their logarithms." (Napier, p. 38) This statement is closely related to the fact that the natural logarithm of 2 is .693147180. Henry Briggs (1561 1630) was an English mathematician and professor of geometry at Gresham House, London at the time he met Napier; he went on to become the first Savilian professor of geometry at Merton College, Oxford. After having read the Descriptio, he went to Edinburgh in 1615 to meet with Napier. It is somewhat unusual in scientific history, but the two men immediately hit it off and in fact worked with each other until Napier's death in 1617. In 1617 with the publication of Logarithmorum chilias prima and in 1624 with Arithmetica logarithmica Briggs implemented the changes in Napier's logarithmic scheme that he and Napier had considered. Quoting from the preface to the Arithmetica logarithmica: That these logarithms differ from those which that illustrious man, the Baron of Merchiston, must not surprise you...concerning that matter I wrote immediately to the author himself...I journeyed to Edinburgh, where, being most hospitably received by him, I lingered for a whole month. But as we talked over the change in the logarithms he said that he had for some time been of the same opinion and had wished to accomplish it; he had however published those he had already prepared until he could construct more convenient ones if his affairs and his health would admit of it. But he was of the opinion that the change should be effected in this manner, that 0 should be the logarithm of unity and 10000000000
12
that of the whole sine; which I could not but admit was by far the most convenient. (NTV, p. 126) It seems clear that Napier had already begun to consider a more accessible approach to his logarithms, and that Briggs' suggestions helped moved him in that direction. (As Napier himself observed at the end of his table of logarithms in the Constructio: "Nothing is perfect at birth." (Napier, p. 87.)) Since the logarithm of 1 becomes 0, this simplifies greatly the computational algorithms that Napier had already discovered, bringing them in line with our present theorems of logarithms. In the Constructio, an appendix was included "On the Construction of another and better kind of Logarithms, namely one in which the Logarithm of unity is 0." In this, Napier suggested but in fact did not do a revision of his own system, reflecting what he and Briggs had already determined would make computations with logarithms easier. Napier also observed in the Constructio that in this case, "if two numbers with known Logarithms be multiplied together, forming a third; the sum of their Logarithms will be the Logarithm of the third" (Napier, p. 50) (in other words, when you multiply, you add) and went on to state all the other familiar laws of logarithms which we use. He went ahead and then observed "whence the only difficulty is in finding the Logarithms of the prime numbers" (Napier, p. 51). Briggs went ahead with his own work after Napier's death, and the Logarithmorum chilias prima and the Arithmetica logarithmica contain what is for all intents and purposes the first table of common logarithms i.e., base 10 logarithms. One of the sections of this module discussed how to calculate logarithms somewhat in the method of Napier and Briggs. However, to get a sense of what they went through, consider the following. In constructing his tables, Napier started with the value of his radius 10,000,000 and began successive multiplications of that by 1-10 -7 ; in more modern terms, he generated the terms of a geometric sequence whose first term was 10 7 and whose ratio was 1-10 -7 . Moulton argues that in performing this process some 100 times, Napier became aware that decreasing a number by the same ratio each time was equivalent to repeatedly decreasing a line segment by the same proportional length, possibly giving rise to Napier's definition of a logarithm. It may have also been the case that through the sheer effort of this work, Napier eventually saw the utility of having (in modern terms) log 1 = 0 and log 10 = 1, but that he needed to go through the tremendous labor of the first construction to reach that conclusion. It is therefore tempting now to claim Napier must have therefore seen that he was constructing a system where the logarithm corresponded to the necessary power of 10 needed to obtain the given number, but that again is our view from the mountain, not Napier's. After working through the first 100 terms (his first table), Napier apparently became aware of the tremendous work left, so he changed to a geometric series whose first term was 10 7 and whose ratio was 1-10 -5 . He carried this out for 50 more terms, giving a second table. He then created two other series in a third table, one having ratio 1- 1/2000 and one having ratio 1 - 1/100. Using the data he had accumulated with his three tables, he was then able to calculate the logarithm values of numbers from 10 7 to almost exactly 5 x 10 6 . It is small wonder that Napier worked on his tables for 20 years before he published them! In the Logarithmorum chilias prima and the Arithmetica logarithmica, Briggs approached the problem differently. He began with log 10 = 1 and then began taking successive square roots, using well-known algorithms for the computations needed for the roots. (For
13
example, if 10 = 3.162277, then log 3.162277 = .5.) Once Briggs obtained the logarithms of the primes, it was easy, using the laws of logarithms, to then generate the values of the logarithms of composite numbers. It may be somewhat astonishing to realize the work required by Briggs to obtain, in Arithmetica logarithmica, the logarithms of all natural numbers from 1 to 20,000 and 90,000 to 100,000, carried out to 14 places. (For example, in his calculations to obtain the logarithm of 2 he took 47 successive square roots.) Adriaan Vlacq a few years later completed Briggs' work and completed a table for all integers from 1 to 100,000 to 10 decimals places. (Until recently, these tables formed the basis of all modern logarithm tables.) Napier also used his idea to construct what were called "Napier's bones", a mechanical multiplication table. Within 7 years, William Oughtred used Napier's ideas to construct the first rectilinear slide rules, basically the first pocket calculator. Some two hundred years after Napier's publication, the French mathematician Pierre-Simon Laplace noted that "by shortening the labors, (Napier) doubled the life of the astronomer." Logarithms also present one of the instances when a given discovery must have been in the air; it now seems clear that roughly at the same time Napier was working on his own logarithms, the same essential concept occurred to the Swiss astronomer and watchmaker Joost Brgi (1552 - 1632). While the exact timing of Brgi's work is not entirely clear, he may have in fact worked on the concept before Napier did; however, it is the case that he did not publish his own tables (actually, his were tables of anti-logarithms) until 1620 in Arithmetische und geometrische Progress-Tabulen. Kepler's statement on the priority issue may be of some interest: "...(this) led the way for Justus Byrgius to the very same logarithms many years before Napier's edition, although this man, a procrastinator and guardian of his secrets, abandoned his child at birth and did not rear it for publicity." (Goldstine, p. 22) Brgi generated the values in his own table by starting with 10 8 and then multiplying by 1 + 10 -4 to obtain the successive terms. The ease with which the discovery of logarithms supported the massive calculations needed for astronomy and navigation generated intense interest among mathematicians. The work of Gregory of St. Vincent and his student A. A. de Sarasa by 1647 made the connection between the natural logarithm of a number a and the area bounded by the x-axis, the vertical a 1 lines x = 1 and x = a (for any a > 0) and the curve y = 1/x, what we would now call dx . x 1 (Natural logarithms are contrasted with common, or Briggsian logarithms by their base. If we use Euler's interpretation of a logarithm as the exponent of a given base, then natural logarithms use the number e = 2.718281828... as their base, while common logarithms use 10.) The Danish mathematician Nicolaus Mercator is usually credited with discovering the formula for the x2 x3 x4 expansion of the natural logarithm of x + 1 as the power series x - + - +... by 1668. 2 3 4 As early as 1664 Newton used this same idea to compute the value of the natural logarithm of 1.1 to 68 decimal places. The ground-breaking work of Newton and Leibniz in the calculus quickly suggested applications of logarithms very far indeed from Napier's original view of the technique.
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It turned out that the method that was developed to help with computational drudgery held a vital key in the new mathematics of calculus. The logarithm took on a life of its own, separate from the astonishing levels of sheer, brute arithmetic that it both replaced and which were required to establish the first logarithm tables. The logarithm continued to play a major role in computational sciences, but it had already begun to move away from its roots. As the 18th century progressed, mathematicians became increasingly aware of the necessary changes which calculus had brought to the techniques available to them, and also to the changes which were needed in their more basic structures in order to give the methods of calculus a concrete logical base. One of the more central concepts which evolved was that of a `function', namely the ability to regard an operation as the pairing of the number which was to be operated on with the number which resulted from the operation itself. In more abstract terms, it allowed mathematicians to regard the operation as an object itself, more divorced from the immediate numerical relationships. For example, while mathematicians had worked with the numerical relationships expressed in 2 2 = 4, 2 3 = 8, and so forth, they could now consider the relationship between the variables x and y in y = 2 x . This is an example of an `exponential' function, where the variable is the exponent of some positive base. Leonhard Euler, the enormously influential mathematician of the later 18th century, formulated the relationship between exponential functions and logarithmic functions when he defined log a b = c to be the case if and only if a c = b; therefore, y = a x was equivalent to the statement x = log a y. This was a long way indeed from Napier's original vision of a logarithm as involving the distances traveled by two moving points, yet it has proved to again be of critical use in many situations. It has provided a gateway into the solutions and analysis of many types of problems that would otherwise be extremely difficult to solve otherwise. An important observation therefore can be made that despite the view of many, both in the past and in present times, mathematics is indeed a changing science. A method that began as a strictly computational device had a significant impact on techniques of calculus, was featured in an important relationship of functions, and surfaced as an important problem solving technique, far from its beginnings. Until fairly recently, high school students versed in math and science were given a significant exposure to the computational methods which logarithms opened up and the mechanical devices which logarithms helped create. With the advent of inexpensive hand-held calculators, the computational roots of logarithms have become obsolete, yet the concepts which logarithms have led to continue to be of great utility. We need to be cautious whenever we decide a particular method or approach has reached a dead end; the history of logarithms provides an excellent example of how mathematics grows and develops new relationships for what may appear to be concepts and techniques which have been outgrown, but which can continue to provide important mathematics. It is often the case that when one finally does make it to the top of a mountain, after a long and difficult climb, that you find things up there very different from what you might have imagined. Vistas open up which are almost unimaginable until you reach the peak, and not only the path up but the ground below takes on a very different appearance. You can marvel at the difficulties of some paths and the ease of others, but it is difficult to not appreciate the sheer
15
effort required along any path to make it to the top, and especially for those who pioneered a particular path. So too it is with the concepts and computations of logarithms; whether it be Napier, Brgi, Briggs, or any of the multitude who followed them, we must continue to remember and appreciate their contributions and exertions. Questions: 1. In Proposition 5 of the Constructio Napier writes "In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which is unity with as many cyphers after it as there are figures after the period." (Napier, p. 8) What exactly does this appear to refer to? 2. In Proposition 52 Napier states "All sines in the proportion of ten to one will have 23025842.34 for the difference of their logarithms." (Napier, p. 38) What exactly is Napier saying in modern terms? 3. In an Appendix to the Constructio, Napier writes "Finally any common number being formed from other common numbers by multiplication, division, [raising to a power] or extraction [of a root]; its Logarithm is correspondingly formed from their Logarithms by addition, subtraction, multiplication, by 2, 3, & c. [or division by 2, 3, & c.]." (Napier, p. 51) Translate the meaning of this statement into more modern terms. 4. In 1617, Napier published the Rabdologi where he described the construction and use of what are now called `Napier's bones' or `Napier's rods'. Find out what these were and what they were used for.
16
John Napier
17
Lesez Euler, lesez Euler, c'est notre maitre a tous. Read Euler, read Euler! He is the master of us all.
Pierre-Simon Laplace 1749-1827
18
Leonhard Euler
19
Frontispiece from Euler's Introductio in Analysis Infinitorum
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Table of Contents from Euler's Introductio in Analysin Infinitorum
21
I soon found an opportunity to be introduced to a famous professor Johann Bernoulli, whose good pleasure it was to advance me further in the mathematical sciences. True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I cam across some obstacle of difficulty, I was given permission to visit him freely every Saturday afternoon and he kindly explained to me everything I could not understand, which happened with such greatly desired advantage that whenever he had obviated one difficulty for me, because of that ten others disappeared right away, and this undoubtedly, is the best method to succeed in mathematical subjects. Leonhard Euler Autobiography of 1767
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Exponent Notation According to Newton Teacher Notes
Level: This activity can be used in Algebra 1, and higher level courses Objective: This activity is designed to give an example of an original source of the introduction of exponents that Newton used at Cambridge University in England Materials: Historical Background for each student; Student Page for each student Time Frame: One class period How to Use: The teacher may want to distribute the Historical Background to the students the day before the activity is scheduled or present it as background on the day of the activity. The students may work in groups or individually. If the students need extra time to finish the problems, they could be assigned as homework. It is interesting to read the way that Newton introduced exponents and the terminology that he used. It is also interesting to note that this was part of a lecture Newton gave at Cambridge University - so your students are being exposed to the type of mathematics that was available to university students in the 1680's. An additional assignment would be to have the students write the complete passage in the background information using modern terminology. Bibliography: Newton, Isaac, Arithmetica Universalis, translated by D. T. Whiteside, Cambridge University Press, 1972.
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Exponent Notation According to Newton Solutions
Solutions to questions: 1. 2. If a is a quantity, "If a quantity should multiply itself," means a a. There will be various answers to this. Hopefully most will find it sufficient. The solution to a2b4 if a is 4 and b is 3 a2b4 = 42 x 34 = 4 4 3 3 x 3 3 = 1296 It means that the a is to multiplied together 3 times: a a a It means a a a. Since he is naming the powers, like a5 as cube-square, he is adding the powers. So a8 would be square-cube-cube or maybe square-square-square-square.
3. 4. 5.
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Exponent Notation According to Newton Student Page
Historical Background In Sir Isaac Newton's widely read Arithmetica Universalis, many beginning algebraic topics are included. One such topic is that of exponents. This activity comes directly from one of Newton's Arithmetica Universalis, which is a collection of the material covered in his lectures at Cambridge University in England in the 1680's. The following excerpts are from the 1972 translation by D. T. Whiteside from Newton's original Latin. t From page 61: If a quantity should multiply itself, for shortness the number of factors is usually appended. Thus. Here note that a number written immediately between two variables relates always to the preceding one. Thus `3' in the quantity a3b2does not signify the b2 is to be taken 3 times, but denotes that a has twice to be multiplied into itself. Note also that these quantities are said to be of as many dimensions, powers or `dignities' as they comprise factors or quantities multiplying one another, and that the number appended is called the index of their power or dimension. Thus a2 is of two dimensions or powers, while a3, as its appended number `3' indicates is of three. Also a2 is said to be the square, a3 the cube, a4 the square-square (fourth power), a5 the square-cube (fifth power), a6 the cube-cube (sixth power), a7 the square-square-cube (seventh power) and so forth. Exercises 1. Explain what "If a quantity should multiply itself," means. 2. Is the explanation Sir Isaac Newton gave his students in the following enough for you to write terms using exponents? Give your reasons why or why not. Would you be able to evaluate a2b4 if a is 4 and b is 3? "Thus." 3. According to Newton what does the `3' in the expression a3b2 mean? 4. What does the phrase "a has twice to be multiplied into itself" mean? 5. Using Newton's method of naming powers, how would he describe the eighth power of a, a8?
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Properties of Exponential Functions Teacher Notes
This activity is a great way to give students the opportunity to graph and observe a variety of exponential functions and then summarize their findings. Students then compare and contrast their summaries with the one written by Leonhard Euler, the great eighteenth century mathematician. Level: Algebra 2 or Precalculus/Trigonometry Materials: Student Pages for each student; Calculator; Graphing calculator (opt.) Time Frame: One class period plus homework; 15 minutes the next class Objective: This activity is designed to enable students to develop most (if not all) of the properties of an exponential function and compare their list to the paragraph written by Euler. It is also designed to give students the opportunity of gaining the satisfaction of creating mathematical ideas similar to those of a well-known mathematician How to Use: The teacher may want to give out only the first two pages and do Exercise B with the students as a class exercise, then assign Exercise A and C-E to be completed as homework. The next day the teacher may hand out the third page and ask students to do Exercise F. Bibliography: Euler, Leonhard. Introduction to Analysis of the Infinite Book 1. Translated by John D. Blanton. Springer-Verlag, 1988.
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Properties of Exponential Functions Solutions
A. z -3 -2 -1 0 1 2 2z 1/8 1/4 1/2 1 2 4
y
z -3 -2 -1 0 1 2
3z 1/27 1/9 1/3 1 3 9
y
z
z
B. Z
1 2 8 4 2 1 1/2 1/4
z
z
1 3
z
-3 -2 -1 0 1 2
-3 -2 -1 0 1 2 v
27 9 3 1 1/3 9 v
z
z
27
C. z -3 -2 -1 0 1 2
1z 1 1 1 1 1 1
y
z -3 -2 -1 0 1 2
1.1z .75 .83 .91 1 1.1 1.21
y
z
z
D. Answers will vary, but will be similar to the ones in A through C above. Students might choose negative or zero values for a. If they do, this is a good time to discuss why these values give very strange graphs, graphs which we will usually avoid. In general, in precalculus and calculus classes we generally want to talk just about continuous functions and functions whose inverses are also functions. E. Answers will vary, but will include some of the following observations. All the graphs intersect at the point (0,1). None of the graphs has negative y-values. If a > 0, as z increases, y increases. As a gets larger, the values of y increase faster as z increases. If a < 1, as z increases, y decreases. If a = 1, then y = 1; i.e. the function is constant. If a < 0, the graph jumps back and forth between positive and negative values for y as z increases. F. Answers will vary. Many students will find that their answers include some, if not all, of Euler's observations.
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Properties of Exponential Functions Student Pages
Leonhard Euler wrote the first easy-to-read precalculus book in 1748. First, see how you might figure out properties of exponential functions; then compare your summary to his. Look at a general exponential function, y = a z , and see what happens to the graph as the value of a changes. Use the same scale for all graphs. Also, you may want to use the integers from -3 to 2 for the z-values. The equation uses z instead of x because that is what Euler used. A. Graph y = 2 z and y = 3z using a table of values.
z y z y
B. Also, graph y = (2-
z z y
1 z
)
1 1 1 z = and y = (3- ) = . 2 3
z y
z
z
29
C. Graph y = 1z and also y = (1.1)z.
z
y
z
y
D. Graph two other exponential equations of the form y = a z to enhance your knowledge of this type of graph. State the equations you are graphing. y=
z y
y=
z y
E. Compare the graphs in section A with those in B, C, and D. Summarize what you notice.
F. Compare what you have written with the description Euler writes concisely below.
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Euler gives a magnificent and clear description of an exponential function in his precalculus book, Introduction to Analysis of the Infinite (1748). The values of the exponential a z depend primarily on the magnitude of the constant a . If a = 1 , then we always have a z = 1 , no matter what value is given z to z . If a > 1 , then a will have a greater value if the value of z is greater than it was originally and as z goes to infinity, so also a z increases to infinity. If z = 0 , then a z = 1 ; if z < 0 , then the values of a z become less than 1 and as z goes to z - , a goes to 0. On the other hand if a < 1 but still positive, then the values of a z decrease when z increases above 0. The exponential [ a z ] increases as z 1 increases in the negative direction. Since when a < 1 , we have < 1 , and if we a 1 -z z let = b , then a = b . For this reason we can examine the case when a < 1 a from the case when a > 1 .
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Properties of Exponents Teacher Notes
This activity gives students the opportunity to look at how a master mathematician expressed several ideas about exponents in his great precalculus book, Introduction to Analysis of the Infinite, published in 1748. Level: Algebra 2 or Precalculus/Trigonometry, possibly honors Algebra 1 Objective: To afford students the chance to read a master mathematician's own words about exponents Materials: Student Pages for each student; Problem #8 if desired Time Frame: 20 minutes How to Use: The teacher may want to give this activity as an extension of what has been done in Algebra 1 or as an introduction to rational exponents or as a follow-up activity. The problem 8 is tough; many excellent students will find it a challenge and interesting. Bibliography: Euler, Leonhard. Introduction to Analysis of the Infinite Book 1. Translated by John D. Blanton. Springer-Verlag, 1988 N. Chuquet. in The History of Mathematics, a Reader, edited by John Fauvel and Jeremy Gray, The Open University, 1987, pp. 247-249 from Nicolas Chuquet, Renaissance Mathematician, Reidel, 1985, pp. 144, 151-153 tr. and ed. by H. G. Flagg, C. M. Hay, B. Moss. Newton, Isaac. Epistola Prior (1676) in The History of Mathematics, a Reader, edited by John Fauvel and Jeremy Gray, The Open University, 1987, p. 403 from H. W. Turnbull (ed.), The Mathematical Correspondence of Isaac Newton, II. Cambridge, 1960, pp. 332-333.
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Properties of Exponents Solutions
1. 2.
3
a2 a
4
3. a
3
1 4
= (a 3 ) 4 = 4 a 3
1
1 2 4. a number between 4 and 4 , that is, between 4 and 16. 2 3 5. a number between 2 and 2 , that is, between 4 and 8. 0 1 6. a number between 32 and 32 , that is, between 1 and 32. 1 2 7. a number between 9 and 9 , that is, between 9 and 81. 8a. The topic is exponents. 8b. Proportional numbers are powers of the number. For example if the number is a, then the proportional numbers are a2, a3, a4, a5, etc. The denomination of a number is its exponent when it is expressed as a power of a. 8c. 1, 4, 16, 64, 256 since 40 = 1, etc. 8d. To Chuquet 42 means the number corresponding to denomination 2 in the list. To us it means 4 x 4 = 16. 8e. It tells how to multiply numbers with the same base, in this case 2, by adding their denominations (meaning our exponents). 8f. Chuquet's secret is the same as our answer to 8e.
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Properties of Exponents Student Pages
What would happen if exponents were numbers other than natural numbers? Islamic mathematicians of the 11th century were the earliest people to use negative numbers as exponents. Nicole Oresme, a French mathematician associated with the University of Paris around 1350, had some ideas about fractional exponents and even tried to explain what it would mean to have an irrational number as an exponent. In 1655, John Wallis, an English mathematician, gave a clearer explanation of fractional exponents and a few years later, Isaac Newton was able to make use of them with ease. Here, however, we look at the explanation of exponents of various types by the prolific mathematician Leonhard Euler in 1748: Let the exponential to be considered be a where a is a constant and the exponent z is a variable. Since the exponent z stands for all determined numbers, it is clear at least that all positive integers can be substituted for z to give determined 1 2 3 4 5 6 values for a , a , a , a , a , a , etc. If for z we substitute the negative integers -1, -2, -3, etc. we obtain
z
1 1 1 1 1 1 , , , , , , etc. If z = 0, a1 a2 a 3 a4 a5 a6 then we have a0 = 1. If we substitute a fraction for z , for instance 1 1 2 1 3 , , , , , etc., we obtain 2 3 3 4 4
the values more values, since the extraction of roots gives several values. However, we will consider only their primary [now called principal] values, since
1 2 1 3 a, a 3 , a 3 , a 4 , a 4 , etc. These symbols can have two or
a a and 3 a ... In like manner we let z take irrational values, even though it is more 7 difficult to understand this concept. ... Thus a has a value which lies between 2 3 a and a ...
from Introduction to Analysis of the Infinite by Leonhard Euler Some statements in that selection probably look more familiar than other statements. For example, in the first sentence Euler talks about negative exponents. In the second sentence he
1 tells about square roots so that we understand a 2
z
5 2 lies between is to be thought of as a ... function. For this reason. a
2
= a
1 and by implication, a 3
= 3 a , etc.
We can use this to help us rewrite expressions with rational exponents as expressions with radicals instead. Complete the following using Euler's idea to rewrite the expressions using radicals.
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2 1 1 2 3 = a 3 = a 2 3 = _________________ 1. a
( )
1 2. a 4
= _________________
3 3. a 4 =__________________= __________________=__________________
The sixth and seventh sentences refer to situations such as the fact that a given positive number has two different square roots, a positive one and a negative one. But since we want to define an exponential function, we will consider only the positive value.
5 lies between the two 2 7 successive integers 2 and 3. Finally, Euler also suggest that one can estimate the value of a when one knows that 7 lies between the two successive integers 2 and 3.
Use this method to estimate the following by stating the numbers between which the given number lies. Example. 4
5 2 3 2 = _______________________________________________________________ 4. 4 2 3 = a number between 4 and 4 , that is, between 16 and 64.
5 2 , since The next sentence suggests a method to estimate the value of a
5. 2 6. 32
5
= ______________________________________________________________
4 5 = ______________________________________________________________
3 7. 9 2 = ________________________________________________________________
Clearly this last observation by Euler does not give very precise estimates, but it is insightful!
8. Familiar, Yet Different, Yet Related. You are a mathematics historian; you even consider yourself a detective of sorts! As you read the selection below, think about these questions: What familiar concept is being discussed? The notation looks familiar, but wait, does it have the same meaning? Or is it a little different?
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Nicolas Chuquet, a mathematician who taught in Lyons, France, wrote these words sometime during the fifteenth century. He invented the notation, but its meaning has evolved somewhat since then. To understand the reason why denomination of number is added to denomination, ..., it is necessary to set down several proportional numbers beginning with 1 and arranged in a continuous sequence, like 1, 2, 4, 8, 16, 32, etc. or 3, 9, 27, etc. Numbers 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 Denomination 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Now it is necessary to know that 1 represents and is in the place of numbers, whose denomination is 0, 2 represents [...] the first terms, whose denomination is 1. 4 holds the place of the second terms, whose denomination is 2. And 8 is in the place of the third terms, 16 holds the place of the fourth terms, 32 represents the fifth terms, and so for the others. Now whoever multiplies 1 by 1, it comes to 1, and because 1 multiplied by 1 does not change at all, neither does any other number when it is multiplied by 1 increase or diminish, and for this consideration, whoever multiplies a number by a number, it comes to a number, whose denomination is 0. And whoever adds 0 to 0 makes 0. Afterwards, whoever multiplies 2, which is the first number by 1, which is a number, the multiplication comes to 2; then afterwards, whoever adds their denominations, which are 0 and 1, it makes 1; thus the multiplication comes to 21. And from this it comes that when one multiplies numbers by first terms or vice versa, it comes to first terms. Also whoever multiplies 21 by 21, it comes to 4 which is a second number. Thus the multiplication amounts to 42. For 2 multiplied by 2 makes 4 when adding the
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denominations, that is 1 with 1, makes 2. And from this it comes that whoever multiplies first terms by first terms, it comes to second terms. Likewise whoever multiplies 21 by 42, it comes to 83. For 2 multiplied by 4 and 1 added with 2 makes 83. And thus whoever multiplies first terms by second terms, it comes to third terms. Also, whoever multiplies 42 by 42, it comes to 16 which is a fourth number, and for this reason whoever multiplies second terms by second terms, it comes to fourth terms. Likewise whoever multiplies 4 which is a second number by 8 which is a third number makes 32 which is a fifth number. And thus whoever multiplies second terms by third terms or vice versa, it comes to fifth terms. And third terms by fourth terms comes to 7th terms, and fourth terms by fourth terms, it comes to 8th terms, and so for the others. In this discussion there is manifest a secret which is in the proportional numbers. It is that whoever multiplies a proportional number by itself, it comes to the number of the double of its denomination, as, whoever multiplies 8 which is a third number by itself, it comes to 64 which is a sixth. And 16 which is a fourth number multiplied by itself should come to 256, which is an eighth. And whoever multiplies 128 which is the 7th proportional by 512 which is the 9th, it should come to 65536 which is the 16th. from N. Chuquet, pp. 144, 151-153 a. Why is this essay included in this activity? In other words, what is the topic?
b. What does Chuquet mean by "proportional numbers?" "denomination of a number?"
c. What are the first five "proportional numbers" starting with 4?
d. Chuquet employs what we call exponent notation, but has it mean something different. What does 42 mean according to Chuquet?
What does 42 mean to us?
e. In the last sentence of this selection from Chuquet there is a description of a familiar property of exponents. State the property. f. What is Chuquet's "secret?"
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Compound Interest Activity: The Manhattan Purchase Teacher Notes
Introduction: Compound interest is a useful application of exponential functions. Compound interest holds much promise for applying mathematical ideas to relatively important everyday applications, such as the amount of money an investor makes on a particular investment, a notion Americans are familiar with living in a capitalist society. The need to understand compound interest also arises in calculating the interest that accrues on credit card purchases, car, home, and boat loans, and financial investments. In the following activity, a historical example is used as a springboard to discuss compound interest. The historical context is centered on the European colonist's purchase of the island of Manhattan from the Native Americans. Level: Algebra I, Algebra II, Precalculus (Exposure to the compound interest formula is a prerequisite.) Objective: To investigate an application of exponential growth, compound interest, using an situation taken from the historical context of the Dutch "purchase" of the island of Manhattan from the Native Americans. Materials: Student Page; Calculator; Graphing calculator and Internet access (optional) Time Frame: Two 45-minute class periods; extended project would be done outside of class How to Use: To begin the discussion of exponential growth and compound interest embedded in a historical context, provide students with the historical opener on an overhead transparency (see Transparency #1). Have students read and reflect on this historical situation. You might ask students to develop their own questions that arise from the historical situation presented. Be prepared to consider social and historical questions as well as mathematical related questions. A second option would be to use the guiding questions in the student activity to develop the lesson. One way to begin the lesson is to look at question 1 involving compound interest. First, it might be helpful to review the general compound interest formula. You might also show students the derivation for the general compound interest formula. It might be helpful to derive the compound interest formula using an example first. Then show the students the formal algebraic derivation for the compound interest formula. Consider the following example involving compound interest. You have $100. You want to invest your money in a fund that guarantees 8% interest compounded quarterly over six years. You have the following given information: r = 8% = 0.08 annual interest rate t=6 time in years
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n=4 number of compounding periods per year nt = 24 number of compounding periods P = $100 principal amount of money being invested S n = total amount of money at the end of n periods Because we are looking for the total amount of money at the end of 24 periods, we want to find S 24 . Guide the students through the derivation using the above given information. The total amount of money we have at the end of the first compounding period is 0.08 S1 = 100 + 100 = 100(1 + 0.02) . 4 The total amount of money we have at the end of the second compounding period is 0.08 2 S 2 = S1 + S1 = S1 (1 + 0.02) = 100(1 + 0.02)(1 + 0.02) = 100(1 + 0.02) . 4 The total amount of money we have at the end of the third compounding period is 0.08 2 3 S3 = S 2 + S2 = S 2 (1 + 0.02) = 100(1 + 0.02) (1 + 0.02) = 100(1 + 0.02) . 4 Explain to your students that we are looking for S 24 ; that is, the total amount of money we have accumulated at the end of the 24th period. Next guide students to the algebraic derivation of the general compound interest. We have the following given information: r = annual interest rate t = time in years n = number of compounding periods per year nt = total number of compounding periods P = principal amount of money being invested S n = total amount of money at the end of n periods r = interest rate per compounding period n nt = total number of compounding periods. The total amount of money we have at the end of the first compounding period is r r S 1 = P + P = P 1 + n n
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The total amount of money we have at the end of the second compounding period is r r r r r r r S 2 = S1 + S1 = P1 + + P1 + = P1 + 1 + = P1 + . n n n n n n n The total amount of money we have at the end of the nt -th compounding period is S nt = P1 + r . n
nt 2
Using the derivation of the general compound interest formula, lead students to the exponential equation that describes the situation described in the historical opener. Let P = 24, r = 7%, n = 1 (compounded annually) and t = 372 (the number of years since 1626 this is being calculated for the year 1998) r y = S nt = P1 + n 0.07 y = 241 + 1
nt
(1)( 372 )
y = 24(1.07) 372 $2,046,334,096,410
Guide your students to discover that the initial formula for y represents an exponential growth function of the form, f ( x) = ab x where a is the initial amount of money, b is the growth factor, and x is the total number of compounding periods. Stress that because b > 1 in compound interest problems, then the function is an example of exponential growth. Guide students to change a and b in f(x) = abx to see the effect on the curve. Encourage students to examine the function for a > 0, a < 0, b >1, and 0 < b < 1 and to write a complete explanation of the results they found. Have students solve additional examples of exponential growth problems either using variations on the historical opener problem or problems from a textbook. Give students an opportunity to create their own problems and to explain them to the class.
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Compound Interest Activity: The Manhattan Purchase Solutions
1. As indicated above, y = 24(1.07) 372 $2,046,334,096,410 2. The 3.8 billion dollars is less than the amount found in number 1, so perhaps, Minuit paid too much, or the interest rate is too high, or the tax assessment value is too low. 3. Minuit should have paid a great deal more. Land in the Old World was dear and certainly, an equivalent amount of land, in say Holland or England, would have been very expensive. But given that the amount found in number 1 is higher than the tax valuation in number 2, find a payment that would give the tax valuation. This time solve for the initial amount given that the final amount is 3.8 billion dollars and the interest rate is 7% with 372 years (this solution is being done in 1998) with annual compounding yields: 3.8 10 = a(1.07)
9 372
3.8 10 9 a= .044567 or about 4.5cents. (1.07) 372
4. Have students find the interest rate in 1626 via the Internet. The Federal Reserve Board holds past and present interest rates. 5. f ( x) = ab x where a is the initial amount of money, b is the growth factor, and x is the total number of compounding periods. 6. The students' explanation should include cases such as: a > 0, b > 1 exponential growth function a > 0, 0 < b < 1 exponential decay function a < 0, b > 1 exponential decay function a < 0, 0 < b < 1 exponential growth function
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Compound Interest Activity: The Manhattan Purchase Student Page
Historical Opener, the chief of a confederation of the Algonquian tribes of the north Atlantic coast, for the purchase of the island. After the final negotiations were agreed upon, Minuit purchased the island of Manhattan (New York City) by giving them trinkets valued at approximately 60 guilders (currency of the Netherlands), or $24. 1. Suppose that the Native Americans invested the money they received from the purchase of the land at 7% compounded annually. How much money would they have today? 2. Now consider that the current tax assessment value of Manhattan is 3.8 billion dollars. Discuss the implications of the land deal based on this information. 3. Based on question two, what might have been a fair price for Minuit to pay? 4. Find the actual interest rate at the time of the purchase in 1626. Using this interest rate, calculate the amount of money the Native Americans would have today based on that interest rate. 5. What is the general equation of an exponential function? 6. What effect do the variables have on the graph of the function?
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Historical Opener The Manhattan Purchase--the chief of a confederation of the Algonquian tribes of the north Atlantic coast--for the purchase of the island. After the final negotiations were agreed upon, Minuit purchased the island of Manhattan (New York City) by giving them trinkets valued at approximately 60 guilders (currency of the Netherlands), or $24.
Transparency #1
43
Suppose the natives invested the money they received from the purchase of the land at 7% compounded annually. How much money would they have today?
Transparency #2
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What Will Happen If....? Teacher Notes
Introduction: This activity helps give students exposure to the number 2.71828 in a context with which they are already familiar, to make e more plausible to students later. Level: Algebra 2 or Precalculus/Trigonometry (prerequisite: the Compound Interest Activity from earlier in this module or work with the compound interest formula from a textbook) Objective: To relate 2.71828 to the compound interest formula thereby setting the stage for e to be discussed next Time Frame: 10 minutes in class, overnight to research the bonus question Materials: Student Pages for each student; calculator; graphing calculator (opt.) How to Use: The teacher may want to use this activity as an introduction to e . The bonus question should be discussed after the students have a chance to answer it if they choose to do so.
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What Will Happen If...? Solutions
1. N 1 2 3 4 5 10 50 100 1000 10,000 100,000 1,000,000 10,000,000 100,000,000 2 2.25 2.37037 2.44241 2.48832 2.59374 2.69159 2.70481 2.71692 2.71815 2.71827 2.71828 2.71828 2.71828 S
2. Increases (but at an ever slower rate) 3. S seems to get closer and closer to 2.71828, rounded. [Note: some of your students may try x = 10,000,000,000,000, which yields S = 2.7606 or x = 100,000,000,000,000, which yields S = 1; these are aberrations of the calculator. For example, what is happening with x = 1014 is that the calculator evaluates the expression 1 1 + 1014
14
and rounds
1 1014
to zero first, then adds 1 and raises the sum to the 14th power.]
Bonus: The number approaches e as n increases without bound.
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What Will Happen If...? Student Pages
What will happen if we use some very special values in the equation we developed for compound interest? r nt where S = amount of money after t years S = P 1 +
n
P = principal r = annual interest n = number of periods per year t = number of years Substitute interest rate = 100% = 1 time = 1 year principal = $1.
Then, S = 1 1 +
1 n1 = 1 + n
1 n . n
1. Create a table of values for n and the corresponding values of S. (Hint: You may use your calculator or graphing utility to find the values of S to five or six decimal places.) n 1 2 3 4 5 10 50 100 1000 10,000 100,000 1,000,000 S
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2. As n increases, the value of S ________________________(increases or decreases). 3. What do you think will happen to the value of S as n approaches infinity?
Bonus: The answer to #3 is a very famous number. Research its name. [Hint: the first person to name it is Leonhard Euler.]
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Euler and Population Growth Teacher Notes
Introduction: This activity highlights a problem about population growth, which was taken from Introductio in analysin infinitorum (Introduction to Analysis of the Infinite) (1748). This famous mathematics book was written by Leonhard Euler (1707-1783) in the middle of the eighteenth century. The fact that this problem appears in Euler's Introductio in analysin infinitorum shows that issues of population growth were of concern to the people living in the eighteenth century. Euler initially solved the problem using logarithms. However, this activity is designed to solve the problem using exponential functions first. Later, when logarithms and logarithmic functions are developed in the module, you can show students how to solve this problem using logarithms. Showing students how to solve this problem using both exponential and logarithmic functions provides a wonderful opportunity to reveal the inherent connection between exponential and logarithmic functions. Level: Algebra I, Algebra II, Precalculus (Prerequisites: Exposure to exponential functions and their graphs) Objectives: To develop students' understanding of exponential growth functions by using a problem of population growth taken from one of Euler's most famous works, Introductio in analysin infinitorum, which was written in the middle eighteenth century. To show students that mathematics was and is done in specific historical and social contexts. Materials: Student Pages; Calculator; Graphing calculator and Internet access (optional) Time Frame: One 45-minute class period. The extended project is to be completed outside of class. How to Use: Consider using cooperative groups to do this activity. At the very least, consider pairing up students to work on this activity. If students have not had much work with exponential functions, you might need to help them find the exponential Bibliography: Euler, Leonhard. Introduction to Analysis of the Infinite, Book 1, translated. by John D. Blanton (New York: Springer-Verlag, 1988). pp. 75-91. Katz, Victor. A History of Mathematics: An Introduction (New York: HarperCollins, 1998).
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Euler and Population Growth Solutions
1.
y = f ( x) = ab x 1 y = 100,0001 + 30 2. Because we are dealing with people, stress to students that they need to round to the nearest whole number.
x 10 20 30 40 50 60 70 80 90 100 y = 100,000(1+1/30)x 138,805 192,668 267,432 371,208 515,255 715,198 992,729 1,377,955 1,912,668 2,654,874
x
3. a. The graph should represent an exponential growth function. b. If the initial population is increased or decreased, the population is still increasing by the one-thirtieth chosen by Euler. If the initial population is greater than the 100,000, the population after 100 years will be greater than 2,654,874; If the initial population is less than the 100,000, the population after 100 years will be less than 2,654,874. c. If the growth rate is larger than one-thirtieth a year, the population after 100 years will be greater than 2,654,874; if the growth rate is smaller than one-thirtieth a year, the population after 100 years will be less than 2,654,874. d. The relation will now be g(x) = 100,000(1 1/30)x. And the population after 100 years will be g(100) = 100,000(1 1/30)100 3370. Extended Project: One place to start is Euler's biography found at
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Euler and Population Growth Student Pages
Introductio in analysin infinitorum (1748) Example II on page 85 Leonhard Euler (1707-1783) If the population in a certain region increases annually by one-thirtieth and at one time there were 100,000 inhabitants, we would like to know the population after 100 years.
The above population growth problem was taken from Introductio in analysin infinitorum (Introduction to Analysis of the Infinite) (1748), a famous mathematics book written by Leonhard Euler (1707-1783. The fact that this problem appears in Euler's Introductio in analysin infinitorum shows that issues of population growth were of concern to the people living in the eighteenth century. Can you think of reasons why population growth would have been an issue in the eighteenth century? Euler initially solved the problem using logarithms. However, you will solve the problem using exponential functions first. Later, when you learn about logarithms and logarithmic functions, you can solve this problem using logarithms. 1. Based on your knowledge of exponential functions, write the relation described in the above problem as an exponential function. 2. For the exponential function you found in problem 1, make a table of values for years 10, 20, 30, ...100. (If possible, use the table menu of a graphing calculator) x 10 20 30 40 50 60 70 80 90 100 f(x) = ?
51
3. Graph the exponential function. Explore its curve.
4. Answer the following questions based on your exploration of the curve. a. What kind of exponential function does this function represent? b. What happens if the initial population is increased or decreased? c. What happens if the growth rate is increased or decreased? d. What happens if the population decreases by one-thirtieth each year? Extended Project: Discover where Euler was living in 1748. Determine the population of that area during the time he lived there. After you determine the population in 1748, determine today's population of that area based on the above exponential model. Compare with the actual number of people living there now. How close was your approximation?
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If the population in a certain region increases annually by one-thirtieth and at one time there were 100,000 inhabitants, we would like to know the population after 100 years.
Introductio in analysin infinitorum (1748) Example II on p. 85 Leonhard Euler(1707-1783)
Transparency #1
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Logistic Growth Teacher Notes
Introduction: This teacher-guided activity serves as a superb way to show students what can happen as time goes by and more and more data becomes available to use in making predictions. It helps explain why a mathematical model may need to change with the passing decades. Level: Algebra 2 or Precalculus/Trigonometry (Prerequisites: Using various regression equations on a graphing utility to describe data) Objective: To study various regression equations culminating in the need for the logistic equation to model real-world data Materials: Overhead projector; transparency made from the Population Data; Graphing calculators with regression equation capabilities, including the logistic function Time Frame: One class period How to Use: This activity works very well as a teacher-guided discussion with the everexpanding data set being modeled by successive functions. It helps the students see that as more and more data is available a model may need to change in order to make better predictions. 1. Perhaps begin by asking students which statistics about New York City might be valuable and why. Lead them to realize that population trends might prove useful in city planning in such areas as number and size of schools, sewage treatment and garbage collection, road construction and maintenance. 2. Using the historical information in the lesson discuss past efforts in history to solve similar prediction problems. 3. Ask students to look at the initial data set and imagine that they are city planners for NYC after the 1820 census and that they need to predict the population for the next few decades. Have them enter the data into lists on their calculators or computers, make a scatterplot of the data and fit an appropriate regression function to the data. Then have them predict the population for the next decade. 4. Ask students to pretend that a couple decades have passed and ask if their prediction was good for 1840. Now have them add this data to their previous lists, make another scatterplot incorporating all the data from 1790 through 1880, choose an appropriate regression function and predict the population for the next decade. Some students may try a quadratic regression function. It actually appears to fit the data better than the exponential function. You may want to tell them that fitting the data is not the only criterion for choosing a particular regression equation. For example, the quadratic function would imply that at first the population of NYC would decrease during the first decade under consideration. While that is
54
possible, it doesn't fit the real data that is ever increasing. They may point out that the exponential function overestimates the population in the final decade of this data set. That will point the way to needing a different regression function, a suspicion that is confirmed by the next data set. 5. For the third data set, after plotting additional points, students may suggest that it is composed of several parts, such as exponential for the first third, quadratic for the second third, and logarithmic for the final third. You can then suggest that this S-shape is characteristic of the logistic function and students may use their TI-83 graphing calculators to fit a regression function to the data. 6. For the final data set, after plotting the additional points, students may decide it is best described as a cubic function, but you may want to point out that a cubic equation would have predicted a negative population for the first three decades of the 1800's. 7. A discussion of possible reasons why the population growth rate slows down in the final set is a lively finale to this activity. Students may bring up such possibilities as population density, insufficient water/sewage treatment plants, insufficient medical services for a greater population, not enough roads or public transportation, not enough schools to support a greater population. Bibliography: John L. Harper, Population Biology of Plants, Academic Press, 1977 Thomas Malthus, An essay on the principle of population, etc., Penguin, 1985 Raymond Pearl and Lowell Reed, On the rate of growth of the population of the United States since 1790 and its mathematical representation, Proceedings of the National Academy of Sciences, 6 (6): 275-288, 1920. Bonnie Shulman, Using Original Sources to Teach the Logistic Equation, The UMAP Journal, 18 (4): 377-402, 1997 _____________, Math Alive! Using Original Sources to Teach Mathematics in Social Context, PRIMUS, VIII (1): 1-13, 1998 U. S. Bureau of the Census, Population at Selected Ranks from 1st to 100th of the Largest Urban Places: 1790 1990, 1998
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Logistic Growth Student Pages
Overview:
Using census data on the population of New York City from 1790 until 1990, we will consider three different modeling techniques and see how well they fit the given data. We will also consider the entire concept of using mathematics to model a real-world situation. An important use of mathematics is its ability to take current numerical information, or data, and then make predictions of future behavior based on that data. A useful field of investigation is that of population growth, with its obvious impact on city planning, environmental concerns, allocation of resources, and many other areas. Data from previous years can indicate important trends; social scientists, economists, and many other professionals have discovered that one can also use data to predict future growth patterns. We will use data from the U. S. Bureau of the Census on the population of New York City to investigate three different predictive models. One simple possibility for population growth is that the growth is constant; in other words, that the population grows by the same amount each year. The population can then be depicted mathematically with an "arithmetic" sequence, a sequence of numbers where any two successive terms differ by the same amount. An example of an arithmetic sequence is 1, 5, 9, 13, 17... A more complicated model uses the premise that the growth is proportional to the present population; in other words, the larger the population, the more people are added each year. In this situation, we often say that the growth rate, say r%, is constant. The population in this case can be represented by a "geometric" sequence, one where any two successive terms have the same ratio. An example of a geometric sequence is 1, 2, 4, 8, 16... A third model uses the idea that the growth of the population will be proportional to two factors: the current population and also how close the population gets to some ceiling or maximal size. This model implies that initial growth may be faster, but as the constraints of resources come into play, then the growth will slow down. The full mathematical development of these three models requires techniques of calculus, and therefore we will not dwell unnecessarily on the methods used to obtain the equations. However, we can briefly summarize the central mathematical conclusions. In the first model the population y will be a linear function of time t, that is, the population y will be given by the familiar formula y = mt + b. The second model results in an exponential model of growth, in which the population y is expressed by y = y 0 e rt where y 0 is some given initial value, r is the rate, t is the time and e is the familiar 2.718... The third model results in the logistic model of c , where for the additional variables, c represents the maximal population growth, y = 1 + ae ( - bt ) which can be sustained (called the "carrying capacity") and a and b are constants which emerge from a particular set of data.
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Let's consider the use of these three models with the same overall set of data, restricted to different time periods. The U. S. Bureau of the Census contains a myriad of information about populations of various segments of the United States from 1790 until the present. (The Constitution of the United States orders that a count of the population be held every 10 years in order to correctly apportion the seats in the House of Representatives; this raw data can then be used in variety of ways to gain interesting information about patterns of growth in the U. S.) One such set of information is the population of New York City as measured every 10 years in the Census. For the period of 1790 to 1820, the Census gives the following counts: Year Population 1790 33,131 1800 60,515 1810 96,373 1820 123,706
Using your graphing calculator, plot those four points using the Year as the x-coordinate and the Population as the y-coordinate and a suitable viewing rectangle; you should obtain something which resembles Chart 1. It should be pretty clear that these four data points look like they almost fall on a line; you can use the capabilities of your particular calculator to discover the regression equation line which will fit the data which gives you an answer of y = 3075.83x 5,473,441.9. Graph that equation over the set of data to observe how good a fit it is. However, you test a model by not only how well it fits given data but also how well it will predict the future. The count for 1830 was 202,589, and the count for 1840 was 312,710. According to the equation above you should expect the count for 1830 to be 155,327 and the count for 1840 to be 186,085; clearly, the mathematical model is at odds with the real data. The linear model does not account for the complexity of the information and significantly undercounts the real population. Let's consider more of the set of data next; to the data given above for 1790 to 1820, expand the data set to include the following through 1880: Year Pop. 1830 202,589 1840 312,710 1850 515,547 1860 813,669 1870 942,292 1880 1,206,299
Graphing these in a suitable viewing rectangle should yield something resembling Chart 2. This graph should look like the graph of an exponential function, examples of which you have already seen. You can again use the capabilities of your calculator to obtain an exponential function that fits the data, yielding y = (11162 10 -27 )e .040578 x . Graph that function over the data set, and . again note the fit. Using our previous approach, the census count for 1890 was 1,515,301 and the count for 1900 was 3,437,202; our equation predicts that the figure for 1890 should be 2,265,258 and for 1900 it should be 3,398.969. The exponential equation fits the given data well, but again it is less useful when we try to expand it to cover a broader scope. Now we will consider more data; we expand the data set to include the population through 1950:
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1890 1,515,301
1900 3,437,202
1910 4,766,883
1920 5,620,048
1930 6,930,446
1940 7,454,995
1950 7,891,957
Graph the population from 1790 through 1950, and note how the graph initially increases very rapidly, but then levels off and increases much more slowly. Compare your graph to Chart 3. This `S' shape is highly suggestive of the model of logistic growth. If your calculator is advanced enough, you can use the set of data from 1790 to 1950 to obtain a logistic equation that 8,592,944.078 models the information; if not use y = . Again, map this function 1 + (12224 10 50 )e ( -.060438 x ) . over the data set, and observe the fit. To get a better sense of the implication of such a model, extend the domain of your graph to 2050 and the range to 9,000,000 and now consider it. Notice how the curve flattens out. What is the implication of such a graph? Now, let's consider the full range of data through the 1990 census. Include in your data the following: Year Population 1960 7,781,984 1970 7,894,862 1980 7,071,639 1990 7,322,564
The graph of the full set of data is in Chart 4; make sure your graph is similar. Note the strong fit of the curve through the data of 1950, and then how the curve fails to model the data, particularly in 1980 and 1990. Some conclusions: The first matter for us to consider here is the history of some of these models. The simplest concept of growth is one that results in an arithmetic pattern; early scientific consideration of population growth of crops and domesticated animals demonstrated this model was not sufficient to explain the observed growth. Thomas Malthus (1766 1834) was an English clergyman and political economist who wrote An Essay on the Principle of Population in 1798. In there, he wrote that "population, when unchecked, increases in a geometrical ratio." When this idea was investigated mathematically, it resulted in the exponential model expressed above. Pierre Francois Verhulst (1804 1849) was a Belgian mathematician and statistician who is credited with first writing about the logistic model of growth in 1845, when he attempted to take into account factors, such as limited resources, that would also tend to slow down growth after time. (Our logistic equation is written in the form y = f(t); Verhulst wrote it in the form t = h(y). This form requires the use of logarithms, and apparently he coined the word `logistic' to suggest the connection between the curve and the characteristic shape of a logarithm.) Next, we need to consider the whole concept of a model in and of itself. First, one point is to recognize that while a mathematical model may fit a particular set of data, it is a somewhat dangerous perspective to feel that the model will always work. We considered three models linear, exponential, and then logistic which did a reasonable job with one set of data, but then became less accurate as the data encompassed a larger time frame. There is a danger in assuming that we can ever exactly describe a living situation with a mathematical model. There are
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techniques of chaos theory that we could call on to account for the fluctuations in the data we considered in the periods of time following 1950, but what is to say that the future data will fit this model as well? It may be that all we can ever hope for is a rough description of a living entity without ever attaining any sort of permanent exactness. Second, mathematics expands to generate more sophisticated techniques as more data is made available. Malthus had a reasonable perspective, given his time; Verhulst developed a better one, but not necessarily because he was somehow smarter he had access to more information. Complicated models require equally complicated mathematics even to obtain approximate correlation with real-world data. We can even incorporate the new field of chaos theory to further refine our model. Third, we need to be careful about assuming that just because we can describe a set of data mathematically, we have somehow discovered what is creating the data. The Americans Raymond Pearl and Lowell Reed independently rediscovered Verhulst's analysis of population growth in 1920. They cautioned: "No process of empirically graduating raw data with a curve can in and of itself demonstrate the fundamental law which causes the occurring change." In other words, do not mistake an ability to describe a set of data with a mathematical formula with understanding what is creating the data itself. It may be that the mathematics helps us to see some underlying principles, but the ability to extrapolate the future from a set of data is at best an inexact procedure. However, in the article which they wrote to discuss the mathematics of logistic growth and its application to population modeling, Reed and Pearl concluded: "...the hypothesis here advanced as to the law of population growth [that of logistic growth]...so closely describes the known facts regarding the past history of that growth, as to make it potentially profitable to continue the mathematical development and refinement of the hypothesis further." It may still be very useful for us to study the particular model, realizing that we still do not have a full mathematical description of the phenomena being studied. In an article appearing in the March, 1998 edition of the journal Primus on the logistic curve, Bonnie Shulman warns that while the difference between the mathematics of a model and the underlying dynamics may be clear to mathematicians, "important policy decisions are made based on the predictions arising from models." It is perfectly understandable that we would use mathematical models to try to predict the future, but those predictions must always be taken with a sense of caution. Biologists have studied population growth of various species extensively, and as such have repeatedly come across examples of logistic growth. Even at that, in Population Biology of Plants, John L. Harper states: "The logistic curve remains central to population biology, not because it describes how populations behave but rather because it provides a standard base of ideal behavior against which the reality can be judged and measured." (Harper, p. 3.)
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Chart 1:
150000 100000 50000 0 1790 1795 1800
1790-1820
1805 year
1810
1815
1820 Series1
Chart 2:
1400000 1200000 1000000 800000 600000 400000 200000 0 1780 1790 1800 1810 1820
1790-1880
Series1
1830 year
1840
1850
1860
1870
1880
Chart 3:
8000000 7000000 6000000 5000000 4000000 3000000 2000000 1000000 0 1750 1800
1790-1950
Series1
1850 year
1900
1950
60
1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910 1920
33131 60515 96373 123706 202589 312710 515547 813669 942292 1206299 1515301 3437212 4766883 5620084
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Newton's Law of Cooling Teacher Notes
Introduction: In this activity students read a little history behind Newton's Law of Cooling as well as using it. An experiment using the CBL to collect data and then mathematics to analyze it could be incorporated here as well; many books, such as the Texas Instruments book by Chris Brueningsen relating mathematics and science using the CBL collection device, are available. Level: Algebra 2 or Precalculus (Prerequisites: Exponential and logarithmic functions; Solving exponential and logarithmic equations; Best fit curves (optional) Objective: The goal of this activity is to give students an opportunity to learn the history behind Newton's Law of Cooling and then to use this law in several exercises. Materials: Graphing calculator; Internet access (optional) Time Frame: This activity can be completed in one class period with the exercises started in the class, the rest to be assigned as homework. The amount of time needed for this activity may be decreased or increased depending on the mathematical level of your students. How to Use: The teacher may want to discuss the history of Newton's Law of Cooling with the class after the students have had time to read the information. It might be necessary to do several examples of solving exponential and logarithmic equations before assigning this activity. If additional activities are desired, the teacher should visit the cited websites (or others that may be available) to determine the appropriateness for his/her class. Additionally, if the teacher decides to run an experiment using CBL equipment, there would need to be additional preparation on the part of the teacher. Bibliography: 1. Edwards, G. H. and David E. Penney, Elementary Differential Equations with Applications, Third Edition, Prentice Hall, 1994. 2. French, A. P., "Isaac Newton's Thermometry", The Physics Teacher, Vol 31, April 1993, pp208 211. 3. French, A. P., "Newton's Thermometry: The Role of Radiation", The Physics Teacher, Vol 31, May 1993, pp310 312. 4. July 1999 5. July 1999 6. Smith, Karl J, Precalculus Mathematics, A Functional Approach, Fourth Edition, Brooks/Cole Publishing Company, 1990. 7. Stewart, James, Calculus, Early Transcendentals, Third Edition, Brooks/Cole Publishing Company, 1995. 8. Zill, Dennis G, A First Course in Differential Equations with Applications, Fourth Edition, PWS-Kent, 1989.
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Newton's Law of Cooling Solutions
1. a. Substituting T0 = 340, T(1) = 100, and Tm = 70 into Newton's Law of Cooling gives: T = Tm + (T0 - Tm )e kt 100 = 70 + (340 - 70)e k (1) 30 = 270e k 30 = ek 270 1 1 = e k ln = k . 9 9 1 k can be left in this form, or it can be evaluated : k = ln -2.1972 9 kt ln(1/9)t ln(1/9) t t b. First notice that e = e = (e ) = (1/9) . Now substitute the values into Newton's Law of Cooling T = Tm + (T0 - Tm )e kt 1 80 = 70 + (340 - 70) 9 1 10 = 270 9 10 1 = 270 9
t t t
3 1 1 1 1 = = 3 = 2t t = hours. 2 27 9 3 3 2. This time the Tm is the unknown. T0 = 200, T(10) = 150 and k = -0.05. Substituting into Newton's Law of Cooling yields: T = Tm + (T0 - Tm )e kt 150 = Tm + (200 - Tm )e -0.5(10) 150 = Tm + (200 - Tm )e -0.5 Multiply both sides by e0.5 . 150e0.5 = Tm e0.5 + 200 - Tm
t
3
2t
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150e0.5 + 200 = Tm (e0.5 - 1) 150e0.5 + 200 Tm = 72.92 e0.5 - 1 3. a. Using the points (0,0) and (34,100), the slope, m, of the line is m= 100 - 0 50 = . 34 - 0 17
The equation of the line relating the linseed oil temperatures to the centigrade temperatures is 50 C = N 17 b. If we substitute into the previous formula, we get C = 50 (1 2 ) 3 5 .2 9 17
This value is lower than the 37 C that we use. c. Again, substituting the given value for N gives us 50 C= (71) 208.82 17 Newton's value is lower than the actual value. d. There are several possible explanations. One is that Newton's values were approximations; another is that the relationship between the linseed oil temperatures and the Celsius temperatures is not linear. A suggested additional activity would be to use the three points (0,0), (12, 37) and (34, 100) in a different model a best fit line or a quadratic equation, for example. And then see how the new model works with the melting point of tin.
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Newton's Law of Cooling Student Pages
History
Besides discovering calculus, Newton experimented in chemistry and alchemy. One of the Newton results discovered is what is now called Newton's Law of Cooling. He wrote anonymously "A Scale of the Degrees of Heat" in the 1701 volume of the Philosophical Transactions of the Royal Society of London. Due to Newton's fame, Roger Cotes easily deduced that Newton was the author of the anonymous paper. Using a linseed oil thermometer, because linseed oil has a high boiling point of 240 C, Newton studied the melting points of various metals and alloys. Newton devised a method for measuring temperatures of up to 1000 C. his law of cooling. Newton's Law of Cooling states that the rate at which the temperature T(t) changes in a cooling body is proportional to the difference between the temperature in the body and constant temperature Tm of the surrounding medium. If one writes this statement as a differential equation and solves, one gets Newton's Law of Cooling in the form T = Tm + (T0 - Tm)ekt where T0 is the initial temperature of the heated body at time t = 0 and where k is a constant which depends on the item being cooled. Notice that this equation is an example of exponential decay decay since k turns out to be negative because the body's temperature is decreasing.
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Exercises: 1. When a cake is removed from the oven, it is 340. One hour later, it has cooled to 100. If the temperature of the room is 70, a. find the constant k in Newton's Law of Cooling; b. find how long will it take for the cake to cool to 80? 2. Suppose the constant k Newton's Law of Cooling -0.05. If the temperature of a cup of coffee is 200 when it is poured and 150 10 minutes later, how warm is the room? 3. According to Newton's paper, "A Scale of the Degrees of Heat", he chose the following temperatures using his linseed oil thermometer: the freezing temperature of water was 0 and the boiling point of water was 34. The Celsius temperature for the freezing of water is 0 and the boiling point of water is 100. a. Find the linear equation that represents the relationship between the linseed oil temperatures and the Celsius temperatures. Let the linseed oil temperatures be the independent variable. b. Newton said that body temperature using his linseed oil thermometer was 12. Using the linear relationship you found in part a, find the Celsius temperature for body temperature. How does this compare with the 37 C that we use? c. Newton found that the melting point of tin was 71. Using the linear relationship you found in part a, find the Celsius temperature for the melting point of tin. According to the Smithsonian Physical Tables, the melting point of tin is 232 C. How does this compare with Newton's value? d. From the two examples you just did, how would you explain the discrepancies between the linseed oil thermometer values that Newton obtained, and the Celsius values that we use?
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Euler and Exponential Functions: An Investigation Teacher Notes
Introduction: This activity highlights a mathematical problem taken from Leonhard Euler's Introductio in analysin infinitorum of 1748. Level: Algebra I, Algebra II, Precalculus (Prerequisites: Exposure to exponential functions and their graphs) Objective: To develop students' understanding of exponential functions by using a problem taken from one of Euler's most famous works, Introductio in analysin infinitorum. Materials: Student Pages; Calculator; Graphing calculator (optional) Time Frame: One 45-minute class period How to Use: Consider using cooperative groups to do this activity. At the very least, consider pairing up students to work on this activity. If students have not had much work with exponential functions, you might need to help them find the Have students consider the following historical problem taken from Leonhard Euler's highly influential pre-calculus book, Introductio in Analysin Infinitorum, written in 1748: If the progression 2, 4, 16, 256, ... is formed by letting each term be the square of the preceding term, find the value of the twenty-fifth term. This example comes from Chapter VI: On Exponentials and Logarithms of Euler's book.. Euler initially solved the problem using logarithms. We simply want to graph the curve that models this progression. It is preferable to have students graph this curve using a graphing calculator. If there is no access to graphing calculators, have students make a table of values and graph the exponential function on graph paper. A suggested window for graphing is x: (0, 10) and y: (0, 300). Students must first determine the function that models this progression. The students should arrive at
y = f ( x) = 2 2 .
x
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A sample table the students may make is as follows:
x 0 1 2 3 4 5 6 7At this time, students should find the twenty-fifth term is
) f (25 = 22 = 216,777,216.
Students will quickly discover this expression cannot be evaluated on a calculator. It is important to stress that Euler solved the problem using logarithms--a topic that will be discussed later and connected with exponential functions. This example is a prelude to logarithmic functions--the inverse of exponential functions. Once the students have discovered the function and curve that modeled Euler's original problem, you can discuss the relationship between this function and the standard exponential function
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y = f ( x) = 2 x.
Then have the students make a comparison between the two functions. Note: Euler's original solution with logarithms to this example is provided (See Logarithmic Functions: Euler and Exponential Functions: An Investigation Revisited). Later in the unit when you introduce logarithms, it would be appropriate to revisit this problem and show students its connection to logarithms and Euler's logarithmic solution. This will make students aware of the inverse relationship between exponential and logarithmic functions. Bibliography: Euler, Leonhard, translated. by John D. Blanton. Introduction to Analysis of the Infinite, Book 1, pp. 75-91 (New York: Springer-Verlag, 1988). Katz, Victor. A History of Mathematics: An Introduction (New York: HarperCollins, 1998).
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Euler and Exponential Functions: An Investigation Solutions
1.
x 0 1 2 3 4 5 6 7
The graph of f(x) looks like this:The graphing window is x: (0,4) and y: (0, 200). 2. The curve starts off relatively flat but increases very rapidly. Students will notice that the curve shoots up quickly around x = 3. 3. The twenty-fifth term is
) f (25 = 22 = 216,777,216.
4. f(25) cannot be evaluated on a standard calculator because it is a very large number. 5. f ( x) = 2 2
x
24
6. Student answers will vary.
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Euler and Exponential Functions: An Investigation Student Pages
Consider the following historical problem taken from Leonhard Euler's highly influential precalculus book, Introductio in Analysin Infinitorum, which was written in 1748. This example comes from Chapter VI: On Exponentials and Logarithms as seen on page 91 of the original manuscript.
Introductio in Analysin Infinitorum Chapter VI On Exponentials and Logarithms Example on page 91 Leonhard Euler (1707-1783)
If the progression 2, 4, 16, 256, ... is formed by letting each term be the square of the preceding term, find the value of the twenty-fifth term.
1. Graph the curve that models the progression of numbers. First, make a table of values. Then, using your graphing calculator or manually plotting values, graph the curve that models the progression of numbers as seen in Euler's pre-calculus book, Introductio in Analysin Infinitorum.
x
y = f ( x) = ?
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Graph of f(x):
2. Describe what the curve looks like?
3. Based on the table of values, identify the twenty-fifth term?
4. What problem do you encounter when evaluating the 25th term on your calculator?
5. Write a general equation for this function. f(x) = ?
6. See if you can find other functions that look like the curve you graphed.
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Introductio in Analysin Infinitorum(1748) Chapter VI On Exponentials and Logarithms Leonhard Euler(1707-1783)
Example on page 91
If the progression 2, 4, 16, 256, ... is formed by letting each term be the square of the preceding term, find the value of the twenty-fifth term.
Transparency
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Calculating ex Teacher Notes
Introduction: We present a slight modification of Euler's method for calculating e x as a power series, a generalized polynomial. Level: Precalculus. A prerequisite is the Binomial Theorem and the calculation of e as in the section "What Will Happen If..." Objective: To show how to calculate the power series for ex Time Frame: One or two class periods, with a homework assignment at the end. How to Use: This is a teacher-directed lesson. The Student Pages should be given to the students, but the teacher will need to present this material and answer questions about it.
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Calculating ex Student Pages
We present here a method of calculating ex by means of a power series, a polynomial of "infinite degree." We first recall the Binomial Theorem: (a + b) n = a n + na n -1b + n(n - 1) n- 2 2 n(n - 1)n - 2) n -3 3 a b + a b + 1 2 1 2 3 + bn .
You should make up several examples of this for practice. In particular, note that the formula is somewhat simpler when a = 1. We then have (1 + b) n = 1 + nb + n(n - 1) 2 n(n - 1)(n - 2) 3 b + b + 1 2 1 2 3 + bn .
Next, recall the result, discussed in an earlier section of this module, that e can be found by considering the expression 1 1 + n
n
and letting n get larger and larger. Today, we generally write this as 1 e = lim 1 + . n n This means simply that as we let n get larger and larger, the given expression gets closer and closer to a particular number, which we have called e. Euler, however, writing before the notion of limit had been explained, would simply write 1 e = 1 + , n where n is an infinitely large number. Today, we do not generally use such concepts, and probably Euler himself knew that such numbers did not really exist. But what he meant by the phrase "infinitely large number" was simply a very very very large number; he used his intuition then to determine what properties such very very very large numbers would have. We will see some of these properties as we work through his argument.
n n
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1 Since e = 1 + , where n is infinitely large, we can also calculate e r . First, we note n that if n is infinitely large and r is a finite quantity, then n/r is also infinitely large. We can therefore rewrite e as 1 e = 1 + n/r It follows that
n/r 1 . e = 1 + n / r r r n/r
n
.
By one of the basic rules for exponents, we can rewrite this in the form 1 e = 1 + . n/r
r n
But we know that
1 r = .. We therefore have, finally, n/r n r er = 1 + , n
n
again where n is infinitely large. r . Of n course, we know that the binomial theorem is only valid for finite values of n, but we will assume that it is also true when n is infinitely large. After all, we are thinking of such values n simply as very very very large positive integers. However, since we do not know how large n actually is, we will only calculate the first few terms of the binomial expansion and use dots to indicate that this expression simply goes on for a long long time. Technically, this "infinite polynomial" is called a power series. We now apply the binomial theorem to this last expression, taking a = 1 and b = We get
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r e = 1 + n
r
n
r n(n - 1) r n(n - 1)(n - 2) r n(n - 1)(n - 2)(n - 3) r = 1+ n + + + + 1 2 n 1 2 3 1 2 3 4 n n n = 1+ r + n(n - 1)r 2 n(n - 1)(n - 2)r 3 n(n - 1)(n - 2)(n - 3)r 4 + + + 1 2n 2 1 2 3n3 1 2 3 4n 4
2
3
4
We want to simplify the various terms of this expression. So we have to use a property of ``infinitely large numbers.'' In particular, we note that if n is infinitely large, subtracting 1 n -1 from n will have no appreciable result. Thus = 1 . Similarly, if we subtract 2 or 3 or indeed n any finite number from n, this will not really change that infinitely large number. We therefore n-k = 1 for any finite number k. What this means, then, is that in the expression above, have n the quantities n, n 1, n 2, n 3, ... in the numerators always "cancel" the factors of n in the denominators. It follows that our expression reduces to er = 1 + r + r2 r3 r4 + + + 1 2 1 2 3 1 2 3 4 ( k - 1) k is usually called
You may know that if k is any integer, the expression 1 2 3
r
k factorial and is written k! Thus, our expression for e , usually called its power series expansion, is written as er = 1 + r + r2 r3 r4 + + + 2! 3! 4!
The value of the power series expansion for Euler was that it enabled him to calculate values for e r . In particular, he could calculate the value for e1 = e . We can do it as well, by substituting r = 1 into the series. We get e = 1+1+ 1 1 1 1 1 1 1 + + + + + + + 2! 3! 4! 5! 6! 7! 8! 1 1 1 1 1 1 1 = 1+1+ + + + + + + + 2 6 24 120 720 5040 40320 = 1 + 1 + 0.5 + 0.166666667 + 0.041666667 + 0.008333333 + 0.001388889 + 0.000198413 + 0.000024802 + 2.718278771 2.71828
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Your calculator can, of course, calculate e as well as e r , for any r in a fraction of a second. You may wonder how it does it. Although different calculators use different procedures for doing this, one of the ways your calculator may work is, in fact, by using the series for e r ,. Your calculator is basically hard-wired to perform additions and multiplications; since calculating with a power series just involves these operations, all that has to happen is that the calculator must perform very many of these operations very quickly. And that is precisely what it can do. Although a normal graphing calculator can probably only calculate e to 10 decimal places, it has been calculated to far more places by a computer. The first 100 decimal places are as follows: e 2.71828 18284 59045 23535 02874 71352 66249 77572 47093 69995 95749 66967 62772 40766 30353 54759 45713 82178 52516 64274 Like another important number , the number e is irrational and even transcendental. The first statement was proved by Euler in 1737. That e is transcendental, that is, that it cannot be a solution to a polynomial equation with integer coefficients, was proved in 1873 by Charles Hermite. Interestingly, these two important numbers are closely related, as we will see in the next activity. Exercises: 1. Use your calculator to determine the decimal form of the next three terms in the series for e, namely, 1/9, 1/10, and 1/11. Add these values to the value already calculated and check that you now have e calculated correctly to 8 decimal places. 2. Use the power series for e r to calculate e0.1. Namely, replace r by 0.1 in that series and evaluate using your calculator. Check that using the terms through the fifth power term gives you accuracy to 8 decimal places. 3. Use the power series for e r to calculate e0.1 and compare with the answer given directly by your calculator. 4. Use the power series for e r to calculate e1 and compare with the answer given directly by your calculator. How many terms do you need to use to get e1 to 8-place accuracy?
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Equations using e Teacher Notes
Introduction: Students are intrigued by e and want to know more about where it is used. Here are two equations that help to satisfy their curiosity. Level: Algebra 2 or Precalculus/Trigonometry (Prerequisites: discussion of e) Objective: to introduce students to two equations using e Materials: Student Pages for each student Time Frame: 20 minutes How to Use: The teacher may want to use this short activity to amplify students' work involving the transcendental number e Bibliography: Kasner, Edward and Newman, James, Mathematics and the Imagination. New York: Simon and Shuster, 1940. Maor, Eli. e: The Story of a Number. Princeton University Press. 1994.
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Equations using e Solutions
1. Answers may vary. Many students will answer parabola. 2. e is the basis of natural logarithms and is used in many real-life applications, such as population growth or radioactive decay. i is the -1 which is used for complex numbers and in many real-life applications, such as electricity. is used, for example, in the area formula for a circle and volume formulas for cylinders, cones and circles. 1 is the identity element for multiplication, just to name one reason it is special. 0 is a place holder, just to name one reason it is special.
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Equations using e Student Pages
1. Hanging link chains and two brothers Imagine a chain made of identical links, hanging from the two endpoints. You've probably seen such a curve as a barrier around a grassy plot or the wire suspended from telephone poles along a highway. Sketch below the shape of the curve formed by the chain. What shape does the curve look like? ________________________________
You are not alone in thinking about this. Over three hundred years ago two brothers of the famous Bernoulli family of mathematicians tried to figure out the equation that described it. In May, 1691, Jakob Bernoulli stated the problem in a scientific journal, Acta Eruditorum. A year later his younger brother Johann as well as two other famous mathematicians, Christian Huygens and Gottfried Wilhelm Leibniz, correctly solved this problem. In fact Jakob Bernoulli was not able to solve it; he and his brother Johann had a strong rivalry as shown in this letter from Johann to a friend. "You say that my brother proposed this problem; that is true, but does it follow that he had a solution of it then? Not at all. When he proposed this problem at my suggestion (for I was the first to think of it), neither the one nor the other of us was able to solve it; we despaired of it as insoluble, until Mr. Leibniz gave notice to the public in the Leipzig journal of 1690, p. 360, that he had solved the problem but did not publish his solution, so as to give time to other analysts, and it was this that encouraged us, my brother and me, to apply ourselves afresh. The efforts of my brother were without success; for my part, I was more fortunate, for I found the skill (I say it without boasting, why should I conceal the truth?) to solve it in full.... The next morning, filled with Joy, I ran to my brother, who was still struggling miserably with this Gordian knot without getting anywhere, always thinking like Galileo that the catenary was a parabola. Stop! Stop! I say to him, don't torture yourself anymore to try to prove the identity of the catenary with the parabola, since it is entirely false." quoted from e the Story of a Number, p. 141 Actually this curve is called a catenary from the Latin word catena meaning chain. The equation e ax + e - ax that describes the catenary is y = where a is a constant that is determined by the 2a density of the chain and the tension as it is held. Calculus is required to prove that this is correct.
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2. The Most Beautiful Equation
e i + 1 = 0
Many consider this to be one of the most beautiful equations ever discovered since it incorporates five of the special numbers of mathematics. It was discovered by Leonhard Euler, a mathematician who studied and contributed to many different branches of mathematics. Why is each of the five numbers special in mathematics?
e _______________________________________________________________ i _______________________________________________________________ _______________________________________________________________
1________________________________________________________________ 0________________________________________________________________ To Benjamin Pierce, one of Harvard's leading mathematicians in the nineteenth century, this formula of Euler came as something of a revelation. Having discovered it one day, he turned to his students and said: "Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth." (Kasner and Newman, pp. 103-104)
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Introduction to Logarithms Teacher Notes
Level: Algebra 2 or Precalculus/Trigonometry properties of exponents) (Prerequisites: Properties of logarithms,
Objective: To use original sources dealing with logarithms Materials: Transparency of Napier's justification for logarithms. Copy of Paragraphs 102 - 107 from Euler's Introductio in analysin infinitorum for each student; Student Pages for each student Time Frame: 1 to 3 class periods, depending on how used How to Use: The transparency can be used when you first introduce logarithms. Then, consider the excerpt from Euler's Introductio in analysin infinitorum (1748). In these paragraphs, Euler introduces all the information about logarithms that we usually cover in our modern day courses - in two pages! Be sure to find out if the students think that this is a sufficient amount of explanation for understanding. In the student Activity that follows, the students will read and then do examples demonstrating Euler's explanations on logarithms. It would be good after the students finish the activity to point out that there probably is enough information about logarithms here for mastering the basics of logarithmic functions. You may want to use each separate paragraph of the Student Activity on different days, depending on your planned lesson, or the whole Student Activity after you have completed logarithms. Euler uses "log n" to mean "log base a of n." Bibliography: Euler, Leonhard. Introduction to Analysis of the Infinite Book1. Translated by John D. Blanton. Springer Verlag, 1988
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Seeing John Napier, A Description of the Admirable Table of Logarithms, 1616
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Introduction to Logarithms Solutions
Part 1 1. If az = y, then z = log y. 2. Yes 3. "a" is the base of the logarithm and it must be a number greater than 1. 4. The domain is positive real numbers, if the logarithm is to be real also. Part 2 5. If y = 1 in az = y, then az = 1 which implies that z = 0. But a0 = 1 corresponds to loga 1 = 0. 6. By definition, a1 = a loga a= 1, a2 = a2 loga a2 = 2, ..., an = an loga an = n 7. Since a is a positive number, 1/a, 1/a2, ... will also be positive. 1/a, 1/a2,...are less than one since the reciprocal of a positive number greater than one is a positive number less than one. (There are other explanations.) loga 1/a = loga a-1 and by the definition of logarithms, loga a-1 = y ay = a-1. Therefore, y = -1 = loga a-1. The same argument holds for the other logarithms. Part 3 8. Power rule: log yn = nz Product rule: log vy = x + y = log v + log y Quotient rule: log (y/v) = z x = log y log v 9. Let log y = z and log v = x, By the definition of logarithms, y = az and v = ax. vy = axaz = ax+z so log vy = x + y. It follows that log vy = x + y = log v + log y. Hence, the logarithm of the product of two numbers is equal to the sum of the logarithms of the factors. 10. Let log y = z and log v = x, By the definition of logarithms, y = az and v = ax. v/y = ax /az = ax-z so log v/y = x - y. It follows that log v/y = x - y = log v - log y. Hence, the logarithm of the quotient of two numbers is equal to the difference of the logarithms of the factors.
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11. If you know the logarithm of 2 and the logarithm of 3, you know the logarithms of 6 = (2)(3), 9 = 32, 27 = 33,... , 4 = 22, 8 = 23, ..., 1.5 = 3/2, 2/3, ... 36 = 2232,...and an infinite number of other examples using just these two values. See the Student Activity Calculation of Logarithms - Using the Method of Napier and Briggs for more examples. Part 4 12. The only way for the value of a logarithm to be rational is if the number is a power of the base. There are many examples: log4 8 = 3/2 (since 8 = 43/2, i.e. 8 is a power of the base 4). 13. A surd is the square root of a number that is not a perfect square such as 2 , 5 , 7 ,...
14 Part 5 15. The rule usually looks like this: log a n = 16. loga n = p ap = n logb n = q bq = n log10 n log10 a ap = bq a = bq/p
Since b, p and q are all constant, a is also a constant and this value has nothing to do with n, the number whose logarithm we are evaluating. 17. Looking for log2 n = q, log10 n = p (we would know this value, since it is a base 10 logarithm) log102 = 0.3010300, also log2 2 = 1 Then p/q= 0.3010300 and q = p/0.3010300 = 3.3219277p. So, if every common logarithm is multiplied by 3.3219277 then we will have produced a table of logarithms for base 2. Note that this is equivalent to the usual method you gave in number 15: log10 n log10 n = = (3.3219277) log10 n log 2 n = log10 2 .3010300
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Introduction to Logarithms Student Pages
Part 1 Read paragraph 102. Then answer the questions. 1. In this paragraph, Euler gives the definition of the LOGARITHM of y. What is it?
2. Is this the same definition that you have been given in your class?
3. In the paragraph, what is "a"? What properties must "a" have?
4. What does the last sentence say about the domain of the logarithmic function?
Part 2 Read paragraph 103. Then answer the questions. 5. Explain why Euler says that, no matter the base, log 1 = 0. (We would write loga 1 = 0.)
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6. Explain why "log a = 1, log a2 = 2, log a3 = 3, log a4 = 4, etc".
7. Explain why 1/a, 1/a2, ... are positive numbers less than 1 and why their logarithm is negative.
Part 3. Read paragraph 104. Then answer the questions. 8. What does Euler say the product rule, the quotient rule and the power rule for logarithms are?
9. Give the explanation Euler uses for the product rule.
10. Give a similar explanation for the quotient rule.
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11. How can these rules be used to "find the logarithms of many numbers from a knowledge of the logarithms of a few"?
Part 4 Read paragraph 105. Then answer the questions. 12. What does the first sentence mean? Give an example of a logarithm of a number that is rational.
13. What is a surd? You may have to use a dictionary.
14. Why, according to Euler, is the logarithmic function a transcendental function?
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Part 5 Read paragraph 107. Then answer the questions. 15. Give the change of basis rule that you have learned in your class.
16. Translate into your own words: "If".
17. Show that Euler is describing how to change base 10 logarithms to base 2 logarithms his example.
in
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Introduction to Logarithms From Introductio in analysin infinitorum
Taken from Introductio in analysin infinitorum (1748) by Leonhard Euler
Chapter VI. On Exponentials and Logarithms
102. Just as, given a number a, for any value of z, we can find the value of y [= az], so, in turn, given a positive value for y, we would like to give a value for z, such that az = y. This value of z, insofar as it is viewed as a function of y, it is called the LOGARITHM of y. The discussion about logarithms supposes that there is some fixed constant to be substituted for a, and this number is the base for the logarithm. Having assumed this base, we say the logarithm of y is the exponent in the power az such that az = y. It has been customary to designate the logarithm of y by the symbol log y. If az = y, then z = log y. From this we understand that the base of the logarithms, although it depends on our choice, still it should be a number greater than 1. Furthermore, it is only of positive numbers that we can represent the logarithm with a real number. 103. Whatever logarithmic base we choose we always have log1 = 0, since in the equation az = y, which corresponds to z = log y, when we let y = 1 we have z = 0. From this it follows that the logarithm of a number greater than 1 will be positive, depending on the base a. Thus log a = 1, log a2 = 2, log a3 = 3, log a4 = 4, etc. and, after the fact, we know what base has been chosen, that is the number whose logarithm is equal to 1 is the logarithmic base. The logarithm of a positive number less than 1 will be negative. Notice that log 1/a = -1, log 1/a2 = -2, log 1/a3 = -3, etc., but the logarithms of negative numbers will not be real, but complex, as we have already noted. 104. In like manner if log y = z, then log y2 = 2z, log y3 = 3z, etc., and in general log yn = nz or log yn = n log y, since z = log y. If follows that the logarithm of any power of y is equal to the product of the exponent and the logarithm of y. For example logy = (z) = (log y), log1/y = log y-1/2 = -(log y), and so forth. It follows that if we know the logarithms of any number, we can find the logarithms of any power of that number. If we already know the logarithms of two numbers, for example log y = z and log v = x, since y = az and v = ax, it follows that log vy = x + y = log v + log y. Hence, the logarithm of the product of two numbers is equal to the sum of the logarithms of the factors. In like manner log (y/v) = z x = log y log v, that is, the logarithm of a quotient is equal to the logarithm of the numerator diminished by the logarithm of the denominator. These rules can be used to find the logarithms of many numbers from a knowledge of the logarithms of a few. 105. From what we have seen, it follows that the logarithm of a number will not be a rational number unless the given number is a power of the base a. That is, unless the number b is a power of the base a, the logarithm of b cannot be expressed as a rational number. In case b is a power of the base a, then the logarithm of b cannot be an irrational number. If, indeed, log b =
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n, then an = b, but this is impossible if both a and b are rational. It is especially desirable to know the logarithms of rational numbers, since from these it is possible to find the logarithms of fractions and also surds 107. There are as many different systems of logarithms as there are different numbers which can be taken as the base a. It follows that there are an infinite number of systems of logarithms. Given two different systems of logarithms, there is a constant which relates the logarithms of the same number. If. If the logarithms of all numbers have been computed in one system, then it is an easy task, by means of this golden rule for logarithms, to find the logarithms in any other system. For example, we have logarithms for the base 10. From these we can find the logarithms with any other base, for instance the base 2. We look for the logarithm of a number n for base 2, which will be q, while the logarithm with base 10 of the same number n will be p. Since for base 10, log 2 = 0.3010300 and for base 2, log 2 = 1, then p/q = 0.3010300/1 and q = p/0.3010300 = 3.3219277. If every common logarithm is multiplied by 3.3219277 then we will have produced a table of logarithms for base 2.
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Development of Logarithms Using Sequences Teacher Notes
Level: Algebra 2 or Precalculus (Prerequisites: The students should be familiar with the properties of logarithms and exponents and have at least an intuitive understanding of sequences.) Objective: In this activity, the students will explore the ideas and background of the historical invention of the logarithm. Materials/Resources: Student Pages for each student; calculator (optional) Time Frame: This entire activity will probably take the equivalent of three class periods, with part of the activity sheet assigned as homework. Part 4 is the most difficult part and may be omitted if necessary. How to Use: An important thing to keep in mind as you work through this is that John Napier, the inventor of logarithms, was looking for a tool that would simplify complex mathematical calculations which were done laboriously by hand. In this project, you will explore the ideas and background of the historical invention of the logarithm. This is an easy-to-follow activity that provides an excellent development of the meaning of logarithms. The teacher may need to do some of the exercises as guided practice to provide a direction for student answers. Bibliography: Adapted from Lawrence Morales' on line project found at July 1999
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Development of Logarithms Using Sequences Solutions
Part I: Relationships 1. S2: 2 4 8 16 32 S2: 21 22 23 24 25 64 128 26 27 256 28 512 29 1024 210 2048 211 4096 212 8192 213
The sequence S1 gives the power of 2 for the number in S2. 2. (8)(32) = 256 (512)(16) = 2924 = 213 = 8192 (128)( 4) = 2722 = 29 = 512
(There are many other possible examples.)
The product is always another term of S2. This is because each element of S2 is a power of 2, so the product of elements in S2 will also be an element of S2. 3. 8 in S2 corresponds to 3 in S1; 32 in S2 corresponds to 5 in S1; the product 256 corresponds to 8 = 3 + 5 in S1. In the second example, 512 corresponds to 9, 16 corresponds to 4, and the product, 8192, corresponds to 13 = 9 + 4. In the third example, 128 corresponds to 7, 4 corresponds to 2, and the product, 512, corresponds to 7 + 2 = 9. When multiplying the elements of S2, the product will always be an element of S2. This is because 2n2m = 2n+m. In words, multiplication in S2 corresponds to addition in S1. 32/8 = 25/23 = 25-3 = 22 = 8 512/16 = 29/24 = 29-4 = 25= 32 128/4 = 27/22 = 27-2 = 25= 32
4. 5.
(There are many other possible examples.)
The quotient is always another term of S2. This is because each element of S2 is a power of 2, so the quotient of elements in S2 will also be an element of S2. 6. Any two terms in S2 can be divided by considering the exponents, that is, the corresponding numbers in S1, subtracting these numbers, and then finding the corresponding number in S2. 322= (25)2 = 210 = 1026 84 = (23)4 = 212 = 4096 163 = (24)3 = 212 = 4096
7.
(There are many other possible examples.)
The power of a term in S2 is always another term of S2. This is because each element of S2 is a power of 2, so the power of elements in S2 will also be an element of S2.
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8.
Any term in S2 can be raised to a power by finding its exponent, that is, the corresponding number in S1, multiplying this number by the power, and then finding the corresponding number in S2. The easiest way would be to rewrite the number or numbers in S2 as powers of 2 and use the rules of exponents to evaluate the answer. When multiplying the numbers, add the exponents; when dividing the numbers, subtract the exponents; when raising to a power, multiply the exponent by the power.
9.
Part 2 Extending the Results 10. Yes the results from part 1 hold here also. The rules of exponents hold no matter what the base is. Let u = bn and v = bm in each of the following: L(u) + L(v) = L(bn) + L(bm) =n+m = L(bn+m ) = L(bn bm ) = L(uv) L(u) - L(v) = L(bn)/L(bm) = n/m = L(bn/m ) = L(bn /bm ) = L(u/v) L(uk) = L((bn)k) = L(bnk) = nk definition of u and v definition of the function l definition of the function l rules of exponents definition of u and v definition of u and v definition of the function l definition of the function l rules of exponents definition of u and v definition of u rules of exponents definition of the function l
11.
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= k L(bn ) = k L(u)
definition of the function l definition of u
Part 3: Terminology and a Formal Introduction to Napier 12. If we read sequence S3 from right to left, each number is 1 smaller than the one to its right. Thus we can extend it to the left by putting 0 to the left of 1 and then the negative integers to the left of 0. If we read sequence S4 from right to left, we see that each number comes from the one to its right by dividing by b. Since the number in S4 under 1 is b, the number to be put under 0 must be b/b = 1. Similarly, the number to be put under -1 must be 1/b. The number under -2 is then (1/b)/b = 1/b2, and so on. Yes the results from number 10 still hold. The rules of exponents hold for all integers. There are many examples that would be appropriate. We write 1/b as b-1; then 1/b2 = b-2. In general, we have 1/bn = b-n, because 1/bn in S4 is under -n in S3. The 1 in S4 under the 0 in S3 can be written as 1 = b0. So a 0 exponent always means 1, while a negative exponent means 1 over the corresponding positive power of the base b. log 31 = __0_ log381 = __4_ log33 = _1__ log31/3 = _-1__ log39 = __2_ log3 1/243 = __-5_ log327 = _3__ log31/2187 = _-7
13. 14.
15.
Part 4: Filling up the sequences 16. 1.0000001 1.0000002 1.0000003 1.0000004 1.0000005
Raising 1.0000001 to a small power seems just to be 1 with 6 decimal places after the 0 and then the small power, although with more decimal places, we would find more distinctions between the numbers. In any case, these numbers are very close together, differing by only .0000001. To get to 3, however, we would need to find n such that (1.0000001)n = 3. The calculator shows that n needs to be something near 11,000,000. It is therefore not very feasible to continue the sequence in this way. 17. We must have tt = 2 because the 1 (from + ) in S1 corresponds to 2 in S2. The solution to the equation t2 = 2 is t = 2. The calculator shows that this is equal approximately to 1.414. And since we use exponents of 2 to designate numbers in S2, we also have t = 21/2. That is, 21/2 = 2. Then log2 t = log2 2 = . Since u3 = 2, we have u = 3 2 = 1.260 = 21/3. Since v3 = 4, because 2/3 + 2/3 + 2/3 = 2 and 2 in S1 corresponds to 4 in S2, we get v = 3 4 = 1.587 = 22/3. We could also find v by
18.
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noting that v = u2, because 1/3 + 1/3 = 2/3. It follows that v = ( 3 2 )2. So there are two different ways of expressing v in radical form. Also, log2 u = 1/3 and log2 v = 2/3. 19. The number corresponding to is 4 2 = 1.189 = 21/4. The number corresponding to is 4 8 = ( 4 2 )3 = 1.682 = 23/4. The number corresponding to 3/2 is 8 = (2)3 = 2.828 = 23/2. The number corresponding to 4/3 is 3 16 = ( 3 2 )4 = 2.520 = 24/3 and the number corresponding to 5/4 is 4 32 = ( 4 2 )5 = 2.378 = 25/4. The logarithms to the base 2 of these numbers are, in each case, the fraction that we began with. The number in S2 which corresponds to the fraction m/n in S1 is 2m/n. Since the nth power of this number is 2m, corresponding to the fact that the product of m/n by n is m, we can write this number as n 2 . Also, since m/n is m times 1/n, we know that 2m/n is 21/n raised to the power m. Since 21/n is equal to n 2 , we can also write 2m/n as ( n 2 )m. The calculator shows that 2317/200 = 3 (to three decimal places) and also that 2306/109 = 7 (also to three decimal places). The sum of 317/200 and 306/109 is 95753/21800 and 2 raised to that power is equal to 21, again to three decimal places. Thus 95753/21800 in S1 corresponds to 21 in S2. Answers may vary.
20.
21.
22.
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Development of Logarithms Using Sequences Student Pages
Part I: Relationships Consider the two following sequences: S1 S2 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256 9 512 10 1024 11 2048 12 4096 13 8192
1. Rewrite the numbers in sequence S2 as powers of a single number. How are the sequences related? 2. Note that when two terms of S2 are multiplied, such as 64 and 128, their product, (64)(128) = 8192, is another term of S2. Pick a few pairs of your own to multiply. Is the product always another term of S2? You may have to extend the terms in each sequence to verify your observations. 3. Note that the number in S1 corresponding to 64 is 6, while the number in S1 corresponding to 128 is 7. The number corresponding to their product, 8192, is 13, the sum of 6 and 7. In other words, multiplication in S2 corresponds to addition in S1. Check that this holds for the other pairs you picked in exercise 2. 4. In a brief paragraph, summarize what you have observed. Most importantly, give a justification for your results using rules of exponents. We will now see if similar laws hold for other mathematical operations. In particular, we will examine the operations of division and taking powers. 5. Take any two terms in S2 and divide the smaller one into the larger. Do at least three cases. In each case, check the corresponding numbers in S1 of your two terms as well as their quotient. 6. After trying several pairs of terms in S2, describe how you could divide any two terms in S2 without using a calculator or doing any computations by hand. 7. Take any term in S2 and raise it to a positive integer power. Do at least three cases. Again, observe the corresponding numbers in S1 of the terms you choose and calculate. 8. After trying several cases, describe how you could raise any term in S2 to a positive integral power without using a calculator or doing any computations by hand.
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9. From the previous problems, we have seen how to multiply, divide and raise to powers by relating terms in S1 and S2 in appropriate ways. Assume that you had no calculator or computer device at your disposal. How would these relationships help you to make complex computations? Explain! Part 2 Extending the Results At this point, we should try to generalize our findings and describe them with some more general mathematical symbols or notation. To do this, let's look at the two following sequences where b > 0. S3: S4: 1 b 2 b2 3 b3 4 b4 5 b5 6 b6 7 b7 8 b8 9 ... b9 ... n bn
We begin by defining a function, L(x), which has as its domain the elements of S4. The "value" of L(x) is its corresponding term in S3. For example L(b6)=6. In general, L(bn)=n for any positive integer n. 10. Review your results from Part 1 of the project. Do you think they hold here? In particular, use the general terms of S3 and S4 to check if the properties hold. Use algebra only to show that they are true. (Do not plug in values for b. That way, you will be showing the general relationships hold.) 11. Verify that your conclusions in number 10 are actually the following in function notation. Let u and v be terms in S4. Use algebra to show the following three properties hold. L(u) + L(v) = L(uv) L(u) - L(v) = L(u/v) L(uk) = k L(u)
Part 3: Terminology and a Formal Introduction to Napier Now extend the Sequence S3 and S4 to the left as follows: S3 ... -5 -4 5 S4 ... 1/b 1/b4 -3 1/b3 -2 1/b2 -1 1/b 0 1 1 b1 2 b2 3 b3 4 b4 5 ... b5 ...
12. Why is it reasonable to extend S4 to the left in the way we did? In particular, why should the number in S4 under 0 in S3 be 1? And why should the numbers in S4 under the negative values in S3 be 1 divided by powers of b? Your explanation may well be by analogy to the development of the sequences initially.
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13. Do the general results from number 11 still hold? Test them out by trying a few terms in S3 and S4. Show your examples. Move on when you're convinced that the rules hold for the extended sequences. 14. How can you write the quotients at the left side of S4 in exponential notation? How would you write the 1 in S4 under the 0 in S3 in exponential notation? Give a brief definition of what a negative exponent means and what a 0 exponent means. At this point, recall our original sequences, S1 and S2: S1 S2 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256 9 512 10 1024 11 2048 12 4096 13 8192
John Napier, the Scottish mathematician who invented logarithms, called the terms of the sequence S1 the "logarithm" of the corresponding terms of the sequence S2. The word "logarithm" is a translation of the Greek word logarithmos which comes from the Greek words meaning ratio and number; it therefore means "the number that counts the ratios." For example, he called 7 the logarithm of 128. This example shows how many ratios are required for the term 128 to be found. (Note that Napier did not use base b = 2 as in S2, but the ideas are the same.) The function developed in number 11 above is a logarithmic function for base b = 2. We now replace our L notation relating numbers in S1 to those in S2 by the new notation "log2" (read as logarithm to the base 2). Thus we have the following results: log21 = 0, log22 = 1, log24 = 2 , log28 = 3, etc.
15. Use this "log" notation above to associate the corresponding terms of the following two sequences: S5 S6 0 1 1 3 2 9 3 27 4 81 5 243 6 729 7 2187
Find the following... log 31 = ___ log381 = ___ In general, log33 = ___ log31/3 = ___ log39 = ___ log3 1/243 = ___ log327 = ___ log31/2187 = ___
logba = c if and only if bc = a.
This is the classic modern definition of a logarithm.
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At this point it is very important to remember that Napier did not use this kind of notation! What we are doing here is trying to see the link between our modern logarithms and the theoretical ideas behind Napier's logarithms. Again, Napier started with two related sequences and then went in a very different direction than we do. Part 4 Filling up the sequences Recall that Napier's original motivation for developing logarithms was to replace multiplication and division by addition and subtraction. The use of these sequences enable us to do this on a very limited scale. We can, for example, by the use of sequences S1 and S2 replace the multiplication 16 times 32 by the addition 4 plus 5. And we can replace the division of 256 by 64 by the subtraction of 6 from 8. But what happens if we want to multiply 3 by 7? Neither of these numbers is in S2 and will not be even if we extend the row further to the left or right. What can we do about this? What we need is to have more numbers in S2. One way to accomplish this and this is essentially Napier's idea is to use a base much closer to 1 than our base of 2. Napier, in fact, used 1.0000001. So the first few terms of his sequences, which we will label S7 and S8 looked like the following: S7 S8 1 1.0000001 2 (1.0000001)2 3 (1.0000001)3 4 (1.0000001)4 5 (1.0000001)5
16. Use your calculator to rewrite the numbers in S8 in decimal form. What do you notice about these numbers? Given that we wanted to multiply 3 by 7, about how many numbers would we need to take in the sequence S8 in order for 3 (or even a number very close to 3) to appear? Is this feasible? Napier used some very clever ideas to shorten what looks like an impossible task. Yet he nevertheless worked over twenty years on his table. We will look a somewhat different method for putting more numbers in S2. We will first fill in S1 and then try to figure out the appropriate correspondences. We begin by putting in between the 0 and the 1. We now must determine what the corresponding number in S2 must be. We will call it t. S1 S2 0 1 1 t 2 2 4 3 8 4 16 5 32 6 64
17. To determine t, we need to use the properties of these sequences. In particular, we know that addition in S1 corresponds to multiplication in S2. Therefore, since + = 1 in S1, we must have tt = 2. (Why?) We can rewrite this equation as t2 = 2. What is t? Write the answer in radical form, in decimal form, and in exponential form. In terms of the log2 notation, what is log2 t? 18. Now put 1/3 and 2/3 in their appropriate positions in S1. Let u be the number in S2 corresponding to 1/3 and v be the number corresponding to 2/3. Again using the property
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that addition in S1 corresponds to multiplication in S2, write an equation which u must satisfy. Solve the equation to determine u in radical form, in decimal form, and in exponential form. Then determine v similarly. (Note that instead of using the property that addition in S1 corresponds to multiplication in S2, you may wish to use the property that multiplication in S1 corresponds to taking powers in S2.) What are log2 u and log2 v? 19. Now put and in their appropriate positions in S1. As in question 18, determine the corresponding numbers in S2 in radical form, in decimal form, and in exponential form. Do the same for the numbers 3/2, 4/3, and 5/4 in S1. Determine the logarithm to the base 2 of each of the numbers you have now found in S2. 20. By analogy to your solutions to problems 17, 18, and 19, determine in exponential form and in radical form the number in S2 corresponding to the fraction m/n in S1. Use explicitly one or more of the properties relating S1 to S2. Since we have now filled in S1 with as many fractions as we like, we have also filled in S2 with lots of numbers. If we want to multiply 3 by 7, we now need to determine the numbers in S1 corresponding to 3 and 7 in S2. 21. The numbers in S1 corresponding to 3 and 7 are log2 3 and log2 7, respectively. Show that log2 3 is approximately equal to 317/200 and that log2 7 is approximately 306/109. You will need to use your calculator and the general definition of a logarithm. Find the sum of 317/200 and 306/109. Show that this number in S1 corresponds to 21 in S2. 22. Find two other numbers to multiply by the method of exercise 21. You may have to experiment a bit with your calculator to do this. 23. Consult at least two reliable sources (other than the Internet) in the library and write a short biography of John Napier. Your account should include information not contained in this document. Also, find at least one major application of logarithms (not discussed in class) and briefly describe it. Your write up must be typed and be no more than two pages doublespaced. See me if you need resources to consult.
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Calculation of Logarithms Using the Method of Napier and Briggs Teacher Notes
Level: Algebra 2 or Precalculus (Prerequisites: Properties of exponents; Properties of logarithms; Definition of logarithm) Objective: The purpose of this activity is to give the students an opportunity to investigate, on a introductory level, one way that logarithms can be calculated. Materials: Student Pages; Calculator (optional) Time Frame: One class period with homework assigned to finish the student activity. How to Use: The teacher may want to discuss briefly the included Historical Background with the class. The teacher may also want to review the definitions and properties of logarithms and exponents. It would be a beneficial for the teacher to work the activity before presenting it to the class especially to think about different ways to get the solutions, since some of the methods of solution are not unique. Additionally, the order in which to calculate the logarithms is not necessarily the numerical order, so the teacher will want to warn the students of this fact and perhaps give some hints about what to do when. Also note that these are base 10 logarithms that are being calculated. Bibliography Katz, Victor J., A History of Mathematics: An Introduction, Addison Wesley, 1998.
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Calculation of Logarithms Using the Method of Napier and Briggs Solutions
Part I: log 1 log 1 = y 10y = 1 (definition of logarithm base 10) 10y = 1 y = 0. Therefore log 1 = 0 (this is true for logarithms of any base) At this point it might be best to find log 10 next since it is also easy to calculate: log 10 = y 10y = 10 (definition of logarithm base 10) 10y = 10 y = 1. Therefore log 10 = 1 log 2 Consider the following relationship: 210 = 1024 1000 = 103 210 103 Take the logarithm of both sides of the approximation: log 210 log 103 Now use the power property of logarithms: 10 log 2 3 log 10 And solve for log 2: log 2 (3 log 10)/10 = (3 1) / 10 = 0.3 So log 2 0.30 log 3 Consider the following relationship: 37 2187 2 1093 2 103 37 2 103
Take the logarithm of both sides of the approximation: log 37 log (2 103) Now use the sum property of logarithms: log 37 log 2 + log 103
103
Now use the power property of logarithms: 7log 3 log 2 + 3log 10 Solve for log 3: log 3 (log 2 + 3 log 10)/7 (0.30 + 31)/7 = 3.3/7 = 0.47 So log 3 0.47 log 4 Since 4 = 22, So log 4 0.6 log 5 Since 5 = 10/2, log 5 = log (10/2) = log 10 log 2 (by the difference property of logarithms.) 1 0.30 = 0.70 So log 5 0.70 log 6 Since 6 = 23 , log 6 = log (23) = log 2 + log 3 (by the sum property of logarithms.) 0.3 + 0.47 = 0.77 So log 6 0.77 log 7 Consider the following relationship: 72 = 49 50 = 102/2 72 102/2 Take the logarithm of both sides of the approximation: log 72 log 102/2 Now use the difference property of logarithms: log 72 log 102 log 2 Now use the power property of logarithms: 2 log 7 2 log 10 log 2 Solve for log 7: log 7 (2 log 10 log 2)/ 2 (21 0.30 )/2 = 1.70/2 = 0.85 So log 7 0.85 log 4 = log 22 = 2 log 2 2(0.3) = 0.6
104
log 8 Since 8 = 23, So log 8 0.90 log 9 Since 9 = 32, So log 9 0.94 log 10 log 10 = 1 (see above) Solutions for Part 2: 1. 2. Since log 1 and log 10 are exact values, an equal sign is used and since the other values are approximations, the approximately equal sign is used. Since the other numbers, such as 4, 6 and 8 are products of these prime numbers, we are able to use the approximate values for the logarithms of 2, 3 and 7. Another reason would be that there is no easy way to use the definition of logarithm to find, say log 2 since by the definition, log 2 = y 10y = 2. (Suggestion for additional activity using a graphing calculator, find the intersection of the graphs of y1 = 10x and y2 = 2, to find a value for log 2.) The values are not the same because of the approximations used in the calculations in the project. The values are not exact since approximations were used for log 2, log 3 and log 7. As an example, log 6 was found using both of the approximations for log 3 and log 2. Using a calculator, log 6 = 0.7781513, while the approximation is 0.77. The difference in values is more than 0.008. Using the calculator, log 3 = 0.4771213 and log 2 = 0.30103 while the approximations were log 3 0.47 and log 2 0.30. Both of these approximations were closer to the calculator values than that of log 6. There are many examples of other relationships such as 72 = 49 = 5 10 and so on. 5. The estimate for log 2 is too large because 210 is larger than 103 log 9 = log 32 = 2 log 3 2(0.47) = 0.94 log 8 = log 23 = 3 log 2 3(0.30) = 0.90
3. 4.
105
The estimate for log 5 is too small because the approximation for log 2 is too large and this value is subtracted from an exact value resulting in an approximation that is too small. 6. There are many approaches. One possible solution is that 112 = 121 120 = 456 Taking the log of both sides gives : 2 log 11 = log 456 = log 4 + log 5 + log 6 0.60 + 0.70 + 0.77 = 2.07 So log 11 2.07/2 log 11 1.035 (the calculator value is 1.0413927)
106
Calculation of Logarithms Using the Method of Napier and Briggs Student Pages
Historical Background Briggs and Napier each spent many years in doing lengthy computations to determine logarithms. Since Napier did not use base 10 for his logarithms, but Briggs did, this project will demonstrate Briggs' approach. Keep in mind though, that we will be finding approximations for only a few common logarithms and will not have nearly the accuracy that Briggs found when he was developing his logarithm tables. Recall the definitions and properties that you know for logarithms: Definition: logb x = y by = x Sum Property of logarithms: logb(xy) = logb x + logb y Difference Property of logarithms: logb(x/y) = logb x - logb y Power Property of logarithms: logb xn = n logb x Part 1: Find approximations for the common logarithms for the positive integers 1 through 10. Recall that common logarithms are base 10 logarithms. To help get started, there are several hints of approaches to use for finding the logarithms of the prime numbers 2, 3 and 7. log 1 = _________ log 2 __________ log 2
The Fundamental Theorem (Part 2)Let f (x) be continuous and nonnegative on [a, b] and let A(x) be the area under the graph of y = f (x) over the interval [a, x]. Then we can express this function as the definite integralxA(x) =af (t) dt(wher
Denite Integrals by u-SubstitutionWe adapt the idea of u-substitution to denite integrals with two methods for taking the limits of integration into account. We wish to evaluate a denite integral of the formbf (g(x)g (x) dx.aMethod 1 Use the m
University of Colorado at Boulder Department of EconomicsProf. Jeffrey S. Zax [email protected] 303-492-8268 Economics 4818 Answers: Problem Set 9 11 April 20099.10Suppose we believe that bigger values of the explanatory variables are associated
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Homework Problem Set #6 ATOC/ASEN 5235, Spring 2009 Due Thursday, 5 March 1. 25 points. Petty 7.5. At a certain wavelength in the visible band, the optical thickness of the cloudfree atmosphere is * = 0.2. Determine the transmittance of sunlight at t
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{# read in the data file as a matrix of 13 columns# first column is the year and the next 12 are the monthly# rainfall values#commands to read data, calculate basic monthly statistics, plot#them for each month and also boxplots and histograms
V ISIONS OF S USTAINABILITY IN 2050Anthony D. Cortese, ScD PresidentThe following is a vision for a healthy, peaceful, socially just, economically secure and environmentally sustainable world. This vision is synthesized from the thinking of a larg |
Courses
Course Details
MATH 210B Concepts of Elementary School Mathematics II
3
hours lecture,
3
units
Grade Only
Description: This course is the second course in a one-year sequence in the study of the mathematical concepts needed for teaching elementary school mathematics with emphasis on geometry, transformational geometry, and measurement. This course also promotes an appreciation of the importance of logical thinking and applications of mathematics in problem solving and critical thinking. It studies the understanding and explanation of the basic mathematical concepts and the connections between them. It is designed especially for students preparing for credentials in elementary education. Analytical reading and problem solving are required for success in this course. |
Intermediate Mathematical Olympiad 2011
These resources from the United Kingdom Mathematics Trust (UKMT) include the Intermediate Mathematical Olympiad and the international Kangeroo challenges.
The Olympiad is open to students in Years 9, 10 and 11 who have been invited to participate due to their being in the top 500 students in each group in the Intermediate Challenge.
The questions for Year 9 and below are to be found in the Cayley paper Year 10 in the Hamilton paper and Year 11 take the Maclaurin version.
The guidance notes are provided as an informal guide and show how it is possible to attack a difficult problem which may be unlike anything the students have ever seen before.
Some general guidance as well as three sample Olympiad questions which are discussed in detail, in an attempt to explain some of the thought processes needed when tackling them - these are not intended to be 'model' solutions!
The European Kangaroo Foundation is an international organisation based in Paris which organises the Europe-wide European Kangaroo. The name sounds strange, but recognises the fact that the European organisation was inspired by the Australian Mathematics Trust. Each year over three million school pupils across Europe (and beyond) take part at various levels. The UKMT has been involved in Kangaroo activities for several years and uses the hour-long multiple-choice papers as an extension to the Intermediate Challenge.
The solutions to all the papers are collected together in the solutions booklet |
Bar Graphs (Lesson 53 of 61 math test prep lesson that explains how to create and interpret bar graphs as part of the Data Analysis and Probability material that many state exams cover.
This is one of 61 lessons available in the workbook titled The Essentials of High School Math from Willow Tree Publishing. Each book contains an answer key to all lessons, but an answer key for just this lesson is not available.
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
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Tips to help make Math Simple
Frustration, stress and anxiety. Do these words come to mind when you hear the word "Math"? Well, for those of you who are experiencing trouble with Math, here are some tips to help make the subject an easier one.
Read the Book
Read the assigned sections over carefully and look closely at the sample problems. Decide if you benefit more by reading before or after the instructor covers the material.
Develop a Sound Math Foundation
Because most math courses are cumulative (in other words, new concepts are added to and build upon previous concepts), it is very important that the early material be mastered thoroughly. Similarly, mastery of material from previous courses makes success in later courses more likely, so continually review and practice concepts from prior math classes.
Time Management
Complete all readings – especially homework assignments – as soon after they are announced as possible. And definitely complete all assignments before new material is covered since math is cumulative. This ensures that the information is fresh in one's mind and linked to prior, more fundamental information. Do your assignments early enough that you can get help with the things you do not understand.
Calculator
Learn how to use your calculator effectively and efficiently, especially if exams are timed and you have trouble completing tests in the allotted time. Check with the instructor about suggestions for the appropriate calculator to purchase for a class. Be sure the machine comes with an instruction manual and read the manual. Learn how to use important function keys. Get in the habit of carrying the calculator with you. It is better in the long run to become proficient with your own calculator rather than borrowing other people's calculators.
Show Your Work
Avoid the temptation to skip steps when solving a problem unless you are quite clear about how to proceed. This is a good habit to get into with your math homework. And definitely don't skip steps on an exam no matter how well you know the material. Why take chances (unless you're running out of time)? Showing your work allows you to locate logical or calculation mistakes more easily, and sometimes partial credit is given for the correct portions of an answer.
Organise Your Work and
Write Legibly
Write all numbers and variables clearly so they may be easily distinguished. Pay particular attention to the distinguishing lines of the numbers 4 and 9, 1 and 7 and the letters x and y.
Spaces are as important in math equations as are the numbers and variables themselves. Allow enough space between different terms in an equation so it is easy to distinguish them.
Be sure to line up the terms in each step of the solution, and write steps one below the other rather than to the right or left. Use lined paper or graph paper to help organise the problems on your page. Don't scrunch! Use plenty of paper to work each problem. Recycle the paper at the end of the term if you are concerned about wasting paper.
Support Services and Materials
Find out about the support services and materials available to you. Support services include workbooks, study groups, self-help videos and cassettes, peer tutors, professional tutors, and instructors' office hours. Using the resources from the start of the course may help your confidence and get you off on the right foot. Minimally, make use of these resources as soon as you feel uncomfortable with the material; do not wait until it is too late!
Preparation and Supplies
• Being prepared for each course involves several important factors:
• Complete any previously assigned homework assignments
• Compile a list of questions about the previous assignments to ask the instructor |
MATH 1: Fundamentals of Mathematics
This course provides the student with the foundation in arithmetic that is necessary for a study of MATH 002. It includes whole number concepts, fractions, decimals, percents, ratios and proportions and signed numbers. May not be used to meet graduation requirements.
Credits:3
Overall Rating:0 Stars
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Thanks, enjoy the course! Come back and let us know how you like it by writing a review. |
Graphs are mathematical entities whose theory facilitates the discussion of the relationships between the elements of a set. Introduced about 1870 by the pioneers of combinatorics who were then still known as geometers, graphs have recently been recognized as the most adaptable tools for certain organizational problems. After first being used in operations research, graphs have now been introduced into information theory where they have proved to ba of use in many different areas. ...
This book stresses the connection between, and the applications of, design theory to graphs and codes. Beginning with a brief introduction to design theory and the necessary background, the book also provides relevant topics for discussion from the theory of graphs and codes. ...Algorithmic Graph Theory and Perfect Graphs, first published in 1980, has become the classic introduction to the field. This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems. It remains a stepping stone from which the reader may embark on one of many fascinating research trails.
As graph theory continues its explosive growth, conjectures are proved and new theorems formed. The techniques involved, which have applications in a broad spectrum of mathematics, ranging from analysis to operations research, have become more sophisticated if not more manageable. This new edition, therefore, includes new theorems (e.g. the Perfect Graph Theorem, due to Lovasz) as well as new proofs of classical results. A number of sections have been significantly revised. ...
Graphs and Matrices provides a welcome addition to the rapidly expanding selection of literature in this field. As the title suggests, the book's primary focus is graph theory, with an emphasis on topics relating to linear algebra and matrix theory. Information is presented at a relatively elementary level with the view of leading the student into further research. In the first part of the book matrix preliminaries are discussed and the basic properties of graph-associated matrices highlighted.The Graphs and Models series by Bittinger, Beecher, Ellenbogen, and Penna is known for helping students "see the math" through its focus on visualization and technology. These books continue to maintain the features that have helped students succeed for years: focus on functions, visual emphasis, side-by-side algebraic and graphical solutions, and real-data applications.
This book is concerned with the relations between graphs, error-correcting codes and designs, in particular how techniques of graph theory and coding theory can give information about designs. A major revision and expansion of a previous volume in this series, this *** includes many examples and new results as well as improved treatments of older material. So that non-specialists will find the treatment accessible the authors have included short introductions to the three main topics. This book will be welcomed by graduate students and research mathematicians and be valuable for advanced courses in finite combinatorics. ... |
Need Help With Physics 3?
(Electricity and Magnetism)
DVDs in this area are focused on building skills in Physics 3, which covers Electricity and Magnetism. This gives many students difficulty because this subject is essentially the combination of math and word problems.
These Physics DVDs focus on building problem solving skills by working many fully narrated example problems in step-by-step detail. In this way the student gains confidence and improves skills immediately.
Every DVD on MathTutorDVD.com comes with a Money Back Guarantee! Our DVDs are guaranteed to raise grades immediately. If you are not fully satisfied, simply send the DVD back for a full refund. |
Schaum's has Satisfied Students for 50 Years.
Now Schaum's Biggest Sellers are in New Editions!,... more...
Trigonometry has always been the black sheep... more...
CliffsQuickReview course guides cover the essentials of your toughest classes. Get a firm grip on core concepts and key material, and test your newfound knowledge with review questions. CliffsQuickReview Trigonometry provides you with all you need to know to understand the basic concepts of trigonometry — whether you need a supplement to your... more...
Part of the ''Demystified'' series, this book covers various key aspects of trigonometry: how angles are measured; the relationship between angles and distances; coordinate systems; calculating distance based on parallax; reading maps and charts; latitude and longitude; and more. more... |
Precalculus Honors is a full year five credit course representing the fourth in a sequence of five Honor math programs. Satisfactory completion of Honors Geometry and Honors Alg II are required for entry into Precalculus Honors. The Precalculus program allows students to develop a solid foundation of trigonometric skills and concepts along with analytical geometry, advanced algebra and discrete mathematics.Major units of study include: Trigonometric functions, graphs and inverses of trigonometric functions, trigonometric identities and equations, polar coordinates and complex numbers, conic sections, exponential and logarithmic functions,seqyence and series, combinatorics and probability, vectors and parametric equations, and statistics and data analysis. Throughout the curriculum an indepth study of each major topic will include: math vocabulary, spelling, reading comprehension. Assessment strategies include: tests, quizes,homework, groupwork, projects, class participation, internet assignments, notebook, and marking period assessments.
Program Purpose:
The primary purpose of the Precalculus Honors program is to develop skills and proficiencies and encourage independent thinking. The students math skills will be refined and accerelated to meet the demands of advanced math courses such as Calculus AP, and related fields such as chemistry, physics,college engineering and actuarial sciences. The nature of this course will give students the ability to view mathematics through applications to real problems outside the classroom. The students will have frequent opportunities to see mathematical rules logically developed and applied. Advanced technology will be included in the majority of the coursework throughout the year. |
Follows Saxon's Algebra 1, 3rd edition text. Includes lectures and visual instruction for each and every Saxon lesson. Arithmetic and evaluation of expressions involving signed numbers, exponents,...
More about Algebra 1 (3rd ed) Dive Into Math CD
A supplemental compact disc that is designed to be used in conjunction with Saxon's Algebra 1/2 - 2nd edition textbook. Includes lectures and visual instruction for each and every Saxon lesson. ...
More about Algebra 1/2 (2nd ed) Dive Into Math CD
Follows Saxon's 3rd edition textbook. Includes lectures and visual instruction for each each and every Saxon lesson. Fractions, decimals, signed numbers and their arithmetic operations, translating...
More about Algebra 1/2 (3rd ed) Dive Into Math CD
For use with Saxon Algebra 2, 2nd edition textbook. Includes lectures and visual instruction for each and every Saxon lesson. Graphical solutions of simultaneous equations; scientific notation;...
More about Algebra 2 (2nd ed) Dive Into Math CD
The new Online Algebra 1 Placement Test (APT) covers nineteen key concepts essential for success in Algebra 1. It is a quick, easy, and inexpensive way to see if your...
More about Algebra Placement Test
A supplemental CD that may be used in conjunction with the Saxon Calculus, (1st edition) text. Includes lectures and visual instruction for each and every Saxon lesson. Covers all material...
More about Calculus (1st ed) Dive Into Math CD
A collection of word problems with Catholic themes written by the Seton Staff and designed for students using level D of MCP Mathematics. More than thirty separate lessons with 10 Catholic...
More about Catholic Word Problems Level D
A collection of word problems with Catholic themes written by the Seton Staff and designed for students using level E of MCP Mathematics. 15 separate lessons with 10 Catholic word problems...
More about Catholic Word Problems Level E
Counting with Numbers introduces the child to the practice of neatness in all work, carefulness in the use of books, and following directions. It continues the presentation of recognition of shapes...
More about Counting with Numbers
E-Z Grader is a hand-held manual computer. It helps make your homeschooling just a little bit easier. It saves time and effort otherwise needed to compute grades. Gives you freedom to use any number...
More about E-Z Grader
NEW from Dive Into Math, this Interactive CD is for use with the new Saxon Geometry 1st edition textbook. The DIVE CD-ROM teaches each of the 120 lessons and 12 Investigations, plus...
More about Geometry (1st ed) DIVE Into Math CD |
Book Description
Release date: August 5, 2008 | Age Range: 11 and upEditorial Reviews
About the Author
Best known for her roles on The Wonder Years and The West Wing, Danica McKellar is also an internationally recognized mathematician. She was chosen as ABC World News Tonight's "Person of the Week" for writing Math Doesn't Suck and has recently been featured in Newsweek and The New York Times, and on the CBS Early Show, and NPR's Science Friday.
--This text refers to an out of print or unavailable edition of this titleThis funny math book teaches girls that it's OK to be smart, and that they are perfectly capable of kicking a little pre-algebra butt.
McKellar takes a lightweight approach to math, but is deadly serious about it. In the prologue, she writes that "lots of people change their majors and abandon their dreams just to avoid a couple of math classes in college." Girls in particular, she emphasizes, often use their fear of math to keep them from learning the skills they'll need to succeed in life, and they start backing away from the subject in middle school.
And it's not just fear. Girls often don't see how they'll use math once they get out of school. Testimonials in Kiss My Math fight this. Stephanie Perry, the finance director for Essence magazine, explains how she uses algebraic formulas to stay on top of the magazine's financial performance. Jane Davis, financial strategist at Polo Ralph Lauren, was hired as an assistant buyer because of her facility with math. She describes determining inventory over time by finding the mean of a list of numbers.
McKellar -- famous for playing Winnie Cooper in the "The Wonder Years" but also a summa cum laude math graduate from UCLA -- uses simple language and lots of illustrations to teach pre-algebra. Each chapter covers a single topic, such as the distributive property or exponents. She clearly explains each topic, and includes problems for the reader to solve (answers are in the back). The author is generous with helpful notes and shortcuts.
A lively, breezy writing style makes it seem as if McKellar is sitting next to the reader. She uses examples girls can relate to, like clothes shopping, working on the school play, blind dates, parties, kissing and breath mints. It's like having the perfect math tutor. (I'm not a middle school girl, of course, but I just got finished having one. My daughter is starting high school this month.)
Especially good are the entries called Danica's Diary, which are true stories from the author's life as a student, actress and mathematician. One is titled: Dumbing Ourselves Down for Guys: Why is it so Tempting? McKellar gives practical advice on how girls can avoid this common pitfall.
I can't think of a better book to buy for a girl taking pre-algebra.
Here's the chapter list:
Part 1: Number Stuff Chapter 1: Breath Mint, Anyone? Adding and Subtracting Integers (Including Negative Numbers). Chapter 2: The Popular Crowd. The Associative and Commutative Properties. Chapter 3: Mirror, Mirror, on the Wall... Multiplying and Dividing Integers (Including Negative Numbers!) Chapter 4: A Relaxing Day at the Spa. Intro to Absolute Value. Chapter 5: Long-Distance Relationships: Are They Worth It? Mean, Median, Mode. You Said: Most Embarrassing Moments in School Poll: What Guys Really Think... About Smart Girls Quiz: Are You a Stress Case?
Part 2: Variable Stuff Chapter 6: The Blind Date. Getting Cozy with Variables. Chapter 7: Backpack Too Heavy? Adding and Subtracting with Variables. Chapter 8: Something Just Went "Squish." Multiplying and Dividing with Variables. Chapter 9: Do You Like Him Like Him? Combining Like Terms. Chapter 10: The Costume Party. The Distributive Property. Chapter 11: Didn't That Guy Say He Was Going to Call? Using Variables to Translate Word Problems. More Than 20 Ways to Beat Stress Math... In Jobs You Might Never Expect!
Part 3: Solving for X Chapter 12: The Art of Gift Wrapping. Solving Equations. Chapter 13: Nope, She Never Gets Off the Phone. Word Problems and Variable Substitution. Chapter 14: Can a Guy Be Too Cute? Intro to Solving and Graphing Inequalities. You Said: Your Horror Stories About Procrastination Poll: What Guys Really Think... About Talented Girls Quiz: Do You Pick Truly Supportive Friends?
Part 4: All About Exponents Chapter 15: Champagne and Caviar. Intro to Exponents. Chapter 16: Excuse Me, Have We Met Before? Intro to Variables with Exponents. You Said: Well... That Didn't Work! Do You Sudoku?
I am a mother that went back to college later in life. One of my classes was algebra. I had math anxiety and tried to find way to wiggle my way out of this class. The algebra class was very difficult for me. I could not understand the instructor or the book. I went to tutors,family members and friends and I could not get algebra. I failed the class. I was embarrassed and angry with my myself. I needed something right away. So my boyfriend and I went to Barnes & Nobles and purchased Kiss My Math & Math Doesn't suck. (I do suggest that you purchase both). So I had a six week break before I had to take the algebra class again. I am happy to say that I passed the algebra class with a B and I am looking forward to starting MATH 209 which is the second part of algebra. Danica was easy to understand and the experiences from other young ladies helped a great deal too. Thanks Danica!
Danica McKellar is a beautiful actress who is probably very well off and successful. So why did she go to UCLA to study math after being a very successful child star on the wonder years and then bother to write a book entitled Math Doesn't Suck. Well it is because she wanted to prove she was more than just a good looking actress. She had a brain and could handle math. The attitude that math is not for the ladies was a horrible prejudice in my high school years and even in this enlightened age we haven't quite gotten over it and many a capable young lady lacks the confidence and courage to try to do math. Danica is a rol model who proves that they can. Her first book was so successful and helped young middle school girls overcome their fears and lkearn that math is not really hard and can be fun and interesting whenit is approached in the riht way. So math does not suck! But in addition to convincing young girls and boys that they can learn it she became encouraged to write another book based on the encouraging emails from young ladies who benefitted from the book. Well love of math should not end with middle school and algebra, geometry and calculus are very different form the kind of math you learn in the elementary and middle schools that a good series of lectures in pre-algebra is needed to help those who become discouraged again in high school. It bothers Danica to see a girlfriend of hers give up on medical school just because calculus is required. So in the same interesting style as her first book Danica interest the high schoolers with concepts like negative numbers, mathematical inequalities, exponential functions and much more. By uncovering the mysteries of pre-algebra Danica unlocks the door to advanced levels of mathematics that students in high school need. This book is good for high school teachers and anyone else with an interest in mathematics. But it is aimed at and can help most high school girls who are capable of doing well in math and nedd it for the careers they seek, like med school. |
MATH
320 Number Systems, Statistics and Probability (5) Continuation of the
development of the real numbers including applications and models of rational
numbers, irrational numbers, percent, and proportional reasoning. Introduction
to the basic notions of chance and probability. Introduction to data analysis
and statistics. This course involves substantial use of 2- and 3-dimensional
concrete materials in a cooperative learning setting. Prerequisites: A grade of
C- or better in MATH 221. (4
units lecture and 1 unit activity).
Objectives:
Mathematics 320 is the second of a sequence of three courses designed to help
students gain the mathematical knowledge, skills, and understanding that are
essential for those pursuing a Multiple Subject Credential as prescribed by the
California Commission on Teacher Credentialing. Students will identify and
prioritize relevant and missing information in mathematical problems. They will
analyze complex problems to identify similar simple problems that might suggest
solution strategies. They will explain their mathematical reasoning through a
variety of methods, such as words, numbers, symbols, charts, graphs, tables,
diagrams, and concrete models. They will use appropriate mathematical notation
with clear and accurate language. They will explain how to derive a result based
on previously developed ideas and explain how a result is related to other
ideas.
Demonstrate an understanding of
functions with domain the rational numbers.
Demonstrate an understanding of
decimals and operations on decimals.
Demonstrate an understanding of
nonterminating decimals.
Demonstrate an understanding of
real numbers.
Demonstrate an understanding of
ratios, proportions and proportional reasoning.
Demonstrate an understanding of
percents.
Demonstrate an understanding of
probabilities.
Perform experiments with tree
diagrams and geometric probabilities.
Using simulations in
probability.
Demonstrate an understanding of
odds, conditional probability and expected value.
Draw sensible conclusions from
displayed data.
Demonstrate an understanding of
measures of central tendency and variation.
Demonstrate an understanding
when statistics is abused.
Note:
Although some attention will be given to how to teach mathematics effectively to
children, this is a content course, not a methods course. Methodology is the
primary focus of the Credential Program.
Class Meetings: MW: 10:50-12:30 pm,
Science III 103.
Th:
10:30-12:30 pm, Science III 103.
Textbook
(required): Custom
edition of A Problem Solving Approach to Mathematics for Elementary Teachers,
10th ed, Part 1, by Billstein, Libeskind, and Lott, and a packet of labs
taken from Mathematics for Elementary School Teachers An Activity Approach,
by Bennett, Burton, and Nelson. Sections of the text being covered in
lecture should be read prior to coming to class. Lectures will not cover all
topics from each section, but students will be responsible for all topics in the
section.
Supplies:
In addition to the textbook, each student must have a scientific calculator,
several colored pencils, and 1/4-inch graph paper. These should be brought to
class with you each day.
Homework:
Mathematics is not a spectator sport. You learn by doing. Although homework
assignments will not be collected, it is assumed that you will do the homework
necessary for success in this class. We will spend considerable time at the
beginning of each lecture working out homework problems. Answers to most of the
assigned problems are in the back of the book. If you are not able to solve a
particular problem, do not hesitate to ask! Your classmates will be
grateful.
Readings:
The student is responsible for reading at least twice each section of the book
covered in class: Before and after the lecture. The student will be told in
advance what sections of the book are to read. If you are not able to
understand something in the book, do not hesitate to ask! Your classmates
will be grateful.
Quizzes:
There will be seven
quizzes, each of them administered most Thursdays at 10:35 pm; the student will
be able to drop the lowest score. Quizzes may not be made-up. If you miss a
quiz you will get a score of "0" (zero) recorded. You may take any quizearly with the instructor's permission. Problems will be similar to those in
the homework or in the preceding activity. The primary purpose of these quizzesis to provide you with frequent evaluation of your content acquisition and
to help you to reduce math anxiety. Quiz work is individual, see Note 2
below.
Activities:
Seven activity
assignments will be collected; the student will be able to drop the lowest
score. Activity attendance is required: Most Thursdays 10:50-12:30.
The student should work in groups of at least 3 people and 4 at most. During
activity sessions the student will work on an activity sheet and each member
of his/her group should hand in a report at the beginning of next Thursday at
the latest. Activity work must be turned in on the activity pages from the
packet. Late reports and reports from students absent from even one meeting
will not be accepted. A report should be neat and readable; solutions should
follow an increasing numerical sequence. Each member of a group will receive the
same number of points, so it is your responsibility to write down in each report
only the names of those who substantially contributed to the activity.
Exams:
There will be two Midterm exams and a cumulative final. Questions and problems
will deal with concepts discussed in lectures, homework, textbook, and
activities. Exchange of information, calculators, and supplies is absolutely
prohibited during exams! To clarify a particular situation, the instructor
reserves the right to a further examination, written or oral. Exam work is
individual, see Note 2 below. Midterm exams may not be made-up. If you miss
an exam you will get a score of "0" (zero) recorded. You may take any exam
early with the instructor's permission.
Partial
Credit: Only substantial
contributions to the solution of a problem will count for partial credit. Mere
restating of a problem or the quoting of an incorrect fact, for example, will
not make you eligible for it. Students must watch out for logical mistakes, and
must make sure that all the hypotheses are met before recalling a particular
theorem.
Grades:
Your final grade is a function of the total of points awarded on the activities
indicated above. Group homework is very helpful and greatly promoted in this
course. However, it is necessary for college graduates to demonstrate individual
competency on the subject. Therefore, regardless of your total of points, in
order to get a D- or better in this course all of the following should be
fulfilled at least:
(a) To have scored at least
200 points in the 2 Midterm exams combined.
(b) To have scored at least
105 points in the final exam.
Generally, the following guidelines
for grades apply:
960 -1000 A
900 - 959 A-
870 - 899
B+ 840 - 869 B 800 - 839 B-
770 - 799
C+ 740 - 769 C 700 - 739 C-
670 - 699
D+ 640 - 669 D 600 - 639 D-
0- 599 F
If
you entered the liberal studies program on Fall 2011, you are required to get a
C or better in all classes that you are taking for your major - this includes
Math 320.
Notes:
1)
It is the student's responsibility
to find out what (s)he missed if (s)he did not attend class. Office hours are
not meant for tutorial courses, but rather to clarify particular situations or
problems occurring during lectures, homework, labs, assignments or readings.
Students are encouraged to make use of the Office Hours.
3)
The instructor will hold graded
papers for one week at most. After this period, he will trash old papers.
Contact him as soon as you foresee a problem picking up your paper(s).
4)
All handouts (with solutions) will
be in the internet (follow the link Math 320 in
5)
Beepers, cell phones, i-pods,
laptops and similar electronic devices must be turned off at all times during
class or lab time. If not, the student will have to leave the room without being
allowed to return.
6)
Students can be at most 10 minutes
late. Students cannot leave the room, unless it is for medical reasons. |
Algebra
Mathcentre provide these resources which cover a wide range of algebraic topics, many of which are suitable for students studying mathematics at Higher Level GCSE, or A Level, as well as those students for whom mathematics is an integral part of their course. Some of the topics covered include completing the square, factorising quadratics, partial fractions, integration, simultaneous linear equations, logarithms and polynomial division.
Comprehensive notes, with clear descriptions, for each resource are provided, together with relevant diagrams and examples. Students wishing to review, and consolidate, their knowledge and understanding of algebraic principles |
HSI STEM
The HSI STEM Grant promotes Math Success and provides a support system that encourages completion and success in Math classes and a smooth transition from LSC-North Harris to STEM academic programs at the University of Houston or other four-year universities. The grant features newly designed Math courses using innovative teaching techniques and cutting edge technology, high quality tutoring services and access to state-of-the-art technology in the Math Achievement Center (MAC), and access to speakers, mentors, and STEM oriented career and college prep events through participation in the Women in STEM group, and much more.
Featured Math Courses
The Math Department faculty has designed innovative, interactive, student-centered Math courses where students can use the latest technology and Math software to complete their assignments. Many of these featured courses also offer one-on-one, in-class assistance from instructors
Flipped Classroom: A combination of face to face and online coursework. Students watch recorded lectures at home. lecture videos are easily accessible on your iPhone, Android phone, iPAD, etc. Homework, assignments, and labs are completed in class with the instructor who is available to give students individual help as needed. All tests are taken in class with the instructor. This class meets on all assigned days.
ALEKS:
Students meet face to face with an instructor, but class work is completed using a computer program called ALEKS. ALEKS uses artificially intelligent adaptive questioning to find out exactly what the student knows, and designs an individualized learning plan specifically for each student. The instructor is available in the classroom to assist students one on one as needed. All tests (except the final exam) are taken on the computer using ALEKS.
Mathematica Software: Class lessons are lecture based with in-class demonstrations using iPADSs and Mathematica applets. Mathematica is a computational software used in scientific, engineering, and mathematical fields. It allows students to manipulate and solve problems with computer code. Lectures are enhanced by lab and home assignments using the Mathematica computer software.
Maple Software/Flipped Classroom: A combination of face to face and online coursework. This software enables students to visualize mathematical concepts and to investigate and solve problems that would be difficult to solve by hand. Students watch videos with lectures and embedded with animation from Maple. Labs and homework are completed in class with computers and instructor guidance. This class meets on all assigned days.
Online Math Courses: The entire course is offered online. All exams are proctored and must be taken on campus.
Several Math 0310 (Intermediate Algebra) and Math 1314 (College Algebra) featured courses are now available in the Summer 2013 schedule. More of these featured courses, including brand new sections of Math 1316 (Trigonometry) and Math 2412 (Pre Calculus) are available in the Fall 2013 schedule.
For more information about the HSI STEM Grant Math courses and activities, please contact Sylvia Martinez at [email protected] or call 281-765-7806. You can also stop by our office in the Winship Building, in WNSP 166. |
* The Final WW Date is the final date from which you can withdraw from the unit without academic penalty, however you will still incur a financial liability (see Withdrawal dates explained for more information).
This unit introduces students to the curriculum and pedagogy associated with teaching secondary mathematics (Grades 7 to 10). The unit models and provides opportunities for students to engage with rich tasks, problem solving and differentiating the curriculum using examples from across all domains of the Australian Curriculum. This unit will also explore and consider ways in which technology can enhance student learning of mathematics. Students will be encouraged to reflect on their own experience of learning mathematics at secondary school. The responsibility of teachers to foster positive attitudes towards mathematics in the classroom, school and wider community will also be considered |
"Math software" is a very general term, which encompasses computer algebra systems like Maxima and Mathematica, numerical computing environments like Matlab, and basic graphing calculators. It sounds like you want the third kind. – Rahul NarainOct 27 '10 at 7:57 |
Product Details:
.
From the Publisher: ""--"Homework Helpers: Algebra, Revised Edition, will help students understand topics such as linear equalities and inequalities, factoring polynomials, graphing functions, rational functions, quadratic equations, and word problems"--
Description:
About the Contents: Pretest Helps you pinpoint where you need
the most help and directs you to the corresponding sections of the book Topic Area ReviewsMath BasicsNumbers (Signed Numbers and Fractions)Linear Equations and Algebraic FractionsPolynomials and FactoringInequalities, Absolute Value ... |
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
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Card Sorting What is Card Sorting?Card Sorting is a technique for exploring how people group items, so that you can develop structures that maximize the probability of users being able to find items. Card Sorting: Is easy and cheap to conduct En
Filters (EC 14) Often need is to remove some frequencies This is done using Frequency Filters 4 basic types, each with a critical frequency co or o Low Pass Filter: Filter passes frequencies well below the critical frequency: <co Used in detecto
ENSC 460/894 Assignment 1 Solutions : (Jan. 20, `09 due Jan. 29, `09)1 Infrared security camera systems often want to detect the presence of living person relative to the background. (a) Assume the ground is just at freezing temperature, and the hum
ENSC 460/894 mid Term (Feb 28, 2008)1 of 1ENSC 460/894 mid Term (Feb. 28, 2008) This test is OPEN BOOK: any book, notes and calculator may be used, but not a computer. Time: 90 minutes NOTE: Do 2 questions in part I 25 marks each for a total of 5
MCS 320Introduction to Symbolic ComputationFall 2005Maple Lecture 4. Algebraic and Complex Numbers The number 3 is often called irrational because we cannot represent it as a rational number. Never5 theless, we know 3 5 as a solution to the eq
CS286r Computational Mechanism Design Homework 2: Mechanism DesignSpring Term 2002 Prof. David Parkes Division of Engineering and Applied Sciences Harvard University Feb 14, 2002Due: Tuesday 2/26/2002, in the beginning of class. You may use any sou
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CS286r Multi-Agent Learning Homework 1: Game TheorySpring Term 2006 Prof. David Parkes Division of Engineering and Applied Sciences Harvard University Feb 6, 2006Due: Monday 2/13/2005, at the beginning of class. You may use any sources that you wan
Introduction to Computer RecordsWith a computerized farm record keeping system, the recordkeeper can use software to store information, summarize data, generate and print reports and sort transactions into categories and subcategories. Storage of da
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First Draft of a Report on the EDVAC*By John von NeumannVon Neumann's computing responsibility on the Los Alamos Manhattan Project led him to Bell Laboratories and Harvard University in search of additional computing power, but neither facility
GRADING CRITERIA Team Rules of Behavior ExerciseIf everyone in the team did not meet face-to- face and participated in the discussion, it is an automatic F. Otherwise, you will be graded on the following five criteria:1. Are the rules of behavior
Lock-In AmplifiersApplication Note #3Lock-in amplifiers are used to detect and measure very small AC signalsall the way down to a few nanovolts. Accurate measurements may be made even when the small signal is obscured by noi.EC431H1 Digital Signal Processing LAB 2. The STFT and FFT. Applications to Signal Filtering and Detection.Introduction The purpose of this experiment is to become familiar with utilization of the Short-time Fourier Transform for discrete time sig
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A Weakness in the 4.2BSD Unix TCP/IP SoftwareRobert T. Morris AT&T Bell Laboratories Murray Hill, New Jersey 07974ABSTRACT The 4.2 Berkeley Software Distribution of the Unix operating system (4.2BSD for short) features an extensive body of softwarNational Aeronautics and Space AdministrationSTS107: Space Research and YouLetting Our Cells Do the FightingFlight-Induced Changes in the Immune Response The organisms that make us ill, such as bacteria, viruses, and fungi, are like attacking ar |
The following mini quizzes are an indicator of only a few of the skills which you are expected to have at the beginning of
a particular
course.The course itself will
offer little or no time for a formal review… assuming that you are truly
prepared right from the beginning of class.Do not use these problems as a personal study guide thinking that they
will adequately prepare you for the course.These particular problems represent just some of the more important
examples from a much larger body of needed objectives, which are taught in the
prerequisite courses.
If
you do well on the quiz, see a math instructor or start at the appropriate level on the college
placement test which should give you your correct placement.
The answers are
found at the bottom of each page.
Click
on the level of quiz that you wish to take. For a course
description, go back to the math homepage. Quizzes are available for Math
20-95. |
Description
Michael Sullivan's time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing with homework, and reviewing the concepts. The Enhanced with Graphing Utilities Serieshas evolved to meet today's course needs by integrating the usage of graphing calculator, active-learning, and technology in new ways to help students be successful in their course, as well as in their future endeavors.
In the Sixth Edition, new worksheets in MyMathLab—developed from the authors' experience in the classroom—provide mixed review for students who having trouble reconciling various topics, and also give students an opportunity to show their work. The "Are You Prepared?" section openers focus on students mastering the prerequisite material before beginning a new topic, and for the first time, those exercises are assignable in MyMathLab. Concept and Vocabulary exercises are also now assignable in MyMathLab as reading quizzes. |
Math Course Takes 'Real Life' Approach to Algebra
Educational courseware publisher American Education Corp. is taking a new approach to answering the age-old question, "What does algebra have to do with real life?" The company has announced the release of a new course for its A+nyWhere Learning System program. Algebra I: A Function Approach Part 1 is the first semester segment of a full-year algebra course geared to grades 9 and 10, and, in addition to the fundamental concepts and tools of algebra, the course aims to relate the material to "real life."
Taking the fundamentals and applying them to real-world situations using exercises in relevant scenarios allows students to realize the practical uses of linear and quadratic equations, graphs and coordinates, functions, and other algebraic concepts.
The A+nyWhere program is computer based, so students taking courses like Algebra I can use a number of tools incorporated into the software to aid in their assignments and overall comprehension of the material. These tools include onscreen standard and scientific calculators, pictures and diagrams, video tutorials, exercises, practice exams, and, for upper-level courses, interactive feedback |
A no-nonsense, practical guide to help you improve your algebra II skills with solid instruction and plenty of practice, practice, practice. Practice Makes Perfect: Algebra II presents thorough coverage of skills, such as handling decimals and fractions, functions, and linear and quadratic equations, as well as an introducing you to probability and... more...
This work is a concise introduction to spectral theory of Hilbert space operators. Its emphasis is on recent aspects of theory and detailed proofs, with the primary goal of offering a modern introductory textbook for a first graduate course in the subject. The coverage of topics is thorough, as the book explores various delicate points and hidden features... more... |
Geometric Sequences & Series describes geometric sequences and series. It shows how to determine if the sequence is geometric, how to find a specific term, and how to find the "rule" for a sequence. It also shows how to find the sum of a geometric series.
This lesson was designed for a Pre-Calculus class, so it has basically everything about Geometric Sequence and series all in one lesson because it should be a review of materials learned in Algebra 2. It could be redesigned for an Algebra 2 class by separating it into two lessons.
Presentation (Powerpoint) File
Be sure that you have an application to open this file type before downloading and/or purchasing.
1224.5 |
Advantage provides an interactive learning experience and the tools students need to gain learning confidence and improve their grades. Help your child build a solid academic foundation with MiddleSchoolAdvantage, a complete student ... more
Advantage provides an interactive learning experience and the tools students need to gain learning confidence and improve their grades. Help your child build a solid academic foundation with Middle School Advantage, a complete student resource center...
MiddleSchoolAdvantage® 2008 was specially developed to supplement classroom curriculum by including award-winning content that support state standards. In-depth lessons with sample problems and questions teach reinforce and track student ... more
Middle School Advantage® 2008 was specially developed to supplement classroom curriculum by including award-winning content that support state standards. In-depth lessons with sample problems and questions teach reinforce and track student progress inSelect overviews and in-depth lessons from more than 175 middleschool topics based on three years of math curriculum. Build your mental math muscle with hundreds of practice problems and examples. Apply math to everyday life with more ... more
Select overviews and in-depth lessons from more than 175 middle school topics based on three years of math curriculum. Build your mental math muscle with hundreds of practice problems and examples. Apply math to everyday life with more than 300 word... ... more algebra II,...
MiddleSchoolAdvantage 2008 was specially developed to supplement classroom curriculum by including award-winning content that support state standards. In-depth lessons with sample problems and questions teach, reinforce and track student ... more
Middle School Advantage 2008 was specially developed to supplement classroom curriculum by including award-winning content that support state standards. In-depth lessons with sample problems and questions teach, reinforce and track student progress in...
More information about middle school advantage 2011 education suite
Educational Software calling your name? Find all of the top Computers & Software gear that you want at BizRate. Compare prices from top brands like Encore |
Sometimes solve blocks cannot find a solution. Read on to see how to resolve some issues. Errors and Problems with No Solutions Sometimes there might be no solution, or Mathcad might not find a solution. In either case, Find displays the error message "No solution was found." The problem asks for numbers u and v that add to both 2 and 3, …
Easy Solutions and Visualizations: Exploring a System of Three Equations in Three Variables Using Mathcad In modern middle school and high school mathematics algebra is a gatekeeper course. Success in Algebra 1 in grade 8 is considered a prerequisite for college preparedness. My own first experience in Algebra 1 was at South Side Junior High School. At the time, a …
If you are visiting Boston this week for the American Mathematical Association of Two Year Colleges (AMATYC) Annual Meeting, stop by Booth 336 to meet the Mathcad Education Team and to learn more about Mathcad 15.0 and its role in PTC's STEM Education Programs. Ned Daniels and I have been working hard to prepare for the conference. We both have |
This handbook is designed as a work of reference, and provides a convenient source of basic definitions and formulae for use throughtout the three blocks. In addition to this it also contains a quick reference guide to Maple commands; and summaries of the main concepts, definitions and techniques of each of the units.
Purchase Mystery Math: A First Book of Algebra by David A Adler,Edward Miller and Read this Book on Kobo's Free Apps. Explore Kobo's Vast Collection of eBooks - Over 3 Million Titles, Including 2 Million Free Books!
Develops algebraic ideas in a logical sequence, and in an easy-to-read manner, without excessive vocabulary and formalism. This text reinforces the following common thread: learn a skill; practice the skill to help solve equations; and, then apply what you have learned to solve application problems.
Lie algebras have many varied applications, both in mathematics and mathematical physics. This book provides a thorough but relaxed mathematical treatment of the subject, including both the Cartan-Killing-Weyl theory of finite dimensional simple algebras and the more modern theory of Kac-Moody algebras. Proofs are given in detail and the only prerequisite is a sound knowledge of linear algebra. The first half of the book deals with classification of the finite dimensional simple Lie algebras and...
A Collection of Problems in Algebra with Solutions, Book, 2 Matrices and Vector Spaces
The Title "A Collection of Problems in Algebra with Solutions, Book, 2 Matrices and Vector Spaces" is written by E.F. Robertson. This book was published in the year 1984. The ISBN number 0521272866|9780521272865 is assigned to the Paperback version of this title. This book has total of pp. 112 (Pages). The publisher of this title is Cambridge University Press . We have about 80746 other great books from this publisher. A Collection of Problems in Algebra with Solutions, Book, 2 Matrices and Vector...
Can one learn linear algebra solely by solving problems? Paul Halmos thinks so, and you will too once you read this book. The Linear Algebra Problem Book is an ideal text for a course in linear algebra. It takes the student step by step from the basic axioms of a field through the notion of vector spaces, on to advanced concepts such as inner product spaces and normality. All of this occurs by way of a series of 164 problems, each with hints and, at the back of the book, full solutions. This book...
This that are...
Product Description Conf...
From radical problems to rational functions — solve equations with ease Do you have a grasp of Algebra II terms and concepts, but can't seem to work your way through problems? No fear — work with linear and quadratic equations...
This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure theory of linear operators on finite-dimensional vector spaces; the authors have included a self-contained treatment of that subject.
DESCRIPTION: This...
Schaum's has Satisfied Students for 50 Years.Now Schaum's Biggest Sellers are in New Editions!Schaum's Outlines-Problem SolvedMore than 500,000 sold! Linear...
In this book the reader is provided with a tour of the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, together with some of their main applications. The author assumes only that the reader has a basic background in functional analysis, and the presentation is self-contained and paced appropriately for graduate students new to the subject. Experts will also want this book for their library since...
DESCRIPTION: This book consists of nine chapters. Chapter 1 is devoted to algebraic preliminaries. Chapter 2 deals with some of the basic definition and results concerning topological groups, topological linear spaces and topological algebras. Chapter 3 considered some generalizations of the norm. Chapter 4 is concerned with a generalization of the notion of convexity called p-convexity. In Chapter 5 some differential and integral analysis involving vector valued functions is developed. Chapter...
They form a part of the field of algebraic logic, and have applications in proof theory, modal logic, and computer science. This research text uses combinatorial games to study the fundamental notion of representations of relation algebras. Games allow an intuitive and appealing approach to the subject, and permit substantial advances to be made. The book contains many new results and proofs not published elsewhere. It should be invaluable to graduate students and researchers interested in relation...
AS Use of Maths: Algebra and Graphs (incorporating Applying Maths): Core Book In stock at one or more sellers.
The mathematics is presented in contexts relevant to students' studies, linking maths to other areas of the curriculum with a strong focus on integrating ICT. This work includes advice on developing the assessed portfolio and practice exam questions at the end of each chapter. Full colour comprehensions provide practice for the Applying Maths exam. |
Can you be a bit more precise about history of exponents in algebra ? I may perhaps be able to help if I knew some more . A proper software can help you solve your problem instead of paying for a algebra tutor. I have tried many math program and guarantee that Algebra Buster is the best program that I have come across . This Algebra Buster will solve any math problem that you enter and it also make clear every step of the – you can exactly reproduce as your homework assignment. However, this Algebra Buster should also help you to learn math rather than only use it to copy answers.
I couldn't agree more with what has just been said. Algebra Buster has always come to my rescue, be it an assignment or be it my preparation for the midterm exams, Algebra Buster has always helped me do well in algebra. It really helped me on topics like equation properties, like denominators and interval notation. I would highly recommend this software.
Algebra Buster is a great software and is surely worth a try. You will find lot of interesting stuff there. I use it as reference software for my math problems and can swear that it has made learning math much more fun.
I guess you can find what you need at this From what I understand Algebra Buster comes at a cost but it has 100% money back guarantee. That's how I got it. I would advise you to try it out. Don't think you will want to get a refund. |
Statistical Mechanics
The theory of knots and models of simple magnets have suddenly become
interwowen over the last two decades. Both fields have benefitted from
this interaction - with ideas in one helping solve problems in the other.
This course shows how progress in pure mathematics
can help understand problems in apparently totally unrelated branches of
science - this
interaction is typical of modern mathematical physics which has blurred the
distinction between
pure and applied mathematics on many fronts. No pre requisites from
applied mathematics or statisics courses are necessary for this module. |
The book I used in my 'proofs' class was "Doing Mathematics: An Introduction to Proofs and Problem Solving" by Steven Galovich, here on Amazon.
The class was called "Mathematical Structures", which is an apt name since the class wasn't solely about learning to prove things. It was learning to prove things in the context of learning about basic mathematical objects. It starts with basic logic, but after it introduces sets, relations, functions, equivlance relations and the like, it goes onto to develop the ideas of cardinality, including Cantor-Bernstein. It also has a couple other topics, like some basic combinatorics, the constructions of number systems, or looking at consequences of the field axioms.
It was a great introduction to what math is "really about" coming after some mostly computational calculus and linear algebra courses. The price is about $50, so it is a little more than you were looking for. But it is absolutely a book worth having. |
Mathematical Experience, Study Edition for an Amazon.co.uk gift card of up to £7.95, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more
Book DescriptionMore About the Authors
Product Description
Review
"This beautifully written book can be recommended to any cultivated person with a certain sophistication of thought, and also to the practicing mathematician who will find here a vantage point from which to make a tour d'horizon of his science." --Publ. Math. Debrecen "This is an unusual book, being more a book about mathematics than a mathematics book. It includes mathematical issues, but also questions from the philosophy of mathematics, the psychology of mathematical discovery, the history of mathematics, and biographies of mathematicians, in short, a book about the mathematical experience broadly considered… The book found its way into "Much for liberal arts students" courses and into education courses directed at future teachers. Term paper topics, essay assignments, problems, computer applications, and suggested readings are included. This new material should greatly enhance the usefulness of this very creative book. The range of topics covered is immense and the contents cannot easily be summarized. The book makes excellent casual reading, would make a good textbook, or could easily be used as a supplement to nearly any course concerned with mathematics." -Zentralblatt Math "This is a people-centered book about mathematics, and as such it provides an opportunity to explore fundamental issues that are typically absent from the experience of most college and university students (as well as their teachers). This new edition provides an excellent initiation of students into some of the more challenging aspects of mathematics…The study edition of The Mathematical Experience will help its readers acquire a real understanding of mathematics." -Notices of the AMS "Two distinguished mathematicians...have written a perfectly marvelous book about the Queen of Sciences, from which one will get a real feeling for what mathematicians do and who they are. The exposition is clear and full of wit and humor..." --The New Yorker (1984 American Book Award Edition)
Back in the early 90's when I was an almost-penniless mathematics student I was standing in front of a bookshelf in my local bookstore and had to choose between this and Gödel, Escher, Bach. I chose this book and I still don't regret it. [I have also subsequently bought GEB :-)] Driven by their obvious love of the subject, the authors do a credible job of tackling just what it is about mathematics that makes mathematicians love it so much, often to the bafflement of the rest of the world. A particular personal favourite is the series of four conversations between an "ideal mathematician" and, respectively, a University Public Information Officer, a philosophy student, a positive philosopher and a sceptical classicist. I would recommend this book to students of mathematics at any level beyond the elementary, especially those with an interest in the foundations of their subject. The authors do however acknowledge that some parts of the book will seem alien to the layman.
If you are the kind of person who enjoys Horizon and other science programs but wishes they were less stripped of technicalities and more in depth: if you are very curious about physics and mathematics, you may find this a very stimulating read. After all you don't need to be a good comedian to enjoy watching comedy, so with little mathematical knowledge you can enjoy this very wide ranging discussion of many exotic fields of maths and gain a sense of the range and significance and methodologies of this field at the outer limits of human abstract thought. This very readable, demanding, though provoking book that can occupy you for weeks and sustain several re-readings.
Around my early to mid teens, finding the mathematics I was being taught easy but also puzzling I started to have philosophical questions that were just not properly addressed. I later hoped (naively I now know) my Science/Engineering degree would answer those questions. It didn't. I had wanted to dig deeper into what mathematics is but instead was taught increasingly obscure and complex "results" that were useful to my discipline. Had I known it at the time this book would have answered my questions (often answering them with more questions, true). It appears to give an excellent framework to the consideration of any uneasiness you might feel with mathematics especially if you find/found school levels maths easy.
If after reading this you still feel uneasy then you know you need to proceed further into philosophy. |
Algebra 2 - Student Text
Help your students discover the logic, order, beauty, and practicality of algebra. Throughout the course, students are encouraged to use their reasoning ability as they work with the axioms, rules, and principles of algebra. Concepts are developed and mastered through an abundance of worked examples and exercises, with an emphasis on word problems that relate to the physical world. Reviews at the end of each unit measure student progress, and special sections challenge the mathematically talented student. This text calls for a scientific calculator that has the trigonometric functions, statistics, powers, and roots |
From time to time, not all images from hardcopy texts will be found in eBooks, due to copyright restrictions. We apologise for any inconvenience.
Description
Building on the foundations laid in the companion text Modern EngineeringMathematics, this book gives an extensive treatment of some of the advanced areas of mathematics that have applications in various fields of engineering, particularly as tools for computer-based system modelling, analysis and design.
The philosophy of learning by doing helps students develop the ability to use mathematics with understanding to solve engineering problems. A wealth of engineering examples and the integration of MATLAB and MAPLE further support students.
Table of contents
Preface
About the authors
Matrix Analysis
Numerical Solution of Ordinary Differential Equations
Vector Calculus
Functions of a Complex Variable
Laplace Transforms
The z Transform
Fourier Series
The Fourier Transform
Partial Differential Equations
Optimization
Applied Probability and Statistics
Answers to Exercises
Index
New to this edition
· New topics are introduced, including new sections on singular value decomposition, direct design of digital filters and windows, and integral solution of partial differential equations
· More references are made to the use of MATLAB and MAPLE with commands and codes introduced
· The order of chapters is updated, giving a more logical progression throughout the book
· Colour is introduced to make the text more accessible and student-friendly
· Updated Lecturer Solutions Manual
Features & benefits
Graded examples and exercises
Increased emphasis on software packages, particularly symbolic algebra packages. Particular emphasis on use of MATLAB and MAPLE, with basic commands introduced and illustrated |
3: Communicate the breadth and interconnections of the
mathematical sciences
Every course should strive to:
Present
key ideas and concepts from a variety of perspectives;
Employ
a broad range of examples and applications to illustrate and motivate
the material;
Promote
awareness of connections to other subjects (both in and out of the
mathematical sciences), and strengthen each student ability to apply the
course material to these subjects;
Introduce
contemporary topics from the mathematical sciences and their
applications, and enhance student perceptions of the vitality and
importance of mathematics in the modern world.
Key
Ideas and Concepts from Varied Perspectives
Project Intermath
is an interdisciplinary mathematics project that is creating curricula at the
United StatesMilitaryAcademy, CarrollCollegeGeorgiaCollege & StateUniversity, HarveyMuddCollege, MacalesterCollege,
University of Redlands, and the Texas Southern
Consortium. By working with professors from science, engineering, mathematics
and computer science departments, the project aims to foster the creation of
interdisciplinary courses that demonstrate the interdependence of mathematics
and science. For example, at the United
StatesMilitaryAcademy , first semester students study the concept of change
from both a discrete and a continuous point of view. At the end of the
semester students must model and solve particular problems by using a
discrete dynamical system and by using a differential equation. Students then
compare and discuss the appropriateness and the results of the two
approaches. Within the four-course core program, students at the United
States Military Academy also examine mathematical topics from the
perspectives of linear versus nonlinear and stochastic versus deterministic.
At Carroll College, a 4-class core consisting of a total of 18 credit hours
covers many of the topics seen in the first two years of a traditional
curriculum, including differential and integral calculus, multivariable
calculus, differential equations, and linear algebra. The core also includes
topics not usually seen early, if at all: discrete dynamical systems, partial
differential equations, probability, and statistics. Concepts
are threaded together in and between classes to help students develop a
deeper understanding of how different branches of mathematics are
intertwined. The website contains complete texts for over 40 modeling
problems developed at the United
StatesMilitaryAcademy site.
Although a number of the textbooks produced during the calculus reform
movement are no longer in print, both mainstream and reform texts now
consider the concepts of calculus from a variety of perspectives: not only
the symbolic, but also the graphical, numerical, and verbal. The
Calculus Consortium at Harvard Newslettersdiscuss issues involved in teaching calculus. Many
calculus texts now come with software to enhance student understanding from a
variety of perspectives.
A good source of ideas on how to teach linear algebra from various
perspectives is Resources
for Teaching Linear Algebra, edited by David Carlson, Charles R.
Johnson, David Lay, Duane Porter, Ann Watkins, and William Watkins, MAA
Notes vol. 42.
The concept of function can be regarded from many different perspectives
and is important in all undergraduate mathematics courses. The editors of The
Concept of Function: Aspects of Epistemology and Pedagogy (Harel & Dubinsky, 1992) contributed to the body of
research on learning the function concept in order to assist in instructional
approaches. Key
Aspects of Knowing and Learning the Concept of Function by Marilyn
Carlson and Michael Oehrtman is a recent online
article that provides a broad view of the subject.
Victor Donnay, BrynMawrCollege, developed a
PowerPoint presentation
describing how computer visualization can be used to give an intuitive
understanding of complex ideas in modern mathematics.
Promote
Awareness of Connections between Mathematics and Other Subjects
Applications
Dan Maki (Indiana University Bloomington) and Bart Ng
(Indiana University-Purdue University Indianapolis) co-direct the NSF-funded
project Mathematics
Throughout the Curriculum. The website includes links to a prototype
course Analytical
Problem Solving and a set of Home Pages for
Developing Courses, which contain additional information about courses
that relate mathematics to the life sciences, business and economics, the
humanities and social sciences, and the physical sciences and engineering. A newsletter
provides additional information about the project.
The MAA'sJournal
of Online Mathematics and its Applications (JOMA) contains
peer-reviewed articles, class-tested, web-based learning materials, and
self-contained, dynamic, single-purpose learning tools. Many of these
illustrate a range of examples and applications and connections between
mathematics and other subjects. Some recent articles are Special
Relativity and Conic Sections, Designing Attribute Acceptance Sampling Plans,
and Art and Design in Mathematics.
DukeUniversity's
Connected Curriculum Project
collects and develops interactive learning materials for mathematics and its
applications, with applications to biology, chemistry, economics,
engineering, environmental sciences, epidemiology, and physics. Each
application is keyed to the level of mathematics used.
The MAA'sDigital Classroom Resources
provides a select collection of learning materials that are available without
charge through the site. These materials have been classroom tested and peer
reviewed. Many items in the library include editorial reviews and links to a
moderated discussion group focused on the materials.
The entry for the Consortium for Mathematics and Its Applications (COMAP)
in the bibliography contains additional information about incorporating
real-world applications into mathematics courses.
History
The MAA's online journal Convergence
is a new online magazine that provides resources to help teach mathematics
using its history. ReinhardLaubenbacher,
David Pengelley, Jerry Lodder,
and others at New MexicoStateUniversity
have developed a large collection
of instructional materials to teach mathematics using original historical
sources. Other books that link mathematical topics with their history
include William Dunham's Journey Through Genius: The Great Theorems of
Mathematics and The Calculus Gallery, Simon Singh's Fermat's
Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem
and The Code Book: The Science of Secrecy from Ancient Egypt to Quantum
Cryptography, Rudy Rucker'sInfinity
and the Mind: The Science and Philosophy of the Infinite, and Marcia Ascher'sEthnomathematics.
A Multicultural View of Mathematical Ideas. Judith Grabiner
of PitzerCollege developed two
general education courses with an emphasis on history: Mathematics,
Philosophy, and the 'Real World,' and Mathematics
in Many Cultures.
Art
In a note to Project NExT participants, Doris Schattschneider,
Moravian College, gave the following list of websites for courses linking
mathematics, art, and design:
* Survey course on mathematics in art
and architecture by Paul Calter at Dartmouth
College
* Course on Mathematics
in Art by HelmerAskalan
* Course in Mathematics
and Art by Marc Franz
* Jill Britton's website on Symmetry and
Tessellations, with annotated links to many other sites on these and related
topics
* Totally
Tessellated (can be accessed from Britton's site)
* A mini-site
on Escher's work and related math
* An exhibit
by artists whose work has been influenced by M.C. Escher featured at the
Escher Centennial Congress in Rome in 1998
* A website about harmony and
proportion by John Boyd-Brent, M.A, Royal College of Art
* This website,
related to a 'technical' paper by D. Schattschneider
and N. Dolbilin, has Java applets that allow users
to manipulate flexible tilings.
Most textbooks for general education mathematics
courses include sections that connect mathematics with other fields. For
instance, The
Heart of Mathematics: An invitation to effective thinking by Edward
B. Burger and Michael Starbird discusses the
mathematics of bar codes, cryptography, geometry and art, fractals and chaos,
and the likelihood of coincidences. A syllabus
for an interesting course that makes some use of this text is from Sarah
Greenwald, Appalachian State University. Another text, Using
and Understanding Mathematics by Jeffrey O. Bennett and William L.
Briggs, University of Colorado at Boulder, contains sections on financial
management, modeling a variety of real-world situations, mathematics and art,
mathematics and music, mathematics and politics, and mathematics and
business. See also the listings in this section under "Introduce Contemporary
Topics."
Introduce Contemporary Topics
Robert Devaney, BostonUniversity, is a leader
in promoting instruction in the contemporary topic of dynamical systems. In
addition to his books and articles, talks, and professional development
institutes, he has been director of the National Science Foundation's Dynamical Systems and Technology
Project since 1989. The goal of this project is to show students and
teachers how ideas from modern mathematics such as chaos, fractals, and
dynamics, together with modern technology, can be used effectively in the
high school and college curriculum.
The University of Maryland University College offers Mathematics
– Contemporary Topics and Applications as both an in-class and
distance-learning first-year course. The course is a survey of contemporary
topics in mathematics, centering on applications and projects. Topics include
measurements, rates of growth, basic statistics, the mathematics of political
power, the geometry of the solar system, and computer arithmetic. The goals
state that after completing this course a student should be able to cite
elements of good statistical design, undertake elementary statistical
analysis, and recognize and explain the shortcomings of unsound methods of
statistical analysis; mathematically analyze situations involving the
weighting of power in various voting structures and implement apportionment
of power strategies; and use the Pythagorean theorem and properties of
similar triangles to calculate sizes of and distance between objects,
including astronomical objects.
StetsonUniversity offers a wide variety of
courses that meet the general mathematics requirement, including many
that discuss contemporary topics such as chaos and fractals, game theory, and
cryptology.
Both of the popular texts For
All Practical Purposes, produced by the Consortium for Mathematics
and Its Applications, and Excursions
in Modern Mathematics by Peter Tannenbaum
and Robert Arnold aim to convey insight about topics in contemporary
mathematics and its applications to undergraduate students who have limited
mathematical backgrounds. Topics in these books include the mathematics of
voting, fair division, and apportionment, applications of graph theory to
management science, fractal geometry, and statistics.
At Mount Holyoke College George Cobb teaches a course on the Markov Chain
Monte Carlo method (MCMC), "a very general and powerful method for
computer simulation of situations that are too complicated to handle using
more conventional mathematical methods. MCMC has become a very active area of
research at the interface of computer science and statistics, and has had a
powerful impact on the practice of data analysis. As a method for computer
simulation, MCMC has very broad applicability. As a branch of mathematics,
MCMC offers a number of compelling surprises – structures that on a concrete
level seem quite different, but, viewed at the right level of abstraction, turn out to be different versions of the same
idea."
In the article "Geometric
Photo Manipulation" Tom Farmer shows how calculus and linear algebra can
be used to manipulate photographs, a contemporary application with which many
students have experience, thanks to currently available software.
G.H. Hardy once proudly asserted that number theory
would never be applied. Yet today number theory has a range of important
applications. Among these are cryptography (see, for example, lecture notes
from two cryptography
courses by Ed Schaefer at Santa Clara University, the RSA website,
and recent textbooks in number theory and discrete mathematics), and error
detection using check digits and error-correcting codes (see, for example, Numbers
and symmetry: An Introduction to Algebraby
Bernard L. Johnston and Fred Richman and Contemporary
Abstract Algebra by Joseph Gallian).
Enhance Perception of Vitality and
Importance of Mathematics
The World Wide Web provides a wealth of examples of the use and
applicability of mathematics, but searching for appropriate illustrations can
be time consuming. There are several sites that focus on providing good
examples for instructors: Plus,
an Internet magazine that aims to introduce readers to the beauty and the
practical applications of mathematics; Mathematical Moments,
an AMS program that offers a series of pdf files
and podcasts to promote appreciation and
understanding of the role mathematics plays in science, nature, technology,
and human culture; and the Math Forum,
a center that provides resources, materials, activities, person-to-person
interactions, and educational products and services to enrich and support the
teaching and learning of mathematics.
Chance
News is a monthly, on-line newsletter that provides abstracts of
articles from current newspapers, the media, and journals, and suggests
discussion questions for class use. It also includes links to related
resources at other web sites. Since 1992, Chance News has been
maintained by J. Laurie Snell of DartmouthCollege. The
examples are current and can be used for student motivation, for class
discussion, and as exercises in an introductory statistics course or
probability course. The website contains all issues of Chance News
as well as information on signing up for the newsletter by e-mail.
Additional Resources
Additional information and resources on communicating the breadth and
interconnections of the mathematical sciences are in Part 2, Section C.3. |
MAT
230
- Discrete Mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. That is, in contrast to the real numbers that vary continuously, the objects of study in discrete mathematics take on distinct, separated values. Topics include operations on sets, logic, truth tables, counting, relations and digraphs, functions, trees and graph theory. A significant goal of this course is to improve students' critical-thinking and problem-solving skills. |
In this area we build the foundation of Algebra as we study the topic of Pre-Algebra. In this online math course, we will learn in detail about negative and positive numbers, exponents, order of operation, basic equations, and much more!
Section 1: Real Numbers and their Graphs
Section 2: The Number Line In this section, the concept of the number line is introduced and explained in detail. The concept of a negative number is illustrated by examples from everyday life and their relationship to positive numbers is shown on the number line. The student practices using the number line through numerous examples in this section, including basic addition and subtraction of integers. . . . View the lesson
Section 3: Greater Than, Less Than, Equal To In this section, the student learns how to properly use the greater than, less than, and equal to symbols in Pre-Algebra. Numerous problems illustrate how to compare positive or negative numbers with these symbols. The number line is used as a graphical reference to reinforce the concept. . . . View the lesson
Section 4: Adding Integers In this section, the student learns how to add two integers together and get the correct answer every time. Numerous examples of adding positive and negative numbers together are presented and by the end of the lesson the student will have memorized the simple rules for integer addition. The number line is also used to reinforce the concept. . . . View the lesson
Section 5: Subtracting Integers In this section, the student learns how to subtract two integers from one another and get the correct answer every time. Numerous examples of subtracting positive and negative numbers together are presented and by the end of the lesson the student will have memorized the simple rules for integer subtraction. The number line is also used to reinforce the concept. . . . View the lesson
Section 6: Multiplying Integers In this section, the student learns how to multiply two or more integers together. We begin the section by explaining the rules of integer multiplication. Next, we work numerous problems which give the student extra practice in multiplying negative and positive numbers together. . . . View the lesson
Section 7: Dividing Integers In this section, the student learns how to divide integers. We begin the section by explaining the rules of integer division. Next, we work numerous problems which give the student extra practice in dividing negative and positive numbers together. . . . View the lesson
Section 8: Powers and Exponents In this section, the student learns about the concept of an exponent and how it relates to pre-algebra. Numerous examples are provided to solidify this concept prior to moving on the the multiplication and division rule of terms that have exponents with the same base. . . . View the lesson
Section 9: Order of Operations In this section, the student learns about the concept of the order of operations in pre-algebra. This deals with understanding what order the student should perform calculations in an algebraic expression. . . . View the lesson
Section 10: Factors and Multiples In this section, the student learns how to calculate the factors of a number and the multiples of a number. These concepts will be central when we move into algebraic expressions later in this course. . . . View the lesson
Section 17: Adding Fractions In this section, the student will learn how to add fractions. We learn how to add regular fractions along with improper and mixed fractions and we learn to simplify the result. . . . View the lesson
Section 18: Subtracting Fractions In this section, the student will learn how to subtract fractions. We learn how to subtract regular fractions along with improper and mixed fractions and we learn to simplify the result. . . . View the lesson
Section 19: Multiplying Fractions In this section, the student will learn how to multiply fractions. We learn how to multiply regular fractions along with improper and mixed fractions and we learn to simplify the result. . . . View the lesson
Section 20: Dividing Fractions In this section, the student will learn how to divide fractions. We learn how to divide regular fractions along with improper and mixed fractions and we learn to simplify the result. . . . View the lesson |
Purchasing Options
Features
Presents a thorough but accessible introduction to real analysis by starting with extensions of calculus before introducing the axiomatic method
Uses a historical perspective to make real analysis more interesting and less intimidating
Supplements the main development with auxiliary topics, such as formal logic, infinite products, continued fractions, rearrangement of infinite series, and root finding
Assists learning with numerous exercises, figures, and tables
Supports an optional one-semester introduction to analysis, allowing stronger students to join in the second semester
Summary
Most volumes in analysis plunge students into a challenging new mathematical environment, replete with axioms, powerful abstractions, and an overriding emphasis on formal proofs. This can lead even students with a solid mathematical aptitude to often feel bewildered and discouraged by the theoretical treatment. Avoiding unnecessary abstractions to provide an accessible presentation of the material, A Concrete Introduction to Real Analysis supplies the crucial transition from a calculations-focused treatment of mathematics to a proof-centered approach.
Drawing from the history of mathematics and practical applications, this volume uses problems emerging from calculus to introduce themes of estimation, approximation, and convergence. The book covers discrete calculus, selected area computations, Taylor's theorem, infinite sequences and series, limits, continuity and differentiability of functions, the Riemann integral, and much more. It contains a large collection of examples and exercises, ranging from simple problems that allow students to check their understanding of the concepts to challenging problems that develop new material.
Providing a solid foundation in analysis, A Concrete Introduction to Real Analysis demonstrates that the mathematical treatments described in the text will be valuable both for students planning to study more analysis and for those who are less inclined to take another analysis class. |
The skills that students are practicing in related rates problems are:
Differentiating a known equation implicitly with respect to time.
Interpreting the time derivative of a quantity as a rate of change.
The main reason that related rates problems feel so contrived is that calculus books do not want to assume that the students are familiar with any of the equations of science or economics. Every related rates problem inherently involves differentiating a known equation, and the only equations that the calculus book assumes are the equations of geometry.
Thus, you can find related rates problems involving various area and volume formulas, related rates problems involving the Pythagorean Theorem or similar triangles, related rates problems involving triangle trigonometry, and so forth. A few of these problems are compelling -- for example, computing the speed of an airplane based on ground observations of its altitude and apparent angular velocity -- but most of them do feel a bit contrived.
The reality, of course, is that students are familiar with many of the basic equations and concepts of science and economics, and there's no rule against using these in problems. For example, you can make up all sorts of compelling related rates problems by starting with any physics or chemistry equation and imagining a situation where you might want to take its derivative:
The kinetic energy of an object is $K = \frac{1}{2}mv^2$. If the object is accelerating at a rate of $9.8 \text{m}/\text{s}^2$, how fast is the kinetic energy increasing when the speed is $30 \;\text{m}/\text{s}$?
An ideal gas satisfies $PV = nRT$, where $n$ is the number of moles and $R \approx 8.314\;\; \text{J}\; \text{mol}^{-1} \text{K}^{-1}$. Give the rate at which the temperature and volume of the gas are increasing, and then ask about the rate of change in pressure when the volume and temperature reach certain amounts.
The total energy stored in a capacitor is $\frac{1}{2} Q^2 / C$, where $Q$ is the amount of charge stored in the capacitor and $C$ is the capacitance. Give the value of $C$ and the rate at which $Q$ is decreasing, and ask about the rate at which the capacitor is losing energy when the energy is a certain amount.
In astronomy, the absolute magnitude $M$ of a star is related to its luminosity $L$ by the formula
$$
M \;=\; M_{\text{sun}} -\; 2.5\; \log_{10}(L/L_{\text{sun}}).
$$
where $M_{\text{sun}} = 4.75$ and $L_{\text{sun}} = 3.839 \times 10^{26} \text{watts}$. (Note that, by convention, brighter stars have lower magnitude.) If the absolute magnitude of a variable star is decreasing at a rate of $0.09 / \text{week}$, how quickly is the luminosity of the star increasing when the magnitude is $3.8$?
It's easy to make these up: just think of any equation in science or economics whose derivative might be interesting. Wikipedia and/or science textbooks can be helpful for finding equations from a wide variety of fields. |
Learning Outcomes
On successful completion of this module, the student should be able to:
Assessed by:
1
design wire-frame representations of 3-dimensional objects
Examination
2
define matrices for 3-dimensional transformations
Examination
3
explain and design algorithms for the viewing and projection of
3-dimensional objects using transformation matrices
Examination
4
apply the relevant concepts of linear algebra and geometry to the
design of computer graphics algorithms (e.g. vector and matrix
operations and trigonometry)
Examination
5
explain and design basic raster conversion algorithms
Examination
6
explain the colour models and image representations
Examination
Restrictions, Prerequisites and Corequisites
Restrictions:
None
Prerequisites:
No specific modules are required as prerequisites. Students will be expected to know or to learn independently the basics of the following mathematical concepts and techniques:
general algebra (basic transformation of equations);
analytical geometry (equations for line, surface, circle, ellipse etc);
vector representation and algebra (addition and multiplication, dot product, cross-product);
matrix representation and algebra (addition and multiplication). |
Cabri II Plus description
Helps you design 2-D and 3-D figures
Cabri II Plus is appreciated for the solid educational foundation and its simplicity of use.
With just a few clicks, students can: · Construct 2-D and 3-D figures, from the simplest to the most complex, by combining fundamental geometric objects such as points, angles, segments, circles, planes, solids and transformations. · Create expressions using fundamental algebraic concepts, such as numbers, variables and operations. · Connect geometry and algebra by measuring length, angles, area and volume and then attaching these numeric values directly to the figure to use them in calculations or in algebraic expressions. · Explore a figure's properties by manipulating its variable elements. Observe the effects of dynamic transformations like shrinking and enlarging. Make conjectures about algebraic and geometric properties, and then verify relationships among various parts of a figure.
The teacher can: Create activities that: · facilitate the introduction and understanding of new concepts promote the discovery of theorems, instead of just showing them help model real-life situations. · Generate classroom resources by inserting text or pictures in a figure, modifying graphical elements, copying/pasting into other software and producing high-quality printouts. · Present activities to students, have them manipulate figures, observe and guide them. Using Cabri allows you to better assess individual student comprehension. · Expand online by integrating figures that can be manipulated on web pages or by incorporating Microsoft Office documents. · Have students solve problems directly linked to the NCTM standards with possible interdisciplinary connections to physics, geography and the arts.
Advanced tools:
· place and trace: display the paths of geometric figures when they are distorted.
· redefine objects: add or subtract constraints on the relationship between the objects in the figure.
· Numeric and algebraic tools: coordinate points, equations of lines, graphical representations of functions and tables
Tools that can be personalized to meet each teacher's needs:
· Save the history of figures: simultaneously follow the work of several students by replaying each of their actions step-by-step.
· Create your own tools in order to remake frequently used constructions in just a few clicks. Design « black box » tools whose functions are to be discovered by the students.
· Modify the toolbar to fi t the level of the class or the planned activity.
Link between the calculator and computer:
· Import/export files between the teacher's computer and the students' graphing calculators, from Cabri II Plus to the free application Cabri Jr. (available on TI-83 Plus and TI-84 Plus calculators).
· Say Hello to the Cabri Assistant
· Thanks to the new built-in assistant discover the full pedagogical power of Cabri II Plus! Unleash it in your classroom right away. It comes with videos to get started and with more than 60 ready-to-teach activities for your classroom. The assistant guides you for instance on doing algebra, trigonometry, coordinate geometry, data exploration...
· One-click publishing
· Immediate export to png format images, ideal to prepare lectures notes or activities.
· Immediate export to web, ideal to share activities with your students through a web site.
· Cabri II Plus on the Internet and in MS Office
· With the new Plug-in for Cabri II Plus (both PC and Mac), you can insert full Cabri II Plus construction files in a Word or PowerPoint document as well as in an Internet page (html format). Then, the visitor on the web can visualize the Cabri II Plus files and manipulate objects, even if they do not have a Cabri II Plus license.
· Cabri made free for students
· The new licens...
Softpedia guarantees that Cabri II Plus 1.4.2 is 100% CLEAN, which means it does not contain any form of malware, including spyware, viruses, trojans and backdoors. [read more >] |
Understanding real estate math is essential to passing the licensing exam and being a successful agent. This new edition of Real Estate Math focuses on basic math concepts with examples, formulas, calculations, and practice problems. It contains a format that allows students to use it as a workbook. Real Estate Math can be used either as a supplement to your prelicensing courses or as a book for a math review course.
Highlights:
NEW - Additional section and new post-test designed for broker candidates
NEW - Over 75 new problems added to the book's chapters, allowing more practice on every concept
NEW - Explanation and examples of multi-step problems in key concept areas |
Mathematics
Many of our students are destined to be leaders in fields where analytical and problem-solving skills are invaluable and specific applications of mathematics are often indispensable. In the Mathematics Department, we give each student the necessary tools to understand theories and appreciate applications. True problem-solving strength calls for a wide repertoire of knowledge. Students acquire a strong knowledge of mathematics through the study of algebra, geometry, trigonometry, and calculus.
This ability to solve problems is further strengthened by learning to discern relationships between particular skills and concepts and the fundamental principles that unify them. Students learn to formulate key questions, analyze data, and apply learned strategies to new situations. In doing so, students are equipped not only to solve mathematics problems, but also to use an analytical, well-thought-out approach in seeking solutions in other areas of life.
Algebra I
This class introduces the basic principles of future mathematics courses. Students transition from the concrete to the abstract through a wide range of problem-solving situations. The class emphasizes the concept of functions and covers the real number system, operations with positive and negative numbers, simplifying algebraic expressions, solving and graphing linear equations and inequalities, applying rules of exponents, understanding operations involving polynomials, simplifying rational expressions and square roots, and solving both rational and quadratic equations.
Algebra II
This second-year course in algebra aims for a mastery of the manipulative skills in real-number algebra as well as further develop problem-solving skills. Students review and extend their ability to manipulate polynomial and rational expressions and to solve linear, quadratic, fractional, and radical equations and inequalities. The course includes the study of irrational and complex numbers and concludes with an introduction to exponential and logarithmic functions. Placement in this course is based on teacher recommendations following the completion of Algebra I and Geometry.
Algebra II Trigonometry
This second-year course in algebra aims for a mastery of the manipulative skills in real-number algebra. Students review and extend their ability to manipulate polynomial and rational expressions and to solve linear, quadratic, fractional, and radical equations and inequalities. An introduction to the concepts of rational exponents, functions, logarithms, sequences, and series is provided. The course concludes with instruction in elementary trigonometry.
Algebra II Honors
Emphasis in this course is on step-by-step solutions. The pace is set to challenge students with high mathematical abilities. To participate, students must have a strong recommendation from their previous mathematics teachers. This second year of algebra begins with a review of the principles learned in Algebra I and proceeds into greater complexities, with subsequent introduction of the elements of exponents, functions, logarithms, trigonometry, sequences, and series.
Geometry
This course introduces students to concepts in geometry and teaches how to write a two-column proof. Basic algebraic skills are reinforced throughout this course with special emphasis on applications of congruence and similarity of triangles, properties of circles, areas of plane figures, and volumes of solids.
Geometry Honors
Geometry Honors is a course for students who have genuine interest and high aptitude in mathematics. In this course, students gain a greater appreciation of the nature of a mathematical system through the study of mathematical proofs. Other topics emphasized are application of congruence and similarity of triangles, properties of circles, areas of plane figures, and volumes of solids. Also, inductive and deductive reasoning are discussed, and algebraic concepts are reinforced through the study of coordinate geometry.
Nature of Math
This course will provide students with the opportunity to broaden their mathematic experience, to learn what mathematicians do, and to understand how mathematical thinking may be useful to them in their daily lives. This course is designed for students who have successfully completed Algebra II and wish to further their mathematical studies. Though an interactive exploration of mathematics, students will study and appreciate various aspects of mathematics: the internal structure, the history, problem-posing and problem-solving techniques, and the inherent connections with other disciplines. In addition, students will study the properties of geometric structures through building polyhedra. This will enrich their spatial intuition, and enable them to communicate the joy of artistic creation through mathematics.
Statistics
The aim of the Statistics course is to introduce students to the major concepts or traditional and modern statistics, to gain the tools for collecting, properly analyzing and drawing conclusions from statistical data, and to gain an understanding of statistical methods relevant to an interdisciplinary application. Descriptive statistics, elementary probability theory; laws of probability, and random variables are among the themes explored in the class. The course is paced in such a way that solid math students will be challenged.
Pre-Calculus
This course is an introduction to analysis. The intent is to utilize all the mathematical concepts developed in previous mathematics courses and to sum up the basic concepts of mathematics. In preparation for college-level calculus, Pre-Calculus integrates a number of topics, including functions, inverse functions, theory of logarithms, functional trigonometry, polynomial equations, probability, and statistics.
Pre-Calculus Honors
Pre-Calculus Honors offers a challenging introduction to the study of analysis after a brief review of basic mathematical concepts. As a sequel to Algebra II Honors, prerequisites include successful completion of the preceding honors courses or a strong recommendation from the algebra teacher in collaboration with the Mathematics Department. This course integrates a number of topics, including an in-depth study of functions, theory of logarithms, trigonometry, polynomial equations, and statistics. Students are also introduced to limits and the interpretation of computation of derivatives.
Calculus
Calculus provides students with solid preparation for college-level mathematics courses. It begins with a brief reinforcement of skills learned through Pre-Calculus. Major concentration for the rest of the year centers on the study of limits, derivatives, integrals, and their respective applications.
Calculus AP AB
Calculus Advanced Placement AB offers students the opportunity to take a college-level mathematics course and receive college credit with success on the Advanced Placement examination. Using both a calculus textbook and a syllabus set forth for Calculus AB in the College Board's Acorn Book, this course challenges students at a level that prepares them to take advanced mathematics courses in college.
Calculus AP BC
Calculus Advanced Placement BC offers students the opportunity to take a college-level mathematics course and receive college credit with success on the Advanced Placement examination. Using both a calculus textbook and a syllabus set forth for Calculus BC in the College Board's Acorn Book, this course challenges students at a level that prepares them to take advanced mathematics courses in college. |
Simplex Tutor 1.1
the author of this app says:
The Simplex Tutor helps you solve simple linear programming problems. Arithmetic for pivoting can be tedious and mistakes can be hard to catch, so use the tutor to check your work.
Hints will guide you to a solution Tutor completes pivots for you Exact answers are shown as fractions Review linear programming, with examples Review simplex method, with examples 32 sample problem templates included Enter your own problems Enter data as fractions or decimals Custom keyboard for easy data entry Export matrices as HTML or CVS Save matrices as pictures Work problems with up to six constraints and six initial variables
If you are stuck and not sure how to proceed with a problem, you can request a hint guiding you to the solution. The more hints you ask for, the more guidance you will receive. Eventually the tutor will tell you a correct location to pivot on if the problem is solvable, or it will tell you that there is no solution. In this way the rules of the simplex method are continually reinforced and soon you will not need to ask for hints.
For help solving two dimensional linear programming problems graphically or to plot linear inequalities, see the Linear Program Plotter.
For practice with row operations, the Gauss-Jordan method, or finding matrix inverses, see the Row Operations Tutor. |
Differential Equations Problem Solver (Problem Solvers)
Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. All your questions can be found in one convenient source from one of the most trusted names in reference solution guides. More useful, more practical, and more informative, these study aids are the best review books and textbook companions available. Nothing remotely as comprehensive or as helpful exists in their subject anywhere. Perfect for undergraduate and graduate studies.
Here in this highly useful reference is the finest overview of differential equations currently available, with hundreds of differential equations problems that cover everything from integrating factors and Bernoulli's equation to variation of parameters and undetermined coefficients. Each problem is clearly solved with step-by-step detailed solutions.
DETAILS - The PROBLEM SOLVERS are unique - the ultimate in study guides. - They are ideal for helping students cope with the toughest subjects. - They greatly simplify study and learning tasks. - They enable students to come to grips with difficult problems by showing them the way, step-by-step, toward solving problems. As a result, they save hours of frustration and time spent on groping for answers and understanding. - They cover material ranging from the elementary to the advanced in each subject. - They work exceptionally well with any text in its field. - Each PROBLEM SOLVER is prepared by supremely knowledgeable experts. - Most are over 1000 pages. - PROBLEM SOLVERS are not meant to be read cover to cover. They offer whatever may be needed at a given time. An excellent index helps to locate specific problems rapidly.
Customer Reviews:
I doubt that there's a better supplemental DE book out there
By Fugazi - September 22, 2004
I used this book for an introductory differential equations course. With the help of this book, I was completely prepared for everything the professor threw at us and more.
Some of the problems are simple and should help to reinforce the basic concepts learned from another book. Learn the fundamentals from some other book (or even a professor), and refine your thought processes using this one. There are some basic explanations of concepts and problem types, but most of the space is devoted to problems and solutions, as you may have guessed. If I had trouble with a problem from another book, I could almost always find a similar one in the REA book. A few problems are extremely difficult or involved, but the book leads you through every step of every problem. I was blown away by the depth of a few problems, but following the reasoning in this book really helped me to develop a better "feel" for the subject.
Excellent
By A Customer - July 1, 2001
This is an excellent supplement to any Differential Equations class. It starts out with easy and straight forward problems and progressively gets to the more advanced problems. All steps are worked out in an easy to understand way. Most importantly this book shows the algebraic steps in obtaining solutions, which is ussually where students have trouble. This book is a must have for any science major.
An Excellent Resource!
By A Customer - June 15, 2001
If you are studying DEs, get this book! It beats the usual text books that only contain solutions to selected problems, namely the ones that the professor does not assign or examine you on. This book is comprehensive and contains stratagies for attacking the various problems you will encounter in undergraduate DE courses. It even contains problems and solutions to Partial Differential Equations, Fourier Series and Boundary Value Problems, not just ODEs!
Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. All your questions can be found in one convenient source from one of the ...
Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. All your questions can be found in one convenient source from one of the ...
Transform methods provide a bridge between the commonly used method of separation of variables and numerical techniques for solving linear partial differential equations. While in some ways similar ...
Domain decomposition methods are divide and conquer computational methods for the parallel solution of partial differential equations of elliptic or parabolic type. The methodology includes iterative ... |
This activity book is designed to help teachers prepare students to use technology appropriately to solve everyday problems. The 15 activities include topics in number theory, algebra, probability and statistics |
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What's inside... Feature Math and Science Magnet Prepares Students for Algebra and Beyond What's New
Math and Science Magnet Prepares Students for Algebra and Beyond Algebra is an important foundation for building the critical thinking skills we need for solving everyday problems. Picture yourself at the local video rental store getting ready to pay for your selection. The clerk tells you that you have a choice of paying a $25 annual membership fee, plus $1.50 per rental, or paying no membership fee and $2.75 per rental. Would you have imagined that an understanding of algebra and linear equations could help you decide which is the better deal? Or perhaps you have a job offer that requires you to move across the country from Buffalo, N.Y. to San Francisco, Calif., but you would have to cover the cost of gas for approximately 2600 miles in a moving van. If the national average for gasoline is $3.25 per gallon, how much money would you need to save to cover the cost of the move? Would you have believed that when you learned to solve algebraic expressions it would help you find the answer to this unknown variable, too?
There is concern throughout the country that many American students lack the knowledge and skills necessary to succeed in algebra. Those students may not only have greater difficulty solving some of the "real world" problems listed above, but they also may need remedial course work in college and may have a lesser chance of becoming the next generation of American scientists, inventors, and engineers. And with research showing that students who complete Algebra II in high school are more than twice as likely than students with less mathematical preparation to earn a 4-year college degree, we must ensure that students are ready to tackle the more advanced mathematics courses in high school and beyond.
To compete in the 21st century global economy, proficiency in mathematics is crucial. To help ensure our nation's future competitiveness and economic viability, President George W. Bush created the National Mathematics Advisory Panel in April 2006. The Panel was charged with making recommendations on the best use of scientifically based research to advance the teaching and learning of mathematics. During the past two years, the Panel held meetings around the country, reviewed more than 16,000 research studies, received public testimony from 110 individuals, and considered written commentary from numerous organizations and individuals. In addition, the Math Panel conducted, in partnership with the National Opinion Research Center (NORC), a national survey of Algebra I teachers to determine what practices will best prepare American students to succeed in algebra. On March 13, 2008, the 24 expert panelists, including educators, cognitive psychologists, and leading mathematicians, released a report with actionable steps, containing 45 findings and recommendations on numerous topics. Some of these topics included instructional practices, materials, professional development for teachers, learning processes, assessments and research policies, and mechanisms. The report calls for students to attain a strong foundation in basic mathematical skills and for Americans to redefine how they view mathematics, shifting from a belief that particular people cannot learn mathematics to a belief that hard work and effort can pay dividends in achievement.
Some of the report's key findings include: 1.) there should be a systematic progression in mathematics curricula from pre-kindergarten through eighth grade with an emphasis on student mastery of each step; 2.) it is critical to understand and be able to work with fractions (including decimals, percents, and negative fractions), for such proficiency is foundational for algebra; 3.) it is crucial for students to demonstrate quick recall of computational facts if they are to be successful in mathematics; 4.) a student's effort in the learning process is an important factor to ensuring achievement; and 5.) teachers must have a strong understanding of mathematics both prior to and beyond the level they instruct, if students are to succeed.
The K J Clark Middle School of Mathematics, Science & Technology in Chickasaw, Ala., is a magnet school that provides a curriculum rich with many of these recommendations, and the school is producing impressive results for its students. Clark offers a rigorous and relevant mathematics curriculum with a multitude of hands-on activities to get students excited about learning. As the National Math Panel report recommends, the school's approach is systematic and emphasizes conceptual understanding, computational fluency, and problem-solving skills.
Under the leadership of Principal Dianne McWain, a 2007 U.S. Department of Education Terrell H. Bell award recipient, fourth through eighth grade students throughout the Mobile County are being prepared to succeed in mathematics and science in high school and beyond. McWain notes, "We accelerated learning a few years ago. Now our students are so much better prepared. We integrate mathematics into the curriculum everyday and in every class."
The mathematics program allows students to see what is important, according to Math Department Chair Julie Boren. For example, in one algebra class, the students work in groups as they tackle a question involving which one of three candidates won the school's student council election, and by how many votes. "They know we are not going to skip the word problems just because they are difficult. We meet challenges head-on," she said.
The school prepares students for algebra by providing a "core plus" curriculum. The "core plus" takes place in grades four through six, during which time the mathematics instructors teach the grade level county curriculum but add skills from the next grade level as well. By accelerating instruction, all seventh grade students are prepared for the foundations of algebra and all eighth grade students are taught Algebra I for high school credit. Clark also offers geometry for more advanced eighth graders. "We add skills in the sixth and seventh grade that students may need to ensure they take and pass Algebra I in eighth grade," said Boren.
Clark also offers an after-school tutoring program, an in-house tutoring program that removes students from their scheduled classes to obtain extra help, and one-on-one sessions during class with the teacher to ensure that all students, even those who are struggling initially, succeed in the rigorous math program. "It is critical that our students be competitive - it opens doors for them so they can take calculus and upper level math in high school," Boren asserts.
Student enrollment at Clark is determined by a lottery in which there are no academic requirements for admission other than passing the grade the student is in at the time of application. Students come to Clark from parochial or private schools and as many as 60 public elementary schools across the county. The students also arrive with very different backgrounds and levels of academic ability. Teachers work collaboratively to bridge the gap between students' initial levels of knowledge and experience and Clark's standards of proficiency required for promotion.
The U.S. Department of Education named Clark a No Child Left Behind (NCLB) Blue Ribbon School in 2007 in part because it is a high achieving school regardless of its student demographic. Although 58 percent of Clark's student population consists of those from disadvantaged backgrounds, all students have improved their performance on state assessments. Beginning in 2003, Clark began disaggregating information on student performance, in alignment with NCLB's accountability measures and focus on data to drive instruction. By looking at the data on student performance, Clark was able to identify subgroups of students that were not performing as well as the school average and implemented strategies detailed in its Title I School Improvement Plan to close the achievement gap. The data showed that the subgroups that needed more attention were their black students and students eligible for free and reduced-priced lunch. Clark faculty members worked diligently to address the educational needs of those students, and data from the 2006 SAT-10 and Alabama Reading and Mathematics Test (ARMT) showed the progress students had made; on those tests there was little difference between the scores of students in the "black" and "free and reduced-priced lunch" subgroups and students in any other subgroup. In some instances, students in the "free/reduced lunch" subgroup outperformed students in the "paid lunch" group and black students outperformed non-black students.
High-performing schools often share similar characteristics. For example, teachers work collaboratively; there are numerous opportunities for professional development; and data drives instruction and further assessment. All of these characteristics are present at Clark, where teachers use a hands-on approach to address the learning needs of all students. Most importantly, the school's faculty has high expectations, an approach that is paying off for teachers and students.
Teacher Knowledge Is Critical
Consistent with the Math Panel's recommendation that teachers must know in detail the mathematical content they are responsible for teaching and its connections to other important mathematics, Clark aims to increase its teachers' knowledge of math to positively influence student achievement. The district provides in-service training for teachers, and Clark's Math Chair Boren encourages her teachers to be active in professional organizations. Recently some teachers took an online course on differentiating learning strategies and used the strategies to help students use their strengths to master concepts. Clark also sends some teachers to conferences sponsored by the National Council of Teachers of Mathematics (NCTM). Those teachers share what they learn with others at departmental meetings. Principal McWain explains that opportunities for professional development abound at Clark. "We are always on the cutting edge. We try to think outside the box. We incorporate this into the curriculum by giving students new techniques and strategies to succeed. The teachers work cooperatively together-including rewriting and enhancing the curriculum."
Clark aims to increase their students' knowledge with each grade level. A good example of early work with the foundations of algebra is apparent in fourth grade when students study fractions. The fourth grade goal is to expose students to equivalent fractions and basic operations with fractions of like denominators. Some of the activities in the classroom might include making fraction bars and grids, and the elementary teachers use different colors with the bars and grids to help students "see" the fractions.
In fifth grade classes, students use operations with like and unlike denominators. Teachers also expose students to canceling when multiplying fractions and putting fractions in lowest terms. Operations with mixed numbers also are introduced, and by the end of fifth grade, teachers expect students to be proficient with operations with fractions of like denominators and to be able to find equivalent fractions. The sixth graders are expected to master these skills, in addition to changing fractions to decimals and then changing decimals to percents. In the "core plus" curriculum, teachers begin the process of teaching students to work with positive and negative fractions and mixed numbers early. In the seventh grade, students aim to master these skills.
Typical classrooms use a hands-on approach to help students understand key concepts. All of the teachers use games with fractions and white boards in the classroom to encourage students to be proficient. Sixth grade math teacher Angela Rocker said that her students enjoy "Fraction Face-Off," in which a small group of students will be given a fraction problem and race to get the correct answer. The winner of the game will face a new group of challengers. Students use white boards to check for understanding. All of the students in the class are required to do a specific problem and hold up their answer on the boards. According to Boren, "This is a quick way to make sure that all students are focused and understand how to complete the problem. Our students enjoy using these boards!"
Parents also see the advantage of Clark's approach to math. As one parent remarked, "My daughter doesn't even realize she's learning math. They integrate it throughout all the subjects and it's important because we can use it at home in real situations, like sewing skirts for our theater group and determining the circumference of the waists without a pattern. They also have everything a parent needs for the tools to help their child and for the child to work and get whatever they want in life."
U.S. Secretary of Education Margaret Spellings announced a new pilot program under No Child Left Behind (NCLB) aimed at helping states differentiate between underperforming schools in need of dramatic interventions and those that are closer to meeting the goals of NCLB. As part of the new pilot program, states that meet the four eligibility criteria may propose a differentiated accountability model. These eligibility criteria are based on the "bright line" principles of NCLB. (March 18)
During testimony before the U.S. House Committee on Education and Labor hearing on "Ensuring the Availability of Federal Student Loans," Secretary Margaret Spellings launched a new brochure, Federal Aid First, a resource for students and families that encourages them to maximize more affordable Federal student aid options before pursuing other options. To access the brochure and additional information about federal student aid, please visit (March 14)
Education Secretary Spellings announced the release of the final report of the National Mathematics Advisory Panel, and the findings were passed unanimously at the panel's meeting at Longfellow Middle School in Falls Church, Va. The panel reviewed the best available scientific evidence to advance the teaching and learning of mathematics and stressed the importance of effort, algebra, and early math education. (March 13)
Secretary Spellings joined Intel Chairman Craig Barrett to honor Intel Science Talent Search (STS) finalists. STS is America's oldest and most prestigious high school science competition. The top prize this year went to Shivani Sud of Durham, N.C, who developed a model that analyzed the specific "molecular signatures" of tumors from patients with Stage II colon cancer. She used this information to identify patients at higher risk for tumor recurrence and propose potentially effective drugs for treatment. (March 13)
Following a visit to Van Duyn Elementary School in Syracuse, N.Y., where Secretary Spellings highlighted progress toward NCLB goals in New York and across the nation, she joined Representative Jim Walsh (R-NY) and school officials at an education roundtable to discuss the state's accountability plan, standards, and assessments. She also discussed the new tool recently released by the Department known as Mapping New York's Educational Progress 2008. (March 10)
Continuing the dialogue on NCLB and priorities for 2008, Secretary Spellings convened an education roundtable at the West Virginia State Capitol Building with Congresswoman Shelley Moore Capito (R-WV), First Lady of West Virginia Gayle Manchin, West Virginia State Superintendent Steve Paine, and state education leaders and policymakers. She also visited Saint Albans High School in Saint Albans, W.V., and delivered remarks recognizing the progress of the school's students under NCLB. (March 7)
Secretary Spellings continued her national tour to discuss No Child Left Behind (NCLB) in North Carolina, where she addressed the North Carolina State Board of Education in Raleigh and discussed how the federal government can support and facilitate further academic gains made by the state's students under the law. She also participated in a roundtable with educators and school administrators. (March 5)
Secretary Spellings delivered remarks at the Reading First State Directors Conference and declared that with the help of the Reading First program, there have been dramatic gains in student and school achievement. She called on Congress to restore funding for the program to $1 billion, as requested in the President's fiscal year 2009 budget. (March 6)
The March edition of Education News Parents Can Use featured the work of the National Mathematics Advisory Panel and included a discussion about the Panel's final report and how its findings will lead to more effective math instruction in classrooms nationwide. The show also spotlighted what the Department and other key partners are doing to promote math and science literacy through the American Competitiveness Initiative and showcased the work of high-performing schools around the country that are excelling in math education and effectively implementing the Panel's recommendations. To find out more about the program, visit the Education News Parents Can Use Web site. The archived webcast of the show may be viewed online at (March 18)
Applications for the Teaching Ambassador Fellowship positions at the Department are due April 7, 2008. These positions offer highly motivated and innovative public school teachers the opportunity to contribute their knowledge and experience to the national dialogue on education. For more information go to the Teacher Fellowship Web site.
From the Office of Innovation and Improvement
The Full Service Community Schools (FSCS) Program is recruiting peer reviewers for its upcoming grant competition. This program encourages coordination of educational, developmental, family, health, and other services through partnerships between public elementary and secondary schools and community-based organizations and public or private entities. Grants are intended to provide comprehensive educational, social, and health services for students, families, and communities. To obtain additional information or to submit resumes, contact the program at [email protected], using the subject "Reviewer Information."
American History
Students at Henry E. Lackey High School in southern Maryland have developed one of the most comprehensive oral history projects of black life in the region. Students interviewed several of the region's oldest black residents and are creating an hour-long DVD that will be aired during Charles County's 350th anniversary celebration this summer. The project is one of several recent efforts to expand students' knowledge about the black population in Maryland's oldest counties. (March 6)
Elizabeth R. Varon, distinguished lecturer with the Organization of American Historians (OAH), writes in the OAH Newsletter about her experience visiting teachers who participate in the OII-funded Teaching American History (TAH) Program in Rockford, Ill. She notes, "The first thing that struck me was the dedication of the 60 or so teachers who were willing to give up their Saturdays… for a day of intensive collaboration." The Rockford Public School system is in its last year of a fiscal year 2004 TAH grant. (February 2008)
Arts Education
March is Arts in the Schools Month, and to bring attention to the importance of the arts in K-12, the American Association of School Administrators is putting the arts at "center stage" in its March edition of The School Administrator. Among the journal edition's features available to online readers are perspectives on the role of the arts in fostering innovation and the acquisition of skills needed in a knowledge-based economy, stories of schools and districts keeping the arts strong as part of leaving no child behind, and suggestions for policy leaders about the complete curriculum. (March 2008)
The Art of Collaboration: Promising Practices for Integrating the Arts and School ReformPDF (1.53 MB) is a new research and policy brief from the Arts Education Partnership. The brief describes promising practices for building community partnerships that integrate the arts into urban education systems. The publication resulted from a roundtable discussion among the directors of eight demonstration sites that are participating in The Ford Foundation's Integrating the Arts and Education Reform Initiative. (March 24)
Findings from studies by neuroscientists and psychologists at seven universities are helping scientists understand how arts instruction might improve general thinking skills. Learning, Arts, and the Brain, a Dana Consortium report on arts and cognition, does not provide definitive answers to the "arts-makes-you-smarter" question, but it does dispute the theory that students are either right- or left-brained learners. It also offers hints on how arts learning might relate to learning in other academic disciplines. (March 2008)
Charter Schools
Synergy Charter Academy in South Los Angeles was named Charter School of the Year at this year's California Charter School Conference. Caprice Young, former president of the Los Angeles Unified School Board who is now chief executive of the California Charter Schools Association, said, "[Synergy Charter] should be credited with not only closing the achievement gap, but eliminating it." The school was the highest-performing school in South Los Angeles in 2006 and 2007, and was named a National Charter School of the Year last year by the Center for Education Reform. (March 3)
Students in South Carolina might be interested in a new virtual charter school that will open this fall. South Carolina Connections Academy will be the state's first virtual charter school, and will enroll 500 students in its online K-12 program. Connections Academy, a company that runs schools enrolling 10,000 students in 14 other states, will manage the new school. (March 3)
The Center for Education Reform (CER), a Washington-based education reform advocacy group, recently ranked each state based on the strength of its charter school laws and found significant differences among the states. For example, Minnesota had the strongest charter laws in the country, while Mississippi had the weakest. Each state received a letter grade, "A" through "F," based on criteria developed by CER. (Feb. 13)
As charter schools across the nation gear up for lotteries, the National Alliance for Public Charter Schools is offering a free PDF (168 KB) "Charter School Lottery Day Tool Kit." Lottery days can present opportunities to: draw media attention to the demand for quality charters; create awareness among families of school choice, and create an opportunity for charters to communicate their success. Charter school staff can use the tool kit to create their own lottery day event. Materials on preparation, messaging, recruitment, media outreach, timelines, and costs are included. (February 2008)
Closing the Achievement Gap
Each year since the 2005 National Education Summit and the founding of the American Diploma Project (ADP) Network, Achieve has issued an annual report based on a 50-state survey of efforts to close the "expectations gap" between high school requirements and the demands of colleges and employers. Closing the Expectations Gap 2008 reveals that while a majority of states have made closing the expectations gap a priority, some states have moved much more aggressively than others. (February 2008)
Education Reform
Publicschoolinsights.org is a new online resource aimed at building a sense of community among individuals who are working at the local level to strengthen their public schools. The site also features a variety of success stories from U.S. schools and districts that have adopted effective strategies for addressing key challenges in education. (March 2008)
Mathematics and Science
Nearly three out of five U.S. teens (59 percent) do not believe their high school is preparing them adequately for careers in technology or engineering, according to the 2008 Lemelson-MIT Invention Index, an annual survey that gauges Americans' attitudes toward invention and innovation. The good news is that 72 percent believe technological inventions or innovations can solve some of the world's most pressing problems, such as global warming and water pollution. Sixty-four percent of those surveyed are confident that they could invent the solutions. (Jan. 16)
Raising Student Achievement
Fifty-nine exemplary middle schools across the country have been
named "Schools to Watch" as part of a recognition program developed by the National Forum to Accelerate Middle-Grades Reform. Each school was selected by state leaders for its academic excellence, responsiveness to the needs and interests of young learners, and commitment to helping all students achieve to high levels. In addition, each school has made a commitment to assessment and accountability to bring about continuous improvement, teachers who work collaboratively, and strong leadership. (March 14)
A nonprofit organization has launched a national campaign called "Ready by 21" that will work to help youth become better prepared for college, work, and life. Run by the Forum for Youth Investment, the initiative is intended to help state and local leaders improve education and social services during the first two decades of children's lives. The initiative urges leaders to work together on interrelated problems such as drug use, teenage pregnancy, and school dropouts. (March 2008)
Legislation under consideration in Maryland and many other states is intended to ease the transition for students whose parents serve in the military. These students change schools an average of six to nine times between kindergarten and 12th grade. A proposed PDF (341 KB) multi-state compact supported by the Pentagon is intended to reduce the complications involved with these school transfers. (March 2008)
California students who fail to earn a high school diploma before they turn 20 years old cost the state $46.4 billion over the course of their lives. Each year, about 120,000 students in the state drop out. The high cost associated with these dropouts is related to greater rates of unemployment, crime, and dependence upon welfare and state-funded medical care, as well as lost tax-revenues, according to a report from the California Dropout Research Project. (February 2008)
Teacher Quality and Development
Attrition would be lessened if schools offered new teachers more support and guidance, according to an Alliance for Excellent Education PDF (93.9 KB) issue brief. The report found that teachers who graduated from very selective colleges, or who had high SAT scores, were more likely to leave the teaching profession before retirement or transfer to higher-performing schools. (February 2008)
Charter Schools A mayoral change in Indianapolis, the only city nationwide in which the mayor's office authorizes charter schools, has not changed support for that city's 17 charter schools. The new mayor, Greg Ballard, voiced strong support for the charter movement created by his predecessor, Bart Peterson, at a recent conference of charter school leaders. The charter schools, according to Mayor Ballard, are in no danger, and they offer an important choice for parents and a way to improve education in the city.
[More—Indianapolis Star] (Feb. 22)
The proposition that teacher quality is a more important variable than class size and other factors will be put to the test next school year, when the Equity Project, a new charter middle school in New York City, is slated to open. Its creator and first principal, Zeke Vanderhoek, plans to pay the school's expected teachers $125,000 annually, plus potential bonuses based on school-wide achievement. Because that is nearly twice as much as the average teacher in the city earns, the experiment will no doubt garner more than just local attention. For their high salaries, Equity Project teachers will work a longer day and year and will accept some duties that fall to administrators in other schools.
[More—The New York Times] (March 7) (free registration required)
Mathematics and Science Two members of the USA Today's 2007 All-USA Teacher Team find ways to inspire their high school students in economics and mathematics. An economics teacher at the California Academy of Math and Science, where many students are the children of Asian or Hispanic immigrants, taps into students' creativity. The teacher uses techniques such as student playwriting to illustrate economic principles to semester-long assignments in which students develop a proposed start-up company. In College Park, Ga., at Benjamin Banneker High School, 63 percent of students are eligible for free- or reduced-priced meals, and many students already have children of their own or wear ankle bracelets that allow law enforcement officials to monitor their movements. It is at this school that one teacher has inspired his students to learn advanced mathematics and use education as a tool to improve their lives. The school's pass rate on the state graduation exam has jumped from 85 percent to 95 percent between 2005 and 2006.
[More—USA Today] (Feb. 25) and [USA Today] (March 3)
In search of answers to the question of why students in Scandinavia scored high on the latest Program for International Student Assessment (PISA), a U.S. delegation led by the Consortium for School Networking (CoSN) toured Finland, Sweden, and Denmark, where educators cited "autonomy, project-based learning, and nationwide broadband Internet access as keys to their success."
[More—ESchool News] (March 3)
Achievement in mathematics and science, rather than more general barometers of education attainments, are critical to the international economic performance of the U.S., according to a new study by professors at Stanford and the University of Munich. Reported in the spring issue of Education Next, the research supports the conclusion that "if the U.S performed on par with the world's leaders in science and math, it would add about two-thirds of a percentage point to the gross domestic product."
[More— Wall Street Journal] (March 3)
Interest in an international robotics competition among Minneapolis schools and the community's technology sector has flourished over the past two years, from two student teams competing in 2006 to 54 teams this year. For Inspiration and Recognition of Science and Technology (FIRST) is a catalyst for both public and private investments in science and technology programs in high schools, not only in Minneapolis, but across the state of Minnesota. Driving the investment among such private-sector contributors as Medtronic, Boston Scientific, and the 3M Foundation is a desire to encourage future engineers. The Minnesota Department of Education has increased its funding for science, technology, engineering, and mathematics (STEM) initiatives statewide as well, providing more than $4 million to school districts between 2006 and 2008.
[More—Minneapolis Star-Tribune] (March 4)
Raising Student Achievement An analysis of recently released College Board data on Advanced Placement tests by Education Week found that while more students are taking the exams, the "percentage of exams that received [the passing score of at least] a three…has slipped from about 60 percent to 57 percent." College Board spokesperson Jennifer Topiel, while noting that test scores often decline with increases in the number of test takers, observed, "Students should not be placed into AP classes without better preparation." The analysis also revealed a widening gap over the past four years between black and white students earning at least a three on the exams.
[More—Education Week] (Feb. 14) (paid subscription required)
First-year results of a federally supported study of two reading interventions for struggling adolescent readers indicate increases in proficiency, but not enough to get students to grade level in a single year. Research firm MDRC conducted the study of the Reading Apprenticeship Academic Literacy and Xtreme Reading programs, with support from the U.S. Department of Education's Institute of Education Science. It is the first of three reports under the Enhanced Reading Opportunities Study. Researchers plan to follow the 9th grade students involved in the two interventions through 11th grade.
[More—Education Week] (Feb. 14) (paid subscription required)
A majority of American parents believe that their children have the "right amount" of homework, according to the findings of a poll commissioned by MetLife. Parents, teachers, and students were surveyed concerning time spent on homework as well as the perceived value of it. Clear majorities of both students (77 percent) and teachers (80 percent) said homework is important or very important. In addition, three quarters of the more than 2,000 K-12 students surveyed reported that they had adequate time to complete their assignments.
[More—Education Week] (Feb. 15) (paid subscription required)
More than 10,000 preschool-aged youngsters in Dallas are expected to benefit from a city-sponsored early reading preparation program that is modeled on Ready to Read. With support from an $8 million grant from the Wallace Foundation, the Dallas Public Library will manage the "Every Child Ready to Read @ Dallas" program, which will focus on parents, teachers, day-care providers, and others in the city who work with young children. In announcing the new program, Dallas Mayor Tom Leppert said, "Everything revolves around reading," and indicated the city's annual costs for the new program will be less than $600,000, with the Wallace Foundation grant helping for the next three years.
[More—The Dallas Morning News] (Feb. 22)
Researchers from the Centers for Disease Control and Prevention (CDC) believe that physical education may be linked to academic achievement. This belief is based on a national study of students' reading and mathematics test scores and the students' degree of involvement in physical education between kindergarten and fifth grade. According to the CDC researchers, the connection was most notable for girls receiving the highest levels of physical education (more than 70 minutes per week), who scored consistently higher on the tests than those who received less than 35 minutes a week in physical education. The study is available online in the Journal of American Public Health.
[More—USA Today] (March 5)
School Improvement Standards for school leaders, originally drafted in the mid-1990s and used or adapted by more than 40 states, have been revisited and revised by a panel of experts convened by the National Policy Board for Educational Administration and managed by the Council of Chief State School Officers. The revised Interstate School Leaders Licensure Consortium (ISLLC) standards, which guide the preparation, licensure and evaluation of principals and superintendents, were approved last December. The two-year revision process was supported by the Wallace Foundation, which made the investment, according to its director of education programs, because "there's a lot more known now from the research in terms of understanding what leaders do to impact teaching and learning…"
[More—Education Week] (Feb. 27) (paid subscription required)
A $5 million grant from the Michael & Susan Dell Foundation will enable Dallas educators to have instant access to students' academic records from preschool through high school graduation. The plans for an eventual mega-database of student academic information and other related data will begin with a planned "data warehouse" pilot phase next school year. The new system will provide a "one-stop shop" for local educators and help the Dallas Independent School District with its goal of spotting weaknesses in academic performance under its Dallas Achieves reform plan.
[More—The Dallas Morning News] (Feb. 27)
Houston will have its first public Montessori middle school thanks to the perseverance of the parents of Wilson Elementary, an elementary school currently based on the instructional approach pioneered by Maria Montessori more than a century ago. Parents raised more than $345,000 over five years to expand the current school to grades seven and eight. The 25 seats in the school's inaugural seventh grade will be open to students from several public and private Montessori elementary schools in the area.
[More—The Houston Chronicle] (Feb. 27)
Pay-for-performance initiatives continue to attract the attention of local and national press. The National Center on Performance Incentives released its study of the Texas Educator Excellence Grant program, the largest merit-pay plan in the nation. Texas education department officials were reportedly pleased with the first year's results and the study's findings. An examination of The Teacher Advancement Program (TAP), launched six years ago by the Milken Foundation and with 180 participating schools nationwide produced uneven results, with TAP elementary schools doing better than comparison schools in test-score gains, but those at the middle and high school levels lagging behind their non-TAP counterparts.
[More— The Dallas Morning News] (Feb. 29) [Education Week] (March 3) (paid subscription required)
For more than two decades, Project STAR, a study of class size in Tennessee, has informed thinking about the policy issue of class-size reduction. Now, a Northwestern University professor's review of the study's data is questioning whether there is evidence that reducing class size reduces achievement gaps between groups of students. According to the study's author, the longitudinal data provides weak or no evidence that lower-performing students benefited more than others from small classes.
[More—The Washington Post] (March 10) (free registration required) and [Education Week] (Feb. 21) (paid subscription required)
Teacher Quality and Development Can a single set of standards for accrediting teacher-education institutions be developed? This is the question that a new task force of the American Association of Colleges of Teacher Education (AACTE) will seek to answer this spring. Task force members include representatives of the two national accrediting entities – the longstanding National Council for Accreditation of Teacher Education (NCATE) and the relatively new Teacher Education Accreditation Council (TEAC). While the two entities take very different approaches to granting their seals of approval, AACTE's board of directors is hopeful that the task force can agree on a single set of standards.
[More—Education Week] (Feb. 21) (paid subscription required)
The burgeoning field of online learning has launched its first voluntary national standards that will help policymakers and practitioners judge the credibility and worthiness of virtual teaching and online course work. Released last month by the North American Council for Online Learning, the standards address such topics as teacher prerequisites and licensure, technology skills, and subject matter proficiency, as well as instructional issues like online interaction, intellectual property rights, and learning assessments and program evaluations.
[More—Education Week (Feb. 29) (paid subscription required) |
Editorial Reviews
As far as math goes, McKellar knows her stuff. With two girlcentric, best-selling titles under her belt (Math Doesn't Suck, 2008, and Kiss My Math, 2009) and a degree in mathematics from UCLA, it is almost easy to forget that she was Winnie Cooper on The Wonder Years. Facing down a 432-page book devoted to algebra could give even math whizzes pause, but McKellar makes it work, taking the textbook-meets-Seventeen approach by mixing the explanations and equations with boy talk, quizzes, and testimonials from successful women. While a tutor might use this title as a teaching aid, teen girls will want to explore it on their own. Navigation is easy; students are encouraged to hop from chapter to chapter as their homework demands. The breakdown of equations is effective and certainly unconventional—explaining functions in terms of sausage factories, for example, or exponents in terms of whip-bearing female executives (makes sense in the book, promise)—and while McKellar keeps her focus on how to solve math problems, her approach is both readable and even entertaining. --Courtney Jones
About the Author
BestMore About the Author
Best known for her roles on The Wonder Years and The West Wing, Danica McKellar graduated summa cum laude in mathematics from UCLA, went on to co-write a published math theorem, and continues to be an outspoken role model for young women to excel in math.
McKellar's 3rd and latest math book encourages teenage girls to outclass the boys in algebra. This can be an especially tough subject and she tackles it in a way that invites the outlandish idea of actually enjoying math.
She rejects the notion that excelling in algebra is all about natural gift. Rather, it's about determination and making an effort.
While the sex-appeal of the book might be in question, ultimately this is a book broken down into straightforward language, easy charts and memorable reminders of the process of algebra.
Additionally, you'll find motivating stories from youthful female adults who've struggled with math, but learned to conquer their struggles. Danica proclaims. "I'm here to tell you that giving up on ourselves because of our own stereotypes and limited imaginations is a far more destructive force than any challenge or obstacle 'out there,'"
One bonus in the book is the inclusion of good tables and charts. These can be worth the price of the book alone. These succinctly explain the meaning of math terms such as fractions, coefficients and variables.
A surprising aspect of the book is the authors inclusion of advice on how to choose the right guys to date. While she doesn't provide a mathematic formula for this it does seem she dispenses some basic common sense.
What a great book to help tutor teen girls in math! It is organized well, so it is easy to navigate and you don't have to read it straight through. You can pick your topic and go back and forth with ease. The author is obviously a great teacher and knows her stuff. She combines that with her ability to reach out and grab the attention of teen girls and make math fun. A must have to share with the girls in your life!
At last, a book that teaches middle-school students and their parents in simple terms how to solve quadratic equations by completing the square and to solve the quadratic equation in standard form in X (ax^2 + bx + c = 0) to come up with the quadratic forumla. Most colleges and high schools do not teach this -- they just expect the student to know it.
The charter school I support plans to use Hot X: Algebra Exposed in concert with the first two books to ensure most or all Grade 8 students attain an A (93 or higher) in Algebra I.
Danica McKellar needs to continue writing math books. Since these are such outstanding books, few state departments of education will adopt them for use, which means parents, other major care givers, and supportive businesses will have to come up with the funds to help schools with the purchase of these books. Not only should the schools have these books for use as textbooks, but individual students should have their own copy to help learn the material initially and to refresh key points as they complete Algebra II and beyond.
Danica also has web sites that provide additional information for each book. Check out [...]. If you wish to provide input to Danica, you may send it to her at: [email protected]. |
Offerings by Semester
A World of Mathematics
Please register via INTD 0100A
A World of Mathematics
How long will oil last? What is the fairest voting system? How can we harvest food and other resources sustainably? To explore such real-world questions we will study a variety of mathematical ideas and methods, including modeling, logical analysis, discrete dynamical systems, and elementary statistics. This is an alternative first mathematics course for students not pursuing the calculus sequence in their first semester. The only prerequisite is an interest in exploring contemporary issues using the mathematics that lies within those issues. (This course is not open to students who have had a prior course in calculus or statistics.) 3 hrs lect./disc transformations and matrix representations, eigenvalues and similarity, determinants, the inverse of a matrix and Cramer's rule. (MATH 0121 or by waiver) 3 hrs. lect./disc.
Elements of Math Bio & Ecol
Elements of Mathematical Biology and Ecology
Mathematical modeling has become an essential tool in biology and ecology. In this course we will investigate several fundamental biological and ecological models. We will learn how to analyze existing models and how to construct new models. We will develop ecological and evolutionary models that describe how biological systems change over time. Models for population growth, predator-prey interactions, competing species, the spread of infectious disease, and molecular evolution will be studied. Students will be introduced to differential and difference equations, multivariable calculus, and linear and non-linear dynamical systems. (MATH 0121 or by waiver)
Multivariable Calculus
Multivariable Calculus
The calculus of functions of more than one variable. Introductory vector analysis, analytic geometry of three dimensions, partial differentiation, multiple integration, line integrals, elementary vector field theory, and applications. (MATH 0122 or MATH 0200 or by waiver) 3 hrs. lect./disc.
Statistics
Statistics
An introduction to the mathematical methods and applications of statistical inference. Topics will include: survey sampling, parametric and nonparametric problems, estimation, efficiency and the Neyman-Pearsons lemma. Classical tests within the normal theory such as F-test, t-test, and chi-square test will also be considered. Methods of linear least squares are used for the study of analysis of variance and regression. There will be some emphasis on applications to other disciplines. (MATH 0310) 3 hrs. lect./disc.
Operations Research
Operations Research
Operations research is the utilization of quantitative methods as an aid to managerial decisions. In the course, several of these methods will be introduced and studied in both a mathematical context and a physical context. Topics included will be selected from the following: classification of problems and the formulation of models, linear programming, network optimization, transportation problems, assignment problems, integer programming, nonlinear programming, inventory theory, and game theory. (MATH 0200 or waiver)Senior Seminar
Senior Seminar
Each student will explore in depth a topic in pure or applied mathematics, under one-on-one supervision by a faculty advisor. The course culminates with a major written paper and presentation. This experience emphasizes independent study, library research, expository writing, and oral presentation. The goal is to demonstrate the ability to internalize and organize a substantial piece of mathematics. Class meetings include attendance at a series of lectures designed to introduce and integrate ideas of mathematics not covered in the previous three years. Registration is by permission: Each student must have identified a topic, an advisor, and at least one principal reference source. 3 hrs. lect./disc. |
Just the math skills you need to excel in the study or practice of engineering
Good math skills are indispensable for all engineers regardless of their specialty, yet only a relatively small portion of the math that engineering students study in college mathematics courses is used on a frequent basis in the study or practice of engineering. That's why Essential Math Skills for Engineers focuses on only these few critically essential math skills that students need in order to advance in their engineering studies and excel in engineering practice.
Essential Math Skills for Engineers features concise, easy-to-follow explanations that quickly bring readers up to speed on all the essential core math skills used in the daily study and practice of engineering. These fundamental and essential skills are logically grouped into categories that make them easy to learn while also promoting their long-term retention. Among the key areas covered are:
With the thorough understanding of essential math skills gained from this text, readers will have mastered a key component of the knowledge needed to become successful students of engineering. In addition, this text is highly recommended for practicing engineers who want to refresh their math skills in order to tackle problems in engineering with confidence. |
So I was having a lot of medical issues my freshman year, and I pulled a C+ in second quarter calculus and a C- in third quarter calculus (integration and multivariable calculus, respectively.) I still have one more quarter of calculus entitled "Vector analysis." I'm wondering if my poor performance (and poor understanding of the material) is going to negatively affect me later on. Will I have a chance to relearn some of the material in my other classes, or should I self study to get myself up to par? Just so you know, as of right now I'm leaning towards a Mechanical Engineering major, but I may switch into Civil (though highly unlikely.) This upcoming quarter I'm taking Vector Analysis, first quarter Calculus-based Physics, and a MATLAB course (I have absolutely zero programming experience.) Physics and MATLAB both require integral calculus as a prerequisite. I need a B in physics to switch into MechE. Should I be self studying the calculus that I don't quite understand?
Also, will these grades kill my chances of getting into grad school? They're my only grades below a B, and I hopefully won't have many (any?) more C's in my college career.
To understand the computational part of engineering you need to understand differential equations, which is where you use the calculus skills. You then use DEs to describe all kinds of physical phenomena. So, the basic calculus skills are very important to understanding engineering.
Thanks HPuck. So would it suffice to get down integration and partial derivatives and their applications? Or should I learn other things like Taylor Polynomials and Convergence/Divergence (the stuff that tripped me up most)?
Calculus is one of the few subjects you really need to put in the effort to learn well. Study all of it until you have a strong conceptual understanding of it. Bad grades are bad grades, but calculus is important.
Calculus will be used in physics. Multivariable calculus will become important when you take the physics course with electricity and magnetism. Differential equations will become important in many places.
Try and improve on basic integration/misc techniques (integration by parts, partial fraction decomp, trig int, partial derivations, etc) because these will be very important in DE. A lot of questions will have a lot of nested integrals/nasty integrals and you don't want to get stuck in the middle of the question. Freshen up on Taylor and Maclaurin series. When you take linear alg practice matrix techniques, translation matrices, etc. (will be used in systems of differential equations)There are only two subjects you absolutely have to know, without constantly consulting a reference: calculus and physics.
Yeah, it will come back to haunt you if you don't put in the effort to get the hang of it. Get on that, no matter how much work it takes.1 year left of Engineering school. Taylor series rarely comes into play. The only time it did show up, it was just mentioned as part of the proof and we didn't have to do anything with it. But yes, calculus is EXTREMELY important, if not for the pure concept of what it means to take the integral/derivative. Also, so so many people don't have a good grasp on all the material that came before calculus too, especially trig. Know your trig, it will help you out, a lot. |
If you are not using algebra on a daily basis, you should probably kill yourself. It's a pretty basic skill |
Specification
Aims
The unit aims to introduce the basic ideas of differentiable manifolds.
Brief Description of the unit
Differentiable manifolds are among the most fundamental notions of modern mathematics. Roughly, they are geometrical objects
that can be endowed with coordinates; using these coordinates one can apply differential and integral calculus, but the results
are coordinate-independent.
Examples of manifolds start with open domains in Euclidean space R^n , and include "multidimensional surfaces"
such as the n-sphere S^n and n-torus T^n , the projective spaces RP^n and
CP^n , and their generalizations, matrix groups such as the rotation group SO(n), etc. Differentiable
manifolds naturally appear in various applications, e.g., as configuration spaces in mechanics. They are arguably the most general objects
on which calculus can be developed. On the other hand, differentiable manifolds provide for calculus a powerful invariant geometric language,
which is used in almost all areas of mathematics and its applications.
In this course we give an introduction to the theory of manifolds, including their definition and examples; vector fields and differential
forms; integration on manifolds and de Rham cohomology.
Learning Outcomes
On completion of this unit successful students will be able to:
deal with various examples of differentiable manifolds and smooth maps;
have familiarity with tangent vectors, tensors and differential forms;
work practically with vector fields and differential forms;
appreciate the basic ideas of de Rham cohomology and its examples;
apply the ideas of differentiable manifolds to other areas.
Future topics requiring this course unit
Differentiable manifolds are used in almost all areas of mathematics and its applications, including physics and engineering.
The following course units are specifically based on MATH31061/MATH41061 Differentiable Manifolds:
MATH41122 Differential Geometry; MATH41101 Geometric Cobordism Theory
Syllabus
Manifolds and smooth maps. Coordinates on familiar spaces. Charts and atlases. Definitions of manifolds and smooth maps.
Products. Specifying manifolds by equations. More examples of manifolds.
Integration. Orientation. Integral over a compact oriented manifold. Independence of atlas and partition of unity. Integration over singular manifolds and chains. Stokes theorem.
De Rham cohomology. Definition of cohomology and examples of nonzero classes. Poincaré Lemma. Examples of calculation.
Textbooks
No particular textbook is followed. Students are advised to keep their own lecture notes. There are many good sources available
treating various aspects of differentiable manifolds on various levels and from different viewpoints. Below is a list of texts that
may be useful. More can be found by searching library shelves. |
$29.99Algebra 1 Word Problems
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Present algebra topics in a logical order. The text develops an understanding of algebra by justifying methods and by explaining how to do the problems. It introduces graphing, solving systems of equations, operations with... more7.61Reg.$8.95Algebra 1 Flipper26Reg.$20.30Saxon Math Algebra 1 4th Edition Home School Testing Book
This test book is part of Saxon's Algebra 1, 4th Edition Homeschool curriculum. Perfect for students who already have the texts, this book contains tests, a testing schedule, test answer forms, test analysis form, and test... more |
will provide activists with self-standing lessons and online support so they can apply and share mathematical thinking to address globalization and its consequences, notably climate change.
We will create a set of 30 self-standing math lessons for self-learners . Each lesson will present a classic math problem and a deep idea which it illustrates. We will organize an online support center for tutoring and consulting, but especially for supporting "math in action" for analysis and communication of challenges and solutions in our changing world, focusing on climate change. We will publish a paperback, worksheets, a wiki and YouTube videos, all in the Public Domain.
Our age of globalization is propagating rapid changes and dramatic instabilities. Will we match this with an ability to think and talk and act decisively and independently? We need to appreciate the connection between our local choices and the global picture. Mathematics is a language for talking about the subtle but profound distinctions of the models that may be at play. It helps us care about each other's situation around the world. We're all potential climate refugees. Math help us learn to share with each other. It helps us exchange our wisdom as climate patterns shift and disrupt our agriculture, our water supply, our built environment. We need to think deeply and learn rapidly. We need ideas and problems that are to the point and can be spread freely. We need math lessons on-demand!
Andrius Kulikauskas is since childhood a lifelong learner on a quest to know everything and apply that knowledge usefully. At the University of California at San Diego, he studied mathematics as a tool for conceptual thinking. He was appointed senior teaching assistant, responsible for training the first year teaching assistants. He earned his Ph.D. in 1993 for this thesis "Symmetric Functions of the Eigenvalues of a Matrix".
Andrius taught a Precalculus class at UCSD for students who were ill prepared for Calculus. He knew the dismal state of mathematics teaching. High school teachers generally don't think mathematically. Truly, mathematics should be a tool for thinking, where the point is first to understand the relevance of various models, and only then to calculate. Unfortunately, textbooks are full of contrived, irrelevant problems which hinder mathematical intuition. Andrius taught from his own notes: He identified the key ideas leading up to Calculus, and surveyed them all in thirty one-hour lessons. His students did very well!
In 2008, Andrius taught algebra as a Professor at American University in Bosnia and Herzegovina. He taught 90 business students, many of whom were unmotivated because they thought math was irrelevant, and yet they rated him an outstanding teacher. He concentrated on teaching them a few dozen classic problems which were difficult but insightful. For example, suppose that the price of a barrel of oil goes up by one-third this week, but then goes down by one-third next week. What is the overall change in price? Many people suppose that the two changes cancel out and the price is the same. But actually, if the barrel of oil costs $90, then it goes up to $120, but down to $80! And, in general, it goes up from X to 4/3 X and then down to 8/9 X. This is a very thoughtful problem and it illustrates a very deep idea, which is that algebra is the study of thinking step-by-step. If you don't think step-by-step, you can't solve this problem, because you won't know what "one-third" means. Most books and teachers don't and can't teach step-by-step thinking because it's a lot of work to grade. But if the problem is truly relevant, then the effort is worth it!
Here are some of Andrius's favorite problems:
* The world's population is doubling every thirty years. Suppose that was always true. When was Adam created? and Eve? It may be shocking to learn that they appeared only 1,000 years ago! This shows the power of exponential growth, but especially, is a great illustration of how a model (and perhaps every model) breaks down at a certain point.
* What is 10 + 4? The answer is 2! At least the clock says it is! Ten o'clock plus four o'clock is two o'clock. This problem shows that there is more than one mathematical system, but also suggests that we can't write down everything we need to know about a system.
* All parabolas have the same shape! Some may look skinny and others fat. Yet by zooming in and out, flipping them upside down, moving left or right or up or down, we can see that they have the same shape. This teaches us the basics of transforming any graph.
* "A right triangle is half a rectangle". Thus the shape of a right triangle is given by the ratio of the two sides of the rectangle. But you also know the shape if you know one of the right triangle's acute angles. We can switch back and forth between these two ways of looking at the shape. This "bijection" is the basis for trigonometry!
* "Four times a right triangle is the difference of two squares." This geometric fact makes for a quick proof of the Pythagorean theorem, but also shows the theorem's relevance for switching coordinate systems, which is to say, points of view.
Imagine if all of math education was organized around such classic problems. Andrius will write an essay for every such problem to show why and how it's interesting for a mathematician and philosopher. Each lesson can include many variants of the problem along with applications and exercises. A few dozen problems are enough to survey all of algebra. They could fit in a paperback of 150 pages or so.
Such a paperback would allow parents to master the mathematics which they'd like to help their children learn. After reading such a paperback, they would feel confident taming the monstrous 500 page textbooks used at school.
Imagine if people could use such materials without restriction! Ours will be in the Public Domain so they might be shared, adapted and improved! A teacher in Africa might print out one lesson at a time, think up a game that teaches the concept, and then write a letter how it went. People could share examples and applications from around the world!
Indeed, we can set up a wiki to collect classic problems, ideas and examples in math and other subjects. A wiki is an online tool with which individual problems could be easily edited and linked together into a pattern language, as in Christopher Alexander's "A Timeless Way of Building". Yet each lesson can also stand on its own.
We're living in a world with many new issues linking people of very different cultures. We need to be able to understand subtle but profound implications of our decisions. Activists need math to analyze and communicate the choices we have. We can support them online with a chat room and related venues. We can help them find the right lesson to apply to their challenge and to teach the public and the powers-that-be. They can present their lesson with a short video of "math in action".
We have many opportunities to apply math in our world. We can develop math intuition based on real problems, not contrived exercises. We can organize an online community to create learning materials in the Public Domain that encourage self-learners to understand mathematics in terms of its deep ideas and practical applications. As we face global challenges, including our current economic crisis, we find social and business opportunity in recognizing what is truly worth learning, and making that available for all to share freely. We offer a Baltic and Nordic team to start up this activist approach to education.
Andrius will start by identifying noteworthy math problems from his own notes, from websites, from discussion groups, from colleagues online and from mathematics textbooks, especially intuitive books such as "Applying Arithmetic: A Handbook of Applications of Arithmetic" by Zalman Usiskin and Max Bell and "How to Lie with Statistics" by Darrell Huff. He will continuously look for new problems and invite others to contribute them. He will relate each problem with a deep idea that it illustrates. He will select what he thinks are the thirty or so pairs of problems and ideas which most completely and profoundly present mathematics. He will write an essay for each problem/idea.
Andrius will also create a general format for writing up each problem, idea and related examples, illustrations, applications and exercises. He will adapt the wiki at worknets.org with templates and navigation which accomodate the contributing, polishing and organizing of these problems for self-learners.
Early in our project, Andrius will travel to Tampere, Finland to work for one month with filmmaker Teemu Takatalo at the Hirvitalo community centre. Andrius will work with activists to understand what math ideas might be relevant for their campaigns. Teemu will coach them on using video to communicate these ideas.
Andrius will also travel to Latvia to work with open source publisher Didzis Veinbergs. Didzis will find one or more designers and/or illustrators to help with designing the website and illustrating the mathematical ideas. They will also provide images for our video makers in Finland.
Mihkel Pilv and Miksike will find adults who are interested in learning mathematics, perhaps in order to help their children learn. Andrius will visit them in Estonia for one month and tutor them. He will also organize online tutoring at by chat, email and Skype to help them and others around the world.
Midway into our project, Andrius will start publishing individual lessons online. Miksike's adult learners will test the lessons and help create related worksheets. Andrius will return to work with them.
As the videos are completed, they will be published online and also made available through our wiki. Didzis Veinbergs will promote them and the activist's campaigns along with the upcoming book. Andrius Kulikauskas will visit him in Latvia and they will publish and promote a short paperback of "Classic Math Problems".
We will adapt the book and translate it from English into Lithuanian, Latvian and Estonian and publish and promote electronic versions.
All of the materials created for our project will belong to the Public Domain.
Description of partnership("who does what")
Our project includes four partners from Lithuania, Estonia, Latvia and Finland.
Minciu Sodas is the coordinator for our project. It is a sole proprietorship registered in Vilnius, Lithuania. Andrius Kulikauskas is the founder, sole proprietor and Direktorius. Minciu Sodas is an online laboratory for serving and organizing independent thinkers around the world. We have about 150 active and 1,500 supportive participants. Our most spectacular project so far was the Pyramid of Peace in 2008 in Kenya. We organized 100 peacemakers on-the-ground and 100 online assistants to avert genocide during the post-election turmoil. We used our wiki, mailing lists and chat room, all in the Public Domain, to coordinate our work.
Andrius Kulikauskas will select the problems and ideas, author the essays, the lessons and the paperback, and code the wiki and help room. This work is the heart of this project. He will be supported by Minciu Sodas's team of online assistants who will collect interesting examples and tutor self-learners.
Miksike is a pioneer in educational crowdsourcing. More than 1,000 volunteers have created 30,000 worksheets used by 120,000 students, see: lefo.net Miksike is a Private Limited Company based in Estonia. Mihkel Pilv is the director. He is an innovative entrepreneur, ever discovering new business opportunity. Mihkel and Andrius met on a flight to the MIT Media Lab ThinkCycle's Development by Design 2002 conference in Bangalore, India to which they both won travel awards for their papers about their work.
Pispalan kulttuuriyhdistys ry (Pispala Cultural Association) runs the Hirvitalo center of contemporary art in Tampere, Finland. Hirvitalo is a base for artistic and cultural activists, including gardens and permaculture. Markus Petz is now the European Voluntary Service coordinator there. He first met Andrius on a road trip they took together to visit UK independent thinkers. Markus organized a video bridge from Hirvitalo in 2008 with Andrius about his paper "An Economy for Giving Everything Away".
Pispala Cultural Association will organize ten activists addressing the consequences of globalization, especially climate change. Each activist will create a YouTube video illustrating "math in action" in their campaigns. Teemu Takatalo will coach them in video skills.
Ūdenszīmes (Signs of Water) is an NGO from Jēkabpils, Latvia, that is working towards improving the possibilities of education for children and young people in Latvia, thus helping create an educated, ethical and responsible society. It was founded in 2007, and its founding members have been active in other NGOs since 1997. Ūdenszīmes will host Didzis Veinbergs's work to publish our Classic Math Problems as a paperback in the Public Domain. Didzis and Andrius met in 2008 at BarCamp Baltics. Andrius invited Didzis to Vilnius for the workshop "Ethical Public Domain: Debate of Questionable Practices" which Minciu Sodas organized for COMMUNIA, the European Union's thematic network for the Public Domain. Didzis spoke there about his plan to publish books with free licences. Didzis will publish, promote and distribute Andrius's book as well as provide designers and/or illustrators for related websites.
Minciu Sodas will translate the book into Lithuanian, Ūdenszīmes into Latvian and Miksike into Estonian.
Of our total budget of 82,000 EUR, which includes our co-financing and our administrative fee, we ask for 60,000 EUR of which 30,000 EUR is for Minciu Sodas, 10,000 EUR for Miksike, 10,000 EUR for Pispala Cultural Association and 10,000 EUR for Ūdenszīmes.
Each of the partners will contribute co-financing in terms of the work of their many volunteers who will collect examples and staff our help room and translate into Lithuanian (Minciu Sodas), try out our lessons and create worksheets (Miksike), apply our math lessons to climate change and other global challenges (Pispala Cultural Association) and promote and distribute our book (Ūdenszīmes).
Expected outcome(Please describe the expected results of your project, who will be effected and how, directly and/or indirectly)
Our project will result in a wealth of learning materials in the Public Domain.
We will publish a short, readable, popular book "Classic Math Problems" with essays on about 30 math problems and the deep ideas they illustrate, along with applications and exercises, and an introduction that weaves them all together. There is a chance that our book will become a classic. "How to Lie With Statistics" was first published in 1954 and today ranks in the top 7,000 books at Amazon.com
We will organize a thriving where such problems are suggested and polished by completing a template for them. Our wiki will link together the problems by various threads. We will establish a new paradigm for creating learning materials.
We will also include links to 10 videos that show our "math in action". We will have helped 10 activists use math to make their case to the public and relevant institutions. We aim for at least one of them to achieve a success that affects or inspires at least 100,000 people.
We will jumpstart an online help room In our first year we will personally help more than 100 people develop their mathematical thinking.
We will create worksheets for use through Miksike and other venues. In particular, we will create one-page worksheets, each dedicated to one lesson, so that a teacher can print them out on-demand when they prepare for their class.
We will publish CDs and/or USB flash drives with all of the above materials for use in remote areas such as Africa with computers such as the One Laptop Per Child XO.
Our materials will be primarily in English, but also some lessons and worksheets in Estonian and possibly other languages.
All of the content which we create for this project will be in the Public Domain.
We will have a team of enthusiasts in the Nordic and Baltic countries and around the world interested in further developing learning materials for mathematics but also other subjects.
Dissemination of results(Please focus on sharing the results with individuals, organisations or groups who are not your partners this project)
We will have electronic versions of the book and associated learning materials in English, Lithuanian, Latvian and Estonian.
We will print at least 1500 copies of the English version of our book. We will sell at least 1,000 copies in stores in the Nordic and Baltic countries and also by mail.
We will distribute our printed book to at least 500 people around the world who will help promote the printed and/or online versions, bring attention to them and open up business opportunity. We're especially interested in parents, especially homeschoolers, who would like to learn math better so they could teach their children, and also activists who might benefit from applying mathematics and thereby raise public interest in our classic math problems. We expect at least 7,000 people to read or hear lessons from the printed book and that a second printing be profitable.
In the first year that the book is available, we expect at least 10,000 people in the Nordic and Baltic region to download the electronic book in English, Estonian, Latvian or Lithuanian. We expect 100,000 people around the world to download the English version and another 100,000 people to learn from it through the One Laptop Per Child program with which we have links through Edward Cherlin's Earth Treasury, a working group of Minciu Sodas.
We hope to attract media attention to our activists's campaigns in Finland and encourage other activists in the Nordic and Baltic countries to make use of our math lessons.
In Estonia, Latvia and Lithuania, and other Mikiske Learning Folder countries, we would like our math lessons to be a key resource for Miksike's large community of teachers and students.
We expect that Earth Treasury initiative will organize the creation of open source educational software and animations based on lessons in our book.
We expect our wiki to be a growing repository of math lessons and our chat room helproom.org to be the center of lively tutoring and consulting services, for free and for pay, in math and other areas.
Minciu Sodas has many contacts in Africa. We're very excited that they might use and share our math learning materials.
In all of our efforts, our materials will spread more easily and widely because they are in the Public Domain for all to freely share, adapt and improve.
How will you evaluate that your project has met its objectives
We evaluate our project in terms of the relationships which we create, especially the people we include.
We expect, at the end of our project, to have a thriving wiki system with a well structured template, a core team of contributors and a growing collection of lessons and applications in math but also other fields.
We wish to find an eager market for our book so that we can fund a second printing of our own resources. We will note any interest that our book be translated, republished by others, and printed on-demand.
We will be excited to see individual lessons being copied and circulated. We will note the most popular sites for getting our book and note the download statistics.
We will be delighted if our materials are used by teachers and there are students learning math without standard textbooks, but especially in poor countries, such as in Africa.
We would like to see one or more of our lessons receive attention in the press thanks to its significance for some activist campaign. We would like one or more of our videos to be popular at YouTube. We will make up for each lesson a "tag" (like "halfrectangle") which can be used on YouTube, Flickr and other sites to note the uses of our classic math problems that all are encouraged to share. We would be excited to see people using such a tag system.
We plan for an active online help room with regular tutoring activity and potential business opportunity.
We wish to encourage a growing number of people that math's deep ideas and classic problems are truly relevant for thinkers and activists, especially for global issues such as climate change. We will know that we are having an impact on globalization and climate change if our lessons are adapted locally. |
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