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A New Vignette or two: Mathematics for Teachers’ Pleasure

A New Vignette or two: Mathematics for Teachers’ Pleasure. Bill Barton The University of Auckland. The Klein Project …. … seeks to speak to teachers about the mathematics that they deal with on a daily basis. The Klein Project ….

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A New Vignette or two: Mathematics for Teachers’ Pleasure

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  1. A New Vignette or two:Mathematics for Teachers’ Pleasure Bill Barton The University of Auckland

  2. The Klein Project … … seeks to speak to teachers about the mathematics that they deal with on a daily basis.

  3. The Klein Project … … is about contemporary mathematics:its themes,its problems,its excitements,and its applications.

  4. Klein’s Intent “I hope youdraw from mathematics a living stimulus for your teaching.” “This book is designed solely as a mental spur, not as a detailed handbook.”

  5. International Mathematical Union (IMU) What is the Klein Project ? The international Klein Project is a joint project of ICMI and IMU. It seeks to reach ALL upper secondary teachers—not just those who are already enthusiastic mathematicians, but it must also entice those who can rediscover their love for mathematics.

  6. What it is NOT The Klein Project is neutral with respect to the school curriculum: its structure, content, assessment, teaching modes, philosophy. Klein materials are not intended as classroom resources—they are material for teachers. (However we know that some teachers do use these materials in the classroom).

  7. Klein Vignettes A Klein Vignette is a short piece about contemporary mathematics. Vignettes are written with the intention that teachers will: • …want to READ them; • …want to KEEP reading them; • …want to read MORE about the topic; • …want to read ANOTHER one.

  8. For example:

  9. For example:

  10. http://blog.kleinproject.org Klein Project BlogConnecting Mathematical Worlds Home The Klein Project What is a Klein Vignette

  11. Some Calculus Calculators & Power Series Actually this is not (yet) a Vignette because it is not contemporary mathematics, but the application is contemporary, and may well be of interest to Yr 12 or Yr 13 students. It can be found on the Klein Project WEBSITE (not the blog). Google “Klein Project” and it is the second item, then click on “Klein Vignettes”.

  12. Calculators & Power Series How does a calculator know all the values of sin(x) or the exponential or logarithmic functions? Surely it does not store all the values to many decimal places? The answer lies in the field of power series, that is, series of the form:

  13. Calculators & Power Series Provided that |x| < 1 then this series usually converges quickly (depends on an). Hence, if we can find a power series that will approximate sin(x) and other functions sufficiently accurately, then we have a way to evaluate those functions

  14. Calculators & Power Series Let us try it with sin(x). Let us assume that a power series can be found, so we have: Can we find the coefficients ? First, put x = 0. Then we have sin(0) = a0 so a0 = 0

  15. Calculators & Power Series We now have: sin(x) = a1x + a2x2 + a3x3 + … Differentiate: cos(x) = a1 + 2a2x + 3a3x2 + … And again, put x = 0. Then we have: cos(0) = a1 so a1 = 1

  16. Calculators & Power Series If we keep differentiating and putting x = 0, then we can find all subsequent terms:

  17. Calculators & Power Series But this is an infinite series—our calculator surely does not evaluate an infinitude of terms ? It turns our that this series converges quickly for all values of x. Indeed, even evaluating the first two or three terms gives us very good approximations.

  18. Calculators & Power Series Let us name the following partial sums: How good are they as approximations?

  19. Calculators & Power Series y = sin(x) y = S5 y = S3 At x = 1, the error in S3is about 0.008, and S5 is less than 0.001

  20. Calculators & Power Series We can improve these approximations considerably by using Chebyshev Polynomials … but I will leave that for you to read in the Vignette.

  21. Calculators & Power Series If we have a little more time, however, let us look at how we find power series expansions for another function or two. A very simple power series is when every coefficient is equal to 1. This gives us: which, you may remember,

  22. Calculators & Power Series We can create power series for other functions by substituting, for example, -x2 for x, or by differentiating both sides. But look what happens if we integrate both sides…

  23. Calculators & Power Series Start with the simple power series but (for reasons that will come clear) write t instead of x: Now integrate from 0 to x and multiply by -1: So then:

  24. Detecting Fraud Mathematically

  25. Detecting Fraud Mathematically Simon Newcombe (1835-1909) Frank Benford (1883-1948) Invariant under change of scale Invariant under change of base

  26. Invariant under change of base

  27. Is a set of numbers random?

  28. Detecting Fraud Mathematically

  29. Thank you <b.barton@auckland.ac.nz> <www.blog.kleinproject.org> <www.kleinproject.org>

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