## Sample Problems from Chapter 10

**Chapter 10**: Pi and Square Roots

**1. Kohler follows Archimedes**

Kohler, a 5th grader at the time,
started with a 12-dot circle. He
drew the circle, a diameter, and inscribed an equilateral triangle in it.
He measured the
diameter of the circle and perimeter of the triangle in mm. He then found
the ratio of p/d. You try it.

Notice, Kohler got 2 R50 and we had a big dicussion about leaving the answer as
a fraction and decimal, finally. He then did the same thing for
a regular hexagon,
and regular dodecagon. You try it. What if you keep going,
what's happens to this ratio p/d?

Write a program to get the
ratios of
the perimeter of polygons of 3x2^{n} sides to the diameter (d) of the circle.

**2. The Gregory-Leibnitz series**

**3. From Wells..**

**4. Iterating the sqrt of a number.** One day a 7th grader, in a
moment of just playing with a calculator, put in a number, then hit the
square root key again and again. Try that.
Graph the numbers you get vs. the number of iterations. What
do you notice? Try a different number. Write a program to do this.

**5. Finding the square root of 2 by squaring numbers on a
calculator.**

**6.
Finding **^{}by
'iteration of the average' method.

What does ^{} mean? We're looking for a number such that
when you multiply it by itself you get 40 or ^{ }. Suppose we guess 5. 40/5 = 8. 8x5=40, but
the two factors have to be the same. 8 is too big
and 5 is too small. If we find their average, we'll be closer to the
^{}. So we'll do
this= 6.5. Then we'll take 6.5 and find
the average of this and 40/6.5, like this . Notice we have an iteration
problem now where **G** is the guess number ->
**G**! Here's a number line picture of these first 2 iterations:

So we get an infinite sequence 5, 6.5, 6.326923077, 6.324555763,
6.324555320, ... After only 4 iterations we get the ^{ }correct to 9 decimal places. This
method gives a rapidly converging sequence to find the square root of a
number.

Write a program to do this iteration.

**7. Using the binomial expansion to find the square root of 2.**

Ways
young people have solved these kinds of problems

To order Don's
materials

To choose sample problems from
other chapters

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