## Sample Problems from Chapter 8 (2nd part-
iteration)

**Chapter 8**: Solving Equations and Iteration. Don's
videotape #2 shows a 6 and 7 year old working on iterating
functions.

**The idea of
iteration for me, started with trying to solve a quadratic equation
10 different ways! In the process I got into continued
fractions, infinite continued fractions, and iteration- it has been a
very
exciting area of mathematics with applications to population increase,
finding the square root of a number, and extends to the field of chaos
theory and fractals!
And my students find it interesting, which is the most important
thing.**
For my younger students I usually start with a rule or function
like
2**x**. We put a number in for **x**, get the number out, then put
that output in for **x**, and continue that process. We get an
**infinite sequence** of numbers. In this case the sequence diverges,
doesn't go to a number. For example if we put 3->**x**, we get 6. We
then put 6->**x** and we get 12.

We get the infinite sequence 3, 6,
12, 24, ...

Later on, with older students, it is not a big step
to use 1.1**x** as the function and show this is the same problem as
increasing the population of a town 10% each year. A very important
application.

A teacher in one of Don's workshops,
made up this function:. We'll pick a number, say 0,
and put it in for x. What do we get out? 5 + 0/2 = 5. Then we put 5 in
for x. What do we get out this time? 5 + 5/2 = 7.5 Now let's keep track
of the infinite sequence we get: 0, 5, 7.5, 8.75, ... The question is
what's happening? Does this sequence converge? **I ask my students to
do the first 8 or so by hand, to make sure they can divide and write the
answer as a fraction or mixed number and a decimal**. Only then
will I let them use a calulator to do more. Then I'll get
them to
the computer to use *Mathematica* to do 200 iterations and let it
carry the answer to 100 decimal places!

Finish the graph of this sequence, the beginning of which is shown
below:

Start with a new number,
like 100 and see what happens.

Start with -17 and see what happens.
Graph these sequences on the same graph paper. Is there a pattern?

Each infinite sequence has a limit of 10 for . Look at the numbers there. What do you
think would happen if we started with 6 + x/2 ? a + x/2 ?

What
would happen with 5 + x/3 ? 5 + x/4 ? Another interesting function I
do with my younger students is 6/x. Interesting things happen with this
one!

11 ways to solve a quadratic
equation

**Method 1. **By guessing and the sum
and product of the roots (see above)

**Method 2.** Solving x^{2} - 5x + 6 = 0 for
x to get x = .

Jonathan, at
age 7, solved this quadratic equation like this:

**2a.**We can get an infinite continued fraction and
find approximations of the roots of the equation

**2b.**We'll iterate the function starting with different numbers, then graph
these sequences.

**2c.**Graph 3 successive 'pieces' of the infinite
continued fraction

**2d.**Graph y = , then connect points whose coordinates are
consecutive input numbers

**Methods 3., 4., and 5.** You solve x^{2} - 5x + 6 = 0
for x, but in a different way than Jonathan did, (but not one of methods
6-11 below), and do the corresponding
things as in **2a., 2b., 2c. and 2d** above. You might find more than
3 other ways! Please let me know if you do.

**Method 6.** Solving
x^{2}
- x - 1 = 0 using a calculator to hone in on the two solutions.

**Method 7.** By factoring (one of the 'normal' ways)

**Method 8.** By completing the square

**Method 9.** Using the quadratic formula

**Method 10.** Graph x^{2} - 5x + 6 =
y (where it crosses the x-axis will be the roots, if they are real)

**Method 11.** Spiraling in to the intersection of 2 curves

Flash! This just happened (10/26/96): Colleen, a 7th
grader, solved
x^{2} - x - 1 = 0 and got

x = x^{2} - 1. Try iterating this. It's exciting when
something unexpected happens! That's what makes my teaching interesting
and enjoyable. I've spent the last 2 hours working on this in
*Mathematica.*

To some answers to problems above from Ch. 8- part
2, iteration

To problems from Ch. 8 part one-
solving equations

To order Don's
materials

To choose sample problems from
other chapters

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