I'm thinking of a machine or rule. You give me a number (input), I put your number in my machine or use my rule on your number, then I give you a number back (output). I always do the same thing to the number you give me. Your job is to figure out how my machine works or what my rule is doing to your number.

So if you give me 1, I tell you 5. You give me 2, my machine gives out 7 and so on. We'll put the numbers in a table like this:

input output

x y

1 5

2 7

3 9

4 11

10 23

Can you guess my rule? Guessing the rule also means finding the output for any
input number, say 100. Can you do that? Can you write a rule with x in it to =y
?

You make up a rule and send it to me.

To send a rule to Don

**Graph 2x + 3 = y**. Find pairs of number that make this sentence true.
(Don't forget 0 for x)- put them in the table like that below. Look for
patterns. Is there a pattern in the dots? in the numbers? What would happen if
you changed the 3 to a 5? a 6? Graph these on the same graph paper. What do you
see?

What happens if you change the 2 in the original equation to a 3? What do the
two numbers in the equation tell you about the graph?

Now we'll get some functions from different things. The first is the shuttle puzzle.

The object of this puzzle is to interchange the blue and the red pegs (golf
tees). The rules are 1) you can move to a hole that's next to a peg; 2) you can
jump, but **only one** peg and it must be of the **other color**, and 3)
you can't move backwards. You must start with the empty space in the middle and
end that way. You can use golf tees as I do or two different kinds of coins or
bottle caps or pieces of colored paper as the pieces. Try it. It's not easy.

If you have difficulty, try it with 2 on each side of the space in the
middle. Then you can't use the 2 holes on the outside on each side. **Sometimes
when a problem is hard, make up a simpler one, do that, then go to the harder
one.**

When you can interchange 1, 2, 3 and 4 pairs, then make a table like the one
shown above. Fill in the number of **pairs** of pegs and count the number of
moves it takes to interchange the pegs. Put those numbers in the table. Then
find a rule to relate x to y. Graph these pairs of numbers.

**4. The Tower Puzzle**

The object of this puzzle is to move the pile of discs (4 shown here) from any
one peg to either of the other two. The rules are 1) you can only move one disc
at a time; 2) You can't put a bigger disc on top of a smaller one. Look for
patterns.

When you get good at moving 6 discs, then move the discs in the

Find a rule relating these numbers in the table.

Graph these pairs of numbers.

Seashell World - http://www.seashellworld.com

An environmentally educational shopping experience. Exotic sea shells, starfish, sand dollars and unique seashell novelty items. Fun for collectors and crafters alike. Dive on in!

See also:

the IES Java applet of the shell at http://www.ies.co.jp/math/java/misc/oum/oum.html

and Xah Lee's work on the shell at http://xahlee.org/SpecialPlaneCurves_dir/EquiangularSpiral_dir/equiangularSpiral.html

To some answers from ch. 6 problems

To download Don's materials

To choose sample problems from other chapters

Mathman home