**Geoboard Magic**

by Maggie- a 7th grader, Patricia-a 9th grader and Paul-a 6th grader

I started with making a shape with 4 touching nails on the geoboard, 1 unit of area.

I had to figure out a rule for finding the area (y) inside the shape, with different numbers of nails touching the rubberband (x) and the number of nails inside the figure (i).

I wrote down how many units of area would be inside a figure depending on x and made a table, first for no nails inside:

I found patterns in the table, then I wrote the rule as

x - (x+2)/2 = y

Both Patricia and Paul graphed the pairs of numbers (x, y) from their tables, for each i.

They then were able to pick off the slope (1/2 for each one) and the intercept, easily, and write the rules more simply as

x/2 + ^{-}1
= y , when i=0

x/2 + 0 = y , when i=1

x/2 + 1 = y , when i=2, and so on.

The rule with i was simple to see, because the adding number was 1 less than i. They wrote the 3-variable rule as

x/2 + i - 1 = y (also known as Pick's theorem)

They checked this rule with the data they had and it worked!

Don helped Paul, then Maggie, make the 3-D graph of the area y in terms of x and i using the table above. The 3-D graphing board they used was a peg-board and dowel rods, marked off in units. The points were made with rubber bands. This 3-D graphing board was invented by Theo when he was in 5th grade- the same person who years later helped Don with his book "Changing Shapes With Matrices"! It looked like this:

**3D graph **of
the area y (vertical axis) in terms of x, the number of nails touching the
rubber band (coming towards you) and i the number of nails inside (going
to the right). Check out DPGraph.com to graph in 3D on
line- great!

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