**Geoffrey, going into 5th
grade, graphed x = y, x ^{2} = y, x^{3}
= y, and x^{4} = y from x = ^{-}1
to 1 by 0.1's**

Geoffrey had figured out the rule for the shuttle
puzzle P(P+2)=M, and Don asked him to graph this function as x(x+2)=y. He got a
parabola. They looked at the pattern in the parabola- from (0,0) it goes 1
right, 1 up, 1 right then 3 up, 1 right then 5 up, and continued going up the
odd numbers. Don then asked Geoffrey to graph x = y, x^{2} =
y, x^{3} = y, and x^{4} = y from x = ^{-}1
to 1 by .1's, on the same graph paper. He did this below:

Then Don asked him to write about what he found in the graphs.

"When x is negative and y = x raised to an odd power, both
x and y are negative because if you multiply ^{-}.9
by ^{-}.9 by ^{-}.9
you have to get a negative number because a negative times a negative is a
positive , and a negative times a positive is a negative.

When x^{4 }=y the graph
looks more like half a square because when you multiply a number between 0 and 1
and you make it to the fourth power it gets much smaller because it's kind of
like dividing because .1^{2}=.1 x .1=.01".

Geoffrey worked on moving the
parabola y=x^{2}
up 2 units (he found this equation to be y= x^{2}
+2) and to the right 3 units (he found this equation to be y= (x
- 3)^{2 }).

**Fine job
Geoffrey!**

Geoffrey has been working on the
SSAT test off and on, getting ready for the test next year. Don showed Geoffrey
how to multiply 12x13 in his head, and ended up multiplying 22x23 in his head.
Don also showed geoffrey how to square 2 numbers the end in 5, like 25x25
= ? Well the answer has a 25 on the right _ 2 5 . Take the
other number 2, add 1, to get 3, then multiply by 2 by 3 to get 6. The answer to
25x25 = 625. In the process of doing something in school he talked about 5^{8}
= 625^{2} and he proceeded to multiply 625x625 in his head!

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