1-x-x^2

The inverse of 1-x² can be calcuated two ways.

First way: We know 1/(1-y)=(1+y+y²+y³+...) so substitute y=2x, and it comes out.

Second way: Do it directly.

1. The constant coefficent must be 1.
2. The x coefficient must 2.
3. The x² coefficient must 4.
4. The x³ coefficient must 8.

We did a couple more examples of computing an inverse. Then we went to an interesting one.

What is the multiplicative inverse of 1-x-x².

It turns out to be 1+x+2x²+3x³+5x⁴+8x⁵+13x⁶...

The coefficients are the fibonacci series!

1/(1-x)

Treating polynomials as abstract symbolics entities (rather than as a formula into which you substitute a real number for x. Q: What do you get when you multiply (1-x) by (1+x+x²+x³+...) ? A: You get 1. Thus 1/(1-x)=(1+x+x²+x³+...).

5-minute math for a 3-year old

My 3-year old child wanted to do five-minute math. I said, OK, I grabbed the marker, and she said "write a 3". So I wrote a 3. Then she said "write a 2". So I wrote a 2. We spent five minutes with her dictating numbers and I filled up the white board.