Geometric representation of exponents

We are working on a new understanding of exponents. Today we are building them with toothpicks and plasticine. First: X^2

an X^2 is actually going to be a square, with side lengths of x, just like a square meter is a square with side lengths of one meter.Here are 2 squares, and each one has dimensions of x, so this is 2(x)^2Here is a bigger square. The dimensions are 3x, so we call this (3x)^2.

Another way to look at this is to notice that it can be subdivided into pieces of x^2 and that there are 9 of them. We also know that (3x)^2 is the same as (3x)(3x) which is the same as (3)(x)(3)(x) which is the same as (3)(3)(x)(x) which is the same as 9x^2.

This one is in 3 dimensions, a cube with side length x, so it is x^3.

For many of us, this is a new way to explore and express exponents.