Build the Rectangle

You already trust one fact from geometry: a rectangle's area is its width times its height. So to multiply two polynomials, build the rectangle — width one factor, height the other. Its area is the product, before a single calculation. Cut the sides into their terms, and the rectangle falls into pieces you can measure one at a time. Do that with two binomials and something familiar appears in the grid: FOIL turns out to be nothing but the order you read four little rectangles.