Cut a square. On squared paper, draw a square 7 by 7. Cut a strip 2 squares wide off one side, turn it, and lay it along the bottom. The strip is 7 long but the bottom is only 5 wide, so a 2 by 2 piece sticks out. The rest is a 5 by 9 rectangle. That is why 7 × 7 = 5 × 9 + 2 × 2.

Before chapter 4. Danya rounds 47 to 50 and says 2500. Why is he wrong?

The answer. Rounding and stopping gives 291 too many. Squarl goes up 3 and down 3: 44 × 50 = 2200, then adds back 3 × 3 = 9, which gives 2209. When you move the strip, it is 3 squares too long, so a 3 by 3 piece sticks out past the end. The same works for 48 (46 × 50 + 4) and 53 (56 × 50 + 9).

The grown-up layer. This is the difference of two squares, (a + b)(a − b) = a² − b². The area model makes it a picture before it is a formula; IdentityForge builds the same picture.

Pause to think. Why did Danya think "close enough" was fine?

For the ride home. Ask: what is 21 × 19? (Try 20 × 20 first.)