The multiplication rectangle
Choose two numbers from 10 to 99. The rectangle is that many squares wide and that many squares tall, so its area is the answer. Each number is split into tens and ones, which cuts the rectangle into four pieces. Match each piece with a line of the long multiplication beside it.
Thin pieces are drawn a little wider than they should be, so their labels fit.
What's going on
A rectangle 47 squares wide and 23 squares tall holds 47 × 23 little squares. Counting them one at a time would take all evening. So cut it up. 47 is 40 and 7. 23 is 20 and 3. Two cuts make four pieces, and each piece is an easy multiplication.
Long multiplication is the same four pieces, added in two steps. The first line, 47 × 3, is the orange row: 40 × 3 and 7 × 3 together. The second line, 47 × 20, is the blue row: 40 × 20 and 7 × 20 together. Add the two rows and you have the whole rectangle.
That is also why you write a zero at the end of the second line. You are multiplying by 20, not by 2.
Sheet: Why long multiplication works.