Cut the rectangle

About this video

A multiplication is a rectangle: 7 × 12 is a tray of buns in 7 rows of 12. Not many people know 7 × 12 by heart, but this short animation shows how to work it out from sums that are easy. Draw a line down the tray after the tenth column. Now there are two smaller rectangles, 7 rows of 10 and 7 rows of 2, which make 70 and 14. No bun was lost and none was added, so 7 × 12 = 70 + 14 = 84. This is breaking a number apart, which mathematicians call the distributive law.

The cut can go anywhere. Cut the twelve columns into six and six and you get 42 + 42, the same 84, because it is the same tray. So you choose the cut that makes the sums easy, and tens are usually easiest. The video ends with three numbers: two trays of 3 rows of 4 can be counted as two twelves or, stacked up, as six rows of four. Either way it is 24, so three numbers can be multiplied in any grouping. This video goes with the Breaking Numbers Apart lesson of Number Theory Level 1 on Knaph.

Transcript

1. 0:00Seven rows of twelve. How many?
2. 0:04Cut it after ten. Seven tens, and seven twos.
3. 0:10Seventy and fourteen make eighty-four.
4. 0:15Cut it anywhere. Six and six. Still eighty-four.
5. 0:21Two trays of twelve, or six rows of four. Twenty-four.
6. 0:27Breaking a number apart loses nothing.