Multiplication Chart

A multiplication chart is a grid of products. The numbers 1 to 12 run down the left edge and across the top, and the cell where a row meets a column holds their product. To find 7 x 8, go down to the 7 row and across to the 8 column: the cell says 56. Each row is one of the times tables written out in full. The 7 row reads 7, 14, 21, 28 and on, rising by 7 at every step, so any cell can be checked by adding the row number to the cell before it. A times table chart and a multiplication table are the same grid under other names.

Multiplication is commutative: 7 x 8 and 8 x 7 are both 56. On the chart that shows up as a mirror. Fold the grid along the diagonal from the top left corner to the bottom right and every cell lands on its twin, so only the squares on the diagonal appear once. An n x n chart holds n(n + 1)/2 facts: 55 for 10 x 10, 78 for 12 x 12, 120 for 15 x 15 and 210 for 20 x 20. The answers repeat even more often: 24 and 36 can each be made three ways in a 12 x 12 chart.

The square numbers run down the diagonal: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Each one is the last plus the next odd number (4 + 5 = 9, 9 + 7 = 16), because (n + 1)^2 - n^2 = 2n + 1. The squares also anchor the cells beside them. A fact with consecutive factors is n^2 + n, so 7 x 8 = 49 + 7 = 56. A fact whose factors differ by 2 is one less than the square between them, so 7 x 9 = 64 - 1 = 63.

Five rows have patterns that make them quick. The 2s are doubles: 2 x 8 is 8 + 8 = 16, and every product in the row is even. The 10s add a zero: 10 x 7 = 70. The 5s are half the 10s: 5 x 8 is half of 80, which is 40, and a 5s product ends in 0 when the other factor is even and in 5 when it is odd. The 9s are the 10s minus one copy: 9 x 7 = 70 - 7 = 63. From 9 x 1 to 9 x 10 the tens digit is one less than the other factor and the two digits add to 9 (63 has 6 tens, and 6 + 3 = 9). Every multiple of 9 still has a digit sum divisible by 9, which makes a quick check: 9 x 12 = 108, and 1 + 0 + 8 = 9. The 11s repeat the digit from 11 x 1 to 11 x 9: 11, 22, 33 up to 99. For a two-digit number, write its digits apart and put their sum in the middle: 11 x 12 is 1, 1 + 2, 2, which gives 132. When the sum reaches 10, carry the 1: 11 x 57 is 5, 12, 7, which becomes 627.

A sensible learning order starts with the 1s, 2s, 5s and 10s. A fact with any of those as a factor is then known, which covers 42 of the 78. Add the 9s and 11s and 57 are done, leaving 21. Six of those are squares (9, 16, 36, 49, 64 and 144), so 15 remain, and the 3s and 4s account for 9 of them. The last six are 6 x 7 = 42, 6 x 8 = 48, 6 x 12 = 72, 7 x 8 = 56, 7 x 12 = 84 and 8 x 12 = 96. Each one can be built from facts already learned: 7 x 8 is 7 x 5 + 7 x 3 = 35 + 21, and 7 x 12 is 70 + 14. In the US, the Common Core standards (3.OA.C.7) expect students to know all products of two one-digit numbers from memory by the end of grade 3. England's national curriculum asks for recall of the times tables up to 12 x 12 by the end of year 4.

The chart stops at 12 or 20, and larger products come from the distributive property: split a factor by place value and multiply each part. 13 x 7 is 10 x 7 + 3 x 7 = 70 + 21 = 91. For 23 x 47, split both factors: 20 x 40 = 800, 20 x 7 = 140, 3 x 40 = 120 and 3 x 7 = 21, which add to 1,081. That is the area model taught in school, and long multiplication writes the same four partial products in two lines (141 and 940). Each partial product is a one-digit fact with zeros attached, which is why the facts up to 9 x 9 are enough to multiply numbers of any size.