Turning an array

1. This is an array. Rows of squares, and every row the same length. Here there are three rows with eight in each row. You never have to count the squares one at a time. Count ONE row. Then count how many rows there are.
2. Three rows of eight is 24. Count on in eights, once for each row: 8, 16, 24. Twenty-four squares altogether. That is what 3 × 8 means. Three eights.
3. Now turn it. A quarter turn. Watch carefully. Nothing was added and nothing was taken away. These are the same twenty-four squares. But now they read as eight rows with three in each row.
4. Eight rows of three is 24 as well. Count on in threes this time: 3, 6, 9, 12, 15, 18, 21, 24. The same twenty-four. So 3 × 8 and 8 × 3 describe one pile of squares. They cannot come out different.
5. Your turn. Make 4 rows of 6. Drag across the squares to change the array. Make it 4 rows with 6 in each row. Watch what the total does on the way.
6. It is not a trick of that one array. Here is a different one: four rows with seven in each row. Four sevens. 7, 14, 21, 28. Twenty-eight squares.
7. Turn it too. Seven rows with four in each row. Twenty-eight either way. This works for every array there is. Turning something cannot change how many squares are in it.
8. So you only have to learn half the table. Every fact comes with a twin. Learn 4 × 7 = 28 and you get 7 × 4 = 28 for nothing. That is true of every pair in the whole times table.
9. Your turn. Now make 6 rows of 4. Build the turn of the one you just made: 6 rows with 4 in each. The total will be the same 24. Turning an array never adds or takes away a square.
10. Last check. Here is an array of 6 rows with 9 in each row. You know 6 × 9 = 54. What is 9 × 6? 54 · You have to work it out. Yes. It is the same array turned round, so the total is the same.

Turning an array cannot add a square or lose one. So 3 × 8 and 8 × 3 have to be equal. The same goes for every pair. Learn one of each twin and the other is free.