56 appears twice: 7 × 8 and 8 × 7. 24 appears 6 times: 2 × 12, 3 × 8, 4 × 6, 6 × 4, 8 × 3 and 12 × 2. Each one is a factor pair of 24 with both numbers no bigger than 12, counted both ways round.

The 7 row is 7 × 1 to 7 × 12 and the 7 column is 1 × 7 to 12 × 7: the same 12 products, once each way round. They meet at 7 × 7 = 49, so the cross has 23 cells, not 24.

Both are 21. Folding along the diagonal puts every a × b on top of b × a, and they always match: multiplication is commutative. Addition is too, but subtraction and division are not (7 − 3 ≠ 3 − 7).

The digit sums are all 9 up to 9 × 10 = 90. Then 9 × 11 = 99 gives 18, and 1 + 8 = 9 again. The digit sum of any multiple of 9 is a multiple of 9.

The 1, 2, 5, 10 and 11 rows usually come instantly. Listen for derived facts: “7 × 8 is 7 × 4 doubled” or “9 × 6 is 10 × 6 take away 6”. Counting up in sevens is the strategy to move pupils away from.

12 squares, and 12 is written in the corner cell. Every product on the grid is the number of squares in the array that ends at that cell, which is why a multiplication grid is also an area model.

44 products are multiples of 5: the 23 in the 5 cross plus 21 more in the 10 row and column. 5 is prime, so a product is a multiple of 5 only when one of the two numbers is. 7 × 8 = 56 can’t be.

No. 60 products are multiples of 6, but the 6 and 12 crosses hold only 44. 2 × 3, 3 × 4 and 4 × 9 are multiples of 6 because the 2 and the 3 can come from different numbers. That can’t happen with a prime like 5.

The crosses share only 4 × 6 and 6 × 4 (both 24). The products that are multiples of 4 and of 6 are the multiples of 12: 40 cells. Many pupils expect multiples of 24, because 4 × 6 = 24, but 4 and 6 share a factor of 2, so the lowest common multiple is 12.

The repeated digits stop at 11 × 10 = 110. 11 × 11 = 121 and 11 × 12 = 132: the middle digit is the sum of the outer two. The same trick gives 11 × 13 = 143 and 11 × 45 = 495.

Multiplying by 1 leaves a number unchanged: 1 is the identity for multiplication. On the 0 to 12 grid the whole 0 row and 0 column are 0: 25 of the 169 cells.

33 products end in 0. The 10 row and column are full. The others are even numbers times 5 (or times 15, 25… on bigger grids): 5 × 2 × k = 10 × k.

Each row is double the one before: 8 × 7 = 2 × 2 × 2 × 7. Doubling three times is a reliable route to the 8 times table. The 6 row is double the 3 row, and the 12 row is double the 6 row.

The 6 and 12 rows are all multiples of 6 (12 each). The 3 and 9 rows have 6 (the even columns), the 2, 4, 8 and 10 rows have 4 (columns 3, 6, 9, 12), and the 1, 5, 7 and 11 rows have only 2 (columns 6 and 12). A row needs whatever part of 6 it’s missing from the column.

All square numbers, because the row and column are the same number. The gaps are 3, 5, 7, 9…: the odd numbers in order.

22 cells are squares: 12 on the diagonal and 10 off it (1 × 4, 1 × 9, 2 × 8, 3 × 12, 4 × 9 and their turnarounds). The factors pair up: 2 × 8 = 4 × 4 and 3 × 12 = 6 × 6.

78, not 144: the 12 on the diagonal plus half of the other 132, because 7 × 8 and 8 × 7 are one fact. Leaving out the 1, 2, 5 and 10 tables leaves 36.

27. The turnarounds of rows 1 to 3 are columns 1 to 3, and 9 of those cells (3 × 3) are already answered. 36 answers give 27 more for free.

The 12 “row = column” cells are scattered now, but they are still 1, 4, 9 … 144. The diagonal means the row number equals the column number. It only looks like a line when the headers are in the same order.

24 is in both: it is in the 4 row (4 × 6) and in the 6 column. Every product is in the times tables of both its row and its column. Diagonal cells like 6 × 6 are in only one.

108 of 144 are even (3/4) and 36 are odd (1/4). A product is odd only when both numbers are odd. Half the rows are odd and half the columns are, so 1/2 × 1/2 = 1/4 of the cells.

Rows and columns 1, 3, 5, 7, 9 and 11, and both numbers must be odd. The odd products make a 6 × 6 pattern spread over the grid. One even number anywhere makes the product even.

25 on 10 × 10 and 36 on 11 × 11 (the same as on 12 × 12). For an even n it’s (n ÷ 2)², and for an odd n it’s ((n + 1) ÷ 2)².

No and no: 9 and 15 are odd multiples of 3, and 4 and 8 are even but not multiples of 3. The products in both are the multiples of 6: 60 cells.

16/25 (64%), 27/36 (75%), 33/49 (about 67%), 48/64 (75%). Even-sized grids are exactly 3/4 even. Odd-sized grids are below 3/4 and creep towards it.

10, all in the 1 row or column: 2, 3, 5, 7 and 11, each twice. A product of two numbers that are both 2 or more has those numbers as factors, so it can’t be prime.

Every prime from 13 to 139: 29 of them. A prime p only appears as 1 × p, so it needs p to be 12 or less.

5 times: 3 × 12, 4 × 9, 6 × 6, 9 × 4 and 12 × 3. 6 × 6 appears once because it is its own turnaround, so square numbers appear an odd number of times. 1 × 36 and 2 × 18 are missing because 36 and 18 are off the edge.

12 and 24 tie, with 6 each. 36 is next with 5. The winners have lots of factor pairs where both numbers are 12 or less.

48 = 1 × 48 = 2 × 24 = 3 × 16 = 4 × 12 = 6 × 8. Only 4 × 12 and 6 × 8 fit on the grid, and each appears both ways round: 4 times.

59, well under 72. Turnarounds alone bring 144 down to 78, and repeats like 12 = 2 × 6 = 3 × 4 bring it down further.

85 numbers never appear: more than half. They include every prime from 13 upwards, and numbers like 26 = 2 × 13 or 125 = 5 × 25 whose factor pairs all need a number bigger than 12.

8 cells: 1 once, 8 four times (1 × 8, 2 × 4, 4 × 2, 8 × 1), 27 twice (3 × 9, 9 × 3) and 64 once (8 × 8). 8 has more factor pairs that fit on the grid. 125 = 5 × 25 doesn’t fit.

24 squares both ways: 6 rows of 4, then 4 rows of 6. Turning an array round is the reason a × b = b × a, and pupils can draw it for any pair of numbers.

7 × 8 = 7 × 5 + 7 × 3 = 35 + 21 = 56. Splitting after 4 (28 + 28) or after 2 (14 + 42) works too. This is the distributive law, and it is how pupils work out facts they don’t know from ones they do.

(5 + 1) × (5 + 2) = 25 + 10 + 5 + 2 = 42: four parts, one for each pair. A common error is to multiply only 5 × 5 and 1 × 2, which gives 27. The array shows why the other two parts count.

100 + 40 + 30 + 12 = 182. The table is the grid method for long multiplication, and the array shows where each part comes from. It also explains why 13 × 14 isn’t 10 × 10 + 3 × 4.

36 = 25 + 5 + 5 + 1, so 11 more. Going from n × n to (n + 1) × (n + 1) adds 2n + 1, so 7 × 7 is 13 more than 6 × 6. That is why the gaps between square numbers are the odd numbers.

24. Pupils who count in ones can be nudged to count rows of 6 (6, 12, 18, 24) or columns of 4.

2 × 12, 3 × 8, 4 × 6, 6 × 4, 8 × 3 and 12 × 2. 1 × 24 doesn’t fit. The corners lie on a curve: as one side doubles, the other halves.

9 × 6 is one row of 6 fewer than 10 × 6: 60 − 6 = 54. Split after 9, the array shows 9 × 6 + 1 × 6 = 60. Nearly-ten facts (9 × and 11 ×) are quick this way.

It goes up four times, from 12 to 48. The 6 × 8 array holds four copies of the 3 × 4 array. Many pupils expect the product to double.

The facts pupils hesitate on are the ones to work on. Each can be found from a neighbour: 6 × 8 is 6 × 4 doubled, 7 × 9 is 7 × 10 − 7, and 8 × 7 is 7 × 8 turned round.

Only 3: 7 × 6, 8 × 6 and 9 × 6. A turnaround only fills in when that cell is on the grid, and rows 1 to 5 and 10 to 12 aren’t shown here.

You can’t count along a row any more, so each fact has to be recalled or worked out on its own. Every product in a row is still a multiple of that row’s number, so it’s a good check.

25 of the 169 cells are 0: the whole 0 row and 0 column. 0 × 7 means seven lots of nothing (or nothing lots of 7), which is 0. Pupils who write 7 are usually thinking of adding.

The answer page has the same mixed-up headers. Most pupils fill the 1, 2, 5 and 10 rows and columns first. That is a good habit: easy facts first, then work the rest out from them.

Neighbours in the 7 row differ by 7, because each step adds one more 7. In a column they differ by the column number. Only in the 1 row and column do neighbours differ by 1.

47 products are more than 50. The edge follows row × column = 50, which is a curve: if you double the row, you only need half the column. Straight diagonals are where row + column stays the same.

12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12. Each pair adds up to 13, and two numbers with a fixed total have the biggest product when they are as close as possible (6 × 7). It is why a square gives the most area for a given perimeter.

18 products end in 6. The 5 row ends only in 5 or 0, and the 10 row only in 0. Odd last digits are rare: 1 (7 times), 3, 7 and 9 (6 times each), because both numbers must be odd. 0 is the most common (33).

The 1 row is 78, the 2 row is 156, so the 7 row is 7 × 78 = 546. The whole grid is 78 × 78 = 6084, because it is (1 + 2 + … + 12) × (1 + 2 + … + 12).

12 × 20 = 240 and 15 × 16 = 240. The diagonals always match, because each is the same four header numbers multiplied in a different order: (3 × 4) × (4 × 5) = (3 × 5) × (4 × 4).

Not with 0 or 1: 0 × 9 = 0 and 1 × 9 = 9. With numbers between 0 and 1 (½ × 8 = 4) multiplying makes things smaller, so this belief causes trouble later on.

54 is 6 × 9 and sits in the 6 and 9 rows. 7 × 8 must be a multiple of 8, and 54 ÷ 8 isn’t a whole number. From 7 × 7 = 49, one more 7 makes 56.

They are equal. Halving one number and doubling the other keeps the product the same: 2 × 12 = 4 × 6 = 8 × 3 = 24. The six 24s lie on a curve through the grid.

It is four times as big: 48 = 4 × 12. Doubling both numbers doubles the product twice. The split shows four copies of the 3 × 4 array.

1 = 1 × 1, the top-left cell of the diagonal, so it is a square. It is also a cube (1 × 1 × 1). It is not prime, because it has only one factor.

19 on 10 × 10 and 13 on 7 × 7. On an n × n grid it’s always 2n − 1: a row and a column of n that share one cell.

14 of 25 (56%), 42 of 100 (42%), 59 of 144 (about 41%), 152 of 400 (38%). The fraction keeps falling, because bigger grids have more repeats. Mathematicians still study how fast it falls.

The 1, 2 and 3 times tables to 12: 36 products. Rows 1 to 12 and columns 1 to 3 show the same 36 products, turned round.

25 different abundant numbers appear (73 cells), from 12 to 144. All of them are even, so none of the 36 odd products is abundant, and the 1 row has only 12. Numbers with lots of small factors are the ones most likely to be abundant.

Strong questions ask for a prediction before the shading is switched on, then an explanation using factors, turnarounds or arrays. Shared links open the tool exactly as it was set up.