## Subjects

# Patterns of the times table

True or false? It doesn’t matter in what order you multiply two factors, the product will be the same.

## Patterns of the times table

Multiplication. One number times another number, that is one factor times another factor, and a result which we call a product. All these … Factors, Multiplications, And Products … … can be organised in a nice table, like this. This is the times table. You might have practised it and know it all by heart.

When we look at the table we can find several patterns. Let’s look at five of them. Start at this box. Here is the product 20, that is 4 times 5. And the box 20 is also a corner in a larger rectangle.

How many boxes are there in this larger rectangle? "There are 20! It says 20 in the box, and there are 20 boxes in the larger rectangle. Let’s do the same again, but start at the product 35, that is 7 times 5. This is also a corner in a larger rectangle. This rectangle has … … 35 boxes.

It says 35 in the box, and the rectangle has 35 boxes. The pattern is that we can see a multiplication as a rectangle, where one side is one factor, and the other side is the other factor. Let’s look at the next pattern. The numbers along the very top, the factors, are the same numbers we see on the first row. So, if a number is multiplied by 1, the result is the same number.

The same thing happens down the left. Farthest left are the factors, and in the first column in the table there are exactly the same numbers. Because: If a factor, any factor at all, is multiplied by 1, the product equals the factor we started with. Always. This is true for all numbers.

5 times 1? 5. 648 times 1? 648. 52 311 times 1?

52 311. Next pattern, the third: On this row and in this column we see the 2 times table. 2, 4, 6, 8 … all the way up to 20. Is there a pattern here? Yes, all products are even numbers.

And there are more tables with only even numbers. All products are even in the 4 times table too, and in the 6, 8 and 10 times tables. Even products, in every second row and every second column. The last column and the last row are multiplications with the factor 10. Here, too, is a pattern: when you multiply a whole number by 10, you get the product by adding a zero to the original number.

2 times 10 is 20, 3 times 10 is 30, 4 times 10 is 40, All the way to 10 times 10, which is 100. This goes for all whole numbers. And not only those in this times table. 453 times 10? Simple.

Add a zero, and the product is 4530. Here’s the fifth pattern. Let’s circle all the boxes diagonally across the table. These are all factors times themselves. 1 times 1, 2 times 2, 3 times 3, and so on.

Pairs of products are formed. And in each pair, the two products are the same. A multiplication is mirrored over the diagonal. This is the pattern. The reason is that it doesn’t matter in what order you multiply two factors.

5 times 8 is the same as 8 times 5. 7 times 2 Is the same as 2 times 7. We have practised the times tables from 1 to 10, but why stop there? You can expand the table yourself, by adding the 11 times table, and the 12 times table. Draw a table on a piece of paper and use it to practise.

You can practise the same way as before, and all the patterns we have observed will still be there, no matter how large you make the table. Good luck!