Digital roots provide some good opportunities for finding patterns. They also reinforce the children’s knowledge of their multiplication tables and can be used to create some visual patterns.
A worksheet with full instructions can be found below. This involves the children working out the digital roots of the numbers in each of the multiplication tables, starting with the one times table, and progressing to the twelve times table.
The following table shows the digital roots of each of the multipication calculations. Some of the lines have similar patterns. Can you spot them?
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 | 3 |
| 2 | 2 | 4 | 6 | 8 | 1 | 3 | 5 | 7 | 9 | 2 | 4 | 6 |
| 3 | 3 | 6 | 9 | 3 | 6 | 9 | 3 | 6 | 9 | 3 | 6 | 9 |
| 4 | 4 | 8 | 3 | 7 | 2 | 6 | 1 | 5 | 9 | 4 | 8 | 3 |
| 5 | 5 | 1 | 6 | 2 | 7 | 3 | 8 | 4 | 9 | 5 | 1 | 6 |
| 6 | 6 | 3 | 9 | 6 | 3 | 9 | 6 | 3 | 9 | 6 | 3 | 9 |
| 7 | 7 | 5 | 3 | 1 | 8 | 6 | 4 | 2 | 9 | 7 | 5 | 3 |
| 8 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 9 | 8 | 7 | 6 |
| 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
| 10 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 | 3 |
| 11 | 2 | 4 | 6 | 8 | 1 | 3 | 5 | 7 | 9 | 2 | 4 | 6 |
| 12 | 3 | 6 | 9 | 3 | 6 | 9 | 3 | 6 | 9 | 3 | 6 | 9 |
When you have worked out the digital roots of each of the multiplication sums, you can make some interesting visual patterns.
All that you need is the following diagram. You will need one for each multiplication table that you want to draw the pattern for. A page of these diagrams (with six on each page) can be found below, for you to print and photocopy.
Choose your multiplication table (e.g. 5 times table), and make a list of the digital roots (if you haven’t done this already):
5 1 6 2 7 3 8 4 9 5 1 6
Look at the first number – find this on the circle diagram. Now, draw a line from this number to the second number in your list:
Now draw a line from the second number in the list to the third number:
Draw lines from the third number to the fourth, and so on. Continue doing this, until you have reached the end of your list:
The patterns for all of the multiplication tables are shown below:
1, 8 and 10 Times Table:  | 2, 7 and 11 Times Table:  |
3, 6 and 12 Times Table:  | 4 and 5 Times Table:  |
9 Times Table:  |
Here is a display based on this activity.
A variation contributed by Caroline Searle…
As well as the patterns described, other patterns can also be made using these numbers. On squared paper, draw a line the length of the first number in the sequence, then turn the paper 90 degrees to the right, then carry on drawing the next line the length of the next number. Turn 90 degrees to the right again and draw the next line the length of the third number in the sequence. Keep turning and drawing the lines and you will come up with some fantastic patterns. This is also a good activity to do on Logo.
