Patterns With Nines

Tim Butterworth
Mt. Faulkner Primary School

I thought the maths we did was really cool and it was fun doing the work.
I enjoyed it lots. I learnt heaps. It's cool as!

We looked at the patterns we could make with nines. First we counted by nines from 9 to 108. We found there was a pattern with the units decreasing by one and the tens increasing by one for each multiple of nine. When we reached 45, we found that if we reversed the digits then we had the next multiple of nine - 54. We discovered we could then very easily find the next four multiples by reversing the digits of the previous multiples (45/54, 36/63, 27/72, 18/81, 9/90).

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108

After listing the multiples of nine up to 108, we added the digits of each multiple and discovered that each answer was equal to 9 (with 99; 9 + 9 = 18; then 1 + 8 = 9).

We then tested the theory that if we multiplied any number by nine, then the sum of the digits should always equal nine.
1 + 0 + 0 + 9 + 1 + 7 = 18
1 + 8 = 9

We then explored a problem to look at what happened when we multiplied a certain number by multiples of nine.
The number we used was 12 345 679 (8 is not included).
This is what happened.
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 9

It was easy to predict what each digit would be after we had multiplied the first two or three digits and to predict what the number carried would be.

We then multiplied the same number by 18 ...
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 18
1 + 8 = 9

I then predicted what the answer would be when we multiplied the same number by 27 (3 x 9). I predicted that the answer would equal 333 333 333. I then wanted to see if there was a pattern as we multiplied...

When I multiplied 12 345 679 by the 7, there was a pattern of odd numbers, from 3, going up to 9 and then back to 1 (I predicted 11) and then even numbers from 4 to 8. This pattern was not as easy to predict. I was then able to predict the answers if I multiplied the number 12 345 679 by other multiples of 9.