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Adding All Nine

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

Double Digit

Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. Try lots of examples. What happens? Can you explain it?

More Mods

What is the units digit for the number 123^(456) ?

What Is the Question?

Age 11 to 16
Challenge Level

Why do this problem

This problem is based on one of many ways of supporting multiplication. The examples consider numbers from 6 to 9 but does the method work, or can it be modified to work for other numbers? The focus is on the why, rather than any sense that this is a preferable way of calculation. But this is an interesting representation nevertheless and leads to the question, "What sort of person could have thought of this for the first time?".

Possible approach

Rather than show the images on the site it might be good to take a small number of learners in the group into your confidence and ask them to show examples and give the answers without explanation, asking the rest of the group to make sense of the code.

Once the code has been cracked the question has to be asked why it works. This may be something you wish to leave learners to think about over time, returning to the problem if and when any light is shed.

Key questions

Can you find a way of defining the number of fingers you multiply and the numbers you add?

Possible support

Discussion of complements of 10 first might support learners in making the necessary connections.

Possible extension

Could you start at 16 on each hand? What adjustments would you need to make?
How about 26, 36 etc?
Could you use 11 to 15?