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sum: 1abcde multiplied by 3 equals abcde1.
sum: 2fghij multiplied by 3 equals fghij2

Can you replace the letters with numbers?
Is there only one solution in each case?

Why do this problem?

This problem requires learners to think about place value and the way that standard column multiplication works. Although the problem can be done by trial and improvement, it is solved more efficiently if worked through systematically.

Possible approach

You could start by showing the first sum to the whole group and discussing what is required to do it. It would be good for learners to work in pairs, perhaps usingĀ this sheet. (The first page contains the first sum with several lighter versions for children to write on. The second page is the second sum done in the same way.) Alternatively, pairs could use digit cards to move around on a sheet of paper, or mini-whiteboard.

Give the children time to make a start and then after a suitable length of time, bring the group back together to talk about how they are getting on so far. This is a good opportunity to share some initial insights. For example, some pairs may have worked out which digit 'e' must be by looking at the units digit of the answer. Some may have started in a different way, for example by trying digits at random to see what happens when the multiplication takes place. Draw attention to those pairs that have adopted a system in their working and highlight that this means the solution will be found more efficiently. You may need to clarify that all the letters appear in the answer too, in other words that 'e' stands for the same number wherever it appears (the beginnings of algebraic notation). You could then leave learners to continue with the problem.

A discussion of all the steps in the reasoning could take place in the plenary.

Key questions

How does the $1$ (or $2$) in the units column of the answer help?
What are the units digits of the multiples of $3$?

Possible extension

All the Digits makes a good extension activity as the mathematics is similar, but slightly more sophisticated reasoning is required.

Possible support

It might help some children to write out their three times table to begin with, or to be able to refer to a multiplication square as they tackle this problem.