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Trebling


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?

Once you've had a chance to think about it, click below to see how two different pupils began working on the task.

Here is Abdullah's work:

"For each problem I first looked to find a number that would make the units column accurate, then I substituted the number for the answer in the tens column and then continued the process until the calculation was complete."


 

Joshua wrote:

"I wrote out single digit multiples of three up to 9 because each letter was one digit. I noticed that the numbers 1 to 9 only appeared once in the units column of the answers. I looked at the question and realised that 3 x e had to be 21 because it was the only answer ending in 1. This meant that e had to be 7.

I carried the 2 and took it from 7 (the other e) and got 5. So d x 3 had to end in 5 which meant d had to be 5 because 5 x 3 = 15. I then repeated the process."


 

Can you take each of these starting ideas and develop it into a solution?


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.  The richness of the activity comes in the different approaches which could be used to solve it and discussion of these different methods is emphasised in these notes.  You may well need to spend a couple of lessons on this activity.

Possible approach

You could start by showing the first calculation to the whole group.  Give them a few minutes to consider, individually, how they might go about tackling the problem, then pair them up and suggest that they talk to their partner about their ideas so far.  Try to stand back and observe, and resist the temptation to make helpful suggestions!
 
Allow pairs to work on the task so that you feel they have made some progress, but do not worry if they have not completed it or if they report being stuck.  Learners may find this sheet useful for recording. (The first page contains the first calculation with several lighter versions for children to write on. The second page is the second calculation done in the same way.) Alternatively, pairs could use digit cards to move around on a sheet of paper, or mini-whiteboard.  The aim at this stage is for everyone to 'get into' the problem and work hard on trying to solve it, but not necessarily to achieve a final solution.  

At a suitable time, hand out this Word document or this pdf to pairs.  Suggest to the class that when they've finished or can't make any further progress, they should look at the sheet showing two approaches used by children working on this task.  Pose the question, "What might each do next? Can you take each of their starting ideas and develop them into a solution?".  You may like pairs to record their work on large sheets of paper, which might be more easily shared with the rest of the class in the plenary.

Allow at least fifteen minutes for a final discussion.  Invite some pairs to explain how the two different methods might be continued.  You may find that some members of the class used completely different approaches when they worked on the task to begin with, so ask them to share their methods too.  You can then facilitate a discussion about the advantages and disadvantages of each.  Which way would they choose to use if they were presented with a similar task in the future? Why?

Key questions

How does the $1$ (or $2$) in the ones 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.