Digit discovery

Why do this problem?

This problem provides an engaging context in which learners can explore whether the sum or the product of two numbers is larger, and how this depends on the numbers in question. 

Possible approach

Begin by introducing just the first part of the task - what could the two pairs of numbers be, if one pair of numbers multiplies together to give the sum of the other pair of numbers? Allow some time for pupils to work on this.

Before the next part, it may be beneficial to go over the definitions of the words 'sum' and 'product' as a whole class. Introduce the second part of the task and make sure that everybody understands what is being said - when you multiply Rafe's two numbers together, you get the sum of Jasmine's numbers, and vice versa. You might like to give some examples with numbers, but make sure not to give away any solutions!

Provide some time for pupils to work in pairs on this activity. Make it clear that there are two goals of the activity: to find some solutions that satisfy all the conditions, and to be sure that all possible solutions have been found. (Some children might find the solutions very quickly, and will then be focusing on ruling out other possibilities or reasoning about which solutions can't possibly work.) 

Let the children decide how to record their working. You might like to make digit cards available for learners to move around, but duplicate numbers will need to be provided as there's no guarantee that the solution won't involve the same digit multiple times.

After a while, stop the class and share some different ways of working that different pupils have been using. Try to avoid sharing any solutions that have been found (although children will naturally want to share these with each other!) but instead focus on ways of working systematically to check all the possibilities. Are there any ways of narrowing down which pairs of digits to check? Could one pair of digits be 9 and 9, given that the product of these numbers would be 81? Children may have noticed that some pairs of numbers have a sum that can't be factorised using the digits 0-9 - is there any pattern to which pairs of numbers these are?

At the end of the session, bring the class back together and ask pupils to share how they got on with the activity. How can we be sure that there aren't any more solutions?

You might like to spend another session on the extension activity, or you could introduce it as a 'simmering' activity (where pupils could think about it over the course of a week or so, and add their thoughts to a working wall of ideas).

Key questions

What is the sum of those numbers? What is the product?

Is there a different pair of numbers that add up to the product of the first pair? What would the product of that second pair be?

Is the product of two numbers generally bigger or smaller than the sum of the same two numbers? Does this depend on the numbers you're using?

How do we know we don't need to check that pair of numbers? What is it about their sum/product that means it won't be possible to find a solution that includes them?

Possible support

Some children will benefit from having digit cards to move around, and it might be helpful to have times tables grids available so that learners don't get stuck at the multiplication stage of the task.

Possible extension

Children might like to consider what would happen if the cards were allowed to contain numbers of any size rather than just numbers from 0 to 9. Are there any more solutions in that case? (There is an interesting solution when negative numbers are allowed!)

Learners could also focus on coming up with a convincing argument for why there aren't any other possible solutions. They might like to write up their ideas or create a poster going through some of the different things they've learnt in the process of working on this activity, explaining why they think they've found all the possible solutions.

The extension at the bottom of the problem page could also be used as a follow-on for the main activity.