Solving an algebra problem that requires integer solutions provides an opportunity to develop students’ skills…
In this lesson, students have to find a solution to a problem where the unknowns must be non-negative integers. This constraint means that the initial problem, which may appear to be quite open, in fact has a unique solution.
Students have the opportunity to consider related Diophantine equations and determine systematically whether they have one solution, more than one solution or no solutions.
Arriving at a point where they are sure about this allows students to see the power of mathematical thinking.
Why teach this?
Many real-world problems require integer solutions. This adds an interesting constraint to students’ equation solving.
Key curriculum links
- Use algebra to generalise the structure of arithmetic, including to formulate mathematical relationships
- Make and test conjectures about patterns and relationships; look for proofs or counterexamples
Starter activity
Look at this puzzle. Try it in pairs.
A shop sells two sizes of boxes. Small boxes cost £5. Large boxes cost £7. I bought some boxes from the shop and spent a total of £41. How many boxes of each size did I buy?
Students will probably not immediately see how to approach this problem. It may look as though there is insufficient information – the problem is under-specified – and there might be many possible combinations of small and large boxes that could work.
In fact, that is not the case. If students are stuck, encourage them to try some numbers and see what happens.
Colin Foster is an Associate Professor in the School of Education at the University of Leicester. He has written many books and articles for mathematics teachers. Visit his website at foster77.co.uk and follow on X at @colinfoster77. Browse more KS3 algebra resources.
