Challenge Level

This problem supports the development of the idea of generic proof with the children. This is a tricky concept to grasp but it draws attention to mathematical structures that are not often addressed at primary school level. Generic proof involves examining one example in detail to identify structures that will prove the general result. It is possible that only very few children in the class may grasp the idea but this is still a worthwhile activity which provides opportunities for children to explore odd and even numbers and the relationship between them. Proof is a fundamental idea in mathematics and in helping them to do this problem you will be encouraging them to behave like mathematicians.

By addressing the case of adding two odd numbers and an even number, a generic proof that adding two odd numbers and an even number always gives an even answer is developed based on the structure of odd and even numbers. The article entitled Take One Example will help you understand how this problem supports the development of the idea of generic proof with the children. Reading it will help you to see what is involved.

Ask the children to choose two odd numbers and an even number and add them together. It is probably easiest if they choose ones that are easy to model and numbers that they are secure with.

Suggest that they make a model of their numbers using apparatus that is widely available in the classroom. Resist pointing them in specific directions unless they become stuck. If they are stuck then resources such as Multilink cubes, Numicon or squared paper will be helpful.

Invite the class to share any noticings they have about the sum of their three numbers. Encourage learners to respond to each other's suggestions, and having given everyone chance to comment, specifically focus on the fact that two odd numbers and an even number added together make an even number.

Give everyone time to look at the way they have made their particular numbers, and ask whether they can see anything about their model that might help us to understand why this is always the case. You may like to give the class chance to walk around the room looking at all the different representations.

In the plenary, you can share the models that help draw out the general structure. Encourage learners to articulate what it is about the structure of odd and even numbers that means their observations will *always* be true. Using mathematical language to form a convincing argument is an important skill and, alongside the representations, constitutes a proof in the context of Primary
mathematics. (You may like to read our article Why Dialogue Matters in Primary Proof to find out more.)

How would you like to represent these numbers?

What do you notice about the answer?

Can you see anything in your example that would work in exactly the same way if you used three different numbers but still one even and two odds?

Can you say what will happen every time you add any two odd numbers and one even numbers?

Can you convince your friend that this is true?

When adding three numbers there are a number of different combinations of odds and evens that are possible. Ask the children to explore what they are. Get them to identify the possible combinations and the features of those combinations that matter.

Does it matter whether the numbers are odd or even?

How many different cases can you find?

To work on the generic proofs for each case the children will need to consider them separately. Can you create a proof for each case using one example?

A possible extension would be to look at Three Neighbours.

It may be helpful to return to Two Numbers Under the Microscope if the children are struggling with adding three numbers. This might help them to feel more comfortable with the rules they have proved in that problem and so build the foundations for this one.

The children may find it helpful to use representations of numbers such as these to support their thinking.