Take three numbers
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
Problem
Take Three Numbers printable sheet
Choose any two odd numbers and one even number, such as 3, 5 and 2.
How would you like to represent these numbers?
Try adding them together and draw/make the representation of their sum.
What do you notice about the answer?
Look closely at your model.
Would it work in exactly the same way if you used different numbers but still two odds and one even?
Can you use your example to prove what will happen every time you add two odd numbers and one even number?
See if you can explain this to someone else. Are they convinced by your argument?
Once you can convince someone else, see if you can find a way to show the argument on paper. You might draw something or take a photo of things you have used to prove that your result is always true from your example.
Getting Started
What kind of number do you get when you add two odd numbers together?
How about adding an odd number to an even number?
Can you use these results to help you to see what is happening in your addition of two odd numbers and one even number?
Student Solutions
Thank you to everybody who sent in their ideas about this activity. Lots of children thought that the answer would be even, so we've focused here on the solutions which explained why that would be the case.
Dean from Australia tried out a few different sets of three numbers:
5 + 7 + 10 = 22 even
2 + 3 + 5 = 10 even
11 + 7 + 4 = 22 even
2 + 7 + 5 = 14 even
22 + 21 + 27 = 70 even
I notice that the answer is always even no matter how many you do it is always even because adding 2 odd numbers will always end up even then you just add another even which even + even will never be odd so it can not end odd.
102 + 111 + 77 = 290
1002 + 999 + 877 = 2870 does not matter how big it is
Jeffrey from Shenzhen Foreign Language School in China thought about why these facts about odd and even numbers are true:
The sum of the three numbers are always an even number because an odd number is always an even number plus one. Then, we could find out, two odd equals to two even numbers added together plus two, which makes it an even number. Then, add the even number to it. We always know, even plus even equals even, so we'll get two odd numbers plus an even number will always be an even number as the result.
Here are some examples:
111+123+44=278 (even)
5+99+2=106 (even)
77+95+2=174 (even)
I like the way you've thought about odd numbers as just being one more than even numbers! I wonder if we could use this fact to make a drawing or a representation of them?
Om from St Leonard’s Primary School in Exeter, UK sent in this picture to represent adding odd and even numbers together:
This is a very interesting way of representing odd and even numbers, Om. Can you see how they look different from each other when drawn in this way? How could we use this representation to think about what happens when we add together two odd numbers and one even number of any size?
We were sent in several solutions from the children at St Mary's International School in Tokyo, Japan. Take a look at Akira and Yoji's solution to see what they thought was true about adding odd and even numbers together.
Angela from the International School of Geneva - Campus des Nations in Switzerland thought about how we can prove what will always be true for odd and even numbers. In particular, Angela noticed that an even number is just a lot of twos all added together, and an odd number is a lot of twos plus one. Take a look at Angela's full solution to see how these facts can be used to explain what will always be true when we add odd and even numbers together.
Teachers' Resources
Why do this problem?
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.
Possible approach
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 the 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.)
Key questions
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?
Possible extension
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.
Possible support
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.