Two numbers under the microscope
Problem
Two Numbers Under the Microscope printable sheet
Choose any two odd numbers, such as 5 and 9. Add them together.
Draw a picture or make a model to show how the numbers add together.
Adam found some dominoes with 5 and 9 spots on them:
Sarai made a model using Multilink cubes:
Abdul drew a picture of 5 add 9 like this:
What do you notice about the answer?
Look closely at the models and pictures.
Can you see anything in any of them that would work in exactly the same way if you used two different odd numbers?
Can you use your one example to prove what will happen every time you add any two odd numbers?
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 your argument. 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
If you are stuck, try looking at your model or picture in different ways to see if that helps.
Do any of the bits of it match up in pairs?
Think about what you know about the properties of odd and even numbers.
Student Solutions
Well done to everybody who had a go at this activity. We had lots of solutions sent in from the pupils at Halstead Preparatory School in the UK, so thank you all for sending us your ideas!
Lots of children noticed that adding two odd numbers together gives an answer that is an even number. Bea from Halstead Preparatory School had a go at explaning why this is true. She said:
2 odd numbers always to add up to an even.
Odd numbers are one short of an even number. They get that one from the other odd number.
This is a great explanation, Bea - well done!
Pieter from the British International School in Istanbul in Turkey explained the pattern with similar reasoning:
An odd number actually is an "even number plus 1" (or minus 1). So if you add two odd numbers, both "1"s from both numbers together make a "2", which is an even number; and the remaining part of both numbers were even anyway, so the total is always even.
Well done, Pieter!
James from St. John's School in Northwood, England investigated what would happen if we put more numbers under the microscope:
They always add up to an even number because if you add together the even numbers that are one less than each number you'll have an even number. Then if you add together the two that are left you get an even number. If you do this with three odd numbers it would give you an odd number because you'll have three left to add together. If you add up an even number of odd numbers you get an even number but if you have an odd number of odd numbers you get an odd number.
Good ideas, James! I wonder if we could draw a picture to represent what happens when we add odd numbers together?
Teachers' Resources
Two Numbers Under the Microscope
Two Numbers Under the Microscope printable sheet
Choose any two odd numbers, such as 5 and 9. Add them together.
Draw a picture or make a model to show how the numbers add together.
Adam found some dominoes with 5 and 9 spots on them:
Sarai made a model using Multilink cubes:
Abdul drew a picture of 5 add 9 like this:
What do you notice about the answer?
Look closely at the models and pictures.
Can you see anything in any of them that would work in exactly the same way if you used two different odd numbers?
Can you use your one example to prove what will happen every time you add any two odd numbers?
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 your argument. You might draw something or take a photo of things you have used to prove that your result is always true from your example.
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. 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 encouraging them to do this problem you will be helping them to behave like mathematicians.
By addressing the case of adding two odd numbers, a generic proof that adding two odd numbers 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
This problem featured in an NRICH Primary webinar in January 2022.
Ask the children to choose two odd numbers 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, but if they are then resources such as Multilink cubes, Numicon or squared paper will be helpful. After some time exploring they may need some prompting to move them towards looking at the pairing of their dots or cubes. These representations of odd and even numbers may help.
Invite the class to share any noticings they have about the sum of their two 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 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.)
Key questions
How would you like to show 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 two different odd numbers?
Can you say what will happen every time you add any two odd numbers?
Can you convince your friend that this is true?
Possible extension
Even plus even
See what happens if you add two even numbers such as 4 and 12.
Can you 'see' in this example what will happen every time you add two even numbers?
You may find it helpful to work in a similar way to the way you worked for odd numbers.
Adding an odd and an even
See what happens if you add an odd and an even or an even and an odd such as 6 and 9.
Can you 'see' in this example what will happen every time you add an odd and an even or an even and an odd number?
You may find it helpful to work in a similar way to the way you worked before.
You could go on to look at Take Three Numbers.
Possible support
It may be helpful to encourage children to print out and cut out representations of odd and even numbers. In these the 'oddness' of the odd numbers is very clear. Numicon would be an alternative resource to use.