Always, Sometimes or Never?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Problem
Are the following statements always true, sometimes true or never true?
You could cut out the statement cards and arrange them in this grid.
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When you add two even numbers together the answer is even |
When you add two odd numbers together the answer is odd |
If you add an even number to an odd number the answer is even |
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When you multiply by an odd number the answer is odd |
When you multiply by an even number the answer is even |
Doubling a number results in an even number |
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When you multiply a number by itself the answer is even |
The sum of four even numbers is divisible by four |
Adding three consecutive numbers results in an even number |
Can you find examples or counter-examples for each one?
For the “sometimes” cards, can you explain when they are true? Or rewrite them so that they are always true or never true?
Getting Started
Can you think of an example when it isn't true?
How do you know that it is always true?
Is it possible to check all examples? Is there another way of knowing?
Student Solutions
Well done to everybody who investigated these statements and found examples or counter-examples. We haven't received many solutions to this problem yet, so please email us if you have a solution you'd like to share with us.
Usmaan and Mayeda from Marner Primary School in the UK each sent in a solution for one statement. Mayeda said:
When you add two odd numbers together the sum is an odd number?
I think it's never true because if you find the sum of any odd number together like 23 plus 59 it equals 82 and 82 is an even number not an odd number. Also if you do 97 plus 65 it equals 162 and 162 is even so this means that adding two odd numbers equals an odd number is not true.
Usmaan said:
When you multiply a number by itself the answer is even?
It's sometimes true because if you do 4x4 the product is 16 and if you do 9x9 it's 81 and 81 is an odd number so it's sometimes true.
Good ideas - it's easy to show that a statement is 'sometimes' true when you can find both an example and a counter-example for it! I wonder why adding two odd numbers together never gives an odd number?
Teachers' Resources
Why do this problem?
This task is a great opportunity for learners to use reasoning to decipher mathematical statements. We often make mathematical claims that are only true in certain contexts and it is important for learners to be able to look critically at statements and understand in what situations they apply.
The example here only refers to one key topic but similar statements could be created for any area of maths.
Possible approach
You may want to start with one statement and have a class discussion about whether it is true. Ask learners to think of some examples to illustrate the statement and decide whether it is always, sometimes or never true. If they decide it is sometimes true, they could think about what conditions make it true.
Groups of learners could be given the set of statement cards to sort into the grid sheet. Taking each card in turn they could decide if it is always, sometimes or never true. Then they could justify their reasoning. If they think it is always true or never true, they could explain why they think this is. If they think it is sometimes true they could start by coming up with cases for each and trying to generalise.
For learners who have had more experience of reasoning it might be good to ask them to try and write their ideas down in a clear way, perhaps for just one or two of the statements to start with.
It would be worth sharing ideas as a class at the end. You could pick up on a statement that has been problematic or where there does not seem to be a consensus and support a whole class discussion.
Key questions
Can you think of an example when it isn't true?
How do you know that it is always true?
Is it possible to check all examples? Is there another way of knowing?
Possible extension
Learners could be asked to come up with their own statements for things that are always, sometimes and never true within a topic area. Again they should try to justify their reasons and specify the conditions necessary.
If appropriate, pupils could try the problems Always, Sometimes or Never? Number and Always, Sometimes or Never? Shape which use similar ideas.
Possible support
When discussing as a class, suggest types of numbers to try. Learners often need to start with concrete examples to develop their understanding of a particular concept before they can before they can develop their reasoning within that area. Manipulatives such as Numicon can be useful for building an understanding of the difference between odd and even numbers, and can be used by all learners to support their arguments.
Always, Sometimes or Never? KS1 might be a good starting point for pupils who need more support.