Always, Sometimes or Never? Number
Problem
Always, Sometimes or Never? Number statement cards
Are the following statements always true, sometimes true or never true?
How do you know?
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?
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The sum of three numbers is odd |
If you add 1 to an odd number you get an even number |
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Multiples of 5 end in a 5 |
If you add two odd numbers you get an odd number |
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If you add a multiple of 10 to a multiple of 5 the answer is a multiple of 5 |
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What about these more complex statements?
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When you multiply two numbers you will always get a bigger number |
If you add a number to 5 your answer will be bigger than 5 |
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A square number has an even number of factors |
The sum of three consecutive numbers is divisible by 3 |
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Dividing a whole number by a half makes it twice as big |
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You could print off and cut out the statement cards from the top of this page and arrange them in this grid.
Alternatively, you may like to try out your ideas using the interactivities below:
Getting Started
How do you know that it is always true?
Is it possible to check all examples? Is there another way of knowing?
Student Solutions
We had some excellent solutions sent in for this task. Thank you to everybody who shared their ideas.
James from Hamstel Junior School in England made a chart of his answers, explaining his reasoning under each statement. You can click on the picture below to enlarge it.
Well done, James! James has used negative numbers to show that some of these statements aren't always true.
Trisan, Rudy and Marcos from Sahauro in the USA used zero to show that some of the statements aren't always true:
If you multiply 2 numbers you can't always get a bigger answer because you can do zero times five and it will be zero.
Well done for thinking carefully about how to disprove some of these statements, even if they do always work for positive whole numbers.
Prayaan from King's College Junior School in the UK gave clear explanations for his thoughts about each statement, particularly for why a square number must have an odd number of factors:
Each factor usually has a ‘partner’ to multiply to form the original number, so there will be an even number of factors. But for square numbers, the square root doesn’t have a ‘partner’ because it multiplies itself to form a square number. So square numbers never have even number of factors.
Thank you for sending in your explanations, Prayaan. Have a look at Prayaan's full solution and see if you agree with all of his explanations. Most of the solutions we received said that dividing a whole number by a half does always make it twice as big - however, Prayaan thinks that this might not always be true:
This is true for numbers greater than 1, but for negatives, the number becomes smaller. -3 ÷ 0.5= -6
This is a really interesting way of thinking about it, Prayaan. I wonder if -6 is 'twice as big' as -3, or if we can't use that phrase because it would be below -3 on a number line. If you have any ideas about this, please email us.
Teachers' Resources
Why do this problem?
These tasks are 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 examples here only refer to one key topic but similar statements could be created for any area of maths - we have made some similar problems about shape.
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, or they could use the interactivities on a tablet/computer. 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? Are there other ways of knowing?
Possible support
When discussing as a class, suggest types of numbers to consider. Learners often need to start with concrete examples to develop their understanding of a particular concept before they can develop their reasoning within that area. Concrete resources can be useful for developing an understanding of the structure of numbers, and can be used by all learners to support their arguments.
The similar problems Always, Sometimes or Never? KS1 and Always, Sometimes or Never? might be a good starting point for pupils who need more support.
Possible extension
Learners could be asked to come up with their own statements that are always, sometimes and never true within a topic area. Again they should try to justify their reasons and specify the conditions necessary.