Always, Sometimes or Never? KS1
Problem
Are the following statements about number 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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When you add two numbers you
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If you add 10 and take away 1,
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When you add 10 to a number,
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When you subtract one number from another number you can change the order and the answer will be the same |
What about these statements about shapes?
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If you put two squares together
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3D shapes have more than
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When you cut a square in half
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Four sided shapes are called squares |
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Three sided shapes are called triangles
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You could cut out each set of statement cards and arrange them in this grid.
Alternatively, you could use these interactivities to organise your thinking:
Getting Started
How do you know that it is always true?
Is it possible to check all examples? Is there another way of knowing?
You might want to sketch shapes to try out your ideas.
Student Solutions
Thank you to everybody who sent in their thoughts about this task.
Tom from Clifton Hill Primary School in Australia sent in these solutions for the number statements:
Good ideas, Tom! Ayden from St John's College, Cardiff, Wales had some similar solutions, and also sent in their solutions for the shape statements:
The number statement cards are only either "always" and "sometimes". No cards are in the "never" section of the table.
When you add two numbers, you can change the order and the answer will be the same, so that card is true and is in the "always" section of the table. For example, 12+8=20, and 8+12=20. 31+74=105, and 74+31=105 too.
If you add 10 and take away 1, it is the same as adding 9, so that card is true and is in the "always" section of the table. For example, 0+10=10, and 10 - the answer for the previous sum - take away 1 (10-1) = 9. If you add 0 and 9 (0+9) =9, which is the same as the previous answer. In my opinion, adding 9 is quicker than adding 10 and taking away 1.
When you add 10 to a number, the answer is only sometimes a multiple of 10, so that card is in the "sometimes" section of the table. This statement is true if the number you are adding 10 onto is a multiple of 10, but is false if the number is not. For example, 30+10=40, which means this statement is true, but 43+10=53, which means this statement is false. It depends on the situation as it can be true or false.
When you subtract one number from another number, you can only sometimes change the order and the answer will be the same, so that card is in the
"sometimes" section of the table. This statement is true if the two numbers in your equation are equal, but false if they are not. For example, 9-9=0 and 9-9=0, which means this statement is true, but 4-3=1 and 3-4=(-1), which means this statement is false. It depends on the situation as it can be true or false.
Now to the shape statements. The shape statement cards are only either "always" and "sometimes". No cards are in the "never" section of the table.
Three sided shapes are called triangles, so that card is in the "always" section of the table. Proof in file attached, page 1.
Four sided shapes are only sometimes called squares, depending on the situation, so that card is in the "sometimes" section of the table. Proof in file attached, page 2.
When you cut a square in half, you only sometimes get a triangle, so that card is in the "sometimes" section of the table. Proof in file attached, page 3.
If you put two squares together, you only sometimes get a rectangle, so that card is in the "sometimes" section of the table. Proof in file attached, page 4.
3D shapes can only sometimes have more than 4 faces, so that card is in the "sometimes" section of the table. For example, some 3D shapes, like cylinders, have less than 4 faces, which suggests that this statement is false. However, other 3D shapes, like cubes, have more than 4 faces, which suggests that this statement is true. It can be true or false, as it depends on the situation.
You've thought about all of these statements really carefully, Ayden! Ayden's file of evidence goes into more detail about their reasoning on the shape statements.
Kanaa from Ganit Kreeda, Vicharvatika, India sent in these pictures explaining how they decided whether the shape statements were always, sometimes or never true:
Good ideas, Kanaa - I like the way you used examples and counter-examples to decide for each statement!
Thank you as well to the following children who sent in similar solutions to this problem: Ananya, Shivashree, Swara N., Ruhi, Mrunmayee, Aarav, Swara S., Kimaya, Atharva, Shruti, Amey, Avyuktth from Ganit Kreeda; Dean, Lucinda, Marley, Ava from Clifton Hill Primary School in Australia; and Dhruv from The Glasgow Academy in the UK.
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 the key topics of number and shape, but similar statements could be created for any area of maths.
Possible approach
This problem featured in an NRICH Primary webinar in November 2021.
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. The interactivity could be used in pairs on a tablet or computer alongside, or instead of, printed cards. Once all cards are positioned in a cell of the table, a 'Submit' button appears which enables
learners to get feedback on which statements have been correctly placed.
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 at first. This might be in written form, but could be an audio/video recording.
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 that are always, sometimes and never true within a topic area. Again they should try to justify their reasons and specify the conditions necessary. You may particularly wish that learners create their own 'never' statement, as in fact none of the statements given here are 'never true'.
If appropriate, pupils could try Always, Sometimes or Never? which is based around odd and even numbers and uses similar ideas. Older children might like to have a go at the similar problems Always, Sometimes or Never? Number and Always, Sometimes or Never? Shape.
Possible support
When discussing as a class, suggest types of numbers to try or specific shapes to consider. 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. Using a variety of different manipulatives will help all learners to consolidate their understanding and support their arguments.