Always, Sometimes or Never? Shape
Problem
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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A hexagon has six equal length sides
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Triangles have a line of symmetry
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Squares have two diagonals that
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Cutting a corner off a square makes a pentagon |
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The base of a pyramid is a square
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A cuboid has two square faces |
What about these more complex statements?
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When you cut off a piece from
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Triangles tessellate
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The number of lines of symmetry
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Quadrilaterals can be cut into two equal triangles |
You could cut out the 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?
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, such as these similar problems about number.
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 record their ideas clearly, 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 extensions
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.
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 develop their reasoning within that area. Having concrete resources such as 2D and 3D shapes, as well as paper and scissors for children to draw and make their own shapes, can help
children consolidate their understanding 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.