Always, Sometimes or Never? Shape
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:
How do you know that it is always true?
Is it possible to check all examples? Is there another way of knowing?
Thank you to everyone who sent in their solutions and shared their mathematical thinking with our team. We received solutions from Will and Saif who attend Pierrepoint Gamston Primary School in the UK, Ci Hui Minh Ngoc at Kong Hwa School in Singapore, Simran from Maurice Hawk School in the USA, Hunter from APS in Australia and from students at the British International School in Vietnam.
This challenge showed that there is still some more thinking to be done for some of these statements - they were not necessarily as straightforward as they may have seemed at first!
Will, from Pierrepoint Gamston Primary School, shared his really useful approach towards tackling this type of problem:
For this problem, I found the best way to complete it was to picture the shape in my head. For example, to complete the first question, imagine a hexagon in your head. Chances are the lines are equal. However, imagine a hexagon with unequal lines. This is still a hexagon, as long as it has six sides, therefore the answer is 'sometimes'.
A very useful strategy. Thank you, Will!
In the solutions below, there are a few slip-ups. Have a look for yourself - can you find out what is wrong and why?
This first set of answers was sent in by Hunter who attends APS in Australia:
1. A hexagon does have six equal sides.
2. Triangles sometimes have a line of symmetry.
3. A square always has two diagonals that meet at right angles.
4. Cutting a corner off never makes a pentagon.
5. A pyramid sometimes has a square base.
6. A cuboid always has two square faces.
7. When you cut a piece off from a 2D shape it would always reduce the area and perimeter of it.
8. A triangle always can tessellate.
9. Sometimes the lines of symmetry are the same number of sides.
10. A quadrilateral can sometimes be cut into two equal triangles.
We would have loved to know more about the reasoning behind your answers, Hunter. We wonder whether anyone can you suggest any counterexamples for any of his answers?
The next set of answers was submitted by Saif, who also attends Pierrepoint Gamston Primary School:
Thank you, Saif. Again, we'd have loved to learn more about your reasoning. We wonder who spotted any possible tweaks to those answers?
Eira, from Twyford School, shared her solutions which included explanations to justify her answers:
We were wondering that might happen to the overall perimeter if we cut a triangle into the square? Would the perimeter increase or decrease?
This statement prompted some mathematical thinking among our team about the reasons for some shapes, such as triangles, tesselating but not others. Perhaps thinking about angles might be helpful here?
Simran, from Maurice Hawk School shared his answers and reasoning on our worksheet. We have copied his pictures underneath his full set of answers. You might like to compare Simran's reasoning with Eira's for each of the statements. Do they always agee with each other?
What do you notice about Simran's and Eira's answers to question 7? Whose reasoning is most convincing for you?
Here are Simran's pictures which accompanied his answers on our worksheet:
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.