Always, Sometimes or Never?

Always, Sometimes or Never?

Are the following statements always true, sometimes true or never true?

You could cut out the cards from this sheet (word, pdf) and arrange them in this grid.

 
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?

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 statements (word, pdf) to sort into the grid (word, pdf). 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 some problems based around number and 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.

These similar problems might be a good starting point for pupils who need more support.