Always, Sometimes or Never? Number

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 - some similar problems about shape can be found here.

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), 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 found here and here 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.