### Round and Round the Circle

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

### Making Cuboids

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

# Always, Sometimes or Never? Number

## Always, Sometimes or Never? Number

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

How do you know?

What about these more complex statements?

You could cut out the cards from this sheet (wordpdf) 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?

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 (wordpdf) to sort into the grid (wordpdf). 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 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 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.