## How Many Times?

On a digital $24$ hour clock, at certain times, all the digits are consecutive (in counting order). You can count forwards or backwards.

For example, **1:23** or **5:43**.

How many times like this are there between midnight and 7:00?

How many are there between 7:00 and midday?

How many are there between midday and midnight?

### Why do this problem?

This problem will help consolidate children's understanding of the $24$ hour clock notation. It could also be used to focus on ways of working systematically.

### Possible approach

It would be good to have an interactive digital clock on the whiteboard for the duration of this lesson so that you and the class can refer to it whenever necessary. (

This free version would be suitable, for example. Click on the arrows button to switch to a digital display.) You may want to begin by asking a few
oral questions based on the clock before moving on to the problem as it stands.

Explain the challenge to the class and ask children to suggest a few examples so that it is clear what is meant by consecutive. You may need to clarify that all the digits in the time need to be consecutive so, for example, 13:45 wouldn't count, as it only has three consecutive digits. Invite pairs of children to begin working on the first part of the
problem. They could use mini-whiteboards to keep a record of the times they find.

After a short time, draw the group together to share ways of working. Some children may be recording answers as they occur to them, others may have some sort of system - for example starting with the earliest time and working 'upwards'. Discuss the benefit of a systematic approach - it means that we know when we have found all the solutions. Having talked about this, children will be able to
apply a system to the other parts of the question.

In the plenary, as well as sharing solutions, encourage children to articulate reasons for their findings.

### Key questions

Which digits will be possible? Why?

How will you know you've got all the different times?

### Possible extension

Children could also investigate the times which have just three consecutive digits.

5 on the Clock is a problem that requires a similar systematic approach and also involves digital time.

### Possible support

It might be useful for some children to have access to an interactive version of a digital clock themselves, perhaps at a shared computer.