Why do this problem?
will help reinforce telling the time with the help of a digital clock. It also requires children to be systematic in their working.
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. You may want to begin by asking a few oral questions based on the clock before moving on to the problem as it stands.
Once you have introduced the problem, you could ask for some estimates of the total number of times $5$ appears. The problem can be interpreted in slightly different ways, so you may need to reach a consensus as to whether, for example, 05:55 counts as three times, because there are three fives, or just once, because the three fives are all present at just one moment. Give pairs of learners
time to work on the problem - they are likely to need paper or mini-whiteboards for recording.
After a short time, share a few insights they have gained and invite some pairs to explain what they have done so far. Some may have started by looking at the minutes digits, some with the hours, some may have looked at certain times of day. Talk about the different approaches and in particular ask the class how they will make sure they don't miss out any times. If no-one suggests a system,
then you could offer some ideas such as concentrating on the minutes in ascending order, or perhaps beginning with 00:00 and working minute by minute through to 23:59. (You could also discuss which way might be most efficient.)
Leave them to work more on the problem before bringing them together to talk about their final solutions. You could use the $12$-hour display question for discussion in the plenary.
This problem could be presented as a challenge to parents and children while waiting to see teachers at parents' evening! To avoid confusion, specify that, for example, 15.55 counts as three times, and ask families to write their name and answer on a card and post it in a suitably-labelled box. A small prize could go to the family whose correct answer is the first to be drawn by the head at
the next school assembly.
When does $5$ appear in the minutes display?
When does $5$ appear in the hours display?
How will you know that you have got all the different times?
You could ask learners whether each digit will appear the same number of times as the $5$ and invite them to explore this.
It might be useful for some children to have access to an interactive version of a digital clock themselves, perhaps at a shared computer.