# Watch the clock

During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

## Problem

During the third hour after midnight the hands on a clock pointed in the same direction (so one hand was over the top of the other).

At what time, to the nearest second, did this happen?

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## Getting Started

What is the earliest time it could be?

How do you know?

What is the latest time it could be?

How do you know?

Can you get any closer?

How do you know?

What is the latest time it could be?

How do you know?

Can you get any closer?

## Student Solutions

This challenge caused a lot of hard thinking. Some took the time to be around 2.10 and others around 3.15 according to how they interpreted the question. Either way led to some careful working out. The amount the small hour hand moves led to slightly different suggested answers.

Hasa, Javeria and Sarah wrote:

First we estimated an answer. The answer must be between 2 and 3 on the clock. So it must be between 2:10 and 2:15. Then we worked out how many degrees each hand moves in 1 minute.

The minute hand moves $360\div60=6$ degrees per minute.

The hour hand moves $(360\div12)\div60=0.5$ degrees per minute.

If $T$ is the time in minutes after 2:00 then the minute hand has moved $6T$ degrees.

The hour hand has moved $60+0.5T$

When the hands are pointing in the same direction these must be equal

$6T=60+0.5T$

$5.5T=60$

$T=60\div5.5$

$= 120\div11$

$= 10$ min$+10\div11$ x $60$ sec

= $10$ min $55$ sec (to the nearest second) after 2:00

Isobel sent in her suggestion as:

I found out that roughly the answer was 3:17 am. I found this
puzzle quite tricky. So I borrowed my mum's alarm clock and fiddled
with the arms until I found the answer.

## Teachers' Resources

### Why do this problem?

Interpreting and visualising are both important mathematical skills. This problem requires both.

### Possible approach

You may want to introduce this problem by talking about how important it is, in many aspects of mathematics, to be able to visualise. Perhaps ask the children to close their eyes and think of a clock with the hands pointing to 12 o'clock. What would it look like? Now ask them to imagine the hands going round until the clock shows, say, three o'clock. What would have changed? What would the
angle be between the hands? How do you know? Repeat with one or two more examples and then set the problem, perhaps in pairs.

You may wish to encourage them to use large pieces of paper to record their working so that this can be shared with the rest of the group at the end. Give the children a little time to 'get into' the problem and bring them back together to discuss and share any strategies they have found useful so far, before encouraging them to continue.

When appropriate, bring the children back together to share their solutions.

### Key questions

Which hand moves the slower?

How far does it move in an hour?

What about the other hand?

### Possible extension

Children who find a solution and can justify it could investigate how many times the two hands are exactly on top of each either during a 12-hour period, and what those times would be. Is there any pattern to the times?

### Possible support

The problem as written becomes trivial if given a clock, but children who find the original inaccessible may use a clock to help them to write their own questions about where the hands are and the angles between them. You might like to offer the activity Sweeping Hands too.