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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?

Take the Right Angle

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Clock Hands

Age 7 to 11
Challenge Level

Clock Hands

I've been wandering around the shops where I live in Cambridge - there are a lot of shops that seem to be selling interesting looking watches and clocks. Some challenges on NRICH are about digital numbers but lots of people like to use an analogue clock or watch instead. ('Analogue' for our use, means that it is not digital and has a round face with two hands and twelve numbers!)

Well, there are some fascinating ideas. I have a clock in which the hands go around anticlockwise and so do the numbers! There are so many designs, perhaps you've seen some and have been fascinated by them.

I was wondering about different shaped hands. How about a pair of square hands, but we will tilt each hand 45 degrees so that the corner points and the opposite corner is in the centre. I've tried drawing some of these ideas. (However I have not allowed the hour hand to move and the hour is 12, it's only the minute hand that's moving at the moment!). To help to understand the pictures I placed a little dot where the hands are pointing.

So at 12 o'clock it would look like this:-



Every ten minutes later (Remember that I'm not allowing the hour hand to move) we'll have:-



I like the overlap shapes that have been formed.


You could try this with paper squares and a pin stuck very, very close to the corner that's pretending to be the middle of the clock face. I do not have any coloured see-through sheets but I guess that would be great, and even better if you can borrow the OHP in the classroom.

Have a go and see what things occur as the minute hand moves and you hold the hour hand still, look at shapes of overlap and a sort of "Jaw" shape that opens and closes.

We should be used to asking "I wonder what would happen if ...?''

As you can guess I do this frequently and so I used my computer to help me with an equilateral triangular one. Here they are:-


Of course why should we stop there? Circles!:-

pic 4

You can try and explore all different designs. You don't have to just look at the position every ten minutes like I have done; you could try any minutes past/to that you liked.

If you are the sort of person who prefers to work with a clock that the hour hand moves properly with the minutes hand then you might like to look at these:-

pic 6

Good luck.

Why do this problem?

This problem is one that explores spatial awareness more than exploring the pupils' ability to read clock faces.

Possible approach

It would probably be good to have something visual to help children to enter more confidently into this problem. Some cardboard cut-outs that enable the rotation would help.

Key questions

How are you working out where the shapes should go?
Tell me about the overlap you've got here?

Possible extension

As suggested in the problem, exploring different shapes and looking at angles when both hands have moved might be a good next step.

Possible support

Some pupils who find it hard to get into this problem might benefit from observing a more usual clock face with geared hands.