Happy and sad times in #MondayMaths

Monday Maths: Vicki Robinson, Communications Officer

Happy and sad times in #MondayMaths

Posted: 9/01/17

It’s an emotional time with this week’s #MondayMaths, as our puzzling problem looks at ‘happy clocks’ and ‘sad clocks’.

You may have noticed that all watches are advertised with their hands in a ‘happy face’ position (10:10 or 1:50). They can also have sad faces (4:40, 8:20) where the hands form a frown.

At these times on an analogue clock the hands all look like they are set at about the same angles, but are they? And if not, which time has the smallest angle between the hands?

This timely puzzle is a lovely problem to introduce to Year 7 students in Spring 2, when they study angles in 2D Geometry.

If you want to make this puzzle accessible for primary school, Year 6 students could have a go at a simplified version by playing with clock manipulatives.

Time to start

  • Ask your class to identify how far apart each ‘hour’ marker is in degrees (the 360° clock is split into 12 hours so the calculation should be simple)
  • Decide on an easy way to measure the angle between the hands – with the ‘happy clocks’, it might be easier to measure the angle of the hour hand and the minute hand separately from the ’12’ position


The angle of the minute hand will be easier to calculate as they rest exactly on an hour marker, e.g. at 10:10 the minute hand is exactly pointing to the ‘2’.

It’s slightly more complicated with the angle of the hour hand as it will fall somewhere between the hour markers (it is only exactly pointing to an hour when the time is exactly something o’clock).

So how can we work out exactly how far the hand sits between the two hour markers?

If the angle between two consecutive hour markers is X degrees, encourage students to think of the angle between the hands as a whole number of X-degree blocks, plus a fraction of that block.

Mathematical thinking

If your class race ahead of time, how about asking them to investigate some of the relationships between time and angles between hands…

For example, if the minute hand indicates 15 minutes, can the hour hand ever be at right angles to it? (hint: no – but why?)

Best of luck! The answer will be up on our Twitter page this Thursday. Tweet us with your thoughts and comments.

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