# A broken clock

I walked into my classroom this morning and noticed my clock was broken. Not just stopped but really broken, can you see why?

It reminded me of a nice problem solving task which is sort of to do with angles but actually much more to do with ratio and proportion.

I wrote the following on the board:

For a normal clock, what is the angle between the hour hand and the minute hand at the following times:
1) 15:00
2) 13:30
3) 10:15
4) 17:45
5) 9:26

There is a significant range of challenge in these questions.  15:00 – straightforward, right.  As soon as you start moving the minute hand away from 12, you need to consider the fraction of (360/12) degrees that the hour hand moves.  12.30 might be the best option for a question 2 if you really want to scaffold it.  You also might want to squeeze a few more in between Qu 4 and 5.  e.g. 14.40, 15.20.

Next time I do it, I won’t write them up all in one go, but will keep adding to them as I can see learners making progress. Or ask students to challenge themselves by creating their times which might work better in a mixed attainment classroom.

These can all be done without a calculator. It demonstrates how useful it is to have 360 degrees in a circle and 60 minutes in an hour because they have so many factors.

A nice build on this is this question from an OCR Booklet of problem-solving questions.