Clock angle questions in the 11 plus: why twenty past twelve is not 120 degrees

Angles on a clock face are a favourite of 11 plus maths papers, and of grammar school entrance papers generally, because one small fact separates two children who look identical everywhere else.

Try this one on your child. It takes thirty seconds and it is unusually revealing.

The time is twenty past twelve. What is the smaller angle between the hour hand and the minute hand?

Most children who have been taught angles properly will answer 120 degrees, confidently and quickly.

The answer is 110.

Where the 120 comes from, and why it is nearly right

The reasoning behind 120 is good reasoning. A clock face is a circle, so it is 360 degrees all the way round. There are twelve numbers, so each gap from one number to the next is 30 degrees.

At twenty past, the minute hand is on the 4. The hour hand is on the 12. That is four gaps, and four thirties are 120.

Every step of that is correct except the last assumption: the hour hand is not on the 12 any more.

The bit that gets missed

Twenty minutes have gone by. In that time the hour hand has been creeping towards the 1.

In one whole hour, the small hand, called the hour hand, moves from the 12 to the 1. So each hour, the small hand moves 30 degrees. Our question asked for twenty past, which is one third of the way round the whole clock. This means the small hand has moved one third of its way from the 12 to the 1. That is 10 degrees.

So the minute hand is at 120 degrees from the top, the hour hand is at 10, and the angle between them is 110.

That is the entire question. It is not hard maths. This is the bit that children overlook: the hour hand moves all the time, not once an hour.

Why the 11 plus likes this question so much

Because it separates two children who look identical on every other angle question.

One child has learned that a clock face is 360 degrees and the gaps are 30 each. The other has understood that the hands are moving. Both get "three o'clock" style questions right. Only the second one gets this right.

And it is a question that cannot be passed by luck. A child who does not know about the drift does not get 110 by accident. They get a clean, tidy, confident 120.

The same trick is waiting anywhere the hands sit between the numbers. What is the smaller angle at half past four? The hour hand is halfway between the 4 and the 5, so the answer is 45 degrees, not 60.

What to do if your child answers 120

First, do not correct it straight away. Ask this instead:

Where exactly is the hour hand right now?

Most children will say "on the 12". Then ask them what the time will be in ten minutes. They will say half past twelve, which is easy. Now tell them to take a closer look at the small hand: it is plain to see that it has moved to halfway between the 12 and the 1.

They have then taught themselves the thing the question is testing, which sticks considerably better than being told.

Then give them the same clock angle question at a different time, say twenty past three, and see whether the new fact survives contact with a new number. That is the real test of whether anything landed.

And if they answer 110 first time

Then you have learned something genuinely useful about your child, and it is worth more than the one mark: they are not just applying a remembered rule, they are picturing the thing moving.

Children who do that tend to be the ones who cope with the strangest questions in an 11+ paper, the ones nobody has drilled, because they are working from what is actually happening rather than from a method they were handed.

The honest reason we go on about one clock question

Because it is a ten degree mistake that is invisible in every other way.

Your child is not weak at angles. They are not careless. They have a small, specific, teachable gap, and it takes about two minutes to close, without buying anything.

That is what a good diagnostic question does. It does not tell you your child is bad at maths. It tells you exactly which gap nobody has identified yet, and therefore nobody has been able to correct.