Last Friday, just before the bell one of my year 12 students gave me the following problem:

**"How many times a day are the minute and hour-hand perpendicular?"**
I knew the student too well to respond with the first number that came into my head. The solution to this problem was evidently the subject of an intense debate between several students and this carried on down the corridor after the class were dismissed.

I'm sure like many maths teachers, when I'm posed with a problem my first instinct is to turn to algebra. After a bit of head scratching I managed to formulate an equation and soon after I had the solution.

Non of the maths I used was particularly difficult, in fact nothing beyond KS3 but I know how challenging students find it to translate a problem like this into algebraic terms or indeed any line of logical reasoning. I thought it would be interesting to try this problem with my fairly middle of the road year 10 group to see what approaches they would take. I knew they would find it difficult to visualize the solution so I decided to knock up a GeoGebra sketch to help (spoiler alert!):

All of the students came up with 48 pretty quickly and they were pretty resolute when I told them they were wrong;

"There are two in each hour and then 24 hours in a day so there must be 48, how can that be wrong?"

At this point I was glad of my GeoGebra sketch for some back-up as I was struggling to get them to reconsider looking at the problem from another angle and they were starting to lose interest. I actually decided to show them the solution and ask them the more interesting question -

*Why is it 44?*
After some more thinking time and discussion someone worked out that the key thing is how many times the the minute hand passes the hour hand within the 12 hours.

**. This is a neat way of looking at it that seems so obvious once you see it but I also wanted to see if they could think it through algebraically so I encouraged them through my solution:***"Ignore the ticks (the numbers) and just work out how many times the minute hand overtakes the hour hand, its got to be double that"*$t - \frac{t}{12}= \frac{11t}{12}$

Which is a right-angle when equal to any integer in the sequence;

$30n-15$

Substituting;

$t=60×24=1440$

and solving for $n$ gives $n = 44.5$

Again the GeoGebra file helps to clear up where the point 5 comes from and why the answer is 44.

GeoGebra is a fantastic tool for helping students to visualize problems that they find hard to access. This visualization can encourage geometric ways of thinking about and solving problems but it can also be used as a scaffold to the explore algebraic lines of thinking as I chose to do here.

I like this problem; there are many ways you can approach it and anyone who can tell the time on an an analogue clock can have a go at it. The obvious extension task is to ask at what times does 'perpendicular time' occur and then the second hand opens another can of worms. Try it with some of your classes and see what approaches they take.

P.S. Designing a clock in GeoGebra would be a great little problem for further maths students studying polar coordinates and parametric equations.

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