Having now taught this lesson, I’ve edited a few things. My first idea was to do it on squared paper and count squares. But there were too many squares and the totals we got were widely varying due to how students estimated partial squares. Hopefully, it’ll work even better when I do it this way next time…
This problem was posted on Twitter last week by @solvemymaths. I must confess I was pretty slow to solve this! It’s a lovely problem – so easy when you know how. I created this Geogebra file. It didn’t help me solve it directly but it did answer the question of whether this is a fixed shape or not, which I was struggling to visualise from the diagram. It isn’t fixed and that gave me an idea for a lesson. Rather than just give my students the problem, we will draw it first to provide that hook, the hunger to answer the question, “Why does that happen?”. I also provides a bit of practice using a compass, a skill that is needed at GCSE and one which we don’t spend much time practising.
Here’s how I would now run the lesson (key maths words italicised):
1, On plain paper everyone draw a circle using a pair of compasses. You chose the radius, anything between, say 1cm and 6cm. Make sure you write down your radius. Calculate the area of this circle.
2, Then, using a ruler, carefully draw a tangent, doesn’t matter where it is on the circle. Make the line nice and long, using a 30 cm ruler if you have one.
3, Then, using compasses again, mark 5cm either side of where you tangent touches the circle. Label the points A and B.
4, Next draw a second concentric circle, going through A. It should also cut B if you have drawn accurately. You should have a shape looking something like this:
5, Now, calculate the area of the big circle. You’ll obviously need to measure the radius of the big circle first.
6, Now find the area of the orange ring. Compare your area with that of others around you.
Where you go next will depend on the class. I’m not going to give you a full solution here, that would spoil the fun! But the answer is 25π…