I used to struggle teaching circle theorems when I started teaching. Thinking about it, it’s probably because I didn’t understand the common misconceptions students have with them. I can actually remember the first time I taught it. I’m pretty sure that I ended up talking through all of them on mymaths for 45 minutes, by which time they’d forgotten 90% of what I’d said and were bored senseless. Don’t get me wrong, I think the mymaths slides on this topic are great for revision purposes; I just wouldn’t use them as a teaching aid now.
The second time I taught it, I think that I went through 3 theorems during the first lesson, 3 during the second lesson and then gave a mixture of questions in the third lesson. This was slightly better than the first time, but I’m sure it’s infinitely better to get the students searching for connections on applets themselves rather than me fiddling around with them on the board.
Bearing this in mind, I recently taught circle theorems to top set year 9 by flipping the classroom. For homework, I asked the students to try to discover some things about angles inside circles. If they managed to discover any connections, they needed to write these down and draw any diagrams to help to explain this to someone else (Click on the pictures below to see the applets).
Applet 1 Applet 2
Applet 3 Applet 4
When they came back to class we discussed what they’d found out and after tweaking some of the language, we had 6 conjectures about angles inside circles (no one managed to discover the connection between intersecting chords in applet 5 which could be down to rounding errors in geogebra). Whilst they are of course theorems and have been for a long time, to the class they’re conjectures which haven’t been proved yet. I think immediately labelling something as a theorem makes it seem like something that a really clever person worked on ages ago. I prefer to talk about discovering connections with this topic until we’ve proven at least a few of them. If a student discoveries a connection, they can feel a sense of achievement rather than knowing that someone else discovered it way before they did.
The theorem which students often struggle with is from applet 1: The angle at the circumference is half the angle at the centre. This is mainly because they don’t realise that there are 4 seperate cases to consider. Often, you see Case 4 below being presented as a theorem in its own right – I don’t understand why it isn’t presented as a special case of the main theorem?
It seems that the biggest problem is that Case 2 above looks very similar to the theorem from applet 5: angles on the circumference sharing the same arc are equal. This is often called the bow tie theorem for obvious reasons. The problem with calling it the bow-tie theorem is that students see something which looks like a bow-tie and then immediately assume it’s the bow-tie theorem without any understanding of the situation. The reason why the theorem in applet 5 works is because there’s two similar triangles. In Case 2, we certainly don’t have similar triangles.
Another thing about the theorem from applet 5 is that students don’t always know which angles are equal. This is mainly because they don’t realise that the triangles are similar and they don’t look for the angles which lie on the same arc.
I’ve found that there can be misconceptions with the cyclic quadrilateral theorem. Students assume that any quadrilateral inside a circle has opposite angles which add up to 180 degrees. They need to realise that it only works for cyclic quadrilaterals i.e. quadrilaterals in which all of the vertices lie on the circumference of the circle.
So I guess that I keep three things in mind now when teaching circle theorems:
1) The students can discover these theorems easily on geogebra without me having to bore them by talking for 45 minutes.
2) There’s quite a few misconceptions students can have with circle theorems. A few well placed questions here and there should help students to discover these misconceptions for themselves.
3) Calling them theorems straight away tells the students that they’re learning things which other people have discovered instead of discovering things for themselves.
Oh, and one more thing about circle theorems. They can come in really useful with proofs. E.g.
1) Median (Don Steward) – Using the intersecting chord theorem ends up being easier than using pythagoras or similar triangles.
2) Median (Don Steward) – Proving the sine rule using angles on the circumference sharing the same arc are equal.
3) Math and Multiledia (Guilermo Bautista) – Using Case 3 of theorem 1 to prove the tangent half angle formula.
Addition – I recently came across a lovely video explaining how the tangent theorem can be used to find the distance to the horizon on Charlotte King’s blog post.