I was with a Year 10 class doing Pythagoras’ Theorem recently. This was a low attaining group who had encountered it before but were shaky. They had recently been doing rounding to d.p. and s.f. They had made progress with that but were getting a bit bored so the teacher decided to go over Pythagoras which I thought was a nice way of interleaving topics. Calculator answers needed to be rounded (for those triangles which were not pythagorean triples).

It got me thinking about the knowledge and understanding required to apply Pythagoras’ Theorem and also about planning topics. In this lesson, they stuck to finding the hypotenuse and didn’t do any problems where they had to find the shorter side. This decision was made by the teacher because she knew the class and the context. By the end all students had practised it a few times and had built self-belief that they could do it. In other contexts, another teacher might decide to introduce both cases side-by-side.

Is there a “best” approach? Has there been educational research looking into such a finer point of teaching this one topic? If there has, please point me to it because I wouldn’t have the time to look for it. Nor the belief that if I did eventually find something, reading it would actually improve the outcome of my learners. There is no single perfect way of teaching any topic. As teachers, we need to keep our eyes and minds open to approaches that we hadn’t thought of or used before. But we must not agonise over trying to find the holy grail, the golden nugget that will suddenly enlighten our students.

As I wasn’t actually teaching the lesson, just helping out, I had time to draw a quick mindmap in the lesson. I was trying to put myself in one of the students shoes.

What are the things I need to know, understand and be able to do to be successful in this topic?

This, fundamentally is what they care about.

I’ve included a picture of my mindmap here, not because it is a stunningly useful breakdown of this topic, but just to show the complexity of what our students need to learn. And this is without much page space taken up on the “*And then…*” topics of proofs, pythagorean triples, etc., some may say the “interesting bit”.

I am going to try to do this mindmapping exercise more often in my lesson planning. I found it quite easy to do when I was sitting in this lesson, but I find it harder when faced with a blank sheet and maybe some resources and ideas that I might have used before. The point is, I think, not to agonise over the perfect mindmap showing the perfect sequencing of “learning nuggets”. It needs to be done whilst thinking about the class and the context. And if there is ever time for such luxuries, doing it with a colleague surely makes the process more satisfying and enjoyable.

Interesting! I think the side-by-side might be helpful as you could ask “What do you notice is similar? What do you notice is different?” It might even be fun to put up the two cases where the different legs are assigned “x” to see if they realize that for the legs it doesn’t matter (it is the hypotenuse as “x” that matters). Let us know if you come up with anything!

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