This week saw 4 lessons with Year 9 on “Proportional Reasoning”. It’s a skill that pervades lots of topics, obviously ratio but also underlies algebra, fractions, shape, measures, statistics, everything really!
However, I was a bit nervous as it seemed like there were lots of examples that would involve whole class, teacher-led discussion and not enough for students to do. So I had these from Don Steward’s Median, ready on a slide.
It also seemed to me that all the various examples and contexts used fundamentally the same skills and that students (as I had a fairly high achieving class) would “get it” and then quickly get bored.
I was wrong. As we looked at the different examples it became clear that the change of context was not straightforward. In the example above many students initially added 10cm to the 6cm and 7cm to get 16cm and 17cm. Once we had examined it further and introduced the concept of a Double Number Line, they fully appreciated why this was wrong. So then we looked at this one:
Again, many fell into the trap of adding 2 to the 30m to get 32m rather than 35m by taking a multiplicative reasoning approach.
The power of presenting different contexts for the same basic skills both provides interesting ways to practise that skill as well as giving the student (and teacher) an assessment of whether they have mastered it or not yet. Sometimes misconceptions can be strongly engrained, maybe even more so in top set kids who are used to being right most of the time! It takes time to develop the right instincts when approaching these problems and gain that depth of understanding. With topics as fundamental as Proportional Reasoning, that is time well spent.