More than a maths teacher

One of the things I like about returning to topics that I’ve taught many times is that I can predict what misconceptions my students are likely to have and try to address them. Sometimes that’s not so easy, especially if the nature of the misconception isn’t clear to me. This is the case with substitution, I know that a significant number of students have trouble with this. I know that many of these students don’t understand what to do, but why are they unclear? Explaining wasn’t working too well, so it was time to try something else.

Footballers know exactly what substitution means: one player gets swapped for another. In the past I’ve tried telling students that it’s pretty similar with algebra, but instead of footballers, a letter gets swapped for a number. This wasn’t a great success, so I decided to stick with the football metaphor, but try a slightly different approach that involved doing, not telling.

I made some pictures of footballers, I gave a ‘blank’ footballers to the students and stuck some with letters to the board. I told the students that these footballers were a team, which was my slightly spurious justification for putting the addition signs and = T.

The students also had a mini-whiteboard. I asked the students to suggest a substitution. I picked someone who had written their suggestion in the form a=n and held it up (in the style of a substitution at a real football match). I had to look a bit baffled to reach the stage where this happened, but comments like “I can’t see who you want to substitute” or “That looks really complicated, can you make it a bit simpler?” got the students started.

We had a bit of fun with this, blowing a whistle once I picked a suitable substitution, then getting students to come out to the whiteboard and swap their player (now with a number on their shirt) for one of the players with letters.

Obviously we could substitute any numbers (my students picked some really tricky examples!), but here’s a simple example:

I selected a student who held up a = 3, another who held up y = 7 and one who held up c = 5. We ended up with this:

This team is worth 3 + 7 + 5 = 15

We did a few like this until they got the idea – I was very strict about substitutions being written in the form a=n, since that’s what they will encounter in written questions. Then I introduced a few super-players, such as player x, who is so good we had to write a 2 in front of him. We had a couple of injuries as well – at the suggestion of the students – so we ended up with starting formulae like a + y – c = T, or we subbed a negative value (I’m sure every local football team has an off-form player we could nominate for that!)

The football metaphor obviously gets a bit shaky, but it addressed the central problem: my students understood what they were being required to do. Sure, once we moved onto the textbook style questions, we still had some problems, but mainly with BODMAS and minus signs, not with the idea of making a substitution.

At the end of the unit, when I asked students what they had enjoyed, a large number mentioned this activity. I didn’t have the foresight to do any baseline testing, but I was happy with the number of students who could do the substitution questions on the practice exam papers. I’ll be trying this activity again.

Download the files:

Word file with images of footballers (doc format) 

PDF file with images of footballers

It’s that time of year again, GCSE exams are imminent for several year groups, end of year exams loom for others. The revision season is definitely in full swing.

I’ve been using a few starter activities for my classes as a bit of quick revision on some topics, the ones listed below are all on PowerPoint files [pptx format], if you would like any of them, please help yourself – Creative Commons licences apply to them all:

• Algebra terminology – aimed at OCR units A and B on the new modular scheme, where candidates have to distinguish between expressions, equations, formulae and identities. My definitions aim to be straightforward and comprehensible for Y9 and Y10 students. This one uses Microsoft Mouse Mischief for interaction. If you prefer, there’s a non-Mouse Mischief version too – easy to use with mini-whiteboards.
• Pythagoras – straightforward questions. Starter slide with questions, followed by model solutions.
• Circles – area and circumference – starter questions followed by solutions and key formulae to learn
• Prime factors, HCF and LCM – using factor trees and Venn diagrams
• Angles in polygons – the first slide has wordy questions, subsequent slides have diagrams to use as hints, or to help the teacher or a student to explain how to solve the problem
• Percentage changes – starter questions followed by solutions and a key rule to learn
• Volume and surface areas of cuboids
• Decimal calculations, rounding and estimating – not exciting, but it covers some key questions!