Practising Mathematics: Substitution

Over the summer I invested in ATM’s Practising Mathematics by Tom Francome and Dave Hewitt, which is packed with interesting activities aiming to ‘develop the mathematician as well as the mathematics’. Having used several of these tasks during the first half term of the year it has quickly become one of my favourite resources for giving students practice, building on their knowledge by spotting relationships and probing them to develop a deeper understanding through powerful questions. These tasks have been a perfect fit for my mixed attainment classes, and if you haven’t got a copy of the book, I would highly recommend that you invest.

Just last week, whilst most of the country were on their half term break, I taught a lesson on substituting into expressions to a Year 9 class using the Expressions Cards task from Practising Mathematics. I was blown away with just how powerful it was so I wanted to share my experience, but I also wanted to reflect on what I could do even better next time.

The Lesson

The task revolves around a set of cards with expressions written on them. The suggested expressions in the book are:

Substitution 1

I decided to add 4 other cards containing:

Substitution 2

Working in pairs, students were prompted to choose a value for x and substitute it into the expressions. Once they worked out the value for each card they needed to put them in order from smallest to biggest. They then repeated the process and commented on which cards changed position and which stayed the same. Most students started with positive integers less than 10 and were already commenting on what they noticed as I questioned them.

At this point I suggested that students may want to try different types of numbers: large numbers, small numbers, decimals, fractions, negatives, to see what differences there may have been. As they did this I also prompted them to conjecture examples of cards where one would always be smaller than another, no matter what value of x. This prompted even more discussion as I circulated the room. I took several examples from students and we discussed them as a class. This allowed me to probe students even further as well as address any misconceptions. In particular, I was keen to highlight the difference between 2x2 and (2x)2.

The final task I set built on from the last one as I directed students to choose two cards: card A and card B. They had to find values for x so that:

Substitution 3

This was particularly powerful as they had to think of the nature of the expressions on the cards. It also led them to find expressions where one was always bigger than the other.


I was really pleased with the connections that the task allowed the students to make. However, as the lesson went on I found some opportunities to make it even better next time.

  • I may change the cards from expressions to formulae by adding ‘y =’ to the start of each of them. Here I could instruct the students to calculate the size of y in each formula. I could even include a couple of formulae where y is not the subject, so some form or rearranging will need to be done. This resource could then link to linear graphs, simultaneous equations and many more.
  • During the ordering I did not make it clear to everyone that they should write down the order each time so they can compare, although most students were savvy enough to do it anyway.

Here are some photos I took of students’ work after the lesson:

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Author: Gareth Evans

I have been a secondary Maths teacher at a north-west school for 4 years. I am a Head of House, accredited trainer and #mixedattainmentmaths advocate.

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