1 Thinking about the rules for solving equations

To find a possible value (or values) of x, where p(x) is the same as q(x), we need to ‘solve’ the equation p(x) = q(x).

There are certain rules that are sometimes helpful to solve equations. Those rules are often memorised by students – or partially and incorrectly memorised, and many mistakes are made in the process of applying these rules. Part 1 of Activity 1 is intended to make students think about where these rules come from, and why and when they can be used. Part 2 aims to make the students aware of different types of equations. Students’ learn more effectively if they are able to talk to other students about their thinking. Asking them to ‘discuss this with your classmate’ is a good strategy to use.

Before attempting to use the activities in this unit with your students, it would be a good idea to complete all (or at least part) of the activities yourself. It would be even better if you could try them out with a colleague, as that will help you when you reflect on the experience. Trying the activities yourself will mean that you get insights into learners’ experiences that can in turn influence your teaching and your experiences as a teacher. When you are ready, use the activities with your students. After the lesson, think about the way that the activity went and the learning that happened. This will help you to develop a more learner-focused teaching environment.

Activity 1: learning about equations

Part 1: Rules of the solving equations game

Tell your students the following:

When solving equations there are certain rules, or truths, that can be useful. You will probably know these already. The aim of this activity is for you to say where these rules come from, why and when they can be used. To help you make sense of these general statements it is useful to think of examples that fit these general statements.

Which of the following rules can always, sometimes or never be used to solve equations? How do you know?

  • If p(x) = q(x), then p(x) + c = q(x) + c.
  • If p(x) = q(x), then p(x) – c = q(x) – c.
  • If p(x) = q(x), then p(x) .c = q(x) .c.
  • If p(x) = q(x), then p(x)/c = q(x)/c.
  • If [p(x)] 2 = [q(x)]2, then p(x) = – q(x).
  • If [p(x)] 2 = [q(x)] 2, then p(x) = q(x).

Create a graph of each of these rules and compare them with the original p(x) = q(x). Discuss with your classmate what has changed in these graphs and what stays the same.

Part 2: None, one and infinite solutions

Tell your students the following:

Compare the answers for the three equations below. How are your results different? Why are they different?

  • Find x, if 4(x – 8) = 4x – 32.
  • Find x, if 4(x – 8) = 4x – 30.
  • Find x, if 4(x – 8) = x – 32.

Plot the LHS and RHS of each equation on the same graph and interpret your algebraic results graphically. What do you notice?

Consider the equation 2x – 3y = 8. Create a graph of this equation. Now create a graph of another equation of the form ax + by = c, so that the two equations have:

  • the same set of solutions
  • no common solution
  • only one common solution.

Consider again the equation 2x – 3y = 8. Now write another equation of the form ax + by = c, so that the two equations have:

  • the same set of solutions
  • no common solution
  • only one common solution.

During this activity, encourage your students to talk to each other about their ideas and to help each other.

Case Study 1: Mrs Rawool reflects on using Activity 1

This is the account of a teacher who tried Activity 1 with her secondary students.

It was a good idea to have done Part 1 before embarking on the next one. The students at first found it hard in Part 1 to make sense of the general statements. The hint of using examples to understand what the statement was saying worked well.

I walked around the classroom observing how the students tackled Part 2 of the activity. I noticed they had several misconceptions. I thought about how I would deal with this. I took a decision: I asked them all to give their solutions and wrote all of them on the blackboard, whether they were correct or incorrect. I then asked, ‘How do you know this is the correct answer? Discuss it with a classmate.’ Most of the mistakes related to manipulation of the equation. Because we had just done Part 1, this could now be discussed. I noticed, though, that although the students might now know why the ‘rules’ of manipulation are what they are, they needed some practice to recognise these in the setting of solving equations. The other misconception, and this surprised me, was that several students did not know the role of brackets in 4(x – 8). They simply removed the bracket and wrote 4x – 8. So we had a discussion as to why a bracket is given and what it implies.

Creating graphs really helped to make sense of the equations. Renu was the first to finish, so I asked her to help some of the others who were struggling to get a solution. It actually surprised them that the graphs looked different. They became much more confident in their discussions about what was the same and what was different in the three equations.

You may also want to have a look at the key resources ‘Using questioning to promote thinking [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] ’ and ‘Talk for learning’.

Reflecting on your teaching practice

When you do such an exercise with your class, reflect afterwards on what went well and what went less well. Consider the questions that led to the students being interested and being able to get on and those where you needed to clarify. Such reflection always helps with finding a ‘script’ that helps you engage the students to find mathematics interesting and enjoyable. If they do not understand and cannot do something, they are less likely to become involved. Use this reflective exercise every time you undertake the activities, noting, as Mrs Rawool did, some quite small things which made a difference.

Pause for thought

Good questions to trigger this reflection are:

  • How did this activity go with your class?
  • What responses from students were unexpected? Why?
  • What questions did you use to probe your students’ understanding?
  • Did all your students participate in the activity? Were there any students who were less confident about the graphs? How will you support their learning in the next lesson?
  • What points did you feel you had to reinforce at the end of the lesson?

If you have access to the internet, there are some mathematics graphing software packages available such as Autograph. They can help your students to visualise the graphs of equations.

What you can learn in this unit

2 Contextualising equations