3 Conjecturing and generalising

Making conjectures (theories), and then reasoning out whether they are true, sometimes true, or false, is part of developing the ideas of generalising that algebraic thinking depends on.

The ‘additive identity’ – that is, the idea that adding or subtracting zero leaves the original number intact – is relatively easily grasped. However, because of its later application in solving algebraic equations, this identity is well worth exploring.

It is important for students to be able to articulate their understanding of such identities and this can be done by asking the students to develop conjectures. The class can develop statements or conjectures about what happens when zero is added to, or subtracted from, a number.

Students often try out a range of different numbers in order to test their ideas. It is important to encourage students to consider whether their conjecture (or theory) works for all numbers. In this way, your students will begin to generalise about number properties in an algebraic way.

The rules or conjectures developed by a class can be displayed and/or the student who came up with the concept could have their name attached to it, for example ‘Prem’s Rule’.

Prem’s rule: ‘When you add zero with a number it doesn't change the number that you started with.’ (a + 0 = a)

Anisha’s rule: ‘When you subtract zero from a number it doesn't change the number you started with.’ (a – 0 = a)

Jyotsna’s rule: ‘If you take the number you started with away from the same number you get 0.’ (a – a = 0)

Vishal’s rule: ‘It doesn’t matter if the numbers are swapped around on each side of the number sentence. If the numbers are the same, the number sentence will still balance.’ (a + b = b + a)

Simi’s rule: ‘When you add two numbers, you can change the order of the numbers you add, and you will still get the same number.’ (a + b = b + a)

Exploring and conjecturing

The following activity shows how you can help students start to think algebraically by exploring arithmetic statements and making conjectures about whether they are always true, sometimes true or never true. It can sometimes come as a surprise to students that they are allowed to say ‘this is not true’. It is really important that they should not just accept everything they see with numbers in it, but should be willing to think ‘is this always true or can I refute it?’

Activity 2: Conjectures

Preparation

Write several arithmetic statements on the blackboard. Some examples you might use are:

• (3 + 5) + 8 = 3 + (5 + 8)
• (3 + 5) × 8 = 3 + (5 × 8)
• (3 – 5) – 8 = 3 – (5 – 8)
• (3 × 5) + 8 = 3 × (5 + 8)

Note that some of the statements should be true and some not.

There are more examples of statements in Resource 2.

The activity

• Check the validity of each statement.
• For all the correct statements, change one, two or all three numbers to write several similar statements. Are all of these true? If yes, do you feel that these statements would be true for all possible choices of numbers? Write down your thoughts as a conjecture.
• For all the incorrect statements, change one, two or all three numbers to write several similar statements. Are all of these incorrect or can you find a correct statement? Do you feel that these statements would be incorrect for all possible choices of numbers? Write down your thoughts as a conjecture.

This activity provides students with valuable opportunities to learn through talk. You may want to look at the key resource ‘Talk for learning [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] ’ to help your planning for this aspect of the activity.

 Video: Talk for learning

Case Study 2: Mrs Kapur reflects on using Activity 2

I divided the students into groups of five and then gave them ten minutes to discuss the validity of the statements I had written on the blackboard.

There was a lot of discussion among the groups [Figure 3]. This made me extremely happy because, when I listened into their conversations they were all thinking of the reasons why the statements were true or if they could think of numbers that would make them untrue, or the other way round.

There were a few who were not contributing to the thinking in their group, so I told the group to make sure to involve them in the discussion too. One of them had missed some time at school and needed help feeling part of the group again. His arithmetic was particularly good so they soon appreciated his contributions. I told the class that everyone had to do their share of the thinking and that sharing ideas would help everyone. Also I said that I would pick the student who would make the presentation so everyone should be able to report on what was said. So then they all got involved in the exercise.

I asked different students to give the answers from their group’s discussion and say if they thought the statement was always true, sometimes true, or never true. I asked them what numbers they had tried and also to try to explain why they had chosen those particular values. Then I also got other groups to contribute the numbers they had chosen, so that we then had a substantial number of examples.

This took a considerable time, especially with an incorrect statement as several groups were sure they could find a way to make it true. This meant we did not get through all of the statements I had prepared and so I asked them to do the rest as a home assignment, writing their own individual conjectures.

We discussed what they had found out the following day with a lot of contributions from most of them. I noticed that some students at the back of the class were very quiet. When I went to talk to them, some of them said they did not understand what was meant to be done, so I asked them to give the reasons why the conjectures made by others were correct or wrong.

 Pause for thought What do you think about the way Mrs Kapur intervened with the quiet students at the back of the class? What might the possible reasons for them not understanding what they needed to do?Now think about how your students responded to the activity and reflect the following questions: What responses from students were unexpected? Why?What questions did you use to probe your students’ understanding? Did you modify the task in any way? If so, what was your reasoning for this?

2 Thinking algebraically

4 Moving on to more formal generalisations