2 Making connections to understand mathematics

Students study and use factors and multiples from an early age. In secondary school, students are required to study the HCF (highest common factor) and LCM (lowest common multiple) of numbers. They have to be able to apply this knowledge when working with expressions. These topics and concepts are thus studied at different times, and in different years. Therefore, students can fail to see connections between the different aspects that they study, and their knowledge can become fragmented as a result. Students often rely on memorisation at each stage rather than understanding the underlying principles leading to a lack of appreciation of the power of factors and multiples.

Activity 1 aims to address this fragmentation by focusing on the mathematical thinking processes involved in finding factors of numbers and expressions. Students are asked to make connections between what a factor is, between factors of numbers and factors in expressions. The activity requires students to work in pairs or small groups and exchange their ideas with other students.

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 a learner’s 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: Finding factors of numbers and expressions

Figure 1 Students working in small groups.

Tell your students:

  • List the various factors of the following numbers and expressions.
    • 60
    • 3xy
    • 15
    • 12x2y3
    • 3x4 – 27x4
    • 2x2 – 8x + 8
  • Why do you think some numbers only have two factors?
  • Describe your method of finding the factors to the others in your group. Did you all use the same methods? Did all the methods work well with all questions?
  • Write your own definition or description of what a factor is, with examples of where it can be found.

When your students have attempted these tasks in pairs or small groups, bring the class together and ask different students to share their definitions of a factor. Use these definitions as a starting point for exploring their answers to the other questions. Pay careful attention to who answers your questions – is it always the same students? How can you develop the confidence of other students to share their thinking?

Resources 2 and 3, ‘Using groupwork’ and ‘Talk for learning’, have more information on these subjetcs.

Case Study 1: Mrs Kapur reflects on using Activity 1

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

I put students in groups of four so that they could offer mutual support and to give them more opportunities to come up with a rich collection of ideas. I organised the groups by putting a notice on the wall with the names of students in each one. I tried to have a mix in each group of students who were confident with learning maths and students who were not so confident. My aim was that the more confident students would support the others.

The students came up with the factors very easily in the case of pure numbers, but there were a lot of arguments about the ones involving expressions. This happened in most of the groups, so I thought it might be helpful to discuss these questions as a whole class. First I asked them to prepare a short presentation for the whole class; this is what they came up with:

  • They quite accurately identified the factors of 60 as 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 – although some groups missed out a few of them, they quickly wrote down the ones they had missed. This led to a discussion about the need to be systematic, and what approaches could help them to be systematic.
  • In the second question, some groups identified only three factors: 3xy, 3, and x and y.
  • Many groups were quite good at noticing prime factors and explaining that these were special because could only be divided by themselves and by 1.

I really wanted the students to notice the similarity between factors of numbers and expressions. As a first step I asked, ‘In what way are non-prime factors different from prime factors?’

They were a bit baffled by that, so I added, ‘OK, look at the factors of 60 as an example. Now start your sentence with: “Non-prime factors are different because …”.’ There were several volunteers to have a go at answering this, but it sounded neither clear nor concise – which I did not think would encourage good discussions or learning. I decided to give them a ‘speaking frame’, like they sometimes give writing frames in language learning. So I told them, ‘Try saying it first to a partner and then I will ask you to say it to the whole class.’

They practised verbalising their thinking in that way with each other for a few minutes. I also noticed that some were writing what they had said down in their exercise books, probably so as not to forget what to say when asked. The speaking frame really seemed to have helped because when we came to sharing their sentences, their responses were clear and concise and they used mathematical language in a precise way. They had managed to both notice and express that non-prime factors were numbers that could be decomposed further into products of other numbers until only prime factors were used. I was seriously impressed! But they still had not made the connection between these kinds of factors and the factors of expressions.

My first instinct was simply to tell them – to just pass on my knowledge to them. But they were so full of enthusiasm and engagement from being asked to come up with their ideas that I could not just tell them. I wanted them to discover the answer themselves, and experience the joy of discovering – seeing the beauty of mathematical structures and their connectivity. But what question could I pose so that they could become aware of this connection between factors of numbers and factors of expressions? So many questions and approaches came up in my mind, but they were too complicated, or simply disguised ‘telling them what it is’ questions! What if I just told them what I really wanted them to do? So I said:

I really want you to discover the connections and similarities and differences between finding factors of numbers, and finding factors of expressions. For example, between the factors of 60 and the factors of 3xy. And I don’t just want to tell you, I want you to think about it and discover it for yourself. So think for a moment about all these discussions we have had during this lesson, about the difference between prime and non-prime factors, and look at the factors you have found for 60 and 3xy. What is the same, what is different? Have you got all the factors for all? Have a go at it in your groups.

And they did find the missing factors for 3xy. They found the factors for 3x4 – 27x2: from re-writing it as 3x2(x2 – 9) and then going onto identifying the factors as 3, x, x2, x2 – 9, (x2 – 9), x2(x2 – 9) and 3x2(x2 – 9).

Only then did we move on to think about methods used. We had to do that in the next lesson, because we ran out of time. This was not a problem, because in the next lesson I was able to ask them to think back to the previous lesson. I asked them to look in their exercise books at what they had done and what their thinking had been, asking them to re-enter that thinking. Coming up with descriptions of methods went rather smoothly, probably because we had done so much thinking about it already. We shared the different methods with the whole class, and they corrected each other if the method was lacking or overcomplicated. We actually ended up with a ‘Suggested method of Mrs T’s Class’, and wrote that on a large piece of paper that was then displayed on the wall.

Reflecting on your teaching practice

When you do such an activity with your class, reflect afterwards on what went well and what went less well. 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 Kapur did, some quite small things that made a difference.

Pause for thought

After the lesson, try to find time to talk about these questions with a fellow teacher:

  • How did it go with your class?
  • What responses from students were unexpected? Why?
  • What questions did you use to probe your students’ understanding?
  • Did you feel you had to intervene at any point?
  • What points did you feel you had to reinforce?

1 Some issues with learning from textbooks

3 Practising techniques and noticing differences between LCM and HCF