# 3 Practising techniques and noticing differences between LCM and HCF

The next activity develops the practice of asking students to think about the methods they have used. This activity is, again, very similar to an activity that can be found in textbooks. The difference is that it gives students a mixture of problems in terms of having to find common multiples and factors in one activity, and mixing up numbers and expressions. The other difference is the request to make notes on the methods they have used. The aims of these modifications to the textbook activity are to make students aware of connections between topics, noticing the differences and sameness, and for the mathematical thinking processes involved to become explicit. Again, to help students engage with this new way of working, it might help to let them work in pairs or small groups when you facilitate this activity in your classroom.

## Activity 2: Practising techniques and noticing differences between LCM and HCF

• Find common factors and multiples of the following: [Write these problems on the board.]
• 48 and 72
• x2 and 3xy
• and
• (a – b)2 and (a – b)3
• (a2 – b2) and (a3 – b3)
• Write down the methods you used to work these out.
• Now convince your partner that these methods are mathematically correct. If you are working with a partner, try to convince another pair of students.

## Case Study 2: Teacher Faraz reflects on using Activity 2

The students did the first question with great confidence. The second one provoked some discussion but the third one was left by most. For this third question I gave a hint of getting factors within the root sign, and then some of them got the answer almost at once. The fourth question resulted in a bit of discussion, but they came up with an answer. However, for the last question, some pairs came up with a2 – b2 as the common factor and a3 – b3 as the multiple.

They described their methods in terms of an algorithm. They kept repeating that they had given the rule and they had learned that this was the rule and that I had told them so! I tell you, this did cause some soul-searching on my part! But I insisted and kept asking how they knew they were allowed to do each step and why they were doing each step. I asked them to imagine their little sister keeping asking that question, ‘Why?’, and that she would not be happy with ‘Because I tell you’ as an answer.

 Pause for thought 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 Making connections to understand mathematics

4 Learning from the work of fictitious students