# 2 Being able to see what is happening in division

The concept of division can be difficult for students to grasp, partly because there are many ways of talking about it. When you have a problem such as ‘42 divided by 6’, you can read it as:

• ‘How many times does 6 go into 42?’
• ‘How many groups of 6 can be made out of 42?’
• ‘How many would be in each of the six groups?’
• ‘What is one-sixth of 42?’

Although the answer is always 7, the ways you can get to that answer can be very different, and can result in confusion for some students.

Helping students to be aware that there is more than one way to think about division and to be alert to the possibility of ambiguity in everyday language will support their mathematical development. It is important to teach the students to remember to think very carefully about the meaning of every problem they encounter that asks them to divide, such as ‘divide 42 into 6’.

Being able to visualise what is happening is an important step to understanding the algorithm. In the following activity you will ask the students to use division intuitively and in context to find the quotient and the remainder. In later activities you will build on the thinking the students have developed in this activity to more deeply explore the division algorithm.

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 them for yourself will mean you get insights into learners’ experiences that can in turn influence your teaching and your experiences as a teacher.

## Activity 1: Dividing up lengths

### Preparation

This activity sets out to challenge students’ mathematical understanding of division. If your students are younger or have less experience with division, use easier numbers – it is the picture they build and the thinking they do that are important here.

This activity is best done with groups of four to six students working together. You may want to look at Resource 2, ‘Using groupwork’, to help you prepare for this activity.

### The activity

Rajni is to have new ceramic floor tiles in her bedroom (Figure 1). The length of the bedroom floor is 5,273 mm and its breadth is 4,023 mm.

Figure 1 Tiles for Rajni’s bedroom floor.

She has looked at a catalogue and shortlisted two designs of tiles:

• Pink Sparrow
• Rosewood Matte.

Pink Sparrow is a square tile of 600 mm in length and Rosewood Matte is a square tile of 450 mm in length.

• What is the shape of Rajni’s bedroom floor?
• Draw Rajni’s bedroom floor in your notebook. Before you start, think about this: you cannot draw a rectangle of length 5273 mm and breadth 4023mm in a notebook. How will you manage to draw a rectangle that represents Rajni’s bedroom floor? Discuss in your group.
• (Note for the teacher: If your class is younger or you have not yet covered scales, skip to the next question). Now:
• Draw three different-sized diagrams, using different scales of Rajni’s bedroom floor as accurately as you can. (Use just one scale if different scales makes the activity too difficult.) Remember to note the scale used next to the diagram. Describe the differences between the three diagrams.
• In each of the three drawings, cover the floor diagram with Pink Sparrow tiles and Rosewood Matte tiles.
• Is the number of tiles used to cover the diagrams of different sizes the same? Why?
• How many rows of tiles would Rajni use to cover the floor with each type of tile? Would these rows cover the entire floor? Why, or why not?
• How many columns of tiles would Rajni use to cover the floor with each type of tile? Would these columns cover the entire floor? Why, or why not?
 Video: Using groupwork

## Case Study 1: Mrs Agarwal reflects on using Activity 1

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

I tried this activity with a class that was having major problems with the division algorithm. It seemed as though they could not understand what it really meant when they were dividing two quantities.

I reduced the numbers for this activity to three-digit and two-digit numbers so that they could do the arithmetic more easily, as I wanted them to think about and visualise what they were doing. I thought that some of them would have a problem representing the room on paper, so I did some exercises with their atlases where we saw how scales were used to represent large distances. After that, when I actually did this activity, most of them had the idea and could draw the three bedroom floor figures.

Arun wanted to know what I meant by ‘three sizes’. Mita answered his query by saying that maybe we could take different scale sizes. I gave them sheets of squared paper and they all got busy with their drawings. It was taking some time, so I gave them a time limit and said that they should delegate the different sizes to different students in their group.

When they had done their drawings of the bedroom, I asked them if they could make copies so that each student in the group had one copy to use to make the tile layout. I then asked them to cover the floor with either Pink Sparrow tiles or Rosewood Matte tiles.

I remembered to ask: ‘Why is the number of tiles the same, even though the scales are different?’ At first they were unsure of how to answer but eventually someone said ‘Because it’s the same room’ and another said ‘We had to use the scale for the tiles as well’. I was pleased with these answers because they weren’t just mechanically doing what was asked, but were thinking about what the mathematics meant.

Then we had the discussion on the last two questions with contributions from the entire class. They were a lot quicker at coming up with answers to the ‘Why?’ questions this time and I was able to help them see that they were dividing up the space and doing divisions, which helped them a great deal. The remainder meant something here as well – ‘the bit left over’ that you would have to cut a tile to fill was a real concept for them.

## 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 able to progress, and those 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 students 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 Agarwal did some quite small things that made a difference.

 Pause for thought Good questions to trigger reflection are: 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?How well did your students understand the mathematical concept?Did you modify the task in any way? If so, what was your reasoning for this?

1 The formal division algorithm

3 Making connections