5 Working on a bigger scale

Asking the students to work on a real-life scale can be useful in helping them to see division in the world around them. The next activity continues to ask the students to explore the division algorithm and to use symbolic mathematical language, but this time they will be working on a bigger scale.

Thinking about a real situation and preparing an answer to a problem they have identified will help the students further develop their powers of visualisation and give visual meaning to symbolic mathematics.

Activity 4: Using the division algorithm in a real-life situation

Part 1: In the school grounds

Preparation

In this activity students will create a tiling plan for a particular part of their school, for example an area of the playground.

When taking students to work in the school grounds (Figure 2), you should always make sure your students are aware of safety hazards they might encounter, such as moving vehicles or building works, and prepare for changes in the weather.

Figure 2 Using the school ground for an activity.

The activity

Divide the class into groups of two or three students. Ask each group to do the following:

  • Identify some part or portion of the school where they would like to lay tiles. The constraint is that the shape of the portion must be a rectangle. They can choose stairs or steps, hallways, rooms, open areas, floors or walls, etc.
  • Measure the length L and breadth B of the floor or wall they would tile.
  • Choose the length (l) of the square tile with which they want to tile the floor or wall.
  • For the chosen tile, calculate the value of q and r (refer to previous activity) for both L and B.

Once they are done, every group comes back to the classroom to report.

Part 2: Back in the classroom

Preparation

On the blackboard, make a table as shown in Table 1 with as many rows as there are groups.

Table 1 Using the division algorithm in a real-life situation.

L l q r B l q r
Group 1
Group 2
Group 3
Group 4
Group 5
Group …

The activity

Ask the groups to fill in their row in the table on the blackboard with their findings from Part 1 of this activity. Discuss the following with the class:

  • What is the same and what is different between the observations of each group of students?
  • Did some of you get the value of r as 0? Why do you think you got this value?
  • If you wanted to make sure that r = 0, how would you change the values of l?
  • If for both L and B the value of r = 0, what is the relation between L and B?
  • How is the relation between (L, l, q and r) linked to the division algorithm?

Case Study 4: Mr Chadha reflects on using Activity 4

Now this was an extremely interesting activity for the students to take part in. I had to, of course, get tape measures for them to take the measurements. They all had a great time making a plan of the different areas they could measure and they enjoyed discussing where tiles would be a very good idea.

Some of the groups were talking about trying to measure some difficult-shaped areas, so I advised them to keep it simple. They went out quickly once their plans were made because I told them they had just ten minutes to get their measurements and be back in the classroom! I stood outside the classroom with my watch reminding them to be quick about what they had to do.

When they were all back in the classroom they had to decide whether to use big or little tiles; some of them thought that big tiles would be best but then did not like the division sums that they had set themselves so went for smaller ones. After about another five minutes they all had their answers ready to go into the table which I had drawn on the blackboard.

The discussion about whether any of them got r = 0 was interesting and there was a lot of talk about how and when they could get that. It also led to the talk about multiplicands and the divisibility of numbers and when we could say a number can be completely divided by another. We also had a talk about how remainders can be any number left when we remove a certain multiple of the divisor and why we try to remove the maximum numbers of times of the divisor. This included thinking about how the remainder could not be more than the divisor.

I think by the end of this activity I can say that most of my students were able to understand the division algorithm and would know why they were doing what they were doing when using it.

Pause for thought

  • What responses from students were unexpected? What did this reveal about their understanding of the division algorithm?
  • 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?

4 The division algorithm again