# 2 The need for negative numbers

The activities in this unit will help you to develop your students’ understanding of why negative numbers are used and how useful they can be. They will also offer ideas about how to help your students understand how to work with negative numbers rather than just remembering rules. The first activity is designed to help students appreciate the need for negative numbers as part of the numbering system.

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 a learner’s experiences, which can, in turn, influence your teaching and your experiences as a teacher.

## Activity 1: Understanding the need for negative numbers

### Preparation

This activity suggests three different ideas to help students appreciate the need for negative numbers as part of the number system. Using all the ideas, not necessarily in the same lesson, will give the students a wider exposure to thinking about negative numbers.

To illustrate ‘positive’ and ‘negative’, try to find pictures of mountains and deep seas so that the ideas of ‘above’ and ‘below’ can be discussed, along with a zero that is at sea level. Can you think of other situations where positive and negative would be obvious or intuitive to the students?

The activity

### Idea 1: Above and below sea level

Draw a large picture, either on a large piece of paper on the wall or on the blackboard. Your picture should show the sea, mountains above the sea, and space below sea level. Use the pictures that you have collected from magazines or drawn. Suitable items would be a plane, an octopus, a whale, a boat, a house, a car, a fish, etc.

Ask the students where they would place the items on your picture. Encourage them to say ‘above the sea level’ or ‘under the sea level’. When all of the items are stuck on, discuss how high the plane might be and how far under the sea the octopus might be, and so on. Introduce the minus sign to indicate ‘under the sea level’.

### Idea 2: Robot steps

Make a space down the centre of the classroom, ensuring that all the students can see this pathway. Mark its centre with a chalk cross and ask a student to stand on the cross. Tell the class to imagine that the student is a robot who only moves forwards and backwards in a straight line. Use pieces of paper or chalk marks to number paces forward from the cross.

Ask the robot to move to 2, then ask them to move two spaces back. Ask the students to say what number should be put on the cross – hopefully they will say zero.

Ask others to give the robot instructions to move to a certain number and then back to a certain number. Now ask the robot to move to 3 and then move four spaces backwards. They have gone beyond zero! What number can be used to represent one step back from zero? Introduce other numbers beyond zero and ask the students to practise saying negative numbers by telling the robot where to move to.

### Idea 3: A game with benches

Place as many benches as you can across the front of the room and divide the benches into individual seats with chalk lines. Write with chalk a zero on one of the seats (not at the end) and then number the other seats on the benches to the right of zero as 1, 2, 3 and so on. Ask the students how the seats to the left could be numbered. Suggest the negative sign if they do not think of it.

Then play games that involve negative and positive numbers. For example:

• Stand a student behind a seat. The class call out the seat they want the student to move to, for example ‘5’ or ‘–2’ and so on.
• Sit a student on the seat and ask the class to say which seat the student should sit on. Encourage them to use just ‘3’ or ‘5’, and so on for numbers to the right, and ‘negative 2’, ‘negative 4’ and so on for numbers to the left of zero.

Next, make the task more difficult. Sit a student on the seat labelled 5 and ask the class what ‘move’ has to be made to go to seat 2. This is more difficult because ‘negative 3’ can indicate a position relative to zero and can indicate the action of moving three to the left. Make sure you discuss these two meanings.

Now ask the student to make a move and then ask what move would ‘undo’ that move.

Use games like this as often as you can to help build confidence. You could stick the numbers to the wall rather than use chairs. In this way the students will use addition and subtraction of negative numbers naturally for the game.

 Video: Using local resources

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

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

I remember that my classes developed a dislike for negative numbers because there seemed to be so much to remember and it was easy to get mixed up.

I decided to play some of the games in Activity 1 with them. They already knew about negative numbers so they were quick to say that the octopus would be at negative 8 metres. I drew the picture on paper on the wall with a scale marked positive and negative, and left it there after we had done this brief activity. In the morning many of the students arrived with pictures they had drawn so we put them in their correct places on the big picture and had another occasion to think about positive and negative numbers.

Later in the term we played the bench game. They enjoyed this and, although they sometimes found it hard working out a move that is across the zero – from say 5 to –2 – they practised this a lot just because they wanted to keep playing. I definitely think making the actual moves themselves or instructing others to do so helped them to be able to visualise what was happening when we started to do the exercises on negative numbers in the textbook.

To help make the step to using the textbook even easier, I think I will repeat some of these ideas and then also have discussions with the students about how we could record what we are doing in mathematical notation and write this on the blackboard. Hopefully they will then see how the actions relate to the mathematical notation and sums, and to what is asked in the textbooks.

## 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 being 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. When you reflect on how the ideas in Activity 1 went with your class, make a note, as Mrs Kapur did, of some quite small things that made a difference.

 Pause for thought In the case study, Mrs Kapur said that she was thinking about repeating some of the activities and recording the outcomes on the blackboard using mathematical notation and sums. What do you feel might be the advantages of doing this after the students have a lot of experience of the activities and games?Now think about the following questions:How did the activities and games go with your class? What responses from students were unexpected? What did these tell you about their understanding of positive and negative numbers?What questions did you use to probe your students’ understanding? What points did you feel you had to reinforce?What will you do differently next time you use these activities in your teaching?

3 Using number lines to develop understanding of positive and negative numbers