# Investigating Integers

Purpose

This unit begins with Freudenthal’s (1983) annihilation model for demonstrating the addition and subtraction of integers then goes on to introduce other representations. It is designed for students working at Stage 7 of the Number Framework who are able to choose appropriately from a broad range of mental strategies to estimate answers and solve addition and subtraction.

Achievement Objectives
NA4-2: Understand addition and subtraction of fractions, decimals, and integers.
NA4-6: Know the relative size and place value structure of positive and negative integers and decimals to three places.
Specific Learning Outcomes
• Solve simple addition and subtraction equations involving integers.
Description of Mathematics

This unit of work is useful for students working at stage 7 of the Number Framework, Advanced Multiplicative. Students at this stage select from a broad range of strategies to solve addition problems with decimals, subdividing and recombining numbers to simplify problems.

Although students at this stage will be very familiar with the number line as a model for addition and subtraction this unit does not promote the use of the number line as it is considered problematic as a model to show subtraction of negative numbers.

The key teaching points are:

• An integer is a whole number that can be either greater than 0, called positive, or less than 0, called negative.
• Zero is neither positive nor negative.
• Two integers that are the same distance from the origin in opposite directions are called opposites and when added cancel each other out making 0.

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

• Use physical materials, as suggested in the unit, to represent positive and negative values.
• Use calculators in anticipatory ways, predict a result then check on the calculator.
• Contextualise integers in situations that are meaningful to students, particularly possessing and owing money.

The context for this unit is purely mathematical. Ideas are developed using different coloured objects to represent positive and negative one. Integers apply to a range of everyday contexts that might be used to make the mathematics more accessible to students. Useful contexts include possessing and owing money (whole numbers of dollars), height above and below sea level, scores in sports games like golf, and temperatures.

Required Resource Materials
Activity

Getting started

1. Begin with a true/false conjecture on board for students to discuss. Ask students:

Do you think the statement is true or false and why?        1 – -4 = 5
2. Ask students to discuss their ideas with a learning partner. Listen for student explanations. Discuss the idea of positive and negative numbers; introduce the term integer if it does not arise in the discussion.

Ask students to explain what is happening in the equation. Ask how this equation could be demonstrated with equipment.

3. Explain to students that black counters represent positive and red counters represent negative - show the number 2 as 2 black counters, ask what would happen if I added, three more black and three red, what would I have now? Make sure students understand that the extra three black and three red cancel or annihilate each other so the answer is still 2.
4. Ask students to say some numbers that you model, making sure that you include extra red and black counters to cancel out, question students to check that they know there can be any number of extra red and black so long as these sets are equivalent.
5. Pose an equation for students to model with the equipment

3 – 4 =

Discuss the answer and how they used the equipment to model it.

Repeat this process with the following:

-3 + 2 =

4 – -1 =

- 5 + -2 =

6. Ask students to make up three more to swap with their learning partner to solve.
7. Pose another true/false conjecture for discussion.

-3 – 2 = -1

8. Conclude session by asking students to record or discuss what they have noticed about adding and subtracting integers. A summary of these ideas could be written in the mathematics modelling book.

Exploring

Over the next few days introduce the students to a selection of other models and contexts for addition and subtraction of integers from the list below, students could if they wished use the double coloured counters to model some of the calculations. As they work through tasks, add any new ideas to the modelling book.

1. Integer learning experiences from Book 5: Teaching Addition, Subtraction and Place Value
2. Money Matters (copymaster) - Discuss the idea of cash and debts, this task could be followed up with the Figure It Out lesson Money Matters, p21, Number Year 7/8, Book 4.
3. Close to Zero game (copymaster).
4. Bonuses and Penalties game (copymaster) after playing the game get students to make up some of their own cards.
5. Integer Quick Draw game (copymaster).

Reflecting

1. Pose another True/False conjecture

3 – -12 = -15

Ask the students to decide whether this conjecture is true or false and to explain their thinking to their learning partner.

Revisit the summary of ideas where students have recorded what they have noticed about adding and subtracting integers. Ask students to work with their learning partner to develop some guidelines related to adding and subtracting integers for another classroom, for example adding a negative number is just like…, subtracting a negative number is like… or introduce Alistair McIntosh’s thinkboard, show an equation in the middle and model completing the sections.

Give students an equation to solve on their own thinkboard or with their learning partner. These could be differentiated in relation to student abilities.