Mathematics · Year 7 · The Language of Number · Term 1

Introduction to Integers

Students will define integers and represent them on a number line, understanding their use in real-world contexts.

• ACARA Content Descriptions
• AC9M7N02

About This Topic

This topic introduces Year 7 students to the concept of integers, extending the number system beyond zero to include negative values. Under ACARA standards AC9M7N02 and AC9M7N03, students learn to compare and order integers and perform addition and subtraction using the number line. This is a foundational shift from primary school mathematics, moving from counting objects to understanding directed numbers and magnitude. It matters because it provides the mathematical language for real world phenomena like financial debt, changes in temperature, and sea level.

Students often struggle with the abstract nature of 'less than zero' when it is presented only as a set of rules. By using the number line as a visual and physical tool, they can see that subtraction is simply a change in direction or a distance between two points. This topic comes alive when students can physically model the patterns of movement on a large scale number line or through collaborative games that simulate real world gains and losses.

Key Questions

1. Analyze how negative numbers extend the number system beyond whole numbers.
2. Compare the representation of positive and negative integers on a number line.
3. Construct real-world examples where negative integers are essential for description.

Learning Objectives

• Identify integers on a number line, distinguishing between positive, negative, and zero.
• Compare and order sets of integers, justifying their positions relative to zero.
• Construct real-world scenarios that require the use of negative integers for accurate representation.
• Explain the concept of 'opposite numbers' in the context of integers and their position on a number line.

Before You Start

Whole Numbers and Counting

Why: Students need a solid understanding of whole numbers and their order to extend this concept to negative numbers.

Introduction to Addition and Subtraction

Why: Understanding basic addition and subtraction operations provides a foundation for later integer operations and the concept of movement on a number line.

Key Vocabulary

Integer	A whole number (not a fraction or decimal) that can be positive, negative, or zero. Examples include -3, 0, and 5.
Positive Integer	An integer greater than zero. These are the whole numbers we commonly use for counting.
Negative Integer	An integer less than zero. These numbers represent values below zero on a number line.
Number Line	A visual representation of numbers, typically a straight line with markings for integers. It helps in comparing and ordering numbers.
Opposite Numbers	Two numbers that are the same distance from zero on the number line but in opposite directions. For example, 5 and -5 are opposite numbers.

Watch Out for These Misconceptions

Common Misconception

Students believe that a larger digit always means a larger value (e.g., -10 is greater than -2).

What to Teach Instead

Use a vertical number line or thermometer to show that -10 is 'lower' or 'colder' than -2. Peer discussion about 'who owes more money' helps students distinguish between the magnitude of the debt and the actual value of the balance.

Common Misconception

Thinking that 'subtracting' always results in a smaller number.

What to Teach Instead

Model subtracting a negative as 'taking away a debt.' Hands-on modeling with two-colour counters helps students see that removing a negative chip increases the overall value of the set.

Active Learning Ideas

Simulation Game: The Human Number Line

Create a large number line on the floor using masking tape. Students take turns acting as a 'human counter,' physically stepping forward for addition and backward for subtraction, or turning around to face the negative direction when subtracting a negative integer.

30 min·Whole Class

Inquiry Circle: Global Temperatures

In small groups, students research the record high and low temperatures of various cities across the Asia-Pacific region and the world. They must calculate the total temperature range for each city and present their findings on a vertical number line poster.

45 min·Small Groups

Think-Pair-Share: The Debt Dilemma

Provide a scenario where a person has a bank balance of -$20 and receives a bill for $15. Students individually determine the new balance, discuss their reasoning with a partner to check for direction errors, and then share their strategies with the class.

15 min·Pairs

Real-World Connections

• Temperature readings in places like Antarctica or during winter in Canada often require negative integers to describe temperatures below freezing. For example, a temperature of -10 degrees Celsius indicates a significant drop below the freezing point of water.
• Financial records use integers to track money. A bank account balance might show a positive integer for deposits and a negative integer for withdrawals or debts, such as owing $50, represented as -50 dollars.
• Elevation changes in geography are described using integers. Sea level is often represented as zero, with mountains having positive elevations (e.g., Mount Everest at 8,848 meters) and ocean trenches having negative elevations (e.g., the Mariana Trench at approximately -10,984 meters).

Assessment Ideas

Exit Ticket

Give students a card with a number (e.g., -7, 3, 0). Ask them to write one sentence explaining what this number might represent in the real world and to draw its position on a mini number line.

Quick Check

Display a number line with several points labeled A, B, C, D. Ask students to write down the integer represented by each point and then to order them from least to greatest. Review answers as a class.

Discussion Prompt

Pose the question: 'Why do we need negative numbers in mathematics? Can you think of a situation where only positive numbers or whole numbers would not be enough to describe what is happening?' Facilitate a brief class discussion, guiding students to articulate the necessity of integers.

Frequently Asked Questions

How can active learning help students understand integers?

Active learning moves integers from abstract symbols to physical movements. By using strategies like 'Human Number Lines' or 'Two-Colour Counter' simulations, students physically experience the change in direction and magnitude. This tactile approach helps cement the logic of operations, especially the 'double negative' rule, which is often difficult to grasp through lecture alone.

What are some real world examples of integers for Year 7?

Common examples include temperature (above and below zero), altitude (above and below sea level), and finance (credits and debits). You can also use sports scores, such as golf (under par) or goal differences in football, to make the concept more relatable.

Why is the number line so important for this topic?

The number line provides a consistent visual model that works for integers, fractions, and decimals. It helps students understand that numbers have a specific position relative to zero, making it easier to compare values and visualise the distance between points, which leads into absolute value.

How do I explain subtracting a negative simply?

Frame it as 'removing a loss' or 'cancelling a debt.' If you take away a $5 debt, the person is effectively $5 richer. Using physical tokens to represent 'debts' that get physically removed from a pile helps students see why the result is an increase.