Mathematics · Grade 7 · Number Sense and Proportional Thinking · Term 1

Introduction to Rational Numbers

Classifying and ordering rational numbers, including positive and negative fractions and decimals, on a number line.

• Ontario Curriculum Expectations
• 7.NS.A.1
• 7.NS.A.2

About This Topic

The logic of integers is a foundational shift in Grade 7, moving students beyond whole numbers to a full number line that includes negative values. In the Ontario curriculum, this involves understanding integers as positions and as movements or vectors. Students explore how these numbers represent real world contexts like temperature, altitude, and financial debits or credits. This topic is essential because it sets the stage for algebraic thinking and coordinate geometry in later years.

By focusing on the relationship between positive and negative values, students learn that zero is not just 'nothing' but a point of equilibrium. They investigate how operations like adding a negative or subtracting a negative change the direction of movement on a number line. This conceptual leap can be challenging, as it often contradicts earlier intuition about subtraction always making a number smaller. This topic comes alive when students can physically model the patterns through movement and collaborative problem solving.

Key Questions

1. Differentiate between integers, rational numbers, and irrational numbers.
2. Analyze how the position of a rational number on a number line reflects its value.
3. Compare and contrast different forms of rational numbers (fractions, decimals, percents).

Learning Objectives

• Classify given numbers as integers, rational numbers, or irrational numbers.
• Compare and order positive and negative fractions and decimals on a number line.
• Analyze the relationship between the position of a rational number on a number line and its magnitude.
• Explain how fractions, decimals, and percents can represent the same rational number.
• Demonstrate the value of rational numbers using a number line model.

Before You Start

Introduction to Integers

Why: Students need a foundational understanding of positive and negative whole numbers and their representation on a number line.

Fractions and Decimals

Why: Students must be familiar with representing parts of a whole as fractions and decimals before they can extend this to negative values.

Key Vocabulary

Rational Number	A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. This includes all integers, terminating decimals, and repeating decimals.
Integer	A whole number or its negative counterpart. Integers include ..., -3, -2, -1, 0, 1, 2, 3, ...
Irrational Number	A number that cannot be expressed as a simple fraction. Its decimal representation is non-terminating and non-repeating, such as pi or the square root of 2.
Number Line	A visual representation of numbers, typically horizontal, with equally spaced points marked to represent integers and other numbers. It helps visualize the order and magnitude of numbers.

Watch Out for These Misconceptions

Common Misconception

Subtracting always results in a smaller value.

What to Teach Instead

Students often struggle when subtracting a negative number, expecting the result to decrease. Using a physical number line or counters helps them see that removing a 'debt' or a negative value actually increases the total value.

Common Misconception

A negative number with a larger digit is 'bigger' than one with a smaller digit (e.g., -10 > -2).

What to Teach Instead

Students sometimes confuse absolute value with the actual value of the integer. Peer discussion and comparing positions on a vertical thermometer model can help them visualize that -10 is 'lower' or 'less' than -2.

Active Learning Ideas

Human Number Line: Vector Walks

Mark a large number line on the floor with tape. Students take turns acting as 'vectors,' starting at a specific integer and physically walking forward or backward based on addition or subtraction cards drawn by their peers.

30 min·Whole Class

Inquiry Circle: The Zero Pair Challenge

In small groups, students use two-coloured counters to represent integer expressions. They must find multiple ways to represent the same value (e.g., +3) using different numbers of 'zero pairs' and explain their reasoning to the group.

20 min·Small Groups

Think-Pair-Share: Integer Storytelling

Students are given a calculation like (-5) - (-8). They work individually to create a real-world scenario that fits the math, share it with a partner to check for logic, and then present the most creative scenario to the class.

15 min·Pairs

Real-World Connections

• Financial analysts use rational numbers to represent profit and loss, tracking stock market fluctuations and company earnings with fractions and decimals.
• Pilots and air traffic controllers use rational numbers to communicate altitude and flight paths, ensuring safe distances between aircraft using positive and negative values relative to sea level.
• Temperature forecasts use positive and negative decimals to describe conditions, allowing people to plan for weather ranging from hot summer days to freezing winter nights.

Assessment Ideas

Exit Ticket

Provide students with a number line marked from -5 to 5. Ask them to plot the following numbers: -3.5, 2/3, -1, 4.75. Then, ask them to write one sentence explaining why -3.5 is less than -1.

Quick Check

Present students with a list of numbers: 5, -2, 1.5, -3/4, 0, -0.8. Ask them to sort these numbers from least to greatest and identify which are integers and which are rational numbers.

Discussion Prompt

Pose the question: 'Can an integer also be a rational number? Explain your reasoning using examples of both positive and negative integers.' Facilitate a class discussion where students share their ideas and justify their answers.

Frequently Asked Questions

How can active learning help students understand integers?

Active learning allows students to visualize the 'direction' of numbers. Instead of memorizing rules like 'two negatives make a positive,' students use movement or manipulatives to see why the direction flips. This builds a mental model of integers as vectors rather than just static symbols, making the logic of operations much more intuitive and memorable.

What are zero pairs in Grade 7 math?

Zero pairs consist of one positive and one negative counter that together equal zero. They are a vital tool for teaching integer addition and subtraction. By adding or removing zero pairs, students can perform operations like subtracting a larger positive from a smaller one without breaking mathematical logic.

Why do we teach integers using number lines and counters?

Number lines emphasize the linear relationship and distance between numbers, while counters focus on the concept of quantity and balance. Using both models ensures that students understand integers as both a location and an amount, which is necessary for higher-level algebra.

How do integers relate to real-life Canadian contexts?

Integers are perfectly modeled through Canadian weather, where temperatures frequently drop below zero. They also apply to golf scores, changes in the stock market, or tracking the depth of a diver in the Great Lakes. These contexts make the abstract concept of negative numbers concrete.