Mathematics · Grade 6 · The Number System and Rational Quantities · Term 1

Comparing and Ordering Integers

Using number lines and inequalities to compare and order integers.

• Ontario Curriculum Expectations
• 6.NS.C.7.A
• 6.NS.C.7.B

About This Topic

Comparing and ordering integers requires students to locate positive numbers, negative numbers, and zero on a number line. They plot points accurately and use symbols like <, >, and = to show relationships between integers. For example, students determine that -5 is less than 3 because it lies farther left on the number line. This skill connects to everyday contexts such as temperatures below zero or bank balances in the negative.

In the Ontario Grade 6 curriculum, this topic strengthens number sense within The Number System and Rational Quantities unit. Students explain why -2 > -7 and predict comparisons without plotting every time. These abilities prepare them for adding, subtracting, and working with rational numbers later in the year. Class discussions reveal how integers model real-world quantities that increase or decrease.

Active learning shines here because integers can feel abstract at first. When students physically move along floor number lines or sort integer cards into order, they build spatial intuition kinesthetically. Group challenges with inequality statements turn comparisons into collaborative problem-solving, making the concepts stick through movement and peer teaching.

Key Questions

1. Construct a number line to accurately order a set of integers.
2. Explain how inequalities are used to describe relationships between integers.
3. Predict the outcome of comparing two integers based on their position on a number line.

Learning Objectives

• Compare and order sets of integers, including positive numbers, negative numbers, and zero, using a number line.
• Explain the meaning of inequality symbols (<, >, =) when comparing two integers.
• Predict the relative order of integers based on their position on a number line without explicit plotting.
• Represent relationships between integers using inequality statements.
• Analyze the position of integers on a number line to determine their magnitude relative to zero.

Before You Start

Representing Whole Numbers on a Number Line

Why: Students need to be able to accurately place whole numbers and understand their order on a number line before extending this to integers.

Introduction to Positive and Negative Numbers

Why: A basic understanding of what positive and negative numbers represent is foundational for comparing and ordering them.

Key Vocabulary

Integer	A whole number that can be positive, negative, or zero. Examples include -3, 0, and 5.
Number Line	A visual representation of numbers, typically horizontal, with integers ordered from least to greatest from left to right.
Inequality Symbols	Symbols used to show that two numbers are not equal. '<' means 'less than', '>' means 'greater than', and '=' means 'equal to'.
Opposite Integers	Two integers that are the same distance from zero on the number line but in opposite directions. For example, 5 and -5 are opposite integers.

Watch Out for These Misconceptions

Common Misconception

Negative numbers are larger than positive numbers because the negative sign looks bigger.

What to Teach Instead

Students often confuse the sign's appearance with magnitude. Drawing vertical number lines or using temperature examples shows negatives as colder and smaller. Group sorting activities help peers correct each other through visual comparisons on shared lines.

Common Misconception

All numbers to the right of zero are positive, but zero belongs with negatives.

What to Teach Instead

Zero sits between positives and negatives as neither. Hands-on floor number lines let students stand at zero and feel its neutral position. Peer discussions during relays clarify that zero is greater than negatives but less than positives.

Common Misconception

The distance from zero determines order, regardless of direction.

What to Teach Instead

Students might think -7 is greater than -2 because 7 > 2. Card matching games with number lines reveal direction matters more than distance. Collaborative challenges prompt explanations that build correct spatial reasoning.

Active Learning Ideas

Floor Number Line: Human Plotting

Mark a number line on the floor with tape from -20 to 20. Call out integers for students to stand on, then ask pairs to compare their positions and state inequalities aloud. Have them predict where the next number goes before placing it. End with students creating their own sets for classmates.

35 min·Whole Class

Card Sort: Ordering Challenge

Distribute cards with integers like -8, 0, 5, -3. In small groups, students arrange cards on desks from least to greatest, justifying with number line sketches. Switch sets midway and time them for friendly competition. Discuss any errors as a class.

25 min·Small Groups

Inequality Match-Up: Pairs Game

Prepare cards with integers and inequality statements, such as -4 ___ 2. Pairs draw cards, match true statements, and explain using horizontal number lines drawn on paper. Incorrect matches go back; first to 10 wins. Rotate partners halfway.

30 min·Pairs

Real-World Integer Hunt: Individual Scavenger

Students list 10 real-world integers, like golf scores or elevations, then order them on personal number lines. Share in small groups, comparing lists and debating inequalities. Compile class examples on a shared board.

40 min·Individual

Real-World Connections

• Stock market traders compare daily gains and losses, represented by positive and negative integers, to make investment decisions. They need to quickly identify which stocks performed better or worse.
• Meteorologists use integers to report temperatures, comparing daily highs and lows to describe weather patterns. Understanding that -10°C is colder than -2°C is crucial for public safety advisories.
• Scuba divers track their depth using negative integers, with 0 representing the surface. Comparing depths, such as -30 meters and -50 meters, is essential for safety protocols and planning dives.

Assessment Ideas

Exit Ticket

Provide students with three integers, such as -8, 5, and -2. Ask them to: 1. Plot these integers on a provided number line. 2. Write an inequality statement comparing the smallest and largest integers. 3. Explain in one sentence why -8 is less than 5.

Quick Check

Display a number line with several integers marked. Ask students to hold up fingers to indicate the position of a given integer (e.g., 'Show me where -3 goes'). Then, present two integers and ask students to write '<' or '>' on a mini-whiteboard to show their relationship.

Discussion Prompt

Pose the question: 'Imagine you are comparing the scores of two video game players. Player A has a score of -15, and Player B has a score of -7. Who is winning and why? How does the number line help you explain this?' Facilitate a class discussion where students use the vocabulary and number line concepts.

Frequently Asked Questions

How do I introduce number lines for comparing integers in Grade 6?

Start with familiar contexts like temperatures: plot -5°C and 3°C on a horizontal line. Students draw their own lines, mark points, and insert < or >. Gradually add vertical lines for elevations. This builds from concrete to abstract while reinforcing left-to-right order.

What are common errors when ordering mixed integers?

Errors include placing negatives right of positives or ignoring zero's position. Use integer chips: red for negative, black for positive. Students model sets and order them physically. Class anchoring charts with examples prevent repetition of these mistakes.

How can active learning help teach comparing integers?

Active methods like human number lines or card sorts make abstract positions concrete through movement and touch. Students internalize order by physically arranging themselves or peers. Group justifications during games foster verbal reasoning, turning errors into teachable moments and boosting retention over worksheets.

What real-world examples work for integer inequalities?

Use temperatures (-10°C < 5°C), debts ($-20 < $10), or floors (basement -1 < floor 3). Students collect local data, like Winnipeg winters, plot on number lines, and write inequalities. This relevance motivates while showing integers model quantities above and below zero.