Mathematics · Grade 6 · Algebraic Thinking and Expressions · Term 2

Graphing Inequalities on a Number Line

Representing solutions to inequalities on a number line.

• Ontario Curriculum Expectations
• 6.EE.B.8

About This Topic

Graphing inequalities on a number line lets students represent solution sets visually for one-variable inequalities. They plot boundary points with open circles for strict inequalities, such as x > 3, and closed circles for inclusive ones, like x ≥ 3. Arrows indicate the direction of all solutions: right for greater than or equal to, left for less than or equal to. This skill directly addresses key questions on graphical representation, circle types, and symbol direction.

In the algebraic thinking unit, graphing inequalities extends students' work with expressions and equations. It develops number sense by showing ranges rather than single points, a shift from equality solutions. Students connect this to real-world contexts, such as budgeting time where t ≤ 60 minutes or temperatures where T > 0°C. Practicing these graphs strengthens logical reasoning and prepares for multi-step inequalities in later grades.

Active learning shines here because students physically draw, shade, and compare graphs in collaborative settings. Hands-on tasks with manipulatives or digital tools provide instant feedback, while peer discussions clarify misconceptions about circles and arrows. This approach makes abstract relational thinking concrete and boosts retention through movement and dialogue.

Key Questions

1. Explain how to graphically represent the solution set of an inequality.
2. Differentiate between an open circle and a closed circle on an inequality graph.
3. Analyze how the direction of the inequality symbol affects the graph.

Learning Objectives

• Identify the boundary point and direction of the solution set for a given inequality.
• Differentiate between open and closed circles on a number line graph based on the inequality symbol.
• Graph the solution set of one-variable inequalities on a number line with accuracy.
• Analyze how the direction of the inequality symbol (>, <, ≥, ≤) dictates the shading on the number line.
• Compare and contrast the graphical representations of strict inequalities (>, <) and inclusive inequalities (≥, ≤).

Before You Start

Number Lines

Why: Students need to be able to accurately locate and label integers and sometimes fractions or decimals on a number line.

Introduction to Algebraic Expressions

Why: Students should have a basic understanding of variables and how they represent unknown quantities.

Solving One-Step Equations

Why: Familiarity with isolating a variable helps in understanding the concept of a boundary point in inequalities.

Key Vocabulary

Inequality	A mathematical statement that compares two expressions using symbols like <, >, ≤, or ≥, indicating that they are not equal.
Solution Set	The collection of all values that make an inequality true. On a number line, this is represented by a shaded region and specific points.
Boundary Point	The specific number in an inequality that separates the solution set from the non-solution set. It is the value the variable is compared to.
Open Circle	A circle on a number line graph that indicates the boundary point is NOT included in the solution set, used for strict inequalities (<, >).
Closed Circle	A circle on a number line graph that indicates the boundary point IS included in the solution set, used for inclusive inequalities (≤, ≥).

Watch Out for These Misconceptions

Common Misconception

All inequalities use closed circles.

What to Teach Instead

Strict inequalities like x > 2 require open circles to exclude the boundary point. Active pair checks during graphing relays help students verbalize the difference and correct each other's work through immediate peer feedback.

Common Misconception

The arrow always points right.

What to Teach Instead

Less-than inequalities point left from the boundary. Group hunts for real-world examples prompt discussions that reveal how symbol direction matches the solution range, building intuitive understanding.

Common Misconception

Shading is needed on number lines.

What to Teach Instead

Arrows suffice for infinite rays; shading applies more to coordinate planes. Whole-class demos let students test and abandon shading, refining their mental model collaboratively.

Active Learning Ideas

Pairs Relay: Inequality Graphing

Pairs take turns drawing a given inequality on mini number lines, such as x < 4, while their partner checks for correct circle type and arrow. Switch roles after five inequalities. Debrief as a class to share common patterns.

30 min·Pairs

Small Groups: Real-World Inequality Hunt

Provide scenarios like 'ages greater than 12 for a movie.' Groups write inequalities, graph them on shared number lines, and justify choices. Rotate graphs for peer review and revisions.

45 min·Small Groups

Whole Class: Interactive Number Line Demo

Project a large number line. Call out inequalities; students use sticky notes to mark boundaries and arrows as a group. Discuss adjustments based on class input.

20 min·Whole Class

Individual: Inequality Graph Journal

Students solve and graph 10 varied inequalities in journals, coloring solution regions. Pair share to verify before submitting.

25 min·Individual

Real-World Connections

• A city planner might use inequalities to represent acceptable noise levels for residential areas, such as decibels < 60 dB, to ensure peaceful living conditions.
• A baker needs to ensure oven temperatures are within a safe range for baking delicate pastries, perhaps T ≥ 325°F and T ≤ 350°F, to achieve perfect results.
• A coach might set a time limit for a specific drill, stating that the time taken must be less than or equal to 5 minutes (t ≤ 5), to keep the practice session efficient.

Assessment Ideas

Quick Check

Present students with three inequalities: x < 5, y ≥ -2, and z > 0. Ask them to draw a number line for each, correctly placing the boundary point, using the appropriate circle type, and shading in the correct direction. Review their number lines for accuracy.

Exit Ticket

Give students a pre-drawn number line with a boundary point, circle, and shading. Ask them to write the inequality that this graph represents and explain in one sentence why they chose an open or closed circle.

Discussion Prompt

Pose the question: 'Imagine you are explaining how to graph inequalities to a younger student. What are the two most important things they need to remember about the circle and the arrow on the number line, and why?' Facilitate a brief class discussion to consolidate understanding.

Frequently Asked Questions

What does an open circle mean on an inequality graph?

An open circle at the boundary point, like at 3 for x > 3, shows the point is not included in the solution set. Students plot it to mark the cutoff without filling it. This visual cue helps distinguish strict inequalities from inclusive ones during equation solving.

How do you graph x ≥ 5 on a number line?

Place a closed circle at 5 to include that value, then draw an arrow right to show all numbers greater than or equal to 5. Practice reinforces that ≥ combines equality with greater than. Connect to contexts like minimum scores for passing.

How can active learning help students graph inequalities?

Activities like pairs relays and group hunts engage students in drawing, checking, and discussing graphs kinesthetically. This builds confidence with circles and arrows through trial and error, peer teaching, and real-world ties. Collaborative feedback corrects errors faster than worksheets, deepening relational thinking about solution sets.

Why does the inequality symbol direction matter on graphs?

Greater-than symbols (> or ≥) send arrows right; less-than (< or ≤) send them left. This mirrors the number line's order. Analyzing flipped symbols in discussions helps students predict graphs accurately for algebraic patterns.