Linear Inequalities

Active learning works well for linear inequalities because students need to physically manipulate symbols, test values, and visualize solutions to move beyond procedural steps. Moving from abstract symbols to concrete representations helps Year 10 students internalize why operations like sign flips matter and how intervals function.

• Year 10
• Mathematics
• 4 activities
• 25 min–45 min

Learning Objectives

1. Solve linear inequalities in one variable, including those requiring multiplication or division by negative numbers.
2. Represent the solution set of a linear inequality on a number line using appropriate notation.
3. Graph the solution set of a linear inequality in two variables on a coordinate plane, distinguishing between boundary lines and shaded regions.
4. Compare and contrast the process of solving linear inequalities with solving linear equations.
5. Construct a real-world scenario that can be accurately modeled using a linear inequality.

Think-Pair-Share

⏱ 30 min·Pairs

Card Sort: Inequality Solutions

Prepare cards with inequalities, solution steps, number lines, and graphs. In pairs, students match sets correctly, discussing sign flips. Review as a class by projecting matches.

Prepare & details

Explain how solving inequalities differs from solving equations.

Facilitation Tip: During Card Sort: Inequality Solutions, circulate to listen for misconceptions about solution sets and redirect by asking students to explain their card placement using inequality language.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 25 min·Pairs

Number Line Walk: Testing Points

Mark a floor number line from -10 to 10. Pairs select inequalities, walk to test points, and justify why points work or fail. Record correct intervals on mini-whiteboards.

Prepare & details

Analyze the impact of multiplying or dividing by a negative number on an inequality.

Facilitation Tip: For Number Line Walk: Testing Points, ensure students physically mark test points and connect their choices to the inequality’s direction before discussing the sign flip rule.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 45 min·Small Groups

Real-World Modelling: Group Challenges

Small groups create and solve inequalities from scenarios like mobile data limits or temperature ranges. Graph solutions and present to class, critiquing peers' models.

Prepare & details

Construct a real-world problem that can be modelled by a linear inequality.

Facilitation Tip: In Real-World Modelling: Group Challenges, require each group to present their inequality and solution range, using their real-world context to justify why the interval is open or closed.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 35 min·Whole Class

Graphing Relay: Coordinate Grids

Whole class divides into teams. One student per team graphs an inequality segment on a shared grid, tags next teammate. First accurate graph wins.

Prepare & details

Explain how solving inequalities differs from solving equations.

Facilitation Tip: In Graphing Relay: Coordinate Grids, provide grid paper with pre-marked axes to save time and focus attention on shading and boundary lines.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teachers should emphasize testing values early to build intuition about intervals and sign flips. Avoid rushing to the rule; instead, let students discover the pattern through guided exploration. Research suggests using multiple representations—algebraic, number line, graph—strengthens spatial reasoning and algebraic fluency.

What to Expect

Successful learning looks like students confidently solving inequalities, using correct notation on number lines, and shading accurate regions on graphs. They should explain why solutions are ranges and when to flip inequality signs, using language like 'less than' and 'greater than' correctly.

Watch Out for These Misconceptions

Common Misconception

During Number Line Walk: Testing Points, watch for students who ignore the sign flip when multiplying by a negative.

What to Teach Instead

Ask them to test a point on the original inequality, multiply both sides by a negative number, then test the same point again on the new inequality. The shift in shading or validity reveals the rule.

Common Misconception

During Card Sort: Inequality Solutions, watch for students who treat solutions as single values.

What to Teach Instead

Have them place their solutions on a number line or graph to see continuous intervals. Ask them to explain why a range like x > 3 includes all points to the right, not just one.

Common Misconception

During Graphing Relay: Coordinate Grids, watch for students who confuse open circles with equality.

What to Teach Instead

Provide a blank number line and ask them to draw the boundary with a string, then test a point inside and outside the boundary to confirm whether it satisfies the inequality.

Assessment Ideas

Exit Ticket

After Card Sort: Inequality Solutions, provide each student with a blank inequality card and ask them to write a solution set for 4 - 2x ≤ 6, represent it on a number line, and explain why the circle is closed.

Quick Check

During Graphing Relay: Coordinate Grids, ask each team to pause and identify one boundary point, one point in the shaded region, and one point outside. Circulate to check correct shading and boundary notation.

Discussion Prompt

After Real-World Modelling: Group Challenges, ask each group to share their inequality and solution. Then pose: 'How would your solution change if the inequality sign flipped? Show me on your graph what that would look like.' Listen for explanations about boundary lines and shading direction.

Extensions & Scaffolding

• Challenge students to create a real-world inequality problem with a solution they represent in all three formats: algebraic, number line, and graph.
• Scaffolding: Provide partially solved inequalities with missing steps for students to complete before plotting solutions. Use smaller numbers to reduce arithmetic errors.
• Deeper exploration: Introduce compound inequalities like 2 < x + 3 ≤ 7 and ask students to solve, graph, and justify each step using their understanding of isolated inequalities.

Key Vocabulary

Inequality	A mathematical statement that compares two expressions using symbols like <, >, ≤, or ≥, indicating that one expression is not equal to the other.
Solution Set	The collection of all values that satisfy a given inequality. This can be a range of numbers or a region on a graph.
Number Line Representation	A visual method for displaying the solution set of an inequality in one variable, using open or closed circles and shading to indicate the interval of values.
Boundary Line	The line corresponding to an equation (e.g., y = mx + c) that separates the coordinate plane into regions for inequalities in two variables.
Shaded Region	The area on a coordinate plane that represents all the points (x, y) satisfying a linear inequality in two variables.