Understanding Inequalities

Grade 6 Mathematics – Lesson 4 of 4

Unit: Mastering Variables and Expressions
Duration: 60 minutes
Class Size: 25 students
Ontario Curriculum Reference:

• Number Sense and Algebra (6m47): Demonstrate an understanding of inequalities by solving and graphing their solutions involving whole numbers to 100.
• Number Sense and Algebra (6m50): Solve and verify simple one-step and two-step linear inequalities involving whole numbers up to 100.
• Communication (6m22): Communicate mathematical thinking clearly using representations including graphs.

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Learning Goals

By the end of this lesson, students will be able to:

• Solve two-operation inequalities involving whole numbers up to 100.
• Verify their solutions to inequalities through substitution.
• Graph inequality solutions on a number line accurately, understanding open and closed circles.
• Develop problem-solving strategies with inequalities in real-world contexts.

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Materials Needed

• Whiteboard and markers
• Number line templates (to be handed out)
• Inequality signs cards (>, <, ≥, ≤)
• Student notebooks with guided notes template
• Mini-whiteboards and markers for each student
• Counters or small manipulatives
• Exit slips for assessment
• Projector or chart paper showing example problems and graphs

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Lesson Breakdown

1. Activation & Review (10 minutes)

• Start with a quick review quiz/game on solving simple one-step inequalities using mini-whiteboards, e.g., x > 5, x + 3 < 10.
• Discuss how inequalities compare to equations — highlight the difference in solution sets (not one fixed answer).
• Briefly recap concepts from previous lessons: variables, expressions, and one-operation inequalities.

2. Guided Instruction – Two-operation Inequalities (15 minutes)

• Distribute guided notes with step-by-step strategies to solve inequalities involving two operations (e.g., 2x + 5 < 19).
• Teacher models solving inequalities on the board:
  • Identify the operations and isolate the variable using inverse operations.
  • Remind students to reverse the inequality sign if multiplying/dividing by a negative (though here, whole numbers only).
• Introduce verification by substituting solution values back into the inequality.

Curriculum Alignment: Emphasizes understanding and verifying inequalities (6m50).

3. Hands-on Practice & Manipulatives (15 minutes)

• Students work in pairs with counters and inequality sign cards to represent inequalities physically.
• Activity example: Given 3x - 4 ≤ 11, one student sets the expression with counters, and the other tests possible values.
• After physical exploration, students solve similar inequalities independently in their notebooks and verify.
• Circulate and provide targeted support.

Ontario Expectation: Demonstrate understanding through manipulatives and communication (6m22).

4. Graphing Inequalities on Number Lines (10 minutes)

• Explain how to represent inequalities graphically using number lines:
  • Open circle for < and >; closed circle for ≤ and ≥.
• Display examples on the board; model several with different inequality signs and solutions.
• Students practice graphing 4 inequalities on prepared number line templates.

5. Real-World Problem Solving (5 minutes)

• Pose a contextual problem:
  “Maria needs to save at least 50 dollars. She already has 20 and plans to save 5 dollars each week. Write and solve an inequality to find how many weeks she needs to save.”
• Students write inequality, solve it, and graph the solution.

6. Closure & Assessment (5 minutes)

• Collect exit slips where students:
  • Solve one inequality involving two operations.
  • Graph the solution on a number line.
  • Briefly explain their reasoning in one or two sentences.
• Quick class discussion summarizing key points and answering questions.

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Differentiation Strategies

• For students needing support: Provide number lines with labelled increments and additional visual aids. Use step-by-step scaffolding in their guided notes.
• For advanced learners: Challenge with inequalities involving addends or subtrahends greater than 100 or require creating their own real-world inequality problems and graphing solutions.
• For ELL students: Use inequality sign cards with visual cues and paired activities to encourage peer support and oral explanations.

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Assessment and Evaluation

• Formative: Observation during hands-on activities and number line graphing.
• Summative: Exit slips assessing the ability to solve and graph two-operation inequalities correctly and explain their reasoning.
• Learning Skills: Collaboration during pair work, communication clarity in exit slips, and perseverance with challenging problems.

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Teacher Reflection Prompts

• Did students demonstrate clear understanding of inequality concepts and solution strategies?
• Were students able to accurately represent the inequalities visually on number lines?
• How effectively did manipulatives support understanding?
• What adjustments would improve engagement or comprehension in future lessons?

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This lesson plan ensures a rich blend of conceptual understanding, procedural fluency, and application aligned with Ontario’s curriculum expectations for Grade 6 mathematics.