Grade Level

9th Grade

Duration

80 minutes

Common Core State Standards (CCSS)

• HSA-REI.C.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on linear systems in two variables.
• HSA-REI.C.5: Prove that given a system of two linear equations in two variables, one and only one solution, infinitely many solutions, or no solutions exist.

Learning Objectives

• Students will understand the concept of a system of linear equations in two variables.
• Students will graph linear equations and interpret the point(s) of intersection as the solution to the system.
• Students will identify whether systems have one solution, no solution, or infinitely many solutions by analyzing graphs.
• Students will communicate reasoning about the solutions to a system of equations.

Materials

• Graph paper, rulers, colored pencils
• Whiteboard and markers
• Graphing calculators or graphing app (optional)
• Handout with systems of linear equations to graph

Lesson Outline

Introduction (10 minutes)

• Begin with a brief discussion: “What happens when you have two different lines on a graph?”
• Introduce systems of equations by writing two linear equations on the board.
• Explain that solving systems means finding points where the lines intersect — these points satisfy both equations.
• State the learning objectives clearly.

Guided Practice: Graphing Systems (20 minutes)

• Demonstrate graphing two linear equations step-by-step on graph paper:
• Find intercepts or use slope-intercept form.
• Plot points and draw each line clearly.
• Identify the solution to the system as the intersection point.
• Show examples with:
• One solution (intersecting lines)
• No solution (parallel lines)
• Infinitely many solutions (same line)
• Use color coding for clarity.

Student Activity: Hands-on Graphing (25 minutes)

• Distribute handouts with 5 different types of systems.
• Students work in pairs to:
• Graph each system.
• Identify the number of solutions.
• Circulate and provide support.
• Encourage students to explain their reasoning aloud.

Class Discussion & Debrief (15 minutes)

• Ask volunteers to present one graph and explain their solution(s).
• Discuss common mistakes and misconceptions.
• Reinforce how the graph provides a visual solution and verifies algebraic answers.
• Emphasize the connection between graph, solution, and system type.

Exit Ticket (10 minutes)

• Provide 2 quick systems for students to graph independently.
• Students must write their answer and explain: number of solutions and why.
• Collect for formative assessment.

Assessment for Learning

• Observation during pair activity and class discussion.
• Exit ticket review to check individual understanding.
• Use errors and misunderstandings to tailor next lesson on substitution method.

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Substitution Strategy

Grade Level

9th Grade

Duration

80 minutes

Common Core State Standards (CCSS)

• HSA-REI.C.6: Solve systems of linear equations exactly and approximately.
• HSA-REI.C.5: Understand solutions of two variable linear systems — one, none, or infinite solutions.

Learning Objectives

• Students will understand how substitution transforms a system into a single-variable equation.
• Students will solve systems by expressing one variable in terms of the other and substituting.
• Students will analyze solutions for consistency and verify answers.

Materials

• Whiteboard and markers
• Student notebooks
• Worksheets with practice systems
• Calculators (optional)
• Visual substitution flowchart handout

Lesson Outline

Warm-up (10 minutes)

• Review key idea of systems from graphing lesson.
• Pose a simple system and ask, “What if we substitute one part into the other?”
• Introduce substitution as a method where one variable is isolated.

Direct Instruction (15 minutes)

• Model step-by-step substitution on 2-3 systems:

1. Isolate y in one equation.
2. Substitute expression into second equation.
3. Solve single variable equation.
4. Back-substitute to find second variable.

• Emphasize checking solutions by plugging back into original equations.

Guided Practice (20 minutes)

• In pairs, students work through three substitution problems with teacher guidance.
• Prompt explanation of each step verbally or in writing.
• Highlight cases with:
• One solution,
• No solution,
• And infinitely many solutions.

Independent Work (20 minutes)

• Students complete 4 substitution problems in notebooks.
• Differentiated task: 2 easy, 2 challenging problems.
• Teacher circulates for support and clarification.

Group Reflection and Q&A (10 minutes)

• Discuss difficulties encountered.
• Share strategies to avoid common errors (like forgetting to check solutions).
• Ask students how substitution helps avoid “guessing” in graphing.

Formative Assessment (5 minutes)

• Quick quiz: Solve a substitution problem with no guidance.
• Collect for review before next lesson.

Assessment for Learning

• Observation during pair and independent work.
• Quiz to gauge understanding to prepare for elimination lesson.

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Elimination Method

Grade Level

9th Grade

Duration

80 minutes

Common Core State Standards (CCSS)

• HSA-REI.C.6: Solve systems of linear equations exactly and approximately.
• HSA-REI.C.5: Analyze consistency of solutions in systems.

Learning Objectives

• Students will learn to add or subtract equations to eliminate one variable.
• Students will solve systems efficiently using elimination.
• Students will identify solution types through elimination.
• Students will verify solutions to ensure accuracy.

Materials

• Whiteboard and markers
• Student notebooks
• Worksheets with elimination problems
• Colored pencils
• Algebra tiles or manipulatives (optional)

Lesson Outline

Engage & Review (10 minutes)

• Quick verbal review: graphing and substitution.
• Pose question: “What if substitution seems long or tricky?”
• Introduce elimination as a method to combine equations to remove a variable.

Direct Instruction (20 minutes)

• Demonstrate elimination by:
• Aligning equations.
• Multiplying one or both equations to match coefficients.
• Adding/subtracting to eliminate a variable.
• Solving the remaining one-variable equation.
• Back substituting.
• Model at least three examples including:
• One solution,
• No solution,
• Infinite solutions.

Hands-on Practice (25 minutes)

• Students solve elimination systems in pairs.
• Use algebra tiles if available to model elimination visually.
• Encourage annotation of steps.

Student Presentation (15 minutes)

• Volunteers demonstrate solving a problem on the board.
• Class discusses steps, errors, and strategies.
• Teacher provides feedback and models corrections.

Wrap-up & Exit Ticket (10 minutes)

• Exit ticket with two elimination problems, including a no solution case.
• Students submit before leaving.

Assessment for Learning

• Teacher observation.
• Exit ticket review.
• Group discussion participation.

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Choosing Solutions Strategy

Grade Level

9th Grade

Duration

80 minutes

Common Core State Standards (CCSS)

• HSA-REI.C.6: Solve systems of linear equations exactly and approximately.
• HSA-REI.C.5: Understand different solution sets and the methods best suited to solve systems.

Learning Objectives

• Students will analyze systems of equations to decide on the most effective solving method: graphing, substitution, or elimination.
• Students will solve systems using multiple methods and justify their choice.
• Students will evaluate and explain solution types.
• Students will demonstrate understanding by explaining strategies through discussion and written reflection.

Materials

• Whiteboard, markers
• Graph paper
• Calculators
• Student notebooks
• Method choice matrix handout
• Sample systems handout (varied difficulties and structures)
• Exit reflection sheet

Lesson Outline

Warm-up: System Scenarios (10 minutes)

• Present 3 different systems:
• One easy to graph,
• One with fraction coefficients,
• One with coefficients ideal for elimination.
• Ask: Which method would you choose and why?

Mini Lesson: Comparing Methods (15 minutes)

• Review pros and cons of graphing, substitution, and elimination.
• Provide a matrix or flowchart that shows decision points:
• Isolate variable easily? → Substitution.
• Coefficients ready to cancel? → Elimination.
• Want approximate visual? → Graphing.

Collaborative Problem Solving (30 minutes)

• Students in groups of 3-4 receive sets of systems.
• For each system, groups:

1. Decide the best solving method and explain reasoning.
2. Solve the system with chosen method.
3. Verify correctness with a secondary method.

• Groups prepare a brief presentation on one system.

Presentation & Peer Feedback (15 minutes)

• Groups share their chosen system, method, and solution.
• Class asks questions and provides constructive feedback.

Reflection & Exit Ticket (10 minutes)

• Individual written reflection answering:
• How does the choice of method affect efficiency and accuracy?
• Which method do you find most helpful and why?
• Exit ticket with a novel system requiring method choice and solution.

Assessment for Learning

• Group problem-solving observation.
• Presentation clarity and reasoning.
• Written reflection demonstrating conceptual understanding.
• Exit ticket analysis.

EdTPA Tips for Recording

• Capture student engagement in group work.
• Highlight formative questioning and scaffolding.
• Include moments of student explanation and peer critique.
• Show use of visuals (flowchart/matrix) to support decision-making.
• Display varied student understanding and differentiated supports.
• Record exit ticket collection to show individual mastery check.

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These lesson plans scaffold student understanding from visual graphing to algebraic techniques and finally to strategic decision-making, fully aligned with Common Core standards for Algebra in 9th grade.