Solving 3x3 Systems

Lesson Overview

WALT: Solve 3x3 systems using substitution.

Success Criteria:

• Students can identify suitable variables for substitution.
• Students can correctly substitute and reduce 3x3 systems to 2x2 systems.
• Students can solve the resulting equations to find the values of three variables.
• Students can verify their solutions satisfy all original equations.

Class Duration: 60 minutes
Class Size: 30 students
Year Level: 13 (NCEA Level 3, aligned with the New Zealand Curriculum Mathematics and Statistics years 11-13)
Unit: Mastering Simultaneous Equations
Lesson Number: 5 of 16

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Curriculum Links

• Mathematics and Statistics Learning Area – Algebra; solving systems of linear equations in three variables (Year 13, Level 3)
• Achievement Objectives:
  • Algebra: Manipulate and solve systems of equations and inequalities, including those with multiple variables (NZ Curriculum Draft 2025)
• Mathematical Processes: Reasoning, Problem solving, Communicating
• Competencies Addressed:
  • Thinking: Applying algebraic procedures with logical reasoning.
  • Relating to others: Working effectively in pairs for peer support.
  • Using language, symbols, and texts: Accurately expressing steps and reasoning.

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Teaching Style Notes

• High student engagement from Waiheke Island context - integrate collaborative and peer learning.
• Use clear, systematic teaching with scaffolding and multiple entry points for diverse learners.
• Dyslexia-friendly: Use clear fonts, colour coding for variables, and bulleted steps in handouts and whiteboard work.

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Resources Needed

• Whiteboard and markers
• Graphing calculators or algebraic software (optional)
• Printed worksheets with 3x3 system problems and frameworks for substitution
• Visual aids of substitution steps (colour-coded)
• Dyslexia-friendly handouts (large font, spaced, colour-highlighted terms)

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Lesson Plan Breakdown

1. Introduction & Recap (10 minutes)

• Activity: Briefly review simultaneous equations of 2 variables by substitution.
• Teacher’s Actions:
  • Pose a simple 2x2 system and solve by substitution on the board, emphasising the step of isolating a variable.
  • Ask students to share the challenges they face during substitution (addresses misconceptions).
• Purpose: Activate prior knowledge and prepare for 3x3 substitution.
• Differentiation: Provide summary handout sheets for reference. Use graphic organiser showing substitution flow.

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2. Guided Exploration of 3x3 Substitution (15 minutes)

• Activity: Teacher models the substitution method step-by-step on a simple 3x3 system:
  • Identify one equation and isolate one variable.
  • Substitute into other two equations reducing to 2x2 system.
  • Solve the 2x2 system by substitution or elimination.
  • Back-substitute to find the third variable.
• Teacher’s Actions:
  • Use colour coding for variables and equations to help visual tracking.
  • Pause to check understanding through questioning.
• Success Criteria Embedded: Students can follow and explain each step.
• Differentiation: Slower pacing with explicit annotation for students needing extra support; pair stronger students to assist others.
• Dyslexia-Friendly Strategies: Use consistent colours, avoid dense text, and provide steps in a bulleted list.

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3. Paired Practice (20 minutes)

• Activity: Students work in pairs with printed worksheets containing 3x3 systems of varied difficulty.
  • First task: Solve a relatively straightforward system with guided steps.
  • Second task: Challenge problem requiring multiple substitutions with possible fractional answers.
• Teacher’s Role: Circulate, offer targeted support, prompt reasoning, and validate correct strategies.
• Differentiation: Pairs formed combining different ability levels to promote peer learning.
• Advanced learners encouraged to tackle the extension problem with more complex coefficients.
• Formative Assessment: Teacher records observations, prompts students to verbalise reasoning.

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4. Extension Activity for Advanced Learners (10 minutes)

• Create Your Own Problem:
  • Challenge students to design a 3x3 system that requires multiple substitutions and produces either fractional answers or an interesting solution set (e.g., no solution or infinite solutions).
  • Students swap problems and solve peers’ created questions.
• Purpose: Encourage deeper engagement and synthesis of learning.

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5. Reflection, Connection & Summary (5 minutes)

• Activity: Class discussion to summarise key steps and reflect on challenges.
• Teacher’s Prompts:
  • What helped you understand substitution better?
  • What part did you find confusing and how did you overcome it?
  • How does substitution compare to elimination for 3x3?
• Wrap-up: Link this lesson to future lessons on matrix methods or graphing systems visually.
• Highlight: Resilience and mathematical persistence as key competencies.
• Pre-teach: Briefly introduce next lesson focus (e.g., solving 3x3 systems by elimination).

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Assessment Strategies

• Formative checks through questioning during guided practice.
• Monitoring paired work to assess accuracy and process.
• Peer assessment via problem creation and solving.
• Self-assessment checklist on success criteria after practice.

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Additional Notes on Differentiation and Inclusion

• Use flexible grouping based on student needs during practice.
• Visual supports and checklists for step clarity.
• Provide worked examples with errors to analyze and discuss common misconceptions.
• Encourage use of calculators or algebra tools for students with processing difficulties.

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Summary Table

Time	Activity	Focus	Differentiation & Support
10m	Recap 2x2 substitution	Activate prior knowledge	Handouts, graphic organiser, clear language
15m	Teacher-led 3x3 substitution	Model step-by-step solution process	Colour coding, Q&A, peer support
20m	Paired practice	Apply substitution	Pair mixed ability, scaffolded tasks, extension task
10m	Extension / Problem creation	Deepen understanding	Challenge for advanced learners
5m	Reflection and summary	Consolidate learning	Class discussion, reinforcing perseverance

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This lesson plan embodies the principles and recommendations from the New Zealand Curriculum Refresh draft, including explicit teaching, scaffolding, use of rich tasks, peer collaboration, and assessment to inform next steps in learning .

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