No Solutions

Overview

This is Lesson 9 of 16 in the Year 13 unit titled Mastering Simultaneous Equations. The lesson focuses on identifying systems of simultaneous equations that have no solutions. The curriculum and pedagogy align strictly with the New Zealand Curriculum Refresh (Te Mātaiaho), ensuring age-appropriate mathematical rigor and key competencies development. Suitable differentiation and extension activities support diverse learners.

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Learning Intentions

WALT (We Are Learning To):

• Identify systems of simultaneous equations where no solutions exist.

Success Criteria:

• Explain what it means for a system to have no solutions.
• Demonstrate algebraically and graphically why such systems do not have solutions.
• Analyse provided case studies to identify no-solution systems.
• Explore real-world contexts where no solutions might arise.

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Curriculum Alignment

Learning Areas and Achievement Objectives

Aligned to the New Zealand Curriculum Refresh (Years 11–13), specifically:

• Algebra and Statistics Strand — Level 8 (Year 13 focus)

  • Form, solve, and graph systems of two simultaneous equations, one of which may be linear, in two dimensions, and interpret solutions.
  • Connect features of graphs to the solutions of equations and inequations.

• Mathematical Competencies emphasised:

  • Problem solving through algebraic manipulation and graphical interpretation.
  • Reasoning to explain and justify solutions.
  • Representing situations using multiple forms (tables, graphs, equations).
  • Communicating mathematical ideas clearly to peers.

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Timing and Structure (60 minutes, 30 students)

Time	Activity	Description	Differentiation & Extensions
0–10	Starter & Recap	Quick revision on systems with one solution and infinite solutions. Use	Dyslexia friendly summary sheet with key terms and
		a think-pair-share question: “What happens when lines never intersect?”	colour coding; scaffolding for learners needing extra
			support.
10–25	Direct Teaching & Modelling	Teacher-led explanation of no solutions in simultaneous linear systems:	Visual aids showing parallel lines. Use digital graphing
		algebraic form (contradictory equations), graphical form (parallel lines).	tools for dynamic visualisation.
		Emphasise vocabulary: inconsistent system, parallel lines, contradiction.	Highlight vocabulary in accessible format for dyslexic
			students.
25–40	Case Study Analysis (Group Work)	Students work in groups of 3–4. Each group receives a case study involving a	Provide case studies with varying complexity;
		system of equations (some with no solutions, some with one or infinite).	advanced groups can explore systems with parametric forms.
		Task: Identify if no solutions and explain reasoning algebraically and graphically.	Include graphic organiser for reasoning steps.
40–50	Class Discussion & Sharing	Groups present findings. Teacher facilitates clarification. Connect to real-world contexts like incompatible constraints in planning.	Encourage quieter students via think-pair-share before whole class discussion.
50–60	Extension & Reflection	Extension task: Students individually research and prepare a short explanation of a real-life situation involving no solutions in maths, science, or engineering.	For advanced learners, challenge with systems involving inequalities or real parameters.

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Key Resources

• Graph paper and coloured pens.
• Digital graphing tools (e.g., Geogebra).
• Case studies print-outs (prepared beforehand with algebraic and graphic forms).
• Dyslexia-friendly vocabulary and concept summary sheets.
• Reflection and extension handout for research.

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Differentiation Strategies

• For learners with additional needs:

  • Use colour coding and larger fonts on sheets.
  • Provide worked examples with step-by-step annotations.
  • Allow use of calculators and graphing software.
  • Use collaborative learning to support peer explanation.

• For advanced learners:

  • Challenge with systems including linear and nonlinear pairs exhibiting no solutions.
  • Introduce cases involving inequalities and optimisation conflicts as no-solution scenarios.
  • Investigate parametric equations and their interpretations.

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Assessment and Feedback

• Formative assessment through group task and class discussion.
• Teacher observes reasoning and articulation of no-solution concepts.
• Exit reflection question: "How can you tell algebraically and graphically if a system has no solutions?" Written or verbal response.

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Literacy Support: Dyslexia-friendly Reading

• Use clear, sans-serif fonts, increased spacing.
• Key terms highlighted and defined in simple language (e.g., “Inconsistent: No possible values make both equations true at the same time”).
• Visual supports: use diagrams and colour to support understanding.
• Offer oral explanation supplements and peer support opportunities.

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Making Connections (Te Whanaungatanga)

• Relate systems with no solutions to everyday situations—e.g., scheduling conflicts, physical limits in engineering designs, or contradictory business constraints.
• Encourage empathy and resilience (manaaki, manawaroa) by showing how mathematicians persist through challenging problems.

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Teacher Notes

• Ensure students understand the difference between “no solutions” and “infinite solutions” from previous lessons.
• Use questioning to encourage reasoning: “What does it mean graphically if two lines never cross?”
• Highlight the link to the broader curriculum goal of communicating mathematical ideas clearly and supporting reasoning visually and algebraically .

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I trust this lesson plan will engage your Year 13 students and support diverse learning needs effectively. If you want, I can also provide printable student handouts or digital slides to accompany this plan!

If you need me to elaborate on any section or add further resources, just ask.