Algebra in Context

Overview

In lesson 8 of 10, students apply their developing understanding of variables, expressions, equations and formulae to an authentic situation. In groups of three or four, they investigate a scenario, choose an efficient representation and communicate a mathematically justified solution through a creative presentation.

Learning intentions

• WALT choose and connect substitution, simplification, equations and formulae to solve a problem.
• WALT define variables and represent relationships mathematically.
• WALT explain our reasoning and check whether an answer is reasonable.
• WALT work collaboratively and communicate mathematical ideas clearly.

Success criteria

• I can define each variable and identify what it represents.
• I can select a suitable table, diagram, expression, equation or formula.
• I can solve accurately and show how I checked my answer.
• I can explain whether my answer is reasonable in the context.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking questions that challenge advanced learners and deepen understanding.
• Mathematics and Statistics — Mathsteasers / Alignment: connects challenge tasks with familiar textbook topics and mathematical content.
• Mathematics and Statistics — Mathsteasers / Additional resources for advanced learners: provides an appropriate challenge for students ready to extend their reasoning.
• Te Mātaiaho ways of working: students reason, represent, communicate and make connections while solving a meaningful problem.

Lesson structure (45 minutes)

1. 0–5 min · Hook and reconnect. Display the opening problem from the Algebra in the Real World presentation: “A school stall has a $35 set-up cost and earns $4 per item sold. How many items are needed to make $65?” Students estimate, identify possible variables and share which mathematical ideas might help. Briefly revisit that a variable can represent an unknown or changing quantity, and that one situation can be represented in more than one way.

2. 5–10 min · Model the thinking. Use the Algebra in the Real World presentation to model a think-aloud: define (n) as the number of items, write (35+4n=65), solve (4n=30), and discuss why this context does not allow a fractional number of items. Students annotate the real-world algebra planning sheet with the four key moves: define, represent, solve and check. Emphasise that a table, diagram, expression, equation or formula may be useful, but the representation must match the situation.

3. 10–13 min · Choose and plan. Introduce the group challenge and role options using the Algebra in the Real World presentation. Students form mixed-ability groups of three or four and choose one context: a budget, a design, a growing pattern or a measurement problem. Each group assigns roles such as facilitator, calculator/checker, recorder and presenter/producer. Students complete the planning section of the real-world algebra planning sheet, including their question, known information, variables and intended audience.

4. 13–28 min · Investigate and solve. Provide groups with a tiered context card from the real-world algebra planning sheet or allow a group to adapt one with teacher approval. Students research or calculate the relevant information, select representations and solve the problem. Circulate and ask: “What does your variable mean?”, “Why is this representation useful?”, “Could substitution check your answer?”, and “What assumptions have you made?” Offer the optional Algebra Tiles: Linear Expressions Pack to students who benefit from a hands-on way to build and simplify expressions.

5. 28–38 min · Create and share. Groups produce a poster, one-slide explanation, short skit or brief video using their completed working as evidence. Their presentation must include the context, variable definitions, representation, solution, check and a statement about reasonableness. Use the Algebra in the Real World presentation for the presentation checklist. Groups present to the class, or run two-minute presentations in a carousel if time is tight. Audience members record one “clear connection” and one question on their worksheet.

6. 38–45 min · Discuss and assess. Invite two groups using different representations to compare their methods. Discuss how the same relationship can be shown in words, a table, an expression or an equation, and when each is most helpful. Students complete the exit reflection on the real-world algebra planning sheet: “My variable represented…”, “My answer is reasonable because…”, and “One representation I could improve is…”. Collect the sheets to inform the final unit lesson.

Resources

• the Algebra in the Real World presentation
• the real-world algebra planning sheet
• Optional Algebra Tiles: Linear Expressions Pack
• Calculators or tablets
• A4 paper or poster paper
• Coloured pens and markers
• Rulers and sticky notes
• Timer
• Optional device for recording a short video

Assessment

• During modelling, check whether students can distinguish a variable from a fixed value and connect a context to an equation.
• During group work, use questioning to assess representation choice, accurate calculation, substitution checks and reasonableness.
• Review the presentation and exit reflection for defined variables, connected representations, correct solutions and contextual explanations.

Differentiation

• Support: provide tiered contexts with selected numbers, a worked example, sentence starters and the four-step planning template. Allow students to use calculators, diagrams, manipulatives and oral explanations.
• For EAL learners and students needing additional support, pre-teach terms such as variable, constant, expression, equation, substitute and reasonable; pair each term with a symbol, example and plain-language explanation.
• Use mixed-ability roles, but ensure every student contributes mathematical thinking rather than only production work. Offer a quiet planning space and allow presentation by speaking, drawing, acting or recorded explanation.
• Extension: compare two solution methods, such as a table and an equation, and explain which is more efficient. Students ready for further challenge may introduce a second variable while keeping the context and mathematics accessible.

Extension

• Ask groups to change one value in their scenario and predict how the solution will change before recalculating.
• Challenge students to create a second question from the same context that requires substitution or simplification rather than solving an equation.