Algebraic Methods Applications

Overview

In this second lesson of Simultaneous Equations Together, ākonga consolidate substitution and elimination, choose an efficient method, and check solutions algebraically and graphically. They then model real-life situations, interpret solutions, and explain when a system has one, no, or infinitely many solutions through tuakana–teina collaboration.

Learning intentions

• WALT solve simultaneous equations using substitution and elimination.
• WALT choose and justify an efficient method.
• WALT model word problems with simultaneous equations and interpret the solution.
• WALT recognise and explain systems with one, no, or infinitely many solutions.
• WALT communicate mathematical reasoning clearly in te reo Māori and/or the kura’s agreed language practices.

Success criteria

• I can solve a pair of simultaneous equations accurately using substitution or elimination.
• I can check my ordered pair in both original equations and explain what it means.
• I can define variables, form equations, solve a contextual problem, and state the answer with appropriate units.
• I can justify whether two linear relationships have one, no, or infinitely many solutions.

Curriculum links

• Pāngarau: Tau me te Taurangi, with a focus on algebraic relationships, equations, reasoning, and problem solving.
• Te Reo Rangatira: whakarongo, kōrero, pānui, and tuhituhi through precise mathematical explanation and collaborative discussion.
• Marau ā-Kura: use local contexts, community examples, or familiar pūrākau where appropriate when discussing relationships and decision-making.
• Te Reo Pākehā: support mathematical language development where English is being learned as an additional language in the Māori-medium setting.

Pangarau Vocabulary

• tau — number. Example: Identify the tau of students in each group.
• tohu — symbol. Example: Use a tohu such as +, −, or = correctly.
• taurangi — variable. Example: Let x be the taurangi for the number of adult tickets.
• kīanga — expression. Example: Simplify the kīanga (3x+2x).
• whārite — equation. Example: Form a whārite from the information in the problem.
• whakaoti — solve. Example: Whakaoti the pair of simultaneous equations.
• whakangāwari — simplify. Example: Whakangāwari both sides before continuing.
• whakakapi — substitute. Example: Whakarite the value of (x), then whakakapi it into the other equation.
• tauira — pattern or model. Example: Use a tauira to represent the relationship between the quantities.
• otinga — solution. Example: Check that the ordered pair is the correct otinga in both equations.

Encourage ākonga to use these terms when explaining their method and checking their answers.

Lesson structure (90 minutes)

1. 0–8 min · Whakatau and retrieval hook. Display a pair of equations and ask, “Which method would you choose, and why?” Open with the retrieval and hook slides. Students independently recall the steps for substitution and elimination, then share reasoning with a partner. Establish the expectation that a correct answer must include working and a check.

2. 8–23 min · Explicit teaching and comparison. Model one system suited to substitution, such as (y=2x+1) and (3x+y=13), then one suited to elimination, such as (2x+3y=12) and (4x-3y=6). Use the worked-example slides to make the decision points visible. Students annotate their worksheet, identify the most efficient method, and explain why equivalent operations preserve equality.

3. 23–38 min · Guided tuakana–teina practice. Pair ākonga strategically, with a tuakana explaining a step and a teina paraphrasing it before roles change. Distribute the simultaneous equations practice worksheet. Students solve two systems, check each ordered pair by substitution into both original equations, and use the prompt “I chose ___ because ___.” Confer with pairs, questioning rather than correcting.

4. 38–50 min · Graphical meaning and special cases. Draw or display systems representing intersecting, parallel, and coincident lines. Ask: “What does the number of intersection points tell us?” Students use a graphing tool or teacher-prepared grid to connect one solution with an intersection, no solution with parallel lines, and infinitely many solutions with the same line. They justify the cases by comparing gradients and intercepts, or by simplifying the equations.

5. 50–73 min · Context problem investigation. Present a localised scenario, for example: two whānau groups purchase 18 tickets altogether; one group buys adult and rangatahi tickets in a different combination, with each total cost supplied. Students define variables, form two equations, select a method, solve, check, and interpret the ticket quantities and costs. Use the problem-solving and discussion slides. Pairs record a complete explanation on the worksheet, then compare methods with another pair. Invite groups to adapt the context to a kura, sporting, marae, or community setting.

6. 73–84 min · Mathematical communication hui. Groups present one solution, focusing on method choice, checking, and meaning in context. Use the presentation prompts. Class members ask one clarifying question or identify one strength in the explanation. Teacher listens for correct vocabulary and addresses errors such as changing only one side of an equation or accepting an answer that has not been checked.

7. 84–90 min · Exit assessment and reflection. Students complete the final worksheet question: classify a system as having one, no, or infinitely many solutions and justify the classification. They also write one sentence explaining when they would choose substitution over elimination. Collect responses as the exit ticket and close with a brief reflection on how tuakana–teina supported learning.

Resources

• the complete simultaneous equations slide deck
• the simultaneous equations practice worksheet
• Whiteboard, markers, and visualiser
• Graph paper or mini-whiteboards
• Rulers and pencils
• Calculators or a graphing application
• Prepared contextual problem displayed on the board
• Bilingual mathematical language prompts approved by the kura

Assessment

• During retrieval and modelling, check whether ākonga can describe the steps and justify method choice.
• During paired work, assess equation formation, accurate manipulation, checking in both equations, and interpretation of ordered pairs.
• Use the final classification question and method-choice reflection to identify misconceptions for the next teaching sequence.

Differentiation

• Support: provide a substitution/elimination decision guide, colour-code corresponding terms, model one line of working at a time, and offer sentence starters such as “Let (x) represent…” and “This solution means…”.
• Support language learners: pre-teach agreed mathematical kupu and allow rehearsal with a partner before presenting; accept labelled diagrams and oral explanations alongside written reasoning where appropriate.
• Support learners requiring additional scaffolding: use simpler integer coefficients first, provide graph axes already drawn, and check understanding after each procedural step.
• Extend confident ākonga by asking them to create two systems with respectively no and infinitely many solutions, prove their classifications algebraically, and explain how the graphs would appear.