Graphing and Classifying Solutions

Overview

This is lesson 1 of 2 in Simultaneous Equations Together. Ākonga first test ordered pairs by substitution, then represent simultaneous linear equations on coordinate grids and interpret the intersection as a common solution. Collaborative kōrero and contextual problems support mathematical language, reasoning and ako.

Learning intentions

• WALT check whether an ordered pair satisfies one or both equations.
• WALT solve simultaneous equations by graphing.
• WALT interpret the point of intersection in context.
• WALT classify systems as having one solution, no solution or infinitely many solutions.
• WALT explain our mathematical thinking using precise language.

Success criteria

• I can substitute an ordered pair into each equation and decide whether it is a solution.
• I can graph two linear equations accurately using a table of values, intercepts or a gradient and intercept.
• I can identify and explain the intersection point.
• I can classify a system and justify whether it has one, no or infinitely many solutions.

Curriculum links

• Pāngarau: Tau me te Taurangi — relationships, linear equations and representing solutions graphically.
• Pāngarau: Pāngarau Taumata 5, Years 9–10 — graphing, interpreting and classifying solutions; descriptor NZ-TMOA-PANG-FRAME-WAH01.
• Pāngarau: mathematical reasoning, representation, communication and problem-solving.
• Te Reo Rangatira: whakarongo, kōrero, pānui and tuhituhi through mathematical discussion and written explanations.
• Marau ā-Kura: collaborative ako and the use of locally meaningful contexts, names or examples where appropriate.

Pāngarau Vocabulary / Kupu Pāngarau

• pāngarau — mathematics
• whārite — equation
• kīanga taurangi — algebraic expression
• taurangi — variable
• otinga — solution
• huinga o ngā otinga — solution set
• kauwhata — graph
• tuaka x / tuaka y — x-axis / y-axis
• pūwāhi whakawhiti — intercept
• pūwāhi tūtakitahi — intersection point
• rārangi kauwhata — graph line
• whakarōpū — classify
• tirohanga — interpretation

During partner explanations, students should use the Māori terms where appropriate.

Lesson structure (90 minutes)

1. 0–8 min · Pātai whakaoho — finding a shared point. Teacher displays two lines representing two different pricing plans and asks, “When would both plans cost the same?” Open with the hook and learning-intention slides and invite a think-pair-share. Ākonga predict what “same cost” might look like on a graph and share initial ideas.

2. 8–20 min · Check an ordered pair. Teacher models testing ((2, 5)) in (y=2x+1) and (y=-x+7), emphasising substitution, equality and the phrase “satisfies the equation”; pause for hinge questions in the ordered-pair modelling slides. Ākonga use mini-whiteboards to test two further ordered pairs and explain whether each lies on neither, one or both lines.

3. 20–35 min · Graphing a system. Teacher revisits gradient, y-intercept, plotting a table of values and drawing a straight line, then graphs (y=2x+1) and (y=-x+7) on the board. Use the Gradient and Intercept Reference Mat as a desk reference and highlight that the intersection ((2,5)) satisfies both equations. Ākonga plot both equations independently, compare with a partner and state the intersection using ordered-pair notation.

4. 35–58 min · Guided and independent practice. Teacher distributes the graphing simultaneous equations worksheet and completes the first question with the class, checking scale, labels and line accuracy before releasing pairs to work. Ākonga graph three systems, record each intersection and verify one answer by substitution; partners alternate as graph-checker and explainer.

5. 58–73 min · Classifying systems. Teacher uses the classification and discussion slides to show three visual cases: intersecting lines, parallel lines and coincident lines. Introduce the terms one solution, no solution and infinitely many solutions, linking each to the number of shared points. Ākonga analyse three displayed graphs, classify each system and justify their decision using the sentence frame: “This system has ___ because the lines ___.”

6. 73–85 min · Contextual problem-solving. Teacher presents a locally relevant comparison such as two kura fundraising options: (y=4x+20) and (y=6x), where (x) is the number of items and (y) is the amount raised; clarify units and meaning before students begin. Ākonga work in groups of four to graph the equations, identify the intersection and write a short explanation of what it means in the context. One speaker from each group shares the reasoning, while listeners ask one clarifying question.

7. 85–90 min · Exit check and kōrero. Teacher returns to the opening question and asks students to complete the final prompt on the reflection and exit section: graph or interpret a small system, classify it, and explain the meaning of its solution. Ākonga submit their response and share one sentence beginning, “I know the intersection is a solution because…”

Resources

• the Graphing and Classifying Solutions slide deck
• the graphing simultaneous equations worksheet
• the Gradient and Intercept Reference Mat
• Mini-whiteboards, pens and erasers
• Graph paper or squared exercise books
• Rulers, pencils and coloured pens
• Projector or interactive display
• Calculators, if routinely used by the class

Assessment

• Listen for correct use of ordered pair, satisfies, intersection, parallel and coincident during partner and group kōrero.
• Check mini-whiteboard substitutions and circulate during graphing, noting scale, plotting, line accuracy and whether students verify intersections.
• Use the exit response to identify whether students can graph, classify and explain a solution; group students for targeted support in lesson 2.

Differentiation

• Provide partially completed value tables, pre-drawn axes and the reference mat for ākonga who need support; reduce the first graph’s complexity while retaining the same concept.
• Use sentence frames, paired rehearsal and explicit modelling of mathematical language for ākonga developing te reo Māori, English or confidence in academic kōrero.
• Offer enlarged grids, high-contrast colours, ruler guides and a quiet working space for students with visual, motor or sensory needs.
• Extend confident ākonga by asking them to create two systems with a chosen classification, explain the gradients algebraically and identify how changing an intercept changes the graph.
• Use deliberate mixed-ability grouping, rotating roles such as kaiwhakamahere, kaitatau, kaiwhakaatu and kaitaki, so every ākonga contributes within an ako-based environment.