Ngā Rārangi Tōkeke

Overview

Students investigate linear equations through coordinate pairs, graphs, tables and contextual problems. They connect the gradient and intercepts to the equation (y=mx+c), then extend this understanding to standard form, point-gradient form, parallel lines and perpendicular lines. Use local examples, names and contexts from the kura’s Marau ā-Kura where appropriate.

Learning intentions

• WALT test whether coordinate pairs satisfy a linear equation and solve equations for (y), using whārite, taurangi, whakakapi and whakaoti.
• WALT graph and write linear equations in slope-intercept, standard and point-gradient forms, using rārangi tōkeke, rōnaki, tapahi-tuaka and kauwhata.
• WALT interpret the rōnaki, tapahi-tuaka and pūmau in mathematical and contextual situations.
• WALT identify and explain relationships between parallel, perpendicular, horizontal and vertical lines.
• WALT use Pāngarau vocabulary to communicate mathematical reasoning in te reo Māori and English.

Success criteria

• I can substitute a coordinate pair into an equation and justify whether it lies on the line.
• I can identify (m) as the gradient and (c) as the (y)-intercept in (y=mx+c).
• I can find intercepts, graph a line and write its equation from a graph, table or situation.
• I can write an equation for a line parallel or perpendicular to a given line and explain my reasoning.

Curriculum links

• Pāngarau: Tau me te Taurangi, including linear relationships, equations, variables and algebraic representations.
• Pāngarau: Ine me te Āhuahanga, including coordinate graphs, gradient, intercepts and line relationships.
• Te Reo Rangatira: whakarongo, kōrero, pānui, tuhituhi, mātakitaki and whakaatu through mathematical discussion and explanation.
• Marau ā-Kura: use locally meaningful contexts, pūrākau, whenua or community patterns where suitable.

Pangarau Vocabulary

• rārangi tōkeke — straight line. He rārangi tōkeke tēnei i te kauwhata. (This is a straight line on the graph.)
• whārite — equation. Whakaotia te whārite (y=2x+1). (Solve the equation (y=2x+1).)
• taurangi — variable. Ko (x) te taurangi i tēnei whārite. ((x) is the variable in this equation.)
• taupū — coefficient. Ko (2) te taupū o (x) i te (2x+3). ((2) is the coefficient of (x) in (2x+3).)
• pūmau — constant. Ko (3) te pūmau i te (2x+3). ((3) is the constant in (2x+3).)
• otinga — solution. Ko (x=4) te otinga. ((x=4) is the solution.)
• tuaka x — x-axis. Tirohia te wāhi e tapahi ai te rārangi i te tuaka x. (Look at where the line crosses the x-axis.)
• tuaka y — y-axis. Ko te tapahi-tuaka y kei te tuaka y. (The y-intercept is on the y-axis.)
• taunga — coordinates. Whakamahia ngā taunga ((2,5)). (Use the coordinates ((2,5)).)
• pūtake — origin. Ko ((0,0)) te pūtake. (((0,0)) is the origin.)
• rōnaki — gradient/slope. Ko te rōnaki (3), arā, ka piki te rārangi e 3 mō ia 1 ki te taha. (The gradient is 3, so the line rises 3 for every 1 across.)
• tapahi-tuaka — intercept. Ko (4) te tapahi-tuaka y. (The y-intercept is 4.)
• kauwhata — graph. Tuhia te whārite ki te kauwhata. (Plot the equation on the graph.)
• whakakapi — substitute. Whakakapia ngā taunga ki roto i te whārite. (Substitute the coordinates into the equation.)
• whakaoti — solve. Whakaotia te whārite mō (y). (Solve the equation for (y).)
• hononga — relationship. He aha te hononga i waenganui i te rōnaki me te tapahi-tuaka? (What is the relationship between the gradient and the intercept?)

Lesson structure (90 minutes)

1. 0–8 min · Contextual hook. Open with the opening visual and challenge question showing two rārangi tōkeke (straight paths) across a local whenua or community map, with one path rising more steeply than the other. Ask: “How could we describe these paths mathematically?” Students discuss what might represent the rōnaki and tapahi-tuaka, then share initial ideas using terms such as rārangi tōkeke, taunga and kauwhata.

2. 8–23 min · Build the model. Use the worked-example slides to model (y=mx+c), explaining (m) as the rōnaki (the change in (y) for each unit change in (x)), and (c) as the tapahi-tuaka y. Demonstrate whakakapi (substitution) of ((x,y)) into a whārite, whakaoti an equation for (y), and plotting points. Students annotate the examples and complete quick checks: identify the taupū (m), pūmau (c), and whether given pairs satisfy (y=2x-3).

3. 23–40 min · Paired representation task. Distribute the linear equations investigation worksheet to pairs. Students complete tasks involving taunga, tables and kauwhata: test points, complete values, graph (y=2x-1) and (y=-\frac12x+4), and write equations from displayed graphs. Partners must explain each step using the vocabulary rōnaki, tapahi-tuaka, tuaka x, tuaka y, taurangi and whakaoti. Circulate and check that students distinguish the (y)-intercept from the (x)-intercept.

4. 40–53 min · Standard form and special lines. Use the standard-form and special-lines slides to model (Ax+By=C). Show how setting (x=0) finds the tapahi-tuaka y and setting (y=0) finds the tapahi-tuaka x, then graph by joining the intercepts. Students solve and graph (2x+y=6), and identify (y=4) as horizontal and (x=-2) as vertical. Pause for a mini-whiteboard check before students continue.

5. 53–70 min · Point-gradient and line relationships. Model point-gradient form (y-y_1=m(x-x_1)) using the point ((2,3)) and rōnaki 4, then rearrange to (y=mx+c). Students complete paired problems from the worksheet, writing equations through a given point. Introduce that parallel lines have equal gradients, while perpendicular non-vertical lines have gradients whose product is (-1). Students find an equation parallel and one perpendicular to (y=3x+2) through a stated point, explaining the hononga (relationship).

6. 70–82 min · Apply and explain. Display the contextual problem and discussion prompts. Students choose one context, such as the cost of a locally relevant journey, planting rows, or a steady change in height, and represent it with a table, kauwhata and whārite. They identify what the rōnaki and pūmau mean in the context, then compare their line with a partner’s to decide whether the lines are whakarara or hāngai-tika. Invite two pairs to explain different representations.

7. 82–90 min · Exit assessment and reflection. Finish with the three-question exit assessment. Individually, students write a whārite from a kauwhata, find the equation of a parallel line, and explain the meanings of the rōnaki and tapahi-tuaka. Collect responses and ask students to complete the sentence: “The representation that helps me most is … because …”.

Resources

• the Ngā Rārangi Tōkeke teaching deck
• the linear equations investigation worksheet
• Mini-whiteboards, pens and erasers
• Graph paper or exercise books
• Rulers and pencils
• Coordinate grids
• Projector or interactive display
• Calculators for checking, not replacing, algebraic reasoning
• Local context image or map selected by the kura

Assessment

• Listen for correct use of Pāngarau vocabulary and explanations of whakakapi, whakaoti, rōnaki, tapahi-tuaka and hononga during paired work.
• Use mini-whiteboard checks to identify misconceptions about tuaka x, tuaka y, intercepts, negative gradients and vertical lines; reteach immediately using a plotted example and bilingual vocabulary.
• Mark the exit assessment for accurate whārite writing, parallel-line reasoning, interpretation of the rōnaki and pūmau, and appropriate use of the bilingual terms.

Differentiation

• Support students with partially completed tables, labelled axes, a sentence frame (“The gradient is … because …”) and the gradient/intercept reference shown on the slides. Pair students strategically and allow oral explanations before written responses.
• Provide the equation forms (y=mx+c), (Ax+By=C) and (y-y_1=m(x-x_1)) as a choice board, while encouraging students to explain how the forms are connected.
• Extend confident students by asking them to derive the equation of a perpendicular line, compare two standard-form equations without fully graphing them, or create a contextual pair of parallel lines.
• For EAL and SEN learners, pre-teach the bilingual vocabulary with diagrams, read instructions aloud, reduce copying demands and accept a labelled graph plus spoken explanation as evidence of understanding.