Reading Linear Graphs

Overview

Students build on plotting ordered pairs and simplifying expressions to graph linear relationships on the Cartesian plane. They will connect tables, equations and graphs, then interpret gradient, intercepts and points in a practical context.

Learning intentions

Students will:

• plot and label points accurately on the Cartesian plane
• identify the gradient and y-intercept of a linear graph
• interpret what a graph shows about a real situation
• use tables, equations and graphs to solve and check linear problems

Success criteria

• I can choose a suitable scale and plot ordered pairs accurately.
• I can describe a line using its gradient and y-intercept.
• I can explain what a point, intercept and gradient mean in context.
• I can check a graph by substituting values into its equation.

Curriculum links

• Graphing linear relations on the Cartesian plane and solving linear equations using graphical and algebraic techniques.
• Modelling applied problems involving linear relations and interpreting solutions in context.
• Experimenting with linear functions using digital tools to test patterns and conjectures.
• Creating, simplifying and rearranging linear expressions.

Lesson structure (120 minutes)

1. 0–10 min · Hook and diagnostic. Teacher displays two straight-line graphs and asks, “Which situation changes faster, and how can you tell?” using the opening comparison and discussion prompt. Students make an individual prediction, then explain what they notice about steepness, direction and where each line crosses the axes. Teacher collects prior knowledge about axes, coordinates and scale.

2. 10–25 min · Explicit teaching. Teacher uses the graphing and key vocabulary slides to model (y=2x+1): identify the x- and y-axes, plot a table of values, locate the y-intercept, and use the gradient to move up 2 for every 1 across. Students copy the worked example and annotate a graph with the terms gradient, y-intercept, x-intercept, ordered pair and scale. Emphasise that a negative gradient produces a line that falls from left to right.

3. 25–45 min · Guided plotting practice. Teacher distributes the linear graphs practice worksheet and completes the first question with the group. Students complete tables and plot lines such as (y=x+2), (y=3x-1) and (y=-x+4), checking that points satisfy the equation. Pause after each question for students to compare scales, plotted points and line direction. Teacher uses questioning: “What stays constant?” and “How does changing the coefficient of (x) affect the graph?”

4. 45–60 min · Representation matching. Teacher introduces the linear graph matching cards and models matching one graph with its equation, table and descriptive features. In pairs, students match the representations, justify each match using gradient and intercept, and record one explanation on the worksheet. As a group of five, students discuss any disputed match and agree on evidence rather than guessing from appearance.

5. 60–70 min · Break and reset. Students take a short break. Teacher prepares a graphing application or spreadsheet and displays the activity instructions and digital graphing prompt. On return, students independently predict what will happen when the gradient or y-intercept changes.

6. 70–95 min · Digital investigation and modelling. Teacher demonstrates entering (y=mx+c) into the digital tool and changing one parameter at a time. Students investigate (y=2x+1), (y=2x-3), (y=-2x+1) and (y=\frac12x+1), recording observations on the worksheet. They then model a taxi fare: (C=4+2.50d), where (d) is distance in kilometres and (C) is cost in dollars. Students graph the relationship, find the cost for 6 km, determine the distance for a $19 fare, and explain the meaning of 4 and 2.50. Discuss the model’s limitations, including that distance and cost cannot be negative.

7. 95–112 min · Interpret, verify and communicate. Teacher assigns each student one graph or equation from the interpretation and verification questions. Students explain the gradient and intercept, solve one unknown using the graph, then verify the result by substitution. Students present their reasoning to the group using the sentence frame: “The gradient means…, the y-intercept means…, therefore…”. Teacher corrects misconceptions about confusing the x- and y-intercepts or reading an unsuitable scale.

8. 112–120 min · Plenary and exit check. Teacher displays the final review and exit questions. Students answer: “For (y=-3x+6), state the gradient and y-intercept, give two points, and explain what the negative gradient means.” They also write one strategy they will use to check a linear graph. Teacher collects responses to identify next-step support.

Resources

• the complete linear graphs slide deck
• the linear graphs practice worksheet
• the linear graph matching cards
• Graph paper and squared exercise books
• Rulers, pencils and coloured pens
• Mini-whiteboards and markers
• Device for each student or pair
• Graphing application or spreadsheet
• Projector or interactive display

Assessment

• Observe plotting, scale selection and explanations during guided practice; use mini-whiteboards for quick checks of gradient and intercept.
• Review matching-card justifications and digital investigation notes for accurate connections between equation, table and graph.
• Use the final response to assess plotting, interpretation and substitution. Students who confuse gradient and intercept receive a short re-teach using (y=mx+c).

Differentiation

• Support students with a partially completed table, a pre-labelled Cartesian plane, colour coding for (m) and (c), and the sentence frame “The line crosses the y-axis at…”.
• Work with students individually or in pairs, allowing them to use the digital graph while plotting by hand; read instructions aloud and reduce the number of required examples where needed.
• Provide extension students with (y=mx+c) graphs to design a matching real-world situation, compare two taxi companies, or find the break-even point by graphing and solving two linear equations.
• Challenge advanced learners to explain why equal first differences produce a straight-line graph and investigate how restricting the domain changes the interpretation of the taxi model.

Extension

• Students create a linear relationship for a phone plan with a fixed fee and per-unit charge, produce a table and graph, and write three interpretation questions for a peer.
• Students graph two competing plans and determine, algebraically and graphically, when one becomes better value.