Reading Straight Lines

Overview

Students develop a meaning for gradient (slope) and the y-intercept, then connect these features to equations, tables and graphs. They use a graphing tool to test how changing parameters affects a line and apply their understanding to a practical financial context.

Learning intentions

• Students will identify the gradient and y-intercept of a linear graph.
• Students will connect the form (y=mx+c) with the features of a straight-line graph.
• Students will calculate gradient from two points and use substitution to check results.
• Students will use digital tools to investigate and explain how changing (m) and (c) changes a graph.
• Students will model a simple real-life situation using a linear relation.

Success criteria

• I can explain what the gradient and y-intercept mean.
• I can identify (m) and (c) in (y=mx+c).
• I can calculate gradient using rise over run or two points.
• I can describe how a line changes when its gradient or y-intercept changes.
• I can use a graph, equation and context to justify my answer.

Curriculum links

• Linear relations: graphing lines on the Cartesian plane and solving or verifying linear relationships.
• Experimenting with linear functions using digital tools, making conjectures and testing patterns.
• Mathematical modelling of applied linear relations, including financial contexts.
• Creating, rearranging and simplifying linear expressions using algebraic properties.

Lesson structure (120 minutes)

1. 0–10 min · Hook and diagnostic. Teacher displays three lines and asks, “Which line represents the fastest increase, and how can you tell?” using the opening graph comparison. Students individually annotate the lines, then explain their ideas to the group; the teacher records prior vocabulary such as steepness, crossing point and rate.

2. 10–30 min · Explicit teaching. Teacher uses the gradient and intercept explanation to introduce (y=mx+c): (m) is the gradient or rate of change, and (c) is the y-intercept, where the line crosses the y-axis. Model (y=2x+3), identifying (m=2), (c=3), plotting the intercept and moving up 2 and right 1. Students complete guided examples on the guided gradient and intercept worksheet, including (y=-x+4), (y=\frac12x-2) and (y=3).

3. 30–45 min · Gradient from points. Teacher demonstrates (\text{gradient}=\frac{\text{change in }y}{\text{change in }x}) using two labelled points, emphasising positive, negative and zero gradients. Students calculate the gradient of three lines from diagrams, check by counting rise and run, and explain what each value means. The teacher checks understanding through targeted questions: “What does the numerator represent?” and “What would a negative gradient look like?”

4. 45–65 min · Match and justify. Teacher distributes the Linear Graph Matching Cards and asks students to match graphs, equations, tables and descriptions of gradient, intercept and behaviour. Students work as a small team, recording matches and giving a mathematical justification for each. The teacher deliberately challenges misconceptions, including confusing the x-intercept with the y-intercept and reading the gradient as the starting value.

5. 65–85 min · Digital investigation. Teacher opens the digital investigation instructions and demonstrates a graphing tool with sliders or editable equations for (y=mx+c). Students investigate one parameter at a time: compare (y=x+2), (y=3x+2), (y=-x+2), then compare (y=2x-3), (y=2x) and (y=2x+5). They record conjectures on the worksheet, test them digitally, and describe the effect of changing (m) and (c). Each student must state one conjecture and evidence that supports or disproves it.

6. 85–108 min · Financial modelling task. Teacher presents a practical problem on the phone-plan modelling task: Plan A costs a $15 connection fee plus $4 per gigabyte; Plan B costs a $5 connection fee plus $6 per gigabyte. Students define (x) and (y), write (y=4x+15) and (y=6x+5), graph both relations and determine which plan is cheaper for selected data usage amounts. Students explain the meaning of each gradient and intercept, identify the approximate break-even point, and verify one calculated value by substitution.

7. 108–120 min · Plenary and assessment. Teacher uses the final review questions for a short group discussion, then students complete the final questions on the worksheet independently. Students submit: identify (m) and (c) in (y=-2x+7), state the gradient between ((1,3)) and ((4,9)), explain the meaning of the y-intercept, and describe what changes when (c) increases.

Resources

• the complete lesson slide deck
• the guided gradient and intercept worksheet
• the Linear Graph Matching Cards
• Graphing software or an online graphing application
• Laptops or tablets, ideally one per student
• Whiteboard and coloured markers
• Graph paper, rulers and pencils
• Calculators
• the Gradient and Intercept Reference Mat

Assessment

• Listen to explanations during guided examples and matching-card work, checking whether students distinguish gradient from y-intercept.
• Review digital investigation conjectures and the financial modelling equations for correct interpretation of (m), (c), units and graph behaviour.
• Use the final independent questions as an exit check; ask students who finish early to explain one answer using both a graph and an equation.

Differentiation

• Support students with the the Gradient and Intercept Reference Mat, a colour-coded rise/run diagram, partially completed axes and sentence starters such as “The gradient means…” and “The y-intercept shows…”.
• Work beside students who need additional assistance, using integer gradients first and checking each plotted point before introducing fractions or negative gradients.
• For EAL learners, explicitly teach and display “gradient”, “rate of change”, “intercept”, “rise”, “run” and “break-even”, supported by diagrams and repeated oral rehearsal.
• Extension learners can compare two plans algebraically, solve the exact break-even equation (4x+15=6x+5), investigate lines with the same gradient, or create and test a conjecture about negative gradients and inequalities.

Extension

• Ask advanced learners to design a third phone plan that is cheapest for a chosen range of usage, then justify it with a graph and algebra.
• Challenge students to graph (y=2x+1), (y=2x+5) and (y=2x-4), explain the invariant feature, and generalise the pattern.