Linear Graphing Foundations

Overview

In this lesson, students consolidate what makes a relationship linear, and how to identify and interpret the gradient (slope) and intercepts from both equations and graphs. They will practice constructing straight-line graphs and explaining what the gradient and intercepts mean in context.

Learning intentions

Students will:

• define a linear relationship and recognise when a graph represents a straight-line relationship
• determine the gradient, y-intercept, and x-intercept from an equation and from a graph
• construct a straight-line graph for a linear function in slope-intercept form, y = mx + c
• interpret gradient and intercepts in context to make meaningful statements

Success criteria

Students can:

• explain why a linear relationship produces a straight-line graph (constant rate of change)
• correctly identify m (gradient) and c (y-intercept) from y = mx + c
• calculate or read the x-intercept and y-intercept from both an equation and a graph
• plot enough accurate points to draw a correct straight line, and describe what m and c mean for the situation

Curriculum links

• Straight-line graphs: understand and use the slope-intercept form y = mx + c where m is gradient and c is y-intercept
• Construct a straight-line graph using y = mx + c
• Determine slope (gradient), x-intercept and y-intercept from both equation and graph
• Interpret, in context, the slope and intercept of a linear function used to model and analyse a practical situation

Lesson structure (60 minutes)

1. 0–5 min · Retrieval warm-up. Teacher writes three mini-prompts on the board (e.g., “Gradient means…”, “y-intercept is where…”, “A linear graph looks like…”); students answer individually in short phrases.
2. 5–15 min · Direct teach: linear relationships and key features. Teacher models two examples: one linear and one not, then explicitly links linearity to constant gradient; students sketch both quickly and highlight gradient direction and intercepts on the linear example.
3. 15–25 min · Worked example: from equation to graph features. Teacher uses y = mx + c and demonstrates how to find m, c, and intercepts (e.g., substitute x = 0 for y-intercept, set y = 0 for x-intercept); students complete the matching table for one provided equation with teacher checking.
4. 25–40 min · Guided practice: plotting and drawing. Teacher gives students one equation set (2–3 equations) and a coordinate grid; students plot at least 3 points per line using y = mx + c, draw the best straight line, and then read off x- and y-intercepts.
5. 40–50 min · Interpreting meaning in context. Teacher provides a short context problem (e.g., cost or fuel usage) tied to the gradient and intercept; students write two “in context” statements: one about gradient and one about the intercept, using correct units/meanings.
6. 50–58 min · Independent task: equation ↔ graph link. Students complete a worksheet item where they are given either an equation or a graph and must produce the missing features (gradient and intercepts) and justify their reading using “rise/run” or substitution logic.
7. 58–60 min · Exit ticket. Students submit one final response: “For y = mx + c, my gradient is __ and my y-intercept is __, so the intercept means…” (single sentence).

Resources

• Student worksheet for the guided practice and independent task (grid + questions)
• Printed coordinate grids or graph paper
• Whiteboard/markers (or digital equivalent) for worked examples
• Calculator (optional) for solving simple intercept substitutions, if permitted by school policy
• Timer or projected agenda for pacing
• Coloured pencils or highlighters to mark gradient and intercepts
• A context cards sheet (fuel/cost style linear scenario)

Assessment

• Formative checks during Step 2 (quick sketches and definitions) to confirm students distinguish linear vs non-linear
• Teacher questioning and feedback during Step 3 (gradient/intercepts from equation) and Step 4 (whether plotted points align to a straight-line graph)
• Exit ticket at Step 7 assessing understanding of slope-intercept meaning in context

Differentiation

• Support: provide sentence starters for interpretations (e.g., “The gradient tells me the change in ___ for each increase of 1 in ___.”; “The y-intercept represents the starting value when ___ = 0.”)
• Support: offer a partially completed point table (x-values and corresponding y-values) for plotting
• Extension for accuracy: ask students to include a brief justification for an intercept they read from a graph (e.g., “I set y = 0 to find the x-intercept”)
• SEN/EAL: allow working through reading intercepts using highlighting (x-axis/y-axis) and use verbal explanation prompts before writing
• For a class size of 1 student, teacher provides immediate targeted feedback after each plotted point set and after each written interpretation statement