Linear Relationships

Overview

In this lesson (6 of 9), students investigate how linear equations describe straight-line graphs. They interpret and calculate gradient (slope), x-intercept and y-intercept from both an equation and a graph, then practise constructing the graph from the equation.

Learning intentions

Students will:

• determine the slope (gradient), x-intercept and y-intercept of a straight line from its equation and from its graph
• use slope-intercept form, (y = mx + c), to interpret what (m) and (c) mean on a graph
• construct and analyse a straight-line graph from a given linear function
• communicate their reasoning clearly using correct mathematical language (gradient, intercept, equation, graph)

Success criteria

Students can:

• identify (m) as the gradient and (c) as the y-intercept in (y = mx + c)
• calculate the x-intercept and y-intercept accurately for linear equations
• read the gradient and intercepts from a plotted graph using a suitable scale
• draw a correct straight-line graph by plotting two points and joining with a straight edge

Curriculum links

• Linear equations and their graphs: understand and use slope-intercept form (y = mx + c) to link equation to gradient and y-intercept
• Linear equations and their graphs: determine slope (gradient), x-intercept and y-intercept from both equation and graph
• Linear equations and their graphs: interpret slope and intercepts in context (when given a situation statement)
• Linear equations and their graphs: construct a straight-line graph using (y = mx + c)

Lesson structure (60 minutes)

1. 0–6 min · Retrieval warm-up. Teacher writes two equations and asks: “What is the gradient and y-intercept for each?” Students answer on mini-whiteboards, then share one explanation for how they read (m) and (c).

2. 6–15 min · Direct teach: equation → key graph features. Teacher models a worked example: starting with (y = mx + c), identify (m) and (c), then find intercepts by setting (y=0) (x-intercept) and (x=0) (y-intercept). Students follow with a second equation, completing gradient, x-intercept, and y-intercept steps in their notes.

3. 15–26 min · Guided practice: graph → intercepts and gradient. Teacher displays a graph with clear axes and a straight line, plus a recommended scale (e.g., 1 unit per grid). Students, in pairs, read:

• two points on the line
• the rise/run to compute gradient
• where the line crosses the axes to find intercepts Teacher circulates and checks accuracy of point choice and scale use.

4. 26–47 min · Construct graphs: equation → graph, then analyse. Teacher gives each student one linear function card (different but similarly structured), e.g. (y = 2x - 3), (y = -\tfrac{1}{2}x + 4), (y = 3x + 1). Students:

• plot two points using (y = mx + c)
• draw a straight line through the points
• label x- and y-intercepts on the diagram
• write a short interpretation sentence (e.g., “When (x=0), the value is…”). For the final 3 minutes of this section, teacher calls for one “quick check” from each student: gradient and intercepts from their graph.

5. 47–55 min · Consolidation: misconception check (whole class). Teacher shows three brief statements, one correct and two common errors (e.g., confusing x- and y-intercepts; using rise/run with the wrong direction; treating gradient as the y-intercept). Students vote A/B/C with justification: “What would we need to check to fix it?”

6. 55–60 min · Exit ticket. Students complete one combined task on paper: given an equation in slope-intercept form, find x- and y-intercepts, then sketch a graph quickly (two-point method) and state the gradient.

Resources

• Mini-whiteboards and markers
• Printed graph paper (Cartesian plane with labelled axes and consistent grid scale)
• Equation cards (slope-intercept form) and blank response sheets for intercepts
• Worked example sheet (teacher model steps)
• Student calculators (optional; ensure they are not required for integer/half-integer cases)
• Timer for timed practice and exit ticket

Assessment

• Formative checks during the warm-up: gradient and y-intercept answers shown on mini-whiteboards
• Teacher observation during guided practice: accuracy of chosen points and correct rise/run calculation
• Exit ticket: correct intercept calculations and a reasonably accurate sketch demonstrating the correct gradient

Differentiation

• Support:
• Provide a “two-point method” scaffold: choose (x=0) to get the y-intercept, then choose one convenient (x) value to find a second point
• Offer sentence starters for explanations: “The gradient is… because…”, “The x-intercept is when (y=0)…”
• For one student, pre-mark a target scale on graph paper to reduce reading errors
• Extension:
• Add a context statement: “A taxi charges a fixed fee plus a rate per kilometre. Model the cost and interpret the intercepts.”
• Ask students to state how changing (m) versus changing (c) would rotate/shift the line
• EAL/SEN considerations:
• Use consistent language and visual cues (axes labels, colour-coding x-axis, y-axis, intercept dots)
• Check understanding of terms “gradient”, “intercept”, “straight-line” before independent graphing