Intercept and Gradient

Overview

Students revise how to find the gradient of a line on a Cartesian plane, then learn to find the y-intercept using a given rule and interpret what the y-intercept means. They practise with teacher-guided worked examples and a formative worksheet.

Learning intentions

• Students will revise how to calculate the gradient from two points on a Cartesian plane.
• Students will identify the y-intercept as the value of (y) when (x=0).
• Students will use a provided rule to find the y-intercept for linear relationships.
• Students will interpret solutions by linking numeric answers to positions on the graph.

Success criteria

• I can calculate the gradient using rise-over-run (or change in (y) over change in (x)).
• I can state the y-intercept as the (y)-value where the line crosses the y-axis.
• I can apply the given rule to determine the y-intercept correctly.
• I can check reasonableness by comparing my answer to the graph position.

Curriculum links

• Number and algebra: solve and interpret linear relationships, including gradient and intercepts on a Cartesian plane.
• Graphing and interpreting: use Cartesian coordinates to interpret features of linear graphs.
• Mathematical reasoning: justify solutions using connections between equations and graph features.
• Using language of graphs: interpret and communicate meaning of gradient and y-intercept.

Lesson structure (80 minutes)

1. 0–10 min: Warm-up—gradient revision (I do/we do) Teacher displays a graph with two clear points on a line. Students join in to compute rise/run and state the gradient. Teacher emphasises correct direction (left/right and up/down) and units.

2. 10–20 min: Quick check—mini tasks (we do) Students complete 3 short items on gradient (from a table of coordinates and from a graph). Teacher circulates and addresses common errors: sign mistakes and confusing run with rise.

3. 20–30 min: Introduce y-intercept meaning (I do) Teacher draws axes and a line crossing the y-axis. Students identify the point where the line meets the y-axis, then read the corresponding (y)-value. Teacher explicitly states: “y-intercept is what happens at (x=0).”

4. 30–45 min: Use the provided rule for y-intercept (I do/we do) Teacher models one linear relationship where the y-intercept can be found using the given rule (for example, substituting (x=0) into the relationship or using the rule presented in class). Teacher narrates: “Because (x=0), the rule simplifies—this gives the y-intercept directly.” Students then do one paired example with the teacher, marking where the intercept appears on the graph.

5. 45–60 min: Guided practice worksheet (you do) Students complete the teacher’s formative worksheet section:

• Part A: gradient from two points
• Part B: y-intercept from the graph
• Part C: y-intercept using the rule Teacher uses “stop and check” every 5 minutes for quick misconceptions (e.g., giving x-value instead of y-value).

6. 60–70 min: Error analysis and whole-class feedback (we do) Teacher selects 2–3 anonymised responses (printed or projected) showing typical mistakes (wrong sign, confusing intercepts, incorrect rule use). Students discuss what the response did and how to correct it, then re-calculate as a class.

7. 70–80 min: Exit ticket—success criteria check (you do) Students answer 2 questions: one gradient and one y-intercept using the rule, plus a short prompt: “Explain how you know your answer matches the graph.” Teacher collects for formative use.

Resources

• Cartesian plane graph paper (grid) and/or interactive whiteboard graphs
• Gradient reference strategy card: “change in y / change in x” with sign conventions
• Teacher slides showing worked examples (graph + points + axis crossing)
• Formative worksheet with sections A–C aligned to the lesson
• Coloured pencils or highlighters to mark rise/run and the intercept point
• Timer for timed practise segments
• Exit ticket slips or digital form (no links needed)
• Answer key for teacher only (for quick marking and error analysis)

Assessment

• Formative worksheet results used to identify who needs more support with intercept vs gradient.
• Exit ticket checks mastery of both gradient revision and y-intercept using the rule.
• Teacher observation during “stop and check” identifies misconceptions early.

Differentiation

• Support: provide a step-by-step scaffold on the worksheet (number line/rise-run framing, and a prompt “set (x=0)” if the rule uses substitution).
• Support: allow use of a pre-drawn coordinate grid and tracing of the y-axis crossing to read the intercept.
• Extension: include an extra task where students compare two lines with the same gradient but different y-intercepts, predicting which intercept is larger using the rule.
• EAL/SEN: sentence starters for reasoning (“The y-intercept is the y-value when…”; “I know it matches because the line crosses the y-axis at…”). Use consistent language and visual examples with less text.

Extension (optional)

• Advanced learners: give a new linear relationship and ask students to find both gradient and y-intercept, then write the equation in the form that matches the rule taught (and verify by plotting two points).