Homeschool Guide: These lesson plans are a guide for parents. Content may contain errors — always cross-reference with official exam board specifications.

gradients & intercepts

FoundationHigherAll Boards

4 detailed 50-minute lessons with teaching scripts, worked examples, parent guides, and assessment criteria.

Fastmail

Lesson Overview

Total Lessons: 4
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA

Learning Objectives

Prerequisites

Materials & Equipment

Lesson 1: Introduction: gradients & intercepts

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Write down everything you already know about gradients & intercepts. Then check against the key terms: Gradient, Intercepts. Use a mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Gradient, Intercepts.
If stuck: Re-read the revision notes (link above), then break the content into smaller steps.
Extension: See the Stretch & Challenge ideas in Lesson 4.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: gradients & intercepts. By the end you will be able to answer exam questions on it unaided. It connects to the rest of Mathematics because the ideas here recur across the spec."
Mins 5-20 - Direct Instruction: Work through the core ideas below one at a time; after each, ask your student to explain it back in their own words.
Mins 20-30 - Guided Practice: Model the worked example together, then let your student attempt the first practice question with guidance.
Mins 30-35 - Independent Practice: 2-3 practice questions from Lesson 3 below, with immediate feedback.
First Look

Start with the revision notes summary, then attempt: Find the gradient of the line passing through (0, 2) and (4, 10).

Plenary (5 minutes)

Check Out

Your student states one thing they learned and one question they still have about gradients & intercepts.

Lesson 2: Core Concepts: gradients & intercepts

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

Quick recap: write 3 key points from Lesson 1 on gradients & intercepts. Check them against the notes below.

Main Content (35 minutes)

Gradient: The steepness of a line. Found by dividing vertical change by horizontal change.
Intercepts: y-intercept: Where the line crosses the y-axis (when x = 0) x-intercept: Where the line crosses the x-axis (when y = 0)

Practice (10 minutes)

Q: Find the gradient of the line passing through (0, 2) and (4, 10).

Answer: 2

Plenary (5 minutes)

Explain Back

Your student teaches the key points back to you without looking. Fill any gaps immediately.

Lesson 3: Application: gradients & intercepts

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Recall the key terms: Gradient, Intercepts. Define each in one sentence.

Main Content (35 minutes)

Parent/Teacher Guide: Let your student attempt each question alone first, then compare with the model answer. Award method marks for correct working even if the final answer is wrong.

Q1: Find the gradient of the line passing through (0, 2) and (4, 10).

Answer: 2

Q2: Find the y-intercept of y = 5x - 8.

Answer: (0, -8)

Q3: Find the x-intercept of y = 3x + 9.

Answer: (-3, 0)

Q4: A line has gradient -2 and passes through (1, 5). Find its y-intercept.

Answer: (0, 7)

Q5: A distance-time graph shows d = 50t + 100. What does the gradient represent?

Answer: Speed = 50 (units per hour)

Q6: Write the equation of a line with gradient 4 and y-intercept (0, -1).

Answer: y = 4x - 1

Plenary (5 minutes)

Error Review

Review any questions answered incorrectly. Identify whether the error was knowledge, method, or reading the question.

Lesson 4: Exam Practice: gradients & intercepts

Duration: 50 minutes

Starter Activity (5 minutes)

Command Words

Review what these command words require: state (one point), describe (say what happens), explain (say why), compare (both sides), evaluate (judgement).

Main Content (35 minutes)

Extended Answer

Extended question: Extended Answer 6 marks: Two water tanks are being drained. Tank A: d = 120 - 8t. Tank B: d = 100 - 5t. d = depth in cm, t = time in minutes. (a) Which tank starts fuller? (b) Which drains faster? (c) After how many minutes is Tank A empty? (d) Which tank still has water left after 15 minutes? <div class="

(a) Tank A starts at 120 cm, Tank B at 100 cm. Tank A is fuller. (b) Tank A drains at 8 cm/min (gradient = -8). Tank B drains at 5 cm/min. Tank A drains faster. (c) 120 - 8t = 0 → t = 15 minutes (d) After 15 minutes: Tank A = 120 - 8(15) = 0 cm (empty). Tank B = 100 - 5(15) = 25 cm. Tank B still has water. Mark scheme: (a) 1 mark. (b) 1 mark with reasoning. (c) 2 marks. (d) 2 marks for both tanks checked.

Exam Tips: y-intercept: substitute x = 0 into the equation | x-intercept: substitute y = 0 and solve for x | Gradient = change in y ÷ change in x | Positive gradient = uphill; negative = downhill | In context, gradient often means "rate" (price per unit, speed, etc.)
Common Errors: Watch Out! 1. Wrong: The y-intercept of y = 5x - 3 is 3 Correct: The y-intercept is -3 (the sign is included: y-intercept at (0, -3)) 2. Wrong: Gradient = horizontal change ⁄ vertical change Correct: Gradient = vertical change ⁄ horizontal change = Δy ⁄ Δx 3. Wrong: In C = 30t + 50, the gradient means "30 hours per £" Correct: The gradient 30 means £30 per hour (units of y per unit of x)
AO3 - Reasoning & Interpretation: Reasoning and Interpretation A monthly phone bill is B = 0.05t + 12, where B = cost (£), t = number of texts. (a) Explain what the gradient and y-intercept represent. (b) If the bill is £27, how many texts were sent? (c) A rival network charges B = 0.08t + 8. After how many texts is it cheaper to stay with the original network? Answers: (a) Gradient 0.05 = 5p per text. y-intercept 12 = £12 fixed monthly charge. (b) 27 = 0.05t + 12 → 0.05t = 15 → t = 300 texts. (c) 0.05t + 12 133.3. Original network cheaper after 134+ texts.
Stretch & Challenge (Grade 8-9):
  • Synoptic links: explain how gradients & intercepts connects to another Mathematics topic you have studied
  • Real-world: research one real-world use or example of gradients & intercepts
  • Critical: "What are the limitations of the models used in gradients & intercepts?"

Plenary (5 minutes)

Assessment Criteria
  • Got it: Confident explanation + correct worked examples
  • Getting there: Main points OK, needs support with detail
  • Not yet: Confused on key concepts - re-run Lesson 2

Homework & Consolidation

Recommended Resources

🎓 Smart Lesson (Guided)