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

gradients & areas under graphs

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 & areas under graphs

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Write down everything you already know about gradients & areas under graphs. Then check against the key terms: Gradient of a graph, Area under a graph. Use a mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Gradient of a graph, Area under a graph.
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 & areas under graphs. 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: On a distance-time graph, what does the gradient represent?

Plenary (5 minutes)

Check Out

Your student states one thing they learned and one question they still have about gradients & areas under graphs.

Lesson 2: Core Concepts: gradients & areas under graphs

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

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

Main Content (35 minutes)

Gradient of a graph: The rate of change. Calculated by finding the change in y divided by the change in x.
Area under a graph: Can represent important quantities in real contexts (e.g., distance for velocity-time graphs).

Practice (10 minutes)

Q: On a distance-time graph, what does the gradient represent?

Answer: Speed

Plenary (5 minutes)

Explain Back

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

Lesson 3: Application: gradients & areas under graphs

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Recall the key terms: Gradient of a graph, Area under a graph. 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: On a distance-time graph, what does the gradient represent?

Answer: Speed

Q2: On a velocity-time graph, what does the gradient represent?

Answer: Acceleration

Q3: On a velocity-time graph, what does the area under the graph represent?

Answer: Distance travelled

Q4: A car travels 200 miles in 4 hours. Calculate the gradient of the distance-time graph.

Answer: 50 miles per hour

Q5: A velocity-time graph shows a triangle with base 10s and height 30 m/s. Calculate the distance travelled.

Answer: 150 m

Q6: A car accelerates from 10 m/s to 30 m/s in 8 seconds. Calculate the acceleration.

Answer: 2.5 m/s²

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 & areas under graphs

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: A car accelerates uniformly from rest to 25 m/s in 10 seconds, maintains 25 m/s for 30 seconds, then decelerates uniformly to rest in 5 seconds. (a) Calculate the acceleration. (b) Calculate the total distance. (c) If the speed limit is 60 km/h, is the car breaking the limit? Show working. <div class="

(a) Acceleration = 25 ⁄ 10 = 2.5 m/s² (b) Area 1: ½ × 10 × 25 = 125 m. Area 2: 30 × 25 = 750 m. Area 3: ½ × 5 × 25 = 62.5 m. Total = 937.5 m (c) 25 m/s × 3.6 = 90 km/h. This exceeds 60 km/h, so yes, the car is breaking the speed limit. Mark scheme: (a) 1 mark. (b) 3 marks. (c) 2 marks for conversion and conclusion.

Exam Tips: Distance-time: gradient = speed, area = no meaning | Velocity-time: gradient = acceleration, area = distance | Split area into triangles and rectangles for calculations | For curves, count squares or estimate | Always check units in your answer
Common Errors: Watch Out! 1. Wrong: Area under a distance-time graph gives distance Correct: Area under a distance-time graph has no physical meaning — use gradient for speed 2. Wrong: Gradient of a velocity-time graph gives speed Correct: Gradient gives acceleration (rate of change of velocity) 3. Wrong: The area under a curved graph can be found exactly using a triangle formula Correct: For curves, estimate using trapeziums or counting squares — it's an approximation
AO3 - Reasoning & Interpretation: Reasoning and Interpretation A runner's velocity-time graph shows a curve that gradually flattens. (a) What does the flattening curve tell you about the runner's acceleration? (b) How would you estimate the total distance from this graph? (c) Explain why a straight-line graph would be unrealistic for this situation. Answers: (a) The runner's acceleration is decreasing — they speed up more slowly. (b) Count squares under the curve, or divide into trapeziums and sum areas. (c) A straight line means constant acceleration — runners cannot keep accelerating at the same rate; they tire and approach a maximum speed.
Stretch & Challenge (Grade 8-9):
  • Synoptic links: explain how gradients & areas under graphs connects to another Mathematics topic you have studied
  • Real-world: research one real-world use or example of gradients & areas under graphs
  • Critical: "What are the limitations of the models used in gradients & areas under graphs?"

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)