Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Estimate the gradient of a curve with chords
Calculus
CCSS HSF.IF.B.6England: A-level Pure Mathematics
Objective and success criteria
Students will: Estimate the gradient of a curve with chords.
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: Average rate of change over an interval
Leads to: Differentiate from first principles
Materials: Graphing tool or graph paper, Scientific calculators, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 5 min
- Quick retrieval from earlier learning (Average rate of change over an interval). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- x: 0, 1, 2, 3, 4 f(x): 1, 6, 15, 28, 45 Find the average rate of change of f from x = 2 to x = 4. → 15Prompt: Average rate = change in f(x) ÷ change in x.
- f(x) = x² + 4x − 6. Find the average rate of change of f between x = 0 and x = 4. → 8Prompt: Evaluate f at both ends of the interval.
- A ball is thrown upwards. Its height in metres after t seconds is h(t) = 16t − 5t². Find its average speed upwards (rate of change of height, in m/s) from t = 1 to t = 2. A negative answer means it is coming down. → 1Prompt: Find the height at each time.
Hook · 5 min
- Show a distance-time graph of a journey and ask how fast it was going at one exact moment.
- Share the objective: Estimate the gradient of a curve with chords. Read the success criteria together.
I do: model it · 10 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: For y = x², find the gradient of the chord from x = −3 to x = −2.9.
- Step 1: (8.41 − 9) ÷ 0.1
- Step 2: = −5.9
- Answer: -5.9
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 10 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- For y = x², find the gradient of the chord from x = 5 to x = 7. → 12Prompt: Gradient = change in y ÷ change in x.
- For y = x², chords from x = 2 to x = 2 + h have gradients: h = 0.1: 4.1 h = 0.01: 4.01 h = 0.001: 4.001 What is the gradient of the curve at x = 2? → 4Prompt: As h shrinks, the chord gets closer to the tangent.
You do: independent practice · 10 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 8 questions are on the student sheet below, with answers in the key.
Exit ticket · 5 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- For y = x², chords from x = 4 to x = 4 + h have gradients: h = 0.1: 8.1 h = 0.01: 8.01 h = 0.001: 8.001 What is the gradient of the curve at x = 4? → 8Prompt: As h shrinks, the chord gets closer to the tangent.
- For y = x², find the gradient of the chord from x = 1 to x = 3. → 4Prompt: Gradient = change in y ÷ change in x.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Average rate of change over an interval.
- Use concrete materials: graphing tool or graph paper.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Move on to: Differentiate from first principles.
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.
Free lesson plan from eurekami.com/lesson-plan-builder