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Math lesson plan · 12th grade (Year 13 in England) · 30 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 · 4 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.
  1. x: 0, 1, 2, 3, 4 f(x): 2, 3, 6, 11, 18 Find the average rate of change of f from x = 1 to x = 4. → 5Prompt: Average rate = change in f(x) ÷ change in x.
  2. f(x) = −2x² + 3x + 2. Find the average rate of change of f between x = −3 and x = −1. → 11Prompt: Evaluate f at both ends of the interval.
  3. A ball is thrown upwards. Its height in metres after t seconds is h(t) = 30t − 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. → 15Prompt: Find the height at each time.

Hook · 3 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 · 7 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 · 6 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. For y = x², find the gradient of the chord from x = −2 to x = 0. → -2Prompt: Gradient = change in y ÷ change in x.
  2. For y = x², chords from x = 0 to x = 0 + h have gradients: h = 0.1: 0.1 h = 0.01: 0.01 h = 0.001: 0.001 What is the gradient of the curve at x = 0? → 0Prompt: As h shrinks, the chord gets closer to the tangent.

You do: independent practice · 7 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 6 questions are on the student sheet below, with answers in the key.

Exit ticket · 3 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. For y = x², find the gradient of the chord from x = −1 to x = 0. → -1Prompt: Gradient = change in y ÷ change in x.
  2. For y = x³, find the gradient of the chord from x = 3 to x = 3.5. Give 2 decimal places. → 31.75Prompt: Work out both y-values.

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

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Estimate the gradient of a curve with chords

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Date

Score/ 6

  1. Question 1

    For y = x³, find the gradient of the chord from x = 3 to x = 3.1. Give 2 decimal places.

    Answer

  2. Question 2

    For y = x², chords from x = −2 to x = −2 + h have gradients: h = 0.1: −3.9 h = 0.01: −3.99 h = 0.001: −3.999 What is the gradient of the curve at x = −2?

    Answer

  3. Question 3

    For y = x², find the gradient of the chord from x = 3 to x = 4.

    Answer

  4. Question 4

    For y = x³, find the gradient of the chord from x = 2 to x = 2.2. Give 2 decimal places.

    Answer

  5. Question 5

    For y = x², chords from x = −3 to x = −3 + h have gradients: h = 0.1: −5.9 h = 0.01: −5.99 h = 0.001: −5.999 What is the gradient of the curve at x = −3?

    Answer

  6. Question 6

    For y = x², find the gradient of the chord from x = 2 to x = 3.

    Answer

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Answer key: Estimate the gradient of a curve with chords

  1. 1. 27.91

  2. 2. -4

  3. 3. 7

  4. 4. 13.24

  5. 5. -6

  6. 6. 5