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Math lesson plan · 12th grade (Year 13 in England) · 60 minutes

Interpret a derivative as a rate of change in context

Calculus

CCSS HSF.IF.B.6CCSS HSF.IF.B.4England: A-level Pure Mathematics

Objective and success criteria

Students will: Interpret a derivative as a rate of change in context.

  • 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: Find the gradient of a curve at a point

Leads to: the next skill in this area.

Materials: Graphing tool or graph paper, Scientific calculators, Mini whiteboards and pens, Printed student sheet (below).

Warm-up · 6 min

  • Quick retrieval from earlier learning (Find the gradient of a curve at a point). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. Find the gradient of y = −2x³ − 5x² + 2x + 2 at x = 1. → -14Prompt: Differentiate first.
  2. Find the gradient of y = x² + 16/x at x = −2. → -8Prompt: 16/x = 16x⁻¹, which differentiates to −16x⁻².
  3. At what x is the gradient of y = x² + 6x + 6 equal to 16? → x = 5Prompt: Differentiate, then set dy/dx equal to the gradient.

Hook · 6 min

  • Show a distance-time graph of a journey and ask how fast it was going at one exact moment.
  • Share the objective: Interpret a derivative as a rate of change in context. Read the success criteria together.

I do: model it · 12 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: V(t) is the number of users of an app after t days. V′(15) = 32. What does this mean?
  • Step 1: On day 15, users are increasing at about 32 per day
  • Answer: On day 15, users are increasing at about 32 per day
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 12 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. B(h) is a phone's battery level (%) after h hours. B′(6) = −6. What does this mean? (The battery has 6% left / The average rate over the whole period / The total change since the start / At that moment the battery is falling by about 6 percentage points per hour) → At that moment the battery is falling by about 6 percentage points per hourPrompt: A derivative is a rate: units of output per unit of input.
  2. The cost (in dollars) of making x phone cases is C(x) = 2x² + 60x + 200. Use C′(29) to estimate the extra cost of making one more case after 29. → 176Prompt: The derivative gives the cost per extra case at that point (marginal cost).

You do: independent practice · 17 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 · 7 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. V(t) is the number of users of an app after t days. V′(10) = 35. What does this mean? (The total change since the start / There are 35 users on day 10 / The average rate over the whole period / On day 10, users are increasing at about 35 per day) → On day 10, users are increasing at about 35 per dayPrompt: A derivative is a rate: units of output per unit of input.
  2. The cost (in dollars) of making x phone cases is C(x) = 3x² + 60x + 200. Use C′(5) to estimate the extra cost of making one more case after 5. → 90Prompt: The derivative gives the cost per extra case at that point (marginal cost).

Watch out for

  • Forgetting the constant of integration.
  • Differentiating a product by multiplying the derivatives.

Support and stretch

Support

  • Review first: Find the gradient of a curve at a point.
  • 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

  • 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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Interpret a derivative as a rate of change in context

Name

Date

Score/ 8

  1. Question 1

    V(t) is the number of users of an app after t days. V′(10) = 14. What does this mean?

    • A. The average rate over the whole period
    • B. There are 14 users on day 10
    • C. The total change since the start
    • D. On day 10, users are increasing at about 14 per day

  2. Question 2

    The cost (in dollars) of making x phone cases is C(x) = 2x² + 36x + 200. Use C′(13) to estimate the extra cost of making one more case after 13.

    Answer

  3. Question 3

    B(h) is a phone's battery level (%) after h hours. B′(6) = −9. What does this mean?

    • A. The battery has 9% left
    • B. The average rate over the whole period
    • C. At that moment the battery is falling by about 9 percentage points per hour
    • D. The total change since the start

  4. Question 4

    The cost (in dollars) of making x phone cases is C(x) = x² + 34x + 200. Use C′(6) to estimate the extra cost of making one more case after 6.

    Answer

  5. Question 5

    V(t) is the number of users of an app after t days. V′(11) = 31. What does this mean?

    • A. The total change since the start
    • B. On day 11, users are increasing at about 31 per day
    • C. There are 31 users on day 11
    • D. The average rate over the whole period

  6. Question 6

    The cost (in dollars) of making x phone cases is C(x) = 2x² + 30x + 200. Use C′(13) to estimate the extra cost of making one more case after 13.

    Answer

  7. Question 7

    B(h) is a phone's battery level (%) after h hours. B′(2) = −9. What does this mean?

    • A. The total change since the start
    • B. At that moment the battery is falling by about 9 percentage points per hour
    • C. The average rate over the whole period
    • D. The battery has 9% left

  8. Question 8

    The cost (in dollars) of making x phone cases is C(x) = x² + 32x + 200. Use C′(27) to estimate the extra cost of making one more case after 27.

    Answer

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Answer key: Interpret a derivative as a rate of change in context

  1. 1. D. On day 10, users are increasing at about 14 per day

  2. 2. 88

  3. 3. C. At that moment the battery is falling by about 9 percentage points per hour

  4. 4. 46

  5. 5. B. On day 11, users are increasing at about 31 per day

  6. 6. 82

  7. 7. B. At that moment the battery is falling by about 9 percentage points per hour

  8. 8. 86