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Math lesson plan · 12th grade (Year 13 in England) · 45 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 · 5 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 + 3x² − 8x + 6 at x = −2. → -44Prompt: Differentiate first.
  2. Find the gradient of y = x² − 1/x at x = −1. → -1Prompt: 1/x = 1x⁻¹, which differentiates to −1x⁻².
  3. At what x is the gradient of y = x² − 3x + 8 equal to 3? → x = 3Prompt: Differentiate, then set dy/dx equal to the gradient.

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: Interpret a derivative as a rate of change in context. 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: 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 · 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.
  1. V(t) is the number of users of an app after t days. V′(13) = 17. What does this mean? (The total change since the start / There are 17 users on day 13 / On day 13, users are increasing at about 17 per day / The average rate over the whole period) → On day 13, users are increasing at about 17 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) = x² + 41x + 200. Use C′(9) to estimate the extra cost of making one more case after 9. → 59Prompt: The derivative gives the cost per extra case at that point (marginal cost).

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.
  1. The cost (in dollars) of making x phone cases is C(x) = 4x² + 56x + 200. Use C′(35) to estimate the extra cost of making one more case after 35. → 336Prompt: The derivative gives the cost per extra case at that point (marginal cost).
  2. B(h) is a phone's battery level (%) after h hours. B′(3) = −1. What does this mean? (The total change since the start / At that moment the battery is falling by about 1 percentage points per hour / The battery has 1% left / The average rate over the whole period) → At that moment the battery is falling by about 1 percentage points per hourPrompt: A derivative is a rate: units of output per unit of input.

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

    T(m) is the temperature (°C) of a drink m minutes after it is made. T′(27) = −7. What does this mean?

    • A. The average rate over the whole period
    • B. At 27 minutes, the drink is cooling at about 7 °C per minute
    • C. The total change since the start
    • D. The drink is at 7 °C after 27 minutes

  2. Question 2

    The cost (in dollars) of making x phone cases is C(x) = 2x² + 44x + 200. Use C′(8) to estimate the extra cost of making one more case after 8.

    Answer

  3. Question 3

    T(m) is the temperature (°C) of a drink m minutes after it is made. T′(28) = −1. What does this mean?

    • A. The total change since the start
    • B. The average rate over the whole period
    • C. At 28 minutes, the drink is cooling at about 1 °C per minute
    • D. The drink is at 1 °C after 28 minutes

  4. Question 4

    The cost (in dollars) of making x phone cases is C(x) = x² + 34x + 200. Use C′(37) to estimate the extra cost of making one more case after 37.

    Answer

  5. Question 5

    B(h) is a phone's battery level (%) after h hours. B′(6) = −1. What does this mean?

    • A. At that moment the battery is falling by about 1 percentage points per hour
    • B. The total change since the start
    • C. The average rate over the whole period
    • D. The battery has 1% left

  6. Question 6

    The cost (in dollars) of making x phone cases is C(x) = 3x² + 24x + 200. Use C′(26) to estimate the extra cost of making one more case after 26.

    Answer

  7. Question 7

    T(m) is the temperature (°C) of a drink m minutes after it is made. T′(12) = −5. What does this mean?

    • A. At 12 minutes, the drink is cooling at about 5 °C per minute
    • B. The drink is at 5 °C after 12 minutes
    • C. The total change since the start
    • D. The average rate over the whole period

  8. Question 8

    The cost (in dollars) of making x phone cases is C(x) = x² + 14x + 200. Use C′(25) to estimate the extra cost of making one more case after 25.

    Answer

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

  1. 1. B. At 27 minutes, the drink is cooling at about 7 °C per minute

  2. 2. 76

  3. 3. C. At 28 minutes, the drink is cooling at about 1 °C per minute

  4. 4. 108

  5. 5. A. At that moment the battery is falling by about 1 percentage points per hour

  6. 6. 180

  7. 7. A. At 12 minutes, the drink is cooling at about 5 °C per minute

  8. 8. 64