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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 − 4x + 1 at x = 1. → -8Prompt: Differentiate first.
  2. Find the gradient of y = x² − 8/x at x = −2. → -2Prompt: 8/x = 8x⁻¹, which differentiates to −8x⁻².
  3. At what x is the gradient of y = x² − 4x − 3 equal to 2? → x = 3Prompt: 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′(3) = −6. What does this mean? (At that moment the battery is falling by about 6 percentage points per hour / The average rate over the whole period / The total change since the start / The battery has 6% left) → 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) = 4x² + 44x + 200. Use C′(33) to estimate the extra cost of making one more case after 33. → 308Prompt: 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. The cost (in dollars) of making x phone cases is C(x) = 2x² + 33x + 200. Use C′(28) to estimate the extra cost of making one more case after 28. → 145Prompt: 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′(2) = −6. What does this mean? (The average rate over the whole period / At that moment the battery is falling by about 6 percentage points per hour / The battery has 6% left / The total change since the start) → 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.

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

    The cost (in dollars) of making x phone cases is C(x) = 3x² + 21x + 200. Use C′(23) to estimate the extra cost of making one more case after 23.

    Answer

  2. Question 2

    T(m) is the temperature (°C) of a drink m minutes after it is made. T′(17) = −9. What does this mean?

    • A. The drink is at 9 °C after 17 minutes
    • B. The total change since the start
    • C. The average rate over the whole period
    • D. At 17 minutes, the drink is cooling at about 9 °C per minute

  3. Question 3

    The cost (in dollars) of making x phone cases is C(x) = 4x² + 16x + 200. Use C′(26) to estimate the extra cost of making one more case after 26.

    Answer

  4. Question 4

    B(h) is a phone's battery level (%) after h hours. B′(9) = −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 battery has 9% left
    • D. The average rate over the whole period

  5. Question 5

    The cost (in dollars) of making x phone cases is C(x) = 3x² + 47x + 200. Use C′(31) to estimate the extra cost of making one more case after 31.

    Answer

  6. Question 6

    V(t) is the number of users of an app after t days. V′(16) = 33. What does this mean?

    • A. On day 16, users are increasing at about 33 per day
    • B. The average rate over the whole period
    • C. There are 33 users on day 16
    • D. The total change since the start

  7. Question 7

    The cost (in dollars) of making x phone cases is C(x) = 2x² + 12x + 200. Use C′(32) to estimate the extra cost of making one more case after 32.

    Answer

  8. Question 8

    T(m) is the temperature (°C) of a drink m minutes after it is made. T′(24) = −2. What does this mean?

    • A. The total change since the start
    • B. At 24 minutes, the drink is cooling at about 2 °C per minute
    • C. The drink is at 2 °C after 24 minutes
    • D. The average rate over the whole period

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

  1. 1. 159

  2. 2. D. At 17 minutes, the drink is cooling at about 9 °C per minute

  3. 3. 224

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

  5. 5. 233

  6. 6. A. On day 16, users are increasing at about 33 per day

  7. 7. 140

  8. 8. B. At 24 minutes, the drink is cooling at about 2 °C per minute