Back to the lesson plan builder
Lesson length30 min45 min60 min
Questions🔀 New questions

Print the plan for yourself; the student sheet prints on its own page after it.

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 = x³ + 2x² − 4x − 8 at x = 0. → -4Prompt: Differentiate first.
  2. Find the gradient of y = x² + 4/x at x = −2. → -5Prompt: 4/x = 4x⁻¹, which differentiates to −4x⁻².
  3. At what x is the gradient of y = x² + 2x + 5 equal to 12? → 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. V(t) is the number of users of an app after t days. V′(9) = 27. What does this mean? (The average rate over the whole period / On day 9, users are increasing at about 27 per day / The total change since the start / There are 27 users on day 9) → On day 9, users are increasing at about 27 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 + 39x + 200. Use C′(39) to estimate the extra cost of making one more case after 39. → 117Prompt: 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′(13) = 39. What does this mean? (The total change since the start / On day 13, users are increasing at about 39 per day / The average rate over the whole period / There are 39 users on day 13) → On day 13, users are increasing at about 39 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) = 2x² + 13x + 200. Use C′(25) to estimate the extra cost of making one more case after 25. → 113Prompt: 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

eurekami

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 + 32x + 200. Use C′(39) to estimate the extra cost of making one more case after 39.

    Answer

  2. Question 2

    T(m) is the temperature (°C) of a drink m minutes after it is made. T′(30) = −2. What does this mean?

    • A. The drink is at 2 °C after 30 minutes
    • B. The average rate over the whole period
    • C. The total change since the start
    • D. At 30 minutes, the drink is cooling at about 2 °C per minute

  3. Question 3

    The cost (in dollars) of making x phone cases is C(x) = 3x² + 18x + 200. Use C′(33) to estimate the extra cost of making one more case after 33.

    Answer

  4. Question 4

    T(m) is the temperature (°C) of a drink m minutes after it is made. T′(6) = −4. What does this mean?

    • A. The total change since the start
    • B. The drink is at 4 °C after 6 minutes
    • C. At 6 minutes, the drink is cooling at about 4 °C per minute
    • D. The average rate over the whole period

  5. Question 5

    The cost (in dollars) of making x phone cases is C(x) = 3x² + 16x + 200. Use C′(19) to estimate the extra cost of making one more case after 19.

    Answer

  6. Question 6

    B(h) is a phone's battery level (%) after h hours. B′(8) = −7. What does this mean?

    • A. The battery has 7% left
    • B. The total change since the start
    • C. The average rate over the whole period
    • D. At that moment the battery is falling by about 7 percentage points per hour

  7. Question 7

    The cost (in dollars) of making x phone cases is C(x) = 4x² + 58x + 200. Use C′(20) to estimate the extra cost of making one more case after 20.

    Answer

  8. Question 8

    V(t) is the number of users of an app after t days. V′(21) = 28. What does this mean?

    • A. There are 28 users on day 21
    • B. The average rate over the whole period
    • C. On day 21, users are increasing at about 28 per day
    • D. The total change since the start

Free from eurekami.com

for the adult only

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

  1. 1. 266

  2. 2. D. At 30 minutes, the drink is cooling at about 2 °C per minute

  3. 3. 216

  4. 4. C. At 6 minutes, the drink is cooling at about 4 °C per minute

  5. 5. 130

  6. 6. D. At that moment the battery is falling by about 7 percentage points per hour

  7. 7. 218

  8. 8. C. On day 21, users are increasing at about 28 per day