Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Use derivatives for velocity and acceleration
Calculus
CCSS HSF.IF.B.6England: A-level Pure Mathematics
Objective and success criteria
Students will: Use derivatives for velocity and acceleration.
- 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 and use the second derivative
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 and use the second derivative). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find d²y/dx² when y = 2x⁴ − 2x³ − 2x² + 3x + 9. → 24x^2-12x-4Prompt: Differentiate once to get dy/dx.
- f(x) = −x³ − 2x² + 3x − 2. Find f″(1). → -10Prompt: Differentiate twice.
- Find d²y/dx² when y = −3x⁴ + x³ + 2x² + 9x + 2. → -36x^2+6x+4Prompt: Differentiate once to get dy/dx.
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: Use derivatives for velocity and acceleration. 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: A particle moves so that its displacement is s = t³ + t² + 9t metres after t seconds. Find its velocity at t = 5.
- Step 1: v = 3t² + 2t + 9
- Step 2: v(5) = 94 m/s
- Answer: 94
- 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.
- A particle moves so that its displacement is s = t³ − 4t² − 4t metres after t seconds. Find its velocity at t = 4. → 12Prompt: Velocity v = ds/dt.
- s = t³ − 12t² + 36t. At what times is the particle at rest? Separate them with a comma. → t = 2 or t = 6Prompt: At rest means velocity = 0.
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.
- A particle moves so that its displacement is s = t³ + 5t² + t metres after t seconds. Find its velocity at t = 3. → 58Prompt: Velocity v = ds/dt.
- s = t³ − 9t² + 15t. At what times is the particle at rest? Separate them with a comma. → t = 1 or t = 5Prompt: At rest means velocity = 0.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Find and use the second derivative.
- 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