Math lesson plan · 12th grade (Year 13 in England) · 30 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 · 4 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 = 3x⁴ − x³ − 7x² + 7x − 3. → 36x^2-6x-14Prompt: Differentiate once to get dy/dx.
- f(x) = 2x³ − x² − 3x + 9. Find f″(−3). → -38Prompt: Differentiate twice.
- Find d²y/dx² when y = 2x⁴ − 4x³ + 6x² + 4x + 8. → 24x^2-24x+12Prompt: Differentiate once to get dy/dx.
Hook · 3 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 · 7 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 · 6 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 − 5t² metres after t seconds. Find its velocity at t = 5. → 25Prompt: 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 · 7 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 6 questions are on the student sheet below, with answers in the key.
Exit ticket · 3 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- s = t³ + 2t² + t (metres, seconds). Find the acceleration at t = 2. → 16Prompt: Acceleration a = dv/dt = d²s/dt².
- s = t³ − 12t² + 45t. At what times is the particle at rest? Separate them with a comma. → t = 3 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