Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Find and use the second derivative
Calculus
CCSS HSF.IF.B.4England: A-level Pure Mathematics
Objective and success criteria
Students will: Find and use the second derivative.
- 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: Differentiate polynomials term by term
Leads to: Use derivatives for velocity and acceleration
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 (Differentiate polynomials term by term). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find dy/dx when y = 4x³ − x² − 9x + 2. → 12x^2-2x-9Prompt: Differentiate each term separately.
- Differentiate f(x) = x⁴ + 9x² − 7x − 8. → 4x^3+18x-7Prompt: Bring each power down and reduce it by 1.
- Differentiate y = 2x³/3 + 8x²/2 − 6x. → 2x^2+8x-6Prompt: The fractions are just coefficients: x³/3 means (1/3)x³.
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: Find and use the second derivative. 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: Find d²y/dx² when y = 2x⁴ + 4x³ + 5x² + x + 5.
- Step 1: dy/dx = 8x³ + 12x² + 10x + 1
- Step 2: d²y/dx² = 24x² + 24x + 10
- Answer: 24x^2+24x+10
- 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.
- Find d²y/dx² when y = 3x⁴ − 3x³ + 5x² + 2x − 9. → 36x^2-18x+10Prompt: Differentiate once to get dy/dx.
- f(x) = 3x³ + 4x² − 5x − 4. Find f″(−3). → -46Prompt: Differentiate twice.
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.
- f(x) = −x³ + 5x² − 4x − 6. Find f″(1). → 4Prompt: Differentiate twice.
- Find d²y/dx² when y = x⁴ + 5x³ − 2x² − 8x − 5. → 12x^2+30x-4Prompt: Differentiate once to get dy/dx.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Differentiate polynomials term by term.
- 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
- Move on to: Use derivatives for velocity and acceleration.
- 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