Math lesson plan · 12th grade (Year 13 in England) · 30 minutes
Find stationary points and decide their nature
Calculus
CCSS HSF.IF.B.4England: A-level Pure Mathematics
Objective and success criteria
Students will: Find stationary points and decide their nature.
- 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; Solve quadratics by factorising
Leads to: Find where a function is increasing or decreasing; Solve optimisation problems with calculus
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 (Differentiate polynomials term by term; Solve quadratics by factorising). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find dy/dx when y = 5x³ + 2x² + 9x + 9. → 15x^2+4x+9Prompt: Differentiate each term separately.
- Solve: x² + 8x + 15 = 0. Give both solutions, separated by a comma. → x = -5 or x = -3Prompt: Factorise: find two numbers that multiply to 15 and add to 8.
- Differentiate f(x) = x⁴ + 8x² + 6x + 4. → 4x^3+16x+6Prompt: Bring each power down and reduce it by 1.
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: Find stationary points and decide their nature. 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: Use calculus to find the stationary point of y = x² − 10x + 32. Type it like (2, −1).
- Step 1: dy/dx = 2x − 10 = 0 → x = 5
- Step 2: y = 7
- Step 3: (5, 7)
- Answer: (5, 7)
- 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.
- Use calculus to find the stationary point of y = x² − 2x − 5. Type it like (2, −1). → (1, −6)Prompt: At a stationary point, dy/dx = 0.
- Find the x-coordinates of the stationary points of y = x³ − 18x² + 81x + 5. Separate them with a comma. → x = 3 or x = 9Prompt: Differentiate and set dy/dx = 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.
- Use calculus to find the stationary point of y = −x² − 6x. Type it like (2, −1). → (−3, 9)Prompt: At a stationary point, dy/dx = 0.
- y = x³ + 6x² + 9x has a stationary point at x = −1. Is it a maximum or a minimum? (Maximum / Minimum) → MinimumPrompt: Find the second derivative d²y/dx².
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: Find where a function is increasing or decreasing; Solve optimisation problems with calculus.
- 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