Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Find where a function is increasing or decreasing
Calculus
CCSS HSF.IF.B.4England: A-level Pure Mathematics
Objective and success criteria
Students will: Find where a function is increasing or decreasing.
- 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 stationary points and decide their nature
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 stationary points and decide their nature). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Use calculus to find the stationary point of y = 2x² − 20x + 50. Type it like (2, −1). → (5, 0)Prompt: At a stationary point, dy/dx = 0.
- Find the x-coordinates of the stationary points of y = x³ − 12x + 5. Separate them with a comma. → x = -2 or x = 2Prompt: Differentiate and set dy/dx = 0.
- y = −x³ + 12x² − 21x has a stationary point at x = 1. Is it a maximum or a minimum? (Maximum / Minimum) → MinimumPrompt: Find the second derivative d²y/dx².
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 where a function is increasing or decreasing. 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: For which values of x is f(x) = −x² − 12x + 3 increasing?
- Step 1: x < −6
- Answer: x < −6
- 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.
- For which values of x is f(x) = −x² − 6x + 2 decreasing? (x > −3 / x < −3 / x < 3 / x > 3) → x > −3Prompt: Increasing means f′(x) > 0; decreasing means f′(x) < 0.
- Where is f(x) = x³ − 6x² − 15x + 1 decreasing? (x < −1 or x > 5 / −1 < x < 5 / −5 < x < 1 / x > 5) → −1 < x < 5Prompt: Find where f′(x) < 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.
- Where is f(x) = x³ + 6x² + 1 decreasing? (x > 0 / 0 < x < 4 / −4 < x < 0 / x < −4 or x > 0) → −4 < x < 0Prompt: Find where f′(x) < 0.
- For which values of x is f(x) = 2x² + 20x increasing? (x < −5 / x > −5 / x < 5 / x > 5) → x > −5Prompt: Increasing means f′(x) > 0; decreasing means f′(x) < 0.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Find stationary points and decide their nature.
- 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