Math lesson plan · 12th grade (Year 13 in England) · 60 minutes
Differentiate negative and fractional powers
Calculus
CCSS HSN.RN.A.2England: A-level Pure Mathematics
Objective and success criteria
Students will: Differentiate negative and fractional powers.
- 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 axⁿ with the power rule; Fractional and negative indices
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 · 6 min
- Quick retrieval from earlier learning (Differentiate axⁿ with the power rule; Fractional and negative indices). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Differentiate y = −6x⁸. → -48*x^7Prompt: Multiply by the power, then reduce the power by 1.
- Evaluate 81^(2/4). → 9Prompt: The denominator 4 means take the 4th root.
- Kofi differentiated 8x⁶ and wrote 48x⁷. What is the correct derivative? (40x⁵ / 8x⁵ / 48x⁵ / 14x⁵) → 48x⁵Prompt: The power goes down by one, not up.
Hook · 6 min
- Show a distance-time graph of a journey and ask how fast it was going at one exact moment.
- Share the objective: Differentiate negative and fractional powers. Read the success criteria together.
I do: model it · 12 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: Differentiate y = −2/x³.
- Step 1: y = −2x^(−3)
- Step 2: dy/dx = 6x^(−4) = 6/x⁴
- Answer: 6/x^4
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 12 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- Differentiate y = 1/x. → -1/x^2Prompt: Rewrite as 1x^(−1) first.
- Differentiate y = 2x^(3/2). → 3*sqrt(x)Prompt: Multiply by 3/2, then subtract 1 from the power.
You do: independent practice · 17 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 · 7 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- Differentiate y = 10x^(3/2). → 15*sqrt(x)Prompt: Multiply by 3/2, then subtract 1 from the power.
- Differentiate y = −5/x. → 5/x^2Prompt: Rewrite as −5x^(−1) first.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Differentiate axⁿ with the power rule.
- 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