Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Differentiate polynomials term by term
Calculus
CCSS HSF.IF.B.6England: A-level Pure Mathematics
Objective and success criteria
Students will: Differentiate polynomials term by term.
- 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
Leads to: Expand or simplify before differentiating; Find the gradient of a curve at a point; Find stationary points and decide their nature
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 axⁿ with the power rule). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Differentiate y = −x⁷. → -7*x^6Prompt: Multiply by the power, then reduce the power by 1.
- Diego differentiated 5x⁵ and wrote 25x⁶. What is the correct derivative? (20x⁴ / 25x⁴ / 5x⁴ / 25x⁶) → 25x⁴Prompt: The power goes down by one, not up.
- Differentiate y = −3x³. → -9*x^2Prompt: Multiply by the power, then reduce the power by 1.
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: Differentiate polynomials term by term. 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 dy/dx when y = −3x³ − x² − 3x + 6.
- Step 1: dy/dx = −9x² − 2x − 3
- Answer: -9x^2-2x-3
- 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 dy/dx when y = −3x³ + 8x² − 5x − 9. → -9x^2+16x-5Prompt: Differentiate each term separately.
- Differentiate f(x) = −3x⁴ − 5x² + 2x + 5. → -12x^3-10x+2Prompt: Bring each power down and reduce it by 1.
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.
- Differentiate f(x) = −3x⁴ − 4x² − 2x − 5. → -12x^3-8x-2Prompt: Bring each power down and reduce it by 1.
- Differentiate y = 2x³/3 + 4x²/2 + 9x. → 2x^2+4x+9Prompt: The fractions are just coefficients: x³/3 means (1/3)x³.
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
- Move on to: Expand or simplify before differentiating; Find the gradient of a curve at a point; Find stationary points and decide their nature.
- 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