Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Expand or simplify before differentiating
Calculus
CCSS HSA.APR.A.1England: A-level Pure Mathematics
Objective and success criteria
Students will: Expand or simplify before differentiating.
- 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; Multiply binomials (expand double brackets)
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 (Differentiate polynomials term by term; Multiply binomials (expand double brackets)). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find dy/dx when y = −2x³ + 5x² − 4x − 7. → -6x^2+10x-4Prompt: Differentiate each term separately.
- Expand and simplify: (x + 6)(x + 7) → x^2+13x+42Prompt: Multiply each term in the first bracket by each term in the second (FOIL).
- Differentiate f(x) = 4x⁴ + 7x² + 8x + 10. → 16x^3+14x+8Prompt: Bring each power down and reduce it 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: Expand or simplify before differentiating. 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 = (x − 6)(x − 1).
- Step 1: y = x² − 7x + 6
- Step 2: dy/dx = 2x − 7
- Answer: 2x-7
- 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 = (x + 2)(x − 2). → 2xPrompt: Expand the brackets first.
- Find dy/dx when y = (x³ − 3x² − 4x)/x. → 2x-3Prompt: Divide each term on top by x first.
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.
- Find dy/dx when y = (x³ + x² − 4x)/x. → 2x+1Prompt: Divide each term on top by x first.
- Find dy/dx when y = (x − 6)(x + 4). → 2x-2Prompt: Expand the brackets first.
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
- 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