Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Find a curve from its gradient and a point
Calculus
CCSS HSF.BF.B.4England: A-level Pure Mathematics
Objective and success criteria
Students will: Find a curve from its gradient and a point.
- 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: Integrate powers of x (indefinite integrals)
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 (Integrate powers of x (indefinite integrals)). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find ∫ −6x dx. Leave out the + C. → -3*x^2Prompt: Add 1 to the power, then divide by the new power.
- Find ∫ (9x² + 6x + 9) dx. Leave out the + C. → 3x^3+3x^2+9xPrompt: Integrate each term separately.
- Ethan needs ∫ 3/x² dx. Which answer is correct? (−3/x + C / 3·ln x + C / 3/x + C / 3/x³ + C) → −3/x + CPrompt: Rewrite as 3x^(−2).
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 a curve from its gradient and a point. 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: dy/dx = −4x − 5 and the curve passes through (3, 4). Find y in terms of x.
- Step 1: y = −2x² − 5x + C
- Step 2: 4 = −33 + C → C = 37
- Step 3: y = −2x² − 5x + 37
- Answer: -2x^2-5x+37
- 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.
- dy/dx = 6x − 4 and the curve passes through (0, −4). Find y in terms of x. → 3x^2-4x-4Prompt: Integrate to get y, including + C.
- A cyclist's velocity is v = 5 + 4t m/s. Starting from s = 0, how far has she travelled after 6 seconds? → 102Prompt: Displacement is the integral of velocity.
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.
- dy/dx = 2x + 2 and the curve passes through (3, 3). Find y in terms of x. → x^2+2x-12Prompt: Integrate to get y, including + C.
- A cyclist's velocity is v = 5 + 8t m/s. Starting from s = 0, how far has she travelled after 2 seconds? → 26Prompt: Displacement is the integral of velocity.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Integrate powers of x (indefinite integrals).
- 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