Math lesson plan · 12th grade (Year 13 in England) · 30 minutes
Find the gradient of a curve at a point
Calculus
CCSS HSF.IF.B.6England: A-level Pure Mathematics
Objective and success criteria
Students will: Find the gradient of a curve at 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: Differentiate polynomials term by term
Leads to: Find equations of tangents and normals; Interpret a derivative as a rate of change in context
Materials: Graphing tool or graph paper, Scientific calculators, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 4 min
- Quick retrieval from earlier learning (Differentiate polynomials term by term). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find dy/dx when y = −3x³ − 2x² + 5x + 4. → -9x^2-4x+5Prompt: Differentiate each term separately.
- Differentiate f(x) = −2x⁴ − x² + 4x + 4. → -8x^3-2x+4Prompt: Bring each power down and reduce it by 1.
- Differentiate y = 4x³/3 + 2x²/2 − 8x. → 4x^2+2x-8Prompt: The fractions are just coefficients: x³/3 means (1/3)x³.
Hook · 3 min
- Show a distance-time graph of a journey and ask how fast it was going at one exact moment.
- Share the objective: Find the gradient of a curve at a point. Read the success criteria together.
I do: model it · 7 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: Find the gradient of y = 2x³ − 2x² − 6x + 1 at x = 0.
- Step 1: dy/dx = 6x² − 4x − 6
- Step 2: = −6
- Answer: -6
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 6 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 the gradient of y = −2x³ + 3x² − 2x + 4 at x = −2. → -38Prompt: Differentiate first.
- Find the gradient of y = x² − 8/x at x = 2. → 6Prompt: 8/x = 8x⁻¹, which differentiates to −8x⁻².
You do: independent practice · 7 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 6 questions are on the student sheet below, with answers in the key.
Exit ticket · 3 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- Find the gradient of y = x² − 16/x at x = 2. → 8Prompt: 16/x = 16x⁻¹, which differentiates to −16x⁻².
- Find the gradient of y = −3x³ + 3x² − 3x − 1 at x = 2. → -27Prompt: Differentiate 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
- Move on to: Find equations of tangents and normals; Interpret a derivative as a rate of change in context.
- 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