Math lesson plan · 12th grade (Year 13 in England) · 45 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 · 5 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 = 4x³ − 4x² − 9x − 8. → 12x^2-8x-9Prompt: Differentiate each term separately.
- Differentiate f(x) = x⁴ − 3x² − 8x + 6. → 4x^3-6x-8Prompt: 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 · 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 the gradient of a curve at 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: 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 · 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 the gradient of y = −2x³ + 3x² − x − 6 at x = 2. → -13Prompt: Differentiate first.
- Find the gradient of y = x² − 4/x at x = −2. → -3Prompt: 4/x = 4x⁻¹, which differentiates to −4x⁻².
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 the gradient of y = x² − 1/x at x = −1. → -1Prompt: 1/x = 1x⁻¹, which differentiates to −1x⁻².
- At what x is the gradient of y = x² − 3x + 5 equal to 1? → x = 2Prompt: Differentiate, then set dy/dx equal to the gradient.
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