Math lesson plan · 12th grade (Year 13 in England) · 30 minutes
Find equations of tangents and normals
Calculus
CCSS HSG.GPE.B.5England: A-level Pure Mathematics
Objective and success criteria
Students will: Find equations of tangents and normals.
- 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: Find the gradient of a curve at a point; Write the equation of a perpendicular line through a point
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 · 4 min
- Quick retrieval from earlier learning (Find the gradient of a curve at a point; Write the equation of a perpendicular line through a point). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find the gradient of y = −x³ − 2x² + 3x − 3 at x = 0. → 3Prompt: Differentiate first.
- A line has slope 3/2. What is the slope of a line perpendicular to it? Give a fraction or whole number. → -2/3Prompt: Perpendicular slopes multiply to −1.
- Find the gradient of y = x² + 2/x at x = −1. → -4Prompt: 2/x = 2x⁻¹, which differentiates to −2x⁻².
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 equations of tangents and normals. 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 equation of the tangent to y = −x² − 5x + 4 at x = −3. Type the right-hand side of y = … .
- Step 1: point (−3, 10), m = 1
- Step 2: y = x + 13
- Answer: x+13
- 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 equation of the tangent to y = 2x² + x − 6 at x = −2. Type the right-hand side of y = … . → -7x-14Prompt: Find the point: substitute x into the curve.
- Find the equation of the normal to y = x² − 3x − 2 at x = −1. Type the right-hand side of y = … (fractions are fine). → (1/5)x+11/5Prompt: The normal is perpendicular to the tangent.
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 equation of the normal to y = x² + 2x + 2 at x = −3. Type the right-hand side of y = … (fractions are fine). → (1/4)x+23/4Prompt: The normal is perpendicular to the tangent.
- Find the equation of the tangent to y = 3x² − 5x − 6 at x = −3. Type the right-hand side of y = … . → -23x-33Prompt: Find the point: substitute x into the curve.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Find the gradient of a curve at a point.
- 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