Math lesson plan · 12th grade (Year 13 in England) · 60 minutes
Evaluate limits by substitution or by factorising
Calculus
CCSS HSA.APR.D.6England: A-level Pure Mathematics
Objective and success criteria
Students will: Evaluate limits by substitution or by factorising.
- 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: Estimate a limit from a table of values; Simplify algebraic fractions by factorising
Leads to: Differentiate from first principles
Materials: Graphing tool or graph paper, Scientific calculators, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 6 min
- Quick retrieval from earlier learning (Estimate a limit from a table of values; Simplify algebraic fractions by factorising). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- f(x) = (x² − 1)/(x − 1). It has no value at x = 1 itself. x: 0.9 | 0.99 | 0.999 | 1.001 | 1.01 | 1.1 f(x): 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 Estimate lim(x→1) f(x). → 2Prompt: Look at the outputs as x gets closer from both sides.
- g(x) = (x³ − 1)/(x − 1). x: 0.9 | 0.99 | 0.999 | 1.001 | 1.01 | 1.1 g(x): 2.71 | 2.9701 | 2.997 | 3.003 | 3.0301 | 3.31 Estimate lim(x→1) g(x). → 3Prompt: Ignore the value at the point itself; watch the trend.
- x: 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 f(x): 3.9 | 3.99 | 3.999 | 4.001 | 4.01 | 4.1 Does lim(x→2) f(x) exist? (Yes / No: the two sides approach different values) → YesPrompt: Compare the values from the left with the values from the right.
Hook · 6 min
- Show a distance-time graph of a journey and ask how fast it was going at one exact moment.
- Share the objective: Evaluate limits by substitution or by factorising. Read the success criteria together.
I do: model it · 12 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: Find lim(x→−3) (3x² + x − 4).
- Step 1: 3 × (−3)² + 1 × (−3) − 4 = 20
- Answer: 20
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 12 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 lim(x→1) (3x² + 6x − 6). → 3Prompt: Polynomials are continuous: you can substitute directly.
- Find lim(x→3) (x² − x − 6)/(x − 3). → 5Prompt: Substituting gives 0/0, so simplify first.
You do: independent practice · 17 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 · 7 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- Find lim(x→−1) (x² − 1)/(x² − 3x − 4). Give a fraction if needed. → 2/5Prompt: Factorise the top and the bottom.
- Find lim(x→3) (x² + 2x − 8). → 7Prompt: Polynomials are continuous: you can substitute directly.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Estimate a limit from a table of values.
- 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: Differentiate from first principles.
- 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