Math lesson plan · 12th grade (Year 13 in England) · 30 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 · 4 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 − 9)/(x − 3). It has no value at x = 3 itself. x: 2.9 | 2.99 | 2.999 | 3.001 | 3.01 | 3.1 f(x): 5.9 | 5.99 | 5.999 | 6.001 | 6.01 | 6.1 Estimate lim(x→3) f(x). → 6Prompt: Look at the outputs as x gets closer from both sides.
- g(x) = (x³ − 27)/(x − 3). x: 2.9 | 2.99 | 2.999 | 3.001 | 3.01 | 3.1 g(x): 26.11 | 26.9101 | 26.991 | 27.009 | 27.0901 | 27.91 Estimate lim(x→3) g(x). → 27Prompt: Ignore the value at the point itself; watch the trend.
- x: −3.1 | −3.01 | −3.001 | −2.999 | −2.99 | −2.9 f(x): −1 | −1 | −1 | 1 | 1 | 1 Does lim(x→−3) f(x) exist? (Yes / No: the two sides approach different values) → No: the two sides approach different valuesPrompt: Compare the values from the left with the values from the right.
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: Evaluate limits by substitution or by factorising. 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 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 · 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 lim(x→−1) (3x² − x − 4). → 0Prompt: Polynomials are continuous: you can substitute directly.
- Find lim(x→−1) (x² − 3x − 4)/(x + 1). → -5Prompt: Substituting gives 0/0, so simplify first.
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 lim(x→0) (3x² − 4x − 2). → -2Prompt: Polynomials are continuous: you can substitute directly.
- Find lim(x→−5) (x² − 25)/(x² + 3x − 10). Give a fraction if needed. → 10/7 (1 3/7)Prompt: Factorise the top and the bottom.
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