Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Estimate a limit from a table of values
Calculus
CCSS HSF.IF.A.2England: A-level Pure Mathematics
Objective and success criteria
Students will: Estimate a limit from a table of values.
- 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: Function notation
Leads to: Evaluate limits by substitution or by factorising
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 (Function notation). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- f(x) = x² − x + 9. Find f(−2). → 15Prompt: Replace every x with (−2).
- f(x) = −4x − 1. For what value of x is f(x) = −29? → x = 7Prompt: Set the expression equal to the value.
- f(x) = 2x² + 5x + 8. Find f(−4). → 20Prompt: Replace every x with (−4).
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: Estimate a limit from a table of values. 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: 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).
- Step 1: Both sides approach 2.
- Step 2: 2
- Answer: 2
- 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.
- 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 + 1)/(x + 1). x: −1.1 | −1.01 | −1.001 | −0.999 | −0.99 | −0.9 g(x): 3.31 | 3.0301 | 3.003 | 2.997 | 2.9701 | 2.71 Estimate lim(x→−1) g(x). → 3Prompt: Ignore the value at the point itself; watch the trend.
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.
- x: −2.1 | −2.01 | −2.001 | −1.999 | −1.99 | −1.9 f(x): −1 | −1 | −1 | 1 | 1 | 1 Does lim(x→−2) 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.
- g(x) = (x³ + 1)/(x + 1). x: −1.1 | −1.01 | −1.001 | −0.999 | −0.99 | −0.9 g(x): 3.31 | 3.0301 | 3.003 | 2.997 | 2.9701 | 2.71 Estimate lim(x→−1) g(x). → 3Prompt: Ignore the value at the point itself; watch the trend.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Function notation.
- 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: Evaluate limits by substitution or by factorising.
- 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