Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Find vertical and horizontal asymptotes
Algebra
CCSS HSF.IF.C.7.DEngland: A-level Pure Mathematics
Objective and success criteria
Students will: Find vertical and horizontal asymptotes.
- 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: Rational functions and asymptotes; Find limits of rational functions as x → ∞
Leads to: the next skill in this area.
Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 5 min
- Quick retrieval from earlier learning (Rational functions and asymptotes; Find limits of rational functions as x → ∞). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find the horizontal asymptote of f(x) = (5x + 1)/(x + 1). Give the y-value (fractions allowed). → 5Prompt: For large x, compare the leading terms of top and bottom.
- Find the limit as x → ∞ of (−x² + 6x + 3)/(5x² + 3x − 1). Give a fraction if needed. → -1/5Prompt: For huge x, only the highest powers matter.
- What happens to (2x² + 1)/(6x² − 3) as x → ∞? (0 / 1/3 / No finite limit (it grows without bound)) → 1/3Prompt: Compare the degree of the top with the degree of the bottom.
Hook · 5 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Find vertical and horizontal asymptotes. 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) = (4x − 9)/(x − 1). Find the vertical asymptote x = ?
- Step 1: x = 1
- Answer: 1
- 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) = (−2x − 2)/(x + 4). Find the horizontal asymptote y = ? → -2Prompt: For large x, f(x) behaves like the ratio of the x terms.
- What are the vertical asymptotes of f(x) = (5x² + 1)/(4x² − 100)? (There are none / y = 5/4 / x = 5 and x = −5 / x = 25 and x = −25) → x = 5 and x = −5Prompt: Set the denominator equal to zero.
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.
- What are the vertical asymptotes of f(x) = (4x² + 1)/(3x² − 75)? (y = 4/3 / x = 5 and x = −5 / x = 25 and x = −25 / x = 5) → x = 5 and x = −5Prompt: Set the denominator equal to zero.
- f(x) = (−5x − 7)/(x + 6). Find the horizontal asymptote y = ? → -5Prompt: For large x, f(x) behaves like the ratio of the x terms.
Watch out for
- Doing an operation to one side of an equation but not the other.
- Reading 3x as 3 + x, or thinking x always stands for one particular number.
Support and stretch
Support
- Review first: Rational functions and asymptotes.
- Use concrete materials: balance scales or a balance drawing.
- 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