Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Rational functions and asymptotes
Algebra
CCSS HSF.IF.C.7.DCCSS HSA.APR.D.6
Objective and success criteria
Students will: Rational functions and 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 equations; Transformations of functions
Leads to: Find limits of rational functions as x → ∞; Find vertical and horizontal asymptotes
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 equations; Transformations of functions). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Solve: −21/x − 5 = 2 → x = -3Prompt: Isolate the fraction first.
- Solve: (x − 4)/(x − 8) = 2 → x = 12Prompt: Multiply both sides by (x − 8).
- Solve: 28/x + 1 = −3 → x = -7Prompt: Isolate the fraction first.
Hook · 5 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Rational functions and 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: Find the vertical asymptote of f(x) = (3x + 2)/(3x + 6). Give the x-value.
- Step 1: 3x + 6 = 0
- Step 2: x = −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.
- Find the horizontal asymptote of f(x) = (2x + 1)/(x − 3). Give the y-value (fractions allowed). → 2Prompt: For large x, compare the leading terms of top and bottom.
- Find the horizontal asymptote of f(x) = (4x)/(3x − 6). Give the y-value (fractions allowed). → 4/3 (1 1/3)Prompt: For large x, compare the leading terms of top and bottom.
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.
- Find the horizontal asymptote of f(x) = (5x − 3)/(x − 6). Give the y-value (fractions allowed). → 5Prompt: For large x, compare the leading terms of top and bottom.
- Find the vertical asymptote of f(x) = (x + 3)/(2x − 10). Give the x-value. → 5Prompt: The function has no value where the denominator is zero.
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 equations.
- 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
- Move on to: Find limits of rational functions as x → ∞; Find vertical and horizontal asymptotes.
- 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