Free IB Maths AA HL lesson: Improper rationals & oblique asymptotes
A ready-to-teach IB Maths AA HL lesson on Improper rationals & oblique asymptotes (Unit 2 · Functions): editable slides and teacher notes, free; a printable worksheet in 3 levels with answers for school plans. About 55 minutes.
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What's inside
- Slides (PowerPoint, editable) · freeabout 12 slides: do now, objectives, key ideas, a worked example step by step, “your turn”, practice QR code, exit ticket
- Teacher notes (PDF) · freelesson plan with timings, common mistakes, homework links
- Worksheet (PDF) · school plans6 questions, 24 marks: Fluency, Reasoning, Exam-style. Included with school plans and the teacher trial.
- Answers (PDF) · school plansthe answer to every worksheet question, from its mark scheme. Included with school plans and the teacher trial.
Learning objectives
- Rational functions and asymptotes
Key ideas on the slides
- y = (ax + b)/(cx + d) has vertical asymptote x = −d/c (where the denominator is zero) and horizontal asymptote y = a/c.
- Intercepts: y-intercept b/d (put x = 0); x-intercept −b/a (numerator = 0).
- Write asymptotes as equations (x = 2, y = 3), not as numbers.
- HL: when the degree of the numerator is one more than the denominator, divide to find an oblique asymptote y = mx + k.
By the end of the lesson, students should be able to answer a question like: Find the slant asymptote of y = (x^2 + 3)/(x - 1).
Key vocabulary
- vertical asymptote: A line x = k where the function is undefined and the graph goes to ±∞.
- horizontal asymptote: A line y = k the graph approaches as x → ±∞.
- oblique asymptote: A slanted line y = mx + k approached for large |x|.
Common mistakes this lesson tackles
- Graphs and transformations: describing a rational function, and graphs of [f(x)]². Write the rational function in the form a + b/(x − c) first; each of a, b, c is one transformation. For [f(x)]², values between −1 and 1 get smaller and zeros become touching points.
- Giving asymptotes as numbers ("2 and 3") instead of equations.
- Drawing the curve crossing a vertical asymptote.
- Using b/d for the horizontal asymptote.
The worked example
Find the exact equation of the oblique (slant) asymptote for the rational function f(x) = (x² + 5x − 2)/(x + 2).
- Recognize that an oblique asymptote exists because the degree of the numerator (2) is exactly one higher than the degree of the denominator (1).
- Perform algebraic long division or synthetic division. Divide x² + 5x − 2 by x + 2.
- Calculate the first term: x² ÷ x = x. Multiply x(x+2) = x²+2x and subtract to leave 3x − 2.
- Calculate the second term: 3x ÷ x = 3. Multiply 3(x+2) = 3x+6 and subtract to leave a remainder of −8.
- Rewrite the function: f(x) = x + 3 − 8/(x+2).
- Conclude: As x → ±∞, the fractional remainder 8/(x+2) → 0. The oblique asymptote is the linear quotient, y = x + 3.
Tip: Synthetic division is a fast alternative here, but students must remember that the oblique asymptote is given by the quotient alone; the remainder must be entirely discarded when stating the asymptote's equation.
Calculator tips
- Graph it and zoom out to see the asymptotes; the table shows the behaviour near x = 3.
Lesson plan
| 0–5 min | Do now: students write down what they remember about Rational functions & vertical/horizontal asymptotes and Combined transformations |
| 5–8 min | Share the learning objective and success criteria |
| 8–20 min | Teach: the worked example, step by step; students copy the method |
| 20–25 min | Your turn: 1 mini-whiteboard question; check every board before moving on |
| 25–42 min | Practise: the worksheet (6 questions in 3 levels, with school plans and the teacher trial) or the topic's practice questions online; move up a level after two right in a row |
| 42–47 min | Plenary: 1-question exit ticket; collect it to plan the next lesson |
For students: practise this topic
The worksheet's QR code opens the IB Maths AA HL Unit 2: Polynomials and Rational Functions practice questions.
What schools get
- Set this lesson as homework, marked by AI against the mark scheme, with every result in your markbook (with a School plan).
- This lesson's worksheet in three levels, with its answers, ready to print.
- Support, Core and Stretch versions and A/B copies of the worksheet from the full question bank, in the worksheet builder.
- Create lesson: this lesson rebuilt for your class's next scheme-of-work lesson, from the full bank.
School licences from €490 a year. Try everything free for 30 days as a teacher: no card.
Questions teachers ask
Is this lesson free?
The slides and the teacher notes are free to download with a free IB Math Revision teacher account (sign in with Google, Microsoft or email); no card. The printable worksheet in three levels and its answers are included with school plans and the free 30-day teacher trial.
Can I edit the slides?
Yes. The slides are an ordinary PowerPoint file: change, add or delete anything. They also open in Google Slides and Keynote. The logo and web address sit on the slide master, so your edits do not move them.
What do students use for homework?
The practice slide's QR code and web address take students to the IB Maths AA HL Unit 2: Polynomials and Rational Functions practice questions on ibmathrevision.com, where the free questions are open and the full bank comes with a plan.