Math lesson plan · 10th grade (Year 11 in England) · 60 minutes
Composite and inverse functions
Algebra
CCSS HSF.BF.A.1.CCCSS HSF.BF.B.4England: GCSE Higher Algebra
Objective and success criteria
Students will: Composite and inverse functions.
- 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; Rearrange formulas (change the subject)
Leads to: Inverse functions (non-linear)
Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 6 min
- Quick retrieval from earlier learning (Function notation; Rearrange formulas (change the subject)). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- f(x) = 2x² − 3x + 3. Find f(−3). → 30Prompt: Replace every x with (−3).
- Make x the subject of y = 6x − 10. Type the expression for x. → (y+10)/6Prompt: Undo the operations on x in reverse order.
- f(x) = 5x − 8. For what value of x is f(x) = 7? → x = 3Prompt: Set the expression equal to the value.
Hook · 6 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Composite and inverse functions. Read the success criteria together.
I do: model it · 12 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: f(x) = 5x − 4 and g(x) = x² + 1. Find f(g(3)).
- Step 1: g(3) = 3² + 1 = 10
- Step 2: f(10) = 5 × 10 − 4 = 46
- Answer: 46
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 12 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) = 4x − 4 and g(x) = x² + 1. Find f(g(1)). → 4Prompt: Work from the inside out: find g first.
- f(x) = 3x + 3. Find the inverse function f⁻¹(x). → (x-3)/3Prompt: Write y = f(x), then swap x and y.
You do: independent practice · 17 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 · 7 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- f(x) = 7x + 1. Find the inverse function f⁻¹(x). → (x-1)/7Prompt: Write y = f(x), then swap x and y.
- f(x) = x + 1 and g(x) = 2x − 2. Find f(g(x)) and simplify. → 2x-1Prompt: Substitute the whole inner function in place of x in the outer function.
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: Function notation.
- 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: Inverse functions (non-linear).
- 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-10.
Free lesson plan from eurekami.com/lesson-plan-builder