Math lesson plan · 11th grade (Year 12 in England) · 60 minutes
Factorise sums and differences of cubes
Algebra
CCSS HSA.SSE.A.2CCSS HSA.APR.C.4
Objective and success criteria
Students will: Factorise sums and differences of cubes.
- 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: Difference of two squares; Expand three binomials
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 · 6 min
- Quick retrieval from earlier learning (Difference of two squares; Expand three binomials). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Factorise: x² − 9 ((x − 3)² / (x + 3)(x − 3) / (x + 3)² / (x + 9)(x − 1)) → (x + 3)(x − 3)Prompt: This is a difference of two squares: A² − B² = (A + B)(A − B).
- Expand and simplify: (x + 1)(x − 4)(x + 1) → x^3-2x^2-7x-4Prompt: Expand two of the brackets first.
- Use the difference of two squares to work out 84² − 78² without a calculator. → 972Prompt: a² − b² = (a + b)(a − b).
Hook · 6 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Factorise sums and differences of cubes. 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: Factorise: x³ + 8
- Step 1: (x + 2)(x² − 2x + 4)
- Answer: (x + 2)(x² − 2x + 4)
- 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.
- Factorise: x³ − 125 ((x − 5)³ / (x − 5)(x² − 5x + 25) / (x + 5)(x² − 5x + 25) / (x − 5)(x² + 5x + 25)) → (x − 5)(x² + 5x + 25)Prompt: a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²).
- Factorise: 27x³ − 64 ((3x − 4)³ / (3x − 4)(3x² + 4x + 4) / (3x − 4)(9x² − 12x + 16) / (3x − 4)(9x² + 12x + 16)) → (3x − 4)(9x² + 12x + 16)Prompt: 27x³ = (3x)³ and 64 = 4³.
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.
- Factorise: 8x³ − 27 ((2x − 3)³ / (2x − 3)(4x² − 6x + 9) / (2x − 3)(4x² + 6x + 9) / (8x − 27)(x² + 1)) → (2x − 3)(4x² + 6x + 9)Prompt: 8x³ = (2x)³ and 27 = 3³.
- Factorise: x³ + 216 ((x + 6)(x² + 6x + 36) / (x + 6)(x² − 12x + 36) / (x + 6)³ / (x + 6)(x² − 6x + 36)) → (x + 6)(x² − 6x + 36)Prompt: a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²).
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: Difference of two squares.
- 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-11.
Free lesson plan from eurekami.com/lesson-plan-builder