• Quick retrieval from earlier learning (Take out the greatest common factor of a polynomial; Difference of two squares; Factorise quadratics x² + bx + c). Students answer on mini whiteboards.
• Show answers together and note who needs support in the main task.

1. Complete the factorisation: 15x⁵ − 9x⁴ − 27x³ = 3x³( … ). Type what goes inside the bracket. → 5x^2-3x-9Prompt: Divide each term by 3x³.
2. Factorise: x² − 4 ((x + 4)(x − 1) / (x + 2)(x − 2) / (x + 2)² / (x − 2)²) → (x + 2)(x − 2)Prompt: This is a difference of two squares: A² − B² = (A + B)(A − B).
3. Factorise: x² + 9x + 18 ((x + 6)(x + 3) / (x + 18)(x + 1) / (x + 6)(x − 3) / (x + 9)(x + 1)) → (x + 6)(x + 3)Prompt: Find two numbers that multiply to 18 and add to 9.

• Show a "think of a number" puzzle and ask the class to work out the starting number.
• Share the objective: Factorise fully: common factor first, then the bracket. Read the success criteria together.

• Work through the example out loud, saying why you do each step, not just what you do.
• Example: Factorise fully: 2x² + 8x + 6
• Step 1: 2(x² + 4x + 3)
• Step 2: 2(x + 1)(x + 3)
• Answer: 2(x + 1)(x + 3)
• Pause after each step and ask a student to say what comes next.

• Do these together. Students try each step on whiteboards before you reveal it.
• Use the hints as prompts rather than giving the answer.

1. Factorise fully: 2x² + 6x − 20 (2(x + 2)(x − 5) / 2(x − 2)(x + 5) / (2x − 2)(x + 5) / 2(x² + 3x − 10)) → 2(x − 2)(x + 5)Prompt: Every coefficient is a multiple of 2: take it out first.
2. Factorise fully: 4x² − 4 (4(x + 1)(x − 1) / (4x + 1)(x − 1) / 4(x² − 1) / (x + 4)(x − 1)) → 4(x + 1)(x − 1)Prompt: Take out the common factor.

• 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 6 questions are on the student sheet below, with answers in the key.

• Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.

1. Kai says x³ − 81x = x(x² − 81) is fully factorised. What is the full factorisation? (x(x² − 81) / x²(x − 81) / (x + 9)(x − 9) / x(x + 9)(x − 9)) → x(x + 9)(x − 9)Prompt: A bracket can often be factorised further.
2. Factorise fully: 3x² + 3x − 18 ((3x − 2)(x + 3) / 3(x + 2)(x − 3) / (x − 2)(x + 3) / 3(x − 2)(x + 3)) → 3(x − 2)(x + 3)Prompt: Every coefficient is a multiple of 3: take it out first.

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-9.