• 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 − 35x⁴ − 15x³ = 5x³( … ). Type what goes inside the bracket. → 3x^2-7x-3Prompt: Divide each term by 5x³.
2. Factorise: x² − 16 ((x + 4)(x − 4) / (x − 4)² / (x + 16)(x − 1) / (x + 4)²) → (x + 4)(x − 4)Prompt: This is a difference of two squares: A² − B² = (A + B)(A − B).
3. Factorise: x² − 7x + 6 ((x + 6)(x + 1) / (x − 1)(x + 6) / (x + 1)(x + 6) / (x − 1)(x − 6)) → (x − 1)(x − 6)Prompt: Find two numbers that multiply to 6 and add to −7.

• 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: 5x² − 55x + 150 (5(x² − 11x + 30) / 5(x + 5)(x + 6) / 5(x − 5)(x − 6) / (x − 5)(x − 6)) → 5(x − 5)(x − 6)Prompt: Every coefficient is a multiple of 5: take it out first.
2. Factorise fully: 5x² − 20 ((5x + 2)(x − 2) / 5(x² − 4) / 5(x − 2)² / 5(x + 2)(x − 2)) → 5(x + 2)(x − 2)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. Sam says x³ − 16x = x(x² − 16) is fully factorised. What is the full factorisation? (x²(x − 16) / x(x + 4)(x − 4) / x(x² − 16) / (x + 4)(x − 4)) → x(x + 4)(x − 4)Prompt: A bracket can often be factorised further.
2. Factorise fully: 2x² − 16x + 30 ((x − 3)(x − 5) / 2(x² − 8x + 15) / 2(x − 3)(x − 5) / (2x − 3)(x − 5)) → 2(x − 3)(x − 5)Prompt: Every coefficient is a multiple of 2: take it out first.

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