• 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: 12x⁴ + 24x³ + 16x² = 4x²( … ). Type what goes inside the bracket. → 3x^2+6x+4Prompt: Divide each term by 4x².
2. Factorise: 25x² − 4 ((5x + 2)² / (5x − 2)² / (5x + 2)(5x − 2) / 5(x + 2)(x − 2)) → (5x + 2)(5x − 2)Prompt: This is a difference of two squares: A² − B² = (A + B)(A − B).
3. Factorise: x² − 14x + 45 ((x + 45)(x + 1) / (x − 9)(x − 5) / (x + 9)(x + 5) / (x − 14)(x + 1)) → (x − 9)(x − 5)Prompt: Find two numbers that multiply to 45 and add to −14.

• 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: 3x² + 27x + 60 (3(x − 4)(x − 5) / 3(x + 4)(x + 5) / (3x + 4)(x + 5) / (x + 4)(x + 5)) → 3(x + 4)(x + 5)Prompt: Every coefficient is a multiple of 3: take it out first.
2. Factorise fully: 5x² − 45 ((5x + 3)(x − 3) / 5(x² − 9) / 5(x + 3)(x − 3) / 5(x − 3)²) → 5(x + 3)(x − 3)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. Ana says x³ − 4x = x(x² − 4) is fully factorised. What is the full factorisation? (x(x − 2)² / (x + 2)(x − 2) / x²(x − 4) / x(x + 2)(x − 2)) → x(x + 2)(x − 2)Prompt: A bracket can often be factorised further.
2. Factorise fully: 2x² + 8x − 24 (2(x² + 4x − 12) / (x + 6)(x − 2) / 2(x + 6)(x − 2) / (2x + 6)(x − 2)) → 2(x + 6)(x − 2)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.