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

1. Factorise: x² − 6x − 27 ((x + 9)(x − 3) / (x − 9)(x + 3) / (x − 27)(x + 1) / (x − 6)(x + 1)) → (x − 9)(x + 3)Prompt: Find two numbers that multiply to −27 and add to −6.
2. Factorise: x² − 4 ((x + 2)² / (x − 2)² / (x + 2)(x − 2) / (x + 4)(x − 1)) → (x + 2)(x − 2)Prompt: This is a difference of two squares: A² − B² = (A + B)(A − B).
3. Factorise fully: x² − x − 2 → (x-2)(x+1)Prompt: Two numbers: product −2, sum −1.

• Show a "think of a number" puzzle and ask the class to work out the starting number.
• Share the objective: Simplify algebraic fractions by factorising. Read the success criteria together.

• Work through the example out loud, saying why you do each step, not just what you do.
• Example: Simplify: (x² + 7x + 12)/(x + 3)
• Step 1: (x + 3)(x + 4)/(x + 3)
• Step 2: (x + 4)
• Answer: (x + 4)
• 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. Simplify: (x² + 16x + 63)/(x² − 49) ((x + 9)/(x + 7) / (x + 7)/(x − 7) / (x − 9)/(x − 7) / (x + 9)/(x − 7)) → (x + 9)/(x − 7)Prompt: Factorise the top and the bottom.
2. Yuki simplified (x + 9)/x by crossing out the x's and got 1 + 9 = 10. Is this right? (Right / Wrong: only common factors can be cancelled, not terms / Right, but only when x is positive) → Wrong: only common factors can be cancelled, not termsPrompt: Test it: put x = 1 into (x + 9)/x. Do you get 10? Try x = 2.

• 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. Ethan simplified (x + 4)/x by crossing out the x's and got 1 + 4 = 5. Is this right? (Right / Wrong: only common factors can be cancelled, not terms / Right, but only when x is positive) → Wrong: only common factors can be cancelled, not termsPrompt: Test it: put x = 1 into (x + 4)/x. Do you get 5? Try x = 2.
2. Simplify: (x² + 6x + 5)/(x² − 25) ((x + 6)/(x − 25) / (x + 1)/(x − 5) / (x − 1)/(x − 5) / (x + 1)/(x + 5)) → (x + 1)/(x − 5)Prompt: Factorise the top and the bottom.

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-10.