• 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 − 4x − 12 ((x − 12)(x + 1) / (x − 6)(x − 2) / (x − 6)(x + 2) / (x − 4)(x + 1)) → (x − 6)(x + 2)Prompt: Find two numbers that multiply to −12 and add to −4.
2. Factorise: x² − 100 ((x + 100)(x − 1) / (x − 10)² / (x + 10)² / (x + 10)(x − 10)) → (x + 10)(x − 10)Prompt: This is a difference of two squares: A² − B² = (A + B)(A − B).
3. Factorise fully: x² − 10x + 24 → (x-4)(x-6)Prompt: Two numbers: product 24, sum −10.

• 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 + 10x + 24)/(x² − 16) ((x + 6)/(x + 4) / (x − 6)/(x − 4) / (x + 4)/(x − 4) / (x + 6)/(x − 4)) → (x + 6)/(x − 4)Prompt: Factorise the top and the bottom.
2. Kwame simplified (x + 5)/x by crossing out the x's and got 1 + 5 = 6. 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 + 5)/x. Do you get 6? 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 8 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 simplified (x + 2)/x by crossing out the x's and got 1 + 2 = 3. 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 + 2)/x. Do you get 3? Try x = 2.
2. Simplify: (x² − 25)/(x + 5) ((x − 5) / x − 25 / (x + 5) / (x² − 5)) → (x − 5)Prompt: Factorise the top: it is a difference of two squares.

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