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Ignition Learning — Activity Sheet

Algebraic techniques

Mathematics · Year 9

Name: ______________________Date: ____________

Expanding an expression means removing brackets by multiplying every term inside by what's outside — using the distributive property: a(b + c) = ab + ac. Factorising is the reverse: taking a common factor out of every term and writing it outside brackets. These are inverse operations, like addition and subtraction, and are foundational for solving more complex equations.

Example

Expanding: 3(x + 4) = 3x + 12. Factorising the reverse: 3x + 12 = 3(x + 4), since 3 is a common factor of both 3x and 12.

Key terms

Expand:: Remove brackets by multiplying out.
Factorise:: Write an expression as a product by taking out a common factor.
Distributive property:: a(b + c) = ab + ac — multiplying a sum equals the sum of the products.

Questions

1. Expand: 2(x + 3)
- 2x + 3
- 2x + 6
- x + 6
- 2x + 5
2. Expand: 4(y + 1)
- 4y + 1
- 4y + 4
- y + 4
- 4y + 5
3. Factorise: 5x + 5
- 5(x + 1)
- 5(x + 5)
- x(5 + 5)
- 5x(1)
4. Expand: 3(a + 2)
- 3a + 2
- 3a + 6
- a + 6
- 3a + 5
5. Factorise: 6x + 12
- 6(x + 2)
- 6(x + 12)
- x(6 + 12)
- 2(3x + 6)
6. The distributive property states a(b + c) equals:
- ab + ac
- a + b + c
- abc
- a − b − c
7. Expand: 5(x − 2)
- 5x − 2
- 5x − 10
- x − 10
- 5x − 7
8. Expand: 2(3x + 4)
- 6x + 4
- 6x + 8
- 5x + 8
- 6x + 6
9. Factorise: 8x + 12
- 4(2x + 3)
- 2(4x + 6)
- 8(x + 12)
- 4(2x + 12)
10. Expand and simplify: 3(x + 2) + 4
- 3x + 10
- 3x + 6
- 3x + 4
- 7x + 6
11. Factorise: 9x + 15
- 3(3x + 5)
- 5(9x + 3)
- 9(x + 15)
- 3(3x + 15)
12. Expand: −2(x + 3)
- −2x + 3
- −2x − 6
- 2x − 6
- −2x + 6
13. Expand and simplify: 2(x + 1) + 3(x + 2)
- 5x + 8
- 5x + 3
- 6x + 8
- 5x + 6
14. Factorise: 4x + 4y
- 4(x + y)
- x(4 + 4y)
- 4(x + 4y)
- xy(4)
15. Expand and simplify: 4(x + 3) − 2(x + 1)
- 2x + 10
- 2x + 8
- 6x + 10
- 2x + 14
16. Factorise fully: 12x + 18
- 6(2x + 3)
- 2(6x + 9)
- 3(4x + 6)
- 12(x + 18)
17. Expand: 3(2x − 5) + 4(x + 2)
- 10x − 7
- 10x + 23
- 10x − 23
- 7x − 3
18. A rectangle has length (x + 4) and width 3. Its area, expanded, is:
- 3x + 12
- 3x + 4
- x + 12
- 3x + 7
19. Factorise fully: 6x² + 9x (taking out the highest common factor)
- 3x(2x + 3)
- 6x(x + 9)
- 3(2x² + 3x)
- x(6x + 9)
20. Expand and simplify: 5(x − 1) − 2(x − 3)
- 3x + 1
- 3x − 1
- 7x − 7
- 3x − 11
21. A garden bed has area 4x + 20 (in m²). Factorised, its dimensions could be:
- 4 and (x + 5)
- 4 and (x + 20)
- 20 and (x + 4)
- 2 and (2x + 20)

Answer key (parent copy)

1. 2x + 6
2. 4y + 4
3. 5(x + 1)
4. 3a + 6
5. 6(x + 2)
6. ab + ac
7. 5x − 10
8. 6x + 8
9. 4(2x + 3)
10. 3x + 10
11. 3(3x + 5)
12. −2x − 6
13. 5x + 8
14. 4(x + y)
15. 2x + 10
16. 6(2x + 3)
17. 10x − 7
18. 3x + 12
19. 3x(2x + 3)
20. 3x + 1
21. 4 and (x + 5)