10-Week Mathematics Catch-Up Programme (CAPS)

Week 1 – Day 4

Algebra Fundamentals (Grade 8 Review)

Duration: 1 Hour

Learning Outcomes

By the end of today's lesson you should be able to:

Why is this important?

Almost every Grade 11 topic—functions, trigonometry, coordinate geometry, sequences and calculus—requires confident algebra skills. If algebra becomes automatic, Grade 11 maths becomes much easier.

Part A – Study Guide

1. What is Algebra?

Algebra uses letters to represent unknown numbers.

Examples:

Example:

If you have 3 apples and buy x more apples, you now have

3+x3+x\quad

2. Terms

A term is separated by a + or − sign.

Example

4x+7−2y+3x4{x}+7-2y+3x\quad

The terms are:

3. Coefficients

A coefficient is the number in front of the variable.

Example

7x7x\quad

Coefficient = 7

Variable = x

4. Like Terms

Like terms have exactly the same variable.

Examples

✔ 4x and 7x

✔ 9a and 2a

✔ 5mn and 3mn

Not like terms

✘ 3x and 3y

✘ 2a and 2a²

5. Collecting Like Terms

Example

Simplify

5x+3x5x+3x\quad

Add the coefficients.

8x8x\quad

Example

7a−2a+47a-2a+4\quad 5a+45a+4\quad

The constant (4) stays separate.

6. Expanding Brackets

Multiply everything inside the bracket.

Example

3(x+5)3(x+5)\quad

Multiply:

3×x=3x3 × x = 3x\quad 3×5=153 × 5 = 15\quad

Answer

3x+153x+15\quad

Example

5(2a−3)5(2a-3)\quad

Multiply every term.

10a−1510a-15\quad

7. Substitution

Replace the variable with its value.

Example

If

x=4x=4\quad

Find

3x+23x+2\quad

Substitute:

3(4)+23(4)+2\quad 12+212+2\quad 1414\quad

8. Evaluating Expressions

Example

If

a=3,b=5a=3,\quad b=5

Find

2a+b2a+b\quad

Substitute

2(3)+52(3)+5\quad =11=11\quad

Common Mistakes

3x+4=7x3x + 4 = 7x\quad

❌ Wrong!

You cannot add unlike terms.

Correct answer:

3x+43x+4\quad
4(x+2)=4x+24(x+2)=4x+2\quad

❌Wrong.

Multiply every term.

Correct answer

4x+84x+8\quad

❌ Forgetting brackets during substitution.

If

x=−3x=-3\quad

Then

2x2x\quad

is

2(−3)=−62(-3)=-6\quad

Not 2−3.

Part B – Worked Examples

Example 1

Simplify

6x+4x6x+4x\quad

Answer

10x10x\quad

Example 2

Simplify

8y−5y+28y-5y+2\quad

Answer

3y+23y+2\quad

Example 3

Simplify

5a+7−2a+45a+7-2a+4\quad

Collect like terms.

3a+113a+11\quad

Example 4

Expand

2(x+8)2(x+8)\quad

Answer

2x+162x+16\quad

Example 5

Expand

4(3y−2)4(3y-2)\quad

Answer

12y−812y-8\quad

Example 6

If

x=6x=6\quad

Find

2x+92x+9\quad

Substitute

2(6)+92(6)+9\quad 12+9=2112+9=21\quad

Example 7

If

a=2,b=7a=2,\quad b=7

Find

3a+2b3a+2b\quad 3(2)+2(7)3(2)+2(7)\quad 6+14=206+14=20\quad

Example 8

Simplify

7m−4+3m+87m-4+3m+8\quad

Group like terms.

10m+410m+4\quad

Part C – Practice Questions

Section A – Like Terms

Simplify:

  1. (3x+5x)

  2. (9a-4a)

  3. (8y+6y)

  4. (12m-7m)

  5. (10p+5-4p)

  6. (7k+9+2k-3)

  7. (15n-8n+6)

  8. (4x+8+5x-2)

  9. (20a-5a+10)

  10. (3b+7+4b-5)

Section B – Expand the Brackets

  1. (2(x+6))

  2. (5(a+3))

  3. (4(y-7))

  4. (6(p+4))

  5. (3(2x+5))

  6. (2(5m-4))

  7. (7(a-2))

  8. (8(2n+1))

  9. (9(x-3))

  10. (4(3k-6))

Section C – Substitution

If (x=5)

  1. (3x+4)

  2. (2x-7)

  3. (5x+9)

  4. (4x-8)

If (a=4,;b=3)

  1. (a+b)

  2. (2a+b)

  3. (3a-2b)

  4. (5a+b)

If (m=6,;n=2)

  1. (m+n)

  2. (2m-3n)

Part D – Mixed Questions

  1. Simplify
4x+7+6x−54x+7+6x-5\quad
  1. Expand
5(2y+6)5(2y+6)\quad
  1. If
x=4x=4\quad

Find

3x+53x+5\quad
  1. If
a=8a=8\quad

Find

4a−104a-10\quad
  1. Simplify
6p+4−2p+96p+4-2p+9\quad

Challenge Questions

  1. If
x=3x=3\quad

Find

2x2+52x^2+5\quad
  1. If
a=4,b=2a=4,\quad b=2

Find

2a2+b2a^2+b\quad
  1. Simplify
3(x+4)+2x3(x+4)+2x\quad
  1. Expand and simplify
4(2y+3)+5y4(2y+3)+5y\quad
  1. A taxi charges a fixed fee of R25 plus R12 per kilometre.

Write an algebraic expression for the total cost of travelling k kilometres.

Then calculate the cost of travelling 15 km.

Answers

  1. (8x)

  2. (5a)

  3. (14y)

  4. (5m)

  5. (6p+5)

  6. (9k+6)

  7. (7n+6)

  8. (9x+6)

  9. (15a+10)

  10. (7b+2)

  11. (2x+12)

  12. (5a+15)

  13. (4y-28)

  14. (6p+24)

  15. (6x+15)

  16. (10m-8)

  17. (7a-14)

  18. (16n+8)

  19. (9x-27)

  20. (12k-24)

  21. 19

  22. 3

  23. 34

  24. 12

  25. 7

  26. 11

  27. 6

  28. 23

  29. 8

  30. 6

  31. (10x+2)

  32. (10y+30)

  33. 17

  34. 22

  35. (4p+13)

  36. 23

  37. 34

  38. (5x+12)

  39. (13y+12)

  40. Expression: (25+12k)

For 15 km:

25+12(15)=25+180=R20525+12(15)=25+180=R205\quad

Parent's Notes

By the end of Week 1, the student should be able to:

Next lesson (Week 1 – Day 5) will be a 40-mark mixed revision test covering everything from Days 1–4. The questions will be set in the style of a Grade 8/9 CAPS assessment and will help identify any remaining weak areas before moving on to Week 2 (Linear Equations and Formulae).