CSEC MATHEMATICS · MODULE 1 · LESSON A13

Algebra Basics: Symbols, Directed Numbers, Expanding and Substitution

Using letters for numbers, translating words into algebra, directed numbers, simplifying and expanding expressions, substitution and operations other than + − × ÷.

Fundamentals of Secondary Level MathematicsM1 5.1M1 5.2M1 5.3M1 5.4M1 5.5M1 5.6M1 5.7≈ 40 min🎮 3 games
Before you start

By the end of this lesson you can…

1WatchExplainer video 2ReadKey notes & exam tips 3PractisePaper 1 style 4ApplyPaper 2 style
Step 1 · Watch

Explainer video

5 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.

LESSON A13 · EXPLAINER

Algebra Basics: Symbols, Directed Numbers, Expanding and Substitution

Algebra is a language. Once you can read and write it, equations, formulae and graphs stop looking like code.

Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.

Step 2 · Read

Key notes

Copy these into your notebook in your own words — that is how they stick.

Vocabulary

Variable, term, coefficient, constant, expression, equation, identity.

Words → symbols

“Less than” reverses order: 5 less than 2n is 2n − 5. “Product” means ×; “sum” means +.

Directed numbers

Same signs ×/÷ → positive; different → negative. − (−a) = + a.

Like terms

Same letters and same powers can be added or subtracted.

Indices when multiplying

Add powers: x² × x³ = x⁵. When dividing, subtract.

Expanding

Multiply every term in the bracket; for two brackets use FOIL or a grid. (a ± b)² = a² ± 2ab + b².

Substitution

Put values in brackets, follow BODMAS.

New operations

a ∗ b = 3a − 2b: substitute a and b in order. Test commutativity by swapping.

EXAM TIPS & COMMON MISTAKES

  • (x + 4)² is NOT x² + 16. Expand it as (x + 4)(x + 4) and keep the middle term.
  • “n less than m” is m − n. The words and the symbols are in opposite orders.
  • Substitute negatives in brackets: −2² = −4 but (−2)² = 4.
  • For new operations such as a ∗ b, always substitute the FIRST value for a and the SECOND for b.
Play & practise

Games

3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Mathematics Arcade.

Step 3 · Practise

Paper 1 practice — multiple choice

8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.

Adapted from CSEC P1 May/June 2024 (algebra translation item)

1. Seven times the product of two numbers, a and b, may be written as

Answer: B. Product of a and b is ab; seven times it is 7ab.
CSEC Paper 01 style · SO 5.4

2. 4x + 3y − x + 5y simplifies to

Answer: C. 4x − x = 3x and 3y + 5y = 8y.
CSEC Paper 01 style · SO 5.3

3. The value of −3 − (−7) is

Answer: D. Subtracting a negative is adding: −3 + 7 = 4.
CSEC Paper 01 style · SO 5.4

4. 3(x + 4) − 2(x − 1) simplifies to

Answer: A. 3x + 12 − 2x + 2 = x + 14.
CSEC Paper 01 style · SO 5.7

5. (x + 3)(x − 5) is equal to

Answer: B. x² − 5x + 3x − 15.
CSEC Paper 01 style · SO 5.6

6. If a = 3 and b = −2, then 2a² − b is

Answer: C. 2(9) − (−2) = 18 + 2 = 20.
Based on CSEC P1 May/June 2024 (defined-operation item)

7. The operation ∗ is defined by a ∗ b = 3a − 2b. The value of 4 ∗ 5 is

Answer: C. 3(4) − 2(5) = 12 − 10 = 2.
CSEC Paper 01 style · SO 5.2

8. “Five less than twice a number n” is written as

Answer: D. Twice n is 2n; five less than that is 2n − 5.
Step 4 · Apply

Paper 2 practice — structured questions

Write full answers, then reveal the mark scheme and tick the points you earned (17 marks in total). Show all your working — method marks count. Your answers are saved on this device.

CSEC Paper 02 style · Module 1 (Algebra 1)

Question 1 [9 marks]

Answer all parts, showing your working.

(a) Simplify 5x − 3y + 2x + 7y. [2]

MARK SCHEME — (a) (max 2)

(b) Expand and simplify (2x − 3)(x + 4) − x². [3]

MARK SCHEME — (b) (max 3)

(c) Evaluate E = 2p² − 3pq + q when p = −2 and q = 3. [3]

MARK SCHEME — (c) (max 3)

(d) Write as an algebraic expression: “the sum of x and y, divided by 3, then increased by 4”. [1]

MARK SCHEME — (d) (max 1)

CSEC Paper 02 style · Module 1 (Defined operations)

Question 2 [8 marks]

The operation ⊗ is defined by x ⊗ y = x² − 2y.

(a) Evaluate (i) 3 ⊗ 4 (ii) (−2) ⊗ 5. [3]

MARK SCHEME — (a) (max 3)

(b) Find the value of k for which 2 ⊗ k = 0. [2]

MARK SCHEME — (b) (max 2)

(c) Show that ⊗ is not commutative. [3]

MARK SCHEME — (c) (max 3)

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Sources

Where this topic appears in CSEC papers

Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.