Lesson Guide
Explanation
A variable is a slot that holds ANY number. The expression 3n + 2 isn't one calculation — it's infinitely many at once: n = 4 gives 14, n = 100 gives 302. That's algebra's superpower: one line of symbols captures a whole pattern. Convention alert: 3n means 3 × n — the multiplication sign goes invisible so it can't be confused with the letter x. Likewise ab means a × b, and n÷3 is written as the fraction n/3.
Your Turn
Evaluate 5m − 4 when m = 7, and when m = 20. Check: 31 and 96.
Watch Out
3n is 3 × n, NOT a two-digit number. If n = 5, then 3n = 15 — never 35.
Explanation
Vocabulary that everything else builds on. TERMS are the chunks separated by + and − signs: 4x2 − 7x + 9 has three terms. The COEFFICIENT is the number multiplying the variable — and the sign travels with it, so the coefficient of x here is −7, not 7. A CONSTANT is a term with no variable at all (the 9): it never changes. And a hidden convention: x alone has coefficient 1, and −x has coefficient −1 — the 1 is invisible.
Your Turn
For 6a − b + 2ab − 11: how many terms, coefficient of b, and the constant? Check: 4 terms; −1; −11.
Watch Out
The coefficient of x in 5 − 3x is −3. Read the sign IN FRONT of the term as part of it.
Explanation
Translating words into algebra is a skill examiners love. The subtle ones: '5 less than n' is n − 5 (not 5 − n — order flips!). 'Double n then add 3' is 2n + 3, but 'add 3 then double' is 2(n + 3) — brackets capture 'do this first'. Not sure your translation is right? Test it with a number: at n = 10, 2n + 3 = 23 while 2(n + 3) = 26 — different stories. Substitution is your lie detector.
Your Turn
Write an expression: 'Ali has x dollars; Ben has 4 dollars less than triple Ali's amount.' Check: 3x − 4.
Watch Out
'7 less than n' = n − 7. Writing 7 − n is the single most common translation error in Year 8.
Explanation
Substitution gets dangerous when negatives meet powers. The survival rule: wrap the substituted number in BRACKETS. Evaluate 2x2 − 3x at x = −2: write 2(−2)2 − 3(−2). Powers first: (−2)2 = +4 (negative × negative). Then 2(4) − 3(−2) = 8 + 6 = 14. Without brackets you'd write −22 and read it as −4 — wrong answer, lost marks. Brackets first, order of operations second, arithmetic third.
Your Turn
Evaluate 3a2 − ab when a = −3, b = 4. Check: 3(9) − (−3)(4) = 27 + 12 = 39.
Watch Out
(−2)2 = +4 but −22 = −4. The bracket decides whether the minus gets squared. Always bracket your substitutions.
Explanation
Algebra earns its keep by turning patterns into formulas. The matchstick classic: 1 square uses 4 sticks, each extra square adds 3 (it shares a side). Jumps of 3 mean the rule contains 3n; then check the start — n = 1 must give 4, and 3(1) = 3, so add 1: sticks = 3n + 1. The formula isn't just curve-fitting — the 3 IS the three new sticks per square, the 1 IS the extra stick closing the first. Now n = 100 is instant: 301 sticks.
Your Turn
A pattern goes 5, 9, 13, 17, … Find the rule and the 50th term. Check: jumps of 4, rule 4n + 1, 50th term 201.
Watch Out
The jump size gives the coefficient of n — but ALWAYS test your rule on the first term before trusting it.
Explanation
The toolkit: a VARIABLE is a slot for any number; TERMS are the chunks split by + and −; the COEFFICIENT is the number in front, sign included; a CONSTANT has no letter. When SUBSTITUTING, bracket the number first — especially negatives. When building rules from patterns, the jump is the coefficient and the first term is your test. This vocabulary is the grammar of everything that follows: like terms, expanding, factorising all speak this language. Quiz and exercises!
Your Turn
Capstone: for the expression 7 − 2k2, name the terms, the coefficient of k2, the constant — then evaluate at k = −3. Check: terms 7 and −2k2; coefficient −2; constant 7; value 7 − 2(9) = −11.
Watch Out
In −2k2 only k is squared, not the −2 in front. Coefficient stays −2; the power belongs to the letter it touches.
📌 Key Points — this lesson in a nutshell
- A variable is a slot for ANY number; 3n means 3 × n — the × sign is invisible.
- Terms are the chunks split by + and −; the sign in front belongs to its term.
- A constant has no variable; x alone has an invisible coefficient of 1.
- Substitute inside brackets: (−2)2 = +4 but −22 = −4.
- Pattern rules: the jump size is the coefficient of n — always test on the first term.
Step 1: Introduction
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