Letters stand for numbers
Maths · Key Stage 3 · The School
The grammar of algebra
Algebra compresses arithmetic. 3n means 3 × n (the × is invisible). n² means n × n. n/2 means n ÷ 2. And n + n + n is just 3n. The letter is not a mystery — it is a number whose value we have not fixed yet.
Substitution is just replacement
If n = 5, then 3n + 2 = 3 × 5 + 2 = 17, and n² − n = 25 − 5 = 20. Order of operations still rules: in 2n², square first, then double (n = 3 → 2 × 9 = 18, not 36). Negatives are where marks go. Put the value in BRACKETS first and the sign looks after itself: if n = −2, then n² means (−2)² = 4, not −4, because a negative times a negative is positive. But −n² means −(n²) = −4, since the square happens before the minus sign. Two expressions, one keystroke apart, and they differ by 8.
Collecting like terms
4a + 3b + 2a is 6a + 3b — the a-terms combine, the b-term stands alone. Like terms share exactly the same letter part: 3n and 5n are like; 3n and 3n² are NOT. Tidying an expression this way is called simplifying.
The two places substitution goes wrong
Substitution is only replacement, but two things break it. Brackets first. If n = 4, then 2n² means 2 × 4 × 4 = 32, while (2n)² means (2 × 4) squared = 8 × 8 = 64. The bracket decides what gets squared, so read it before you start calculating. Negatives second. If n = −3, then n² is (−3) × (−3) = 9, a positive, while −n² means −(3 × 3) = −9. The habit that closes both traps is to write the substituted value in brackets every time: for 5 − 2n with n = −4, write 5 − 2(−4), which is 5 + 8 = 13, because subtracting a negative adds.