Algebra uses letters to stand for numbers we don't know yet (or numbers that can change). A letter like x is just a placeholder — a mystery box waiting to be opened. Once you know its value, you simply swap it in. If a bag holds x sweets, then two bags hold 2x sweets, whatever x turns out to be.
Why this matters: writing things with letters lets you describe a whole family of situations in one short line — the cost of any number of tickets, the perimeter of any rectangle, the rule behind any pattern. Every equation, formula and graph you meet later is built from this notation, so getting the shorthand and the "collecting" right now saves endless confusion further on.
Collecting like terms: 3a + 5a = 8a (8 of the same tile)
The shorthand rules
Algebra has a tidy way of writing things — we drop the × sign and squash terms together so expressions stay short and readable:
In full
Written as
Why
3 × n
3n
the number goes in front
n × n
n²
"n squared"
a ÷ b
a/b
written as a fraction
1 × n
n
we never write 1n
Terms and like terms
A term is a single part of an expression, like 3a or 5b or 7. Like terms have exactly the same letter part — and only like terms can be added or subtracted together ("collected").
3a + 5a = 8a (both are "a" terms — like 3 apples + 5 apples make 8 apples). But 3a + 5b won't simplify any further — apples and bananas don't combine into one kind of fruit.
Different tiles (a and b) can't merge — only like terms collect.
The fruit analogy is the whole idea: you can only pile together things of the same type. So in a longer expression, sort the terms into groups first, then add each group.
Common mistake: keep each sign glued to the term after it. In 5p + 2 − 3p + 4, the −3p travels with its minus sign: collect 5p − 3p = 2p and 2 + 4 = 6, giving 2p + 6. Dropping that minus is the number-one slip here.
The minus stays glued to 3p — collect p-terms and numbers separately.
Multiplying terms
When terms are multiplied rather than added, the "same letter" rule no longer applies — you can multiply anything. Multiply the numbers, then combine the letters.
Area = 4x × 3x = 12x². Multiply numbers, then the letters.
Adding (collect like terms)
Multiplying (always allowed)
3a + 5a = 8a
3a × 5a = 15a²
3a + 5b = 3a + 5b (stuck)
3a × 5b = 15ab
In a nutshell: letters stand for numbers; write products without ×; only collect like terms (same letters), keeping each sign attached — but when multiplying, any terms can combine.