A pouch holds an unknown number of candies. Add 3 more — how many should you say there are?
A variable is a placeholder for an unknown number — the moment we could compute without knowing, mathematics became a language.
Experiment
Hands-on experiment
🔮 Predict first — you don't know how many candies the pouch holds. Add 3 — how many are there?
🎒 Build an expression
Call the pouch's candy count x. Use the buttons to build.
x
expression so far
Now open the pouch — suppose it held 5.
📖 Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
For ages mathematics handled only known numbers. But real problems mostly begin with unknowns — the candies in the pouch, next year's age, the price we seek.
The breakthrough: give the unknown a name (x) and a seat. Write 'pouch+3' as x+3 and you can build, tidy, and compare without ever knowing the value.
With this invention mathematics became a language. Formulas (area = length×width), equations, functions — everything after is built on letters.
Insight
Insights from the video
“x is a seat, not an answer.”
See x only as 'the answer to find' and algebra feels scary. x is just a chair for a number not yet known — set out the chair and the computation works for whoever sits down later.
“The expression itself can be the answer.”
Arithmetic always ended in a single number; in algebra, 3x+2 is a finished answer. Crossing this gate is the real moment one passes from arithmetic to algebra.
Misconception
Common misconceptions
2x means 'a 2 written next to an x', so if x=5 then 2x=25.
2x is 2×x with the multiplication sign omitted — x=5 gives 2×5=10. The number in front (the coefficient) is always glued on by multiplication.
3x + 2 should compute further, to 5x or 5.
Three pouches and two loose candies can't merge — while x is unknown, 3x+2 IS the final answer. Only like terms combine. Getting comfortable with 'the expression itself is the answer' is algebra's first step.
Formula
Writing it as math
What the pouch experiment confirmed, in mathematical language.
Using letters
Name the unknown and computation starts without it. 2x means 2×x — the multiplication sign is omitted.
Substitution
The moment the pouch opens — put a number in the letter's seat and the value computes.
Collecting like terms
Pouches with pouches, loose candies with loose candies — only like terms combine.
In Real Life
Where you meet it in real life
Every formula is an expression
Area = length × width (ab), speed = distance ÷ time (d/t) — a formula is an expression that works for any numbers. Letters let us write once and reuse forever.
Fares and bills
Taxi fare 0.8x+4.8 (x in km) — behind every bill and rate table sits an expression. The pattern lab's □, grown into x.
Spreadsheets and code
Excel's =A1*2+3, code's price * n — cell names and variables ARE letters. The spreadsheet is humanity's most-used algebra tool.
Age puzzles
'In 10 years, dad will be twice the son's age' — set the son's age to x and the sentence becomes an expression, then an equation.
Try Yourself
Test yourself
Q1Pencils cost $1.20 each. Write the cost of x pencils plus one $0.50 eraser.Show answer ▾
1.2x + 0.5 (dollars). Substitute x=3 and get $4.10 — one expression is a price tag for every quantity.
Q2Simplify 5x − 2x + 7 − 3.Show answer ▾
3x + 4. Pouches with pouches (5−2=3), loose with loose (7−3=4) — collecting like terms.
Q3If x = −2, what is 3x + 10?Show answer ▾
3×(−2)+10 = −6+10 = 4. Wrap negatives in parentheses when substituting and mistakes vanish.
💡 Try answering yourself before revealing it — getting it wrong is where learning starts.
Watch
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Patterns & Correspondence
Writing relations with □ and △ is the soil variables grow from.
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Start asking when two expressions become equal, and you have an equation.
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