Algebraic Notation
Algebra is arithmetic with a blank left in it. The letter is the blank. Everything that looks strange about 3x² − 2x + 5 is a shorthand agreed on so that mathematicians everywhere write the same thing the same way.
National curriculum KS3: use and interpret algebraic notation, including ab for a × b, 3y for y + y + y, a² and a³, a/b for a ÷ b, coefficients as fractions, brackets; substitute numerical values into formulae and expressions; understand and use the concepts of expression, equation, term, factor, formula, identity1 · The letter is a box
Before algebra you wrote ▢ + 3 = 7 and worked out what goes in the box. Algebra replaces the box with a letter: x + 3 = 7. The letter stands for a number that is unknown, or that is allowed to change. A function machine shows the idea: put any number in, the rule does the same thing to it every time.
The algebra version is shorter than the words and, more importantly, it works for every input at once. "Double it and add one" is a sentence; 2x + 1 is a machine.
2 · The conventions
These are the agreed shorthands. None of them is a rule of nature; they are habits that save ink and prevent confusion. Each one below has a reason.
| We write | Instead of | Why |
|---|---|---|
| 3x | 3 × x | The × sign looks like the letter x. Multiplication is shown by writing things next to each other. |
| ab | a × b | Same idea. Letters go in alphabetical order: ab, not ba (they're equal, it's just tidier). |
| 3x | x3 | Number first, then the letter. The number is called the coefficient. |
| x | 1x | A coefficient of 1 is left out. It's still there when you need it (for collecting terms). |
| x² | x × x | Powers show repeated multiplication (see powers and roots). x² is not 2x. |
| a/b or ab | a ÷ b | Division is written as a fraction. The ÷ sign almost disappears after primary school. |
| x/2 or ½x | x ÷ 2 | Halving is the same as multiplying by ½. Both forms are accepted. |
| 3(x + 2) | 3 × (x + 2) | Brackets group things; a number next to a bracket multiplies the whole bracket. |
| −x | −1 × x | A minus in front means the coefficient is −1. |
Translate it
Type an expression in "long" notation and see it tidied. Or type it tidy and see what it secretly means. Try x × 3 + y × y − 1x.
3 · See the difference: 2x versus x²
This is the single most common confusion in KS3 algebra, and a picture fixes it. Use algebra tiles: a small orange square is 1, a teal strip is x (its length is unknown, so it isn't a whole number of oranges), and a plum square is x by x, which is x².
Two strips side by side versus a square with the strip as its side. They're only equal when x = 2 (try it). Everywhere else they're different numbers.
4 · Build an expression from tiles
Any expression in x is a collection of tiles. Set how many of each you have and read off the expression. Negative tiles are shown faded with a dashed edge; a positive and a negative of the same kind cancel (a zero pair).
Words for the parts
| Word | Meaning | In 3x² − 2x + 5 |
|---|---|---|
| Term | A piece separated by + or −. The sign travels with the term. | 3x², −2x, +5 |
| Coefficient | The number multiplying a letter. | 3 (of x²), −2 (of x) |
| Constant | A term with no letter: it never changes. | 5 |
| Variable | The letter. | x |
| Like terms | Terms with exactly the same letters and powers; only these can be added. | 3x and −2x are like; 3x² and 3x are not |
| Expression | Terms joined together, with no = sign. | 3x² − 2x + 5 |
| Equation | Two expressions set equal; true only for certain values. You solve it. | 3x² − 2x + 5 = 9 |
| Formula | An equation that gives one quantity in terms of others. You substitute into it. | A = πr², v = u + at |
| Identity | True for every value; written with ≡. | 2(x + 3) ≡ 2x + 6 |
5 · Substitution
To substitute is to replace the letter with a number and work out the result. Write the number in brackets where the letter was; it prevents nearly every sign error. Remember the order of operations: powers before multiplying, multiplying before adding.
Three traps, all about x = −3. (1) x² = (−3)² = +9, not −9. (2) 2x² means 2 × (x²) = 18; only the x is squared. (3) (2x)² squares everything inside: (−6)² = 36. Try all three above.
6 · Collecting like terms
You can only add tiles of the same shape. 3x + 2x is five strips: 5x. But 3x + 2 is three strips and two small squares, and that's as simple as it gets: you can't combine a strip with a square because you don't know how many squares make a strip.
7 · Why bother with letters?
Formulae you already use
Area of a rectangle: A = lw. Speed: s = d/t. Perimeter of a square: P = 4s. Each one is a rule that works for every rectangle, journey or square ever, written once.
Spreadsheets
=B2*1.2 is algebra: B2 is the variable, 1.2 the coefficient. Fill it down a column and the same rule runs on 1000 rows.
Programming
total = price * quantity is 3x-style algebra with longer names. Python even needs the × written as *, which is why 3x is a syntax error in code but fine on paper.
Science
F = ma, E = mc², V = IR. Physics is written in algebra because the relationships hold for every mass, every current. Rearranging them is next year's topic.
Patterns and sequences
"Add 3 each time, starting at 5" is 3n + 2 for the nth term. The algebra lets you jump straight to the 100th term without writing out 99 others.
Puzzles
"I think of a number, double it, add 7 and get 23." Call it x: 2x + 7 = 23. The notation turns the puzzle into something you can solve mechanically.
Worked examples
Write "multiply a number by 4, then subtract 3" in algebra
Simplify a × b × 3 × a
If p = 4 and q = −2, find 3p − q²
Exam-style: a rectangle has length 2x + 3 and width x. Write an expression for its perimeter and simplify it.
Quick check
What you've learned
A letter stands in for a number, and algebraic notation is a set of agreed shorthands: multiplication by juxtaposition with the number first, powers for repeated multiplication, fractions for division, brackets for grouping. 2x and x² are different shapes, not two ways of writing the same thing. An expression has terms, each with a coefficient; only like terms combine. Substituting means replacing the letter with a bracketed number and following the order of operations. Expression, equation, formula and identity are four different kinds of object, and knowing which one you're looking at tells you what you're allowed to do with it.