Topic 03 · 16 lessons

Algebra

Algebra is arithmetic with a placeholder. Instead of asking "what is 3 times 4?", we ask "which number, multiplied by 3, gives 12?" and call that unknown number $x$ . Once you can reason about unknowns, you can describe patterns that hold for every number at once.

The lessons here move from writing and simplifying expressions, through solving equations, to functions and graphs, the language used across science, economics and computing.

Lessons in this topic

3.1Repeating PatternsA repeating pattern has a short part that is used again and again. Find that part and you can say what comes next, what is missing and what comes far along.
3.2Missing Numbers and EqualitiesThe equals sign says both sides have the same value. A missing number is the one that keeps the two sides balanced.
3.3Number Sequences and RulesA sequence is a list of numbers that follows a rule. Find the rule, and you can continue the list and work out any term in it.
3.4Algebraic ExpressionsAn expression is a recipe for a number, written with letters where the ingredients can vary.
3.5Substitution into ExpressionsSubstituting means replacing each letter with its value and then working out the result in the correct order.
3.6Collecting Like TermsLike terms have the same letters and powers. You can add or subtract their coefficients to simplify an expression.
3.7Expanding Single BracketsTo expand a bracket, multiply each term inside by the term outside. The result has no brackets.
3.8Factorising with a Common FactorFactorising puts brackets back. Find the highest common factor of every term, take it outside and write what is left inside.
3.9Solving EquationsAn equation is a balance. Do the same thing to both sides, undoing operations in reverse order, until the unknown stands alone.
3.10Writing Equations from Word ProblemsTo solve a story problem, call the unknown a letter, write the story as an equation, solve it and check the answer in the story.
3.11Inequalities and Number LinesAn inequality compares two values and has many solutions. Solve it like an equation, but flip the sign if you multiply or divide by a negative.
3.12Coordinates in Four QuadrantsCoordinates (x, y) describe a position from the origin: x says how far across and y says how far up or down.
3.13Linear EquationsA linear equation draws a straight line. Its gradient says how steep; its intercept says where it crosses the y-axis.
3.14Using and Rearranging Simple FormulasA formula links quantities with letters. Substitute to find a value, or rearrange with inverse operations to make another letter the subject.
3.15Functions and GraphsA function is a rule that turns each input into exactly one output. Its graph is a picture of every input–output pair.
3.16Quadratic EquationsA quadratic has an x² term. Its graph is a parabola, and it can have two, one or no real solutions.

The big ideas

a)Letters stand for numbers: A letter is a number we either do not know yet or that is allowed to vary. All the usual arithmetic rules still apply to it.
b)Balance: An equation is a balance. Whatever you do to one side, you must do to the other, and the balance holds.
c)Undo in reverse: To solve an equation, undo the operations applied to $x$ in the reverse order they were applied.
d)Rules become pictures: A function such as $y = 2x + 1$ can be drawn as a graph, letting you see its behaviour at a glance.

Worked example

A number puzzle

I think of a number, multiply it by $4$ and subtract $7$ . The answer is $29$ . What was my number?

Call the number $x$ and write the puzzle as an equation: $4x - 7 = 29$ .
Undo the subtraction by adding $7$ to both sides: $4x = 36$ .
Undo the multiplication by dividing both sides by $4$ : $x = 9$ .
Check: $4 \times 9 - 7 = 36 - 7 = 29$ . ✓

Answer: The number was $9$ .

Where you will meet it

Spreadsheet formulas, the cost of a taxi ride, the dosage rules on medicine and the physics in a video game engine are all algebra: a rule written once and applied to whatever numbers come along.

The wider field

Algebra also covers these ideas, which build on the lessons above:

Inequalities
Simultaneous equations
Polynomials and factorising
Exponents and logarithms
Matrices
Sequences and series