Topic
Algebra
Numbers with names. Letters stand in for unknowns, balance laws keep both sides honest, and graphs make every equation visible.
Try algebra in 10 seconds
How many were in the box?
Some apples in a box, plus 3 more on top. Total of 8 apples.
Box
x = 2
apples
+ 3 = 8
Your total
5
needs 8
Algebra: write it as x + 3 = 8, subtract 3 from both sides.
Algebra in two sentences
Algebra is what arithmetic looks like when you don't yet know all the numbers. You replace the unknown with a letter, write down the relationship, and use balance laws to find what the letter has to be.
Suggested path
The basics
Letters for unknowns, and the moves you keep using to reshape an expression.
Introduction
Letters stand in for unknown numbers. The first idea — and why it works.
Substitution
Swap a letter for a number. The simplest move in algebra.
Expressions
Combine like terms — simplify with a tap.
Like terms
3x and 5x add to 8x. 3x and 5y don't combine. Why like terms matter.
Brackets
Parentheses, square brackets, braces — when each one is used and why.
Expanding
Multiply out the brackets — distribute, FOIL, then simplify.
Simplify
Cancel, combine and rewrite — make the expression as small as possible.
Equations
Balance both sides of a scale and solve for the unknown.
Equations
Balance both sides of a scale and solve for the unknown.
Linear equations
ax + b = c. The cleanest equation — one variable, one solution.
Equations of a line
Slope-intercept, point-slope, two-point — every form for y = mx + b.
Systems of equations
Two equations, two unknowns — substitute, eliminate, or matrix-solve.
Word problems
Turn a sentence into an equation, then solve for the unknown.
Inequalities
Less-than, greater-than, shaded regions — and optimising inside them.
Inequalities
Less-than, greater-than, and number-line shading.
Solving inequalities
Same moves as equations — except multiplying by a negative flips the sign.
Graphing inequalities
Shade the half-plane below or above — and the boundary line tells you which.
Linear programming
Maximise (or minimise) a linear thing under linear constraints.
Exponents, roots & logarithms
Repeated multiplication, its inverse, and the rules that hold them together.
Exponents & roots
2³ is 8. Tap the power, raise the base, see what you get.
Exponent laws
Multiply, divide, raise-to-a-power — three rules that handle every exponent.
Negative exponents
x⁻¹ is 1/x. Negatives in the exponent flip the base over.
Fractional exponents
x^(1/2) is √x. Fractions in the exponent are roots in disguise.
Square roots & surds
Simplify √50 to 5√2 — and rationalise denominators.
Logarithms
log is the inverse of exp — pull the exponent down to a friendly number.
Ratios & proportion
Two ratios that match — direct, inverse, and how to scale them.
Polynomials & factoring
Multi-term expressions: how to grow them, how to break them apart.
Polynomials
Add, subtract and multiply expressions with x², x³ and beyond.
Polynomial division
Long division — but the divisor and dividend are polynomials.
Remainder & factor theorems
If P(a) = 0, then (x − a) is a factor. The fastest factor check there is.
Rational expressions
Polynomials in fractions. Add, subtract, multiply, divide — same rules as fractions.
Partial fractions
Break a complicated fraction into a sum of simple ones.
Factoring
Break x² + 5x + 6 into (x+2)(x+3) — split the middle.
Factoring quadratics
Find two numbers that multiply to c and add to b — the quadratic factor trick.
Completing the square
Rewrite ax² + bx + c into a(x − h)² + k — and solve any quadratic.
Functions & graphs
Rules that turn one number into another — and the pictures of those rules.
Functions & graphs
Slide m and b on y = mx + b and watch the line move.
Evaluating functions
f(3) means: drop 3 in for x, simplify, read the answer.
Odd & even functions
Mirror across the y-axis (even) or rotate around the origin (odd).
Maxima & minima
The highest and lowest points of a curve — without calculus.
Asymptotes
Lines a curve gets close to but never touches — vertical, horizontal, slant.
Quadratics
Parabola playground — change a, b, c and see the curve respond.
Quadratic formula
x = (−b ± √(b² − 4ac)) / 2a — solves every quadratic.
Sequences & series
Spot the pattern, predict the next, and add them all up.
Sequences & series
1, 4, 9, 16… spot the pattern, predict the next.
Arithmetic series
Sum of a sequence with constant difference — and the n(n+1)/2 trick.
Geometric series
Sum of a sequence with constant ratio — and when it converges.
Sigma notation
∑ — the compact way to write a long sum.
Infinite series
Sums that go on forever — and the surprising ones that still settle on a number.
Binomial theorem
Expand (a + b)ⁿ without multiplying — Pascal's triangle does the work.
Mathematical induction
Prove a statement for all n by toppling dominoes — base case + inductive step.
Vectors & matrices
Arrows with size and direction, and grids of numbers that transform them.
Vectors
Arrows with size and direction — add and subtract them visually.
Unit vectors
Vectors with length 1 — the direction without the magnitude.
Dot product
Multiply two vectors to get a number — and read off the angle between them.
Cross product
Multiply two 3D vectors to get a third one perpendicular to both.
Matrices
Numbers in a rectangle — the language of transformations and systems.
Matrix multiplication
Row times column — a rule that's everywhere from graphics to neural nets.
Matrix inverse
The matrix that undoes another — like 1/x but for matrices.
Matrix determinant
A single number that tells you if a matrix is invertible — and how it scales area.
Complex numbers
a + bi where i² = −1 — the numbers that live off the real line.
Trigonometry
sin, cos, tan — the right-triangle ratios and the waves they trace.
Trigonometry
sin, cos, tan and the right-triangle ratios behind every angle.
SOHCAHTOA
The mnemonic every student remembers — sin, cos and tan from a right triangle.
Unit circle
Radius 1, centred at origin — every angle turns into a (cos, sin) coordinate.
Trig identities
sin² + cos² = 1, and the family of equations that always hold.
Sine & cosine laws
Triangles without right angles — two laws that solve them anyway.
Sin, cos, tan graphs
The waves trigonometry draws when you sweep an angle.
Models & reference
Turn real situations into equations — plus the symbols you'll meet everywhere.