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School math

Explanations of individual topics from the school curriculum: the rule in brief, two or three examples with every step explained, and a list of the mistakes that turn a correct rule into a wrong answer.

How the explanations are built#

Each page answers one question: what the rule is, how to use it and where people trip over it. First the rule itself in two or three lines, then worked examples where every transformation is written on its own line with a note on why it is allowed. At the end comes a list of typical mistakes and problems with answer checking.

The explanations are meant both for a student who did not follow the topic in class and for an adult explaining it to a child who wants to put the rule into words rather than only show a finished solution. That is why the examples skip no steps: whenever a new line appears, the text says what was done to it.

Where to start if a topic will not click#

School math is a staircase: percentages rest on fractions, systems of equations on solving a single linear equation, exponents on care with signs. When a problem will not come out, the cause is usually not the topic itself but the step below it. A quick test: take the simplest example from an explanation and solve it on paper without looking. If you get stuck on fractions or on moving a term across the equals sign, start with that page rather than the current topic.

School math Systems of linear equations
A system of two equations in two unknowns is solved with one idea: reduce the two unknowns to one. There are two ways to do it — solve for one variable and substitute, or add the equations so that one variable disappears.
School math Special products and factoring formulas
Seven formulas that save you from multiplying every term by every other term when you expand brackets. Below are the formulas themselves, examples worked line by line, a trick for mental arithmetic and the one mistake that is more common than all the others put together.
School math Percentage word problems
A percent is one hundredth: 1% = 0.01. Almost every school percentage problem comes down to one of three types, and each one takes a single operation — once you have worked out what the problem treats as the whole.
School math Decimals
A decimal is an ordinary fraction whose denominator is always 10, 100 or 1000 — only the denominator is not written, the decimal point stands in for it. Every rule follows from that: where the point goes when you add, how many digits to count off when you multiply, and what to do with the point in the divisor.
School math Laws of exponents
An exponent is shorthand for multiplying the same factor by itself. All five laws of exponents follow from that shorthand in one line, so a forgotten rule can always be rebuilt rather than guessed.
School math How to solve linear equations
A linear equation is one where the variable appears only to the first power — no squares, no variable in a denominator. You solve it with a single idea, the balance: whatever you do to the left side, you do to the right side, and the equation stays true.
School math Dividing fractions
Dividing by a fraction means multiplying by that fraction turned upside down. The rule is short, but it only works under three conditions: you flip the second fraction, mixed numbers are first turned into improper fractions, and a whole number is written as a fraction over one.
School math Area and perimeter
Perimeter is the length of a shape's boundary; area is the size of what is inside. That is also why the units differ: perimeter in centimetres, area in square centimetres. Below are the formulas, three worked problems and a check that catches a mixed-up formula in five seconds.
School math Fractions with unlike denominators
You can only add pieces of the same size — sixths to sixths, twenty-fourths to twenty-fourths. That is why any operation on fractions with different denominators starts with a common denominator, and only then do you add the numerators.