Quarry School

Math, taught from zero.

Free courses that assume nothing. Every lesson says the idea in plain words, shows a picture, gives the reason before the rule, works an example one step at a time, and names the mistake most people make.

Read a sample lesson See the courses

How every lesson runs

The same seven parts, in the same order, in every lesson of every course.

  1. Plain wordsThe idea in everyday language, before any symbol appears.
  2. A pictureA diagram for every idea and for every new word.
  3. WhyThe reason the rule is true, so there is nothing to take on faith.
  4. The ruleStated once, exactly.
  5. HowThe steps, in the order you do them.
  6. A worked exampleEvery step shown with its reason, then the answer checked.
  7. The trapThe common mistake, and how to catch it in your own work.

A sample lesson: what a function is

This is a full lesson at its real length. It needs nothing but arithmetic.

Plain words

A function is a rule that takes an input and hands back exactly one output. Think of a vending machine: press the same button and you get the same snack every time. Two buttons may give the same snack. One button never gives two different snacks.

Picture

4 input multiply by 3, then subtract 2 the rule f 10 output
Put 4 in. The rule gives 3 × 4 = 12, then 12 − 2 = 10. One input, one output.

Why

Because each input has exactly one output, you can name the output before you compute it. That is the whole point of the notation: f(4) names one number, the one the rule produces from 4.

The rule

f(x) is the output of the rule f for the input x.

How

To find f of a number, put that number in parentheses everywhere x appears, then do the arithmetic.

Worked example

f(x) = 3x − 2. Find f(4).

  1. Replace x with 4, in parentheses: f(4) = 3(4) − 2. The input goes everywhere x was.
  2. Multiply first: 3(4) = 12, so f(4) = 12 − 2. Multiplication comes before subtraction.
  3. Subtract: 12 − 2 = 10. One input has produced one output.

f(4) = 10

Check: the machine in the picture turns 4 into 10.

The trap

f(4) does not mean f times 4. The parentheses hold the input. Read it aloud as “f of 4”.

The first two courses

Units go up one at a time, each after a check of its sources and its math. The lessons are free to read. The sample lesson above is the first page.

College Algebra

  • Functions and function notation
  • Domain and range
  • Rates of change and the shape of a graph
  • Composition of functions
  • Transformations
  • Absolute value
  • Linear functions and their graphs

Trigonometry

  • Angles, degrees and radians
  • The six trigonometric functions
  • The unit circle and exact values
  • Signs, identities and reference angles
  • Graphs of sine and cosine
  • Graphs of tangent and the reciprocal functions
  • Sinusoidal models