Proportional relationships
When one quantity is always the same multiple of another, how do you spot it, in a table, a graph or an equation, and what is the one number that does the multiplying?
Where you are 4 inside
What this area is about
Cheese at 18 dollars a kilogram: every weight times 18 gives its price. Two quantities x and y are in a proportional relationship when one fixed number k gives y = kx for every pair. This area names that number and shows how to recognize the pattern in each form it comes in, plus the pattern where a product, not a ratio, stays fixed.
How these fit together
The definition. The constant of proportionality is k, which equals \displaystyle \frac{y}{x} for every pair with x \ne 0: the unit rate. In a table, the test is whether that quotient is the same in every row with a non-zero input, and an input of 0, if there is one, has output 0.
Recognizing it. In a graph, a proportional relationship is a straight line through the origin, and every such line other than the two axes is one. In an equation, it has the form y = kx, with k fixed and not 0, and nothing added on.
The other pattern. Inversely proportional relationships keep the product fixed, xy = k with k \ne 0: when one amount doubles, the other halves.
A path through it
The constant of proportionality comes first. Inverse proportion can follow it at any point. The graph test also follows it, and the equation test follows the graph test.
Finish here and scale factor and direct variation build on it.