Skill: Recognize proportional and non-proportional patterns informally

A pattern is proportional when one quantity is always the same multiple of the other. You can check this by:

1. Finding the unit rate: divide each output by its matching input.
2. Seeing whether the unit rate stays the same.
3. Checking that an input of $0$ gives an output of $0$.

Example: A taxi charges a starting fee of 3$plus$$2$ for each mile.

Miles, $x$	Cost, $y$
$0$	3$
$1$	5$
$2$	7$
$3$	9$

First, check what happens when the number of miles is $0$. The cost is 3$, not$$0$. This means there is a fixed starting fee.

Next, compare the ratios for positive numbers of miles:

$$\frac{5}{1}=5,\qquad \frac{7}{2}=3.5,\qquad \frac{9}{3}=3$$

The ratios are not equal, so there is no constant unit rate from miles to total cost.

Therefore, this is a non-proportional pattern. The cost increases by 2$for each mile, but it also starts with an extra$$3$. A proportional pattern must have a constant ratio and must pair$0$with$0$.