Where Did This Idea Come From?
People have been comparing quantities for thousands of years. Ancient traders needed to know, "If I buy twice as much grain, do I pay twice as much?" That's a question about proportional relationships. But not every situation works that way. Sometimes there's an extra fee, a starting cost, or a flat charge that messes up the simple "multiply" pattern.
Over the centuries, mathematicians and scientists realized they needed two different models. One model handles situations where doubling the input always doubles the output. The other handles situations where there's a head start, a base fee, or some other constant added on. Let's look at how these ideas developed.
The big question this lesson answers is: How do I tell whether a relationship is proportional or non-proportional, and how do I justify my choice?
Core Principles & Definitions
Before you can choose the right model, you need to understand what makes each one tick. Here are the key ideas.
Proportional Relationship
Non-Proportional Relationship
Constant of Proportionality (k)
The Origin Test
Seeing the Difference on a Graph
The fastest way to tell proportional from non-proportional is to look at a graph. A proportional relationship is a straight line that goes through the origin (0, 0). A non-proportional relationship is a straight line that crosses the y-axis somewhere other than zero. Study the diagram below.
Notice how both lines have the same steepness (slope of 2). That means every extra item costs $2 in both situations. The only difference is the starting value. The pink line starts at y = 1 when x = 0. That "+1" makes it non-proportional. When you're deciding which model fits your data, always check: does the line hit the origin?
The Math Behind Each Model
Each model has its own equation form. Recognizing these forms is a quick way to classify a relationship.
You can also use the table method. List all the (x, y) pairs. Compute y ÷ x for each row. If the quotient is always the same and the table includes (0, 0), you have a proportional model. If even one ratio is different—or if y is not 0 when x is 0—pick the non-proportional model.
How to Classify: A Decision Flowchart
Follow this decision flowchart every time you need to decide between a proportional and a non-proportional model. It works whether you're looking at a table, a graph, or a word problem.
The flowchart gives you two checkpoints. First, check the origin. If the data doesn't pass through (0, 0), you can stop—it's non-proportional. If it does pass through the origin, you still need to check that every ratio y ÷ x is the same. Only then can you call it proportional and use y = kx.
Worked Example: Choosing a Model
A local pool charges members for swim lessons. Here is the data a student collected:
| Number of Lessons (x) | Total Cost in $ (y) |
|---|---|
| 0 | 10 |
| 1 | 18 |
| 2 | 26 |
| 3 | 34 |
| 4 | 42 |
Comparing the Two Models Side by Side
| Feature | Proportional | Non-Proportional |
|---|---|---|
| Equation form | y = kx | y = kx + b (b ≠ 0) |
| Passes through origin? | Yes | No |
| Ratio y ÷ x | Same for every pair | Changes from pair to pair |
| Starting value (b) | 0 | Some number other than 0 |
| Real-world example | Buying apples at $2 each (no other fees) | Taxi ride: $3 base fare + $2 per mile |
| Graph shape | Straight line through (0, 0) | Straight line crossing y-axis above or below 0 |
Connection to Algebra & Beyond
In Algebra 1, you'll study these ideas under fancier names. Here's a preview of where proportional and non-proportional models lead.
| What You Know Now | What It's Called in Algebra |
|---|---|
| Proportional model: y = kx | Direct variation: y = kx (special linear function) |
| Non-proportional model: y = kx + b | Slope-intercept form: y = mx + b (general linear function) |
| Constant of proportionality (k) | Slope (m) — the rate of change |
| Starting value (b) | y-intercept (b) — where the line crosses the y-axis |
Beyond linear relationships, you'll eventually explore curves where y is not a straight line at all—things like quadratic and exponential models. But those build on the same skill you're learning right now: looking at data, choosing a model, and explaining why it fits.
Practice Problems
Putting It All Together
A proportional relationship follows the equation y = kx, has a constant ratio y ÷ x = k for every data pair, and its graph is a straight line through the origin (0, 0). A non-proportional relationship follows y = kx + b with b ≠ 0. Its graph is a straight line that does NOT pass through the origin, and the ratios y ÷ x are NOT constant.
To choose the right model, use the origin test (is y = 0 when x = 0?) and the ratio test (is y ÷ x the same every time?). Always justify your choice by stating your evidence—mention whether the data passes through the origin, whether the ratios are constant, and what the equation is. This skill prepares you for slope-intercept form and linear functions in Algebra 1.