Proportional vs. Non-Proportional — I can choose between a proportional model and a non-proportional model and justify the choice. - Pre Algebra

SECTION 1

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.

~1800 BCE • Babylonian Ratios

Babylonian merchants used clay tablets to record ratios (comparisons of two numbers) for trading grain, silver, and other goods at steady rates.

~300 BCE • Euclid's Proportions

The Greek mathematician Euclid wrote rules about proportions in his book Elements, showing how equal ratios connect geometry and arithmetic.

1600s • Linear Equations Appear

Mathematicians like René Descartes began graphing equations on coordinate planes. They noticed some lines pass through the origin and some do not—leading to the idea of proportional vs. non-proportional models.

Today • Everyday Math

You use these models daily—comparing streaming plans, calculating tips, or figuring out shipping costs. Choosing the right model helps you make smart decisions with money and data.

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?

SECTION 2

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

Two quantities are proportional when their ratio (y ÷ x) stays the same for every pair of values. The graph is a straight line through the origin (0, 0).

Non-Proportional Relationship

Two quantities are non-proportional when their ratio changes from row to row, or when the graph is a line that does NOT pass through (0, 0). There is usually a starting value or extra constant.

Constant of Proportionality (k)

In a proportional relationship, the constant of proportionality is the unchanging ratio k = y ÷ x. It tells you how much y changes for each unit of x.

The Origin Test

Ask: "When x = 0, does y = 0?" If yes, the relationship might be proportional. If no, it is definitely non-proportional.

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KEY TAKEAWAY

Think of it like a pizza party. If every pizza costs the same amount and there's no delivery fee, the total cost is proportional to the number of pizzas. But if there's a $5 delivery charge on top, the total cost is non-proportional because even zero pizzas would still cost $5. That extra charge breaks the "always through the origin" rule.

SECTION 3

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.

The cyan line (y = 2x) passes through the origin, so it is proportional. The pink line (y = 2x + 1) crosses the y-axis at 1, so it is non-proportional.

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?

SECTION 4

The Math Behind Each Model

Each model has its own equation form. Recognizing these forms is a quick way to classify a relationship.

PROPORTIONAL MODEL

y = k × x

k = constant of proportionality (the steady rate). There is no added constant. When x = 0, y = 0.

NON-PROPORTIONAL MODEL

y = k × x + b

k = rate of change (slope). b = the starting value (y-intercept). Because b ≠ 0, the line does NOT pass through the origin.

RATIO TEST

y ÷ x = same number every time → proportional

Pick any (x, y) pair from your table. Divide y by x. If the answer is the same for every pair (and the point (0, 0) fits), the relationship is proportional.

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.

SECTION 5

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.

Follow the flowchart from START. First check the origin test, then the constant ratio test. Always state your evidence when you justify your choice.

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.

SECTION 6

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

Swim lesson cost data

Is this proportional or non-proportional? Justify your choice.

1
Step 1 — Check the Origin

Look at the row where x = 0. The total cost y = 10, not 0. Since (0, 0) is NOT in the table, the relationship cannot be proportional.

When x = 0, y = 10 ≠ 0 → NOT proportional
2
Step 2 — Confirm with the Ratio Test

Even though we already know, let's double-check. Compute y ÷ x for each row (skip x = 0 since we can't divide by zero): 18 ÷ 1 = 18, 26 ÷ 2 = 13, 34 ÷ 3 ≈ 11.3, 42 ÷ 4 = 10.5. The ratios are all different.

Ratios are NOT constant → confirms non-proportional
3
Step 3 — Identify the Equation

The cost goes up by $8 for each lesson (18 − 10 = 8, 26 − 18 = 8, etc.). The starting cost is $10. So the equation is y = 8x + 10.

y = 8x + 10 (non-proportional, b = 10)
4
Step 4 — Write the Justification

"I chose the non-proportional model because when x = 0, y = 10 (the line does not pass through the origin). There is a $10 membership fee before any lessons are purchased. The equation y = 8x + 10 fits the data."

Justification complete ✓

SECTION 7

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

Key differences between proportional and non-proportional linear relationships

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KEY TAKEAWAY

Think of a garden hose. A proportional relationship is like turning on a hose with no water already in the bucket—the total water is directly tied to how long you run the hose. A non-proportional relationship is like a bucket that already has some water in it before you even turn on the hose. That head start means the total is never just "rate × time"—you always have to add the extra amount.

SECTION 8

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

From pre-algebra to algebra vocabulary

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.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Maria says, "A relationship is proportional just because the graph is a straight line." Is she correct? Explain why or why not.

Show answer

Maria is not correct. A straight line is necessary for a proportional relationship, but it is not enough. The line must also pass through the origin (0, 0). For example, y = 3x + 5 is a straight line, but it is non-proportional because the y-intercept is 5, not 0. A proportional relationship requires both a straight line AND a pass through the origin.

Question 2: BASIC CALCULATION

A table shows: x = 1, y = 4; x = 2, y = 8; x = 3, y = 12; x = 4, y = 16. Is this proportional or non-proportional? Justify using the ratio test.

Show answer

Compute y ÷ x for each pair: 4 ÷ 1 = 4, 8 ÷ 2 = 4, 12 ÷ 3 = 4, 16 ÷ 4 = 4. The ratio is always 4. Also, when x = 0, y would be 0 (since y = 4x). This means the relationship is proportional with a constant of proportionality k = 4. The equation is y = 4x.

Question 3: INTERMEDIATE

An online music service charges $5 per month plus $1.50 per song downloaded. Write the equation for total monthly cost y in terms of songs downloaded x. Is this proportional or non-proportional? Justify your answer with two pieces of evidence.

Show answer

The equation is y = 1.50x + 5. This is non-proportional for two reasons. First, the $5 monthly fee means that when x = 0 (no songs), y = 5, not 0, so the graph does not pass through the origin. Second, the ratios y ÷ x are not constant: for 1 song, y ÷ x = 6.50 ÷ 1 = 6.50; for 2 songs, y ÷ x = 8 ÷ 2 = 4. Different ratios confirm the non-proportional model.

Question 4: APPLIED

A lemonade stand sells cups for $2 each. On Saturday they made $0 before any cups were sold. On Sunday a different stand charges $2 per cup but also has a $6 table rental. Write an equation for each day. Which day's model is proportional? If you sell 10 cups on each day, what is the total for each?

Show answer

Saturday: y = 2x (proportional, passes through the origin with constant ratio 2). Sunday: y = 2x + 6 (non-proportional, starts at $6 when x = 0). For 10 cups on Saturday: y = 2 × 10 = $20. For 10 cups on Sunday: y = 2 × 10 + 6 = $26. Saturday's model is proportional because the only cost is the per-cup price with no extra fees. Sunday's $6 table rental makes it non-proportional.

Question 5: CRITICAL THINKING

A table shows: x = 2, y = 6; x = 4, y = 12; x = 5, y = 15. Carlos says it's proportional because y ÷ x = 3 every time. Priya says she isn't sure because the table doesn't include x = 0. Who makes the stronger argument, and what would you add to settle the debate?

Show answer

Priya makes the stronger argument because a constant ratio alone doesn't guarantee proportionality—you also need to confirm that (0, 0) belongs to the data. Carlos correctly showed that y ÷ x = 3 for every given pair, which is a good start. To settle the debate, you should check whether the pattern gives y = 0 when x = 0. If the equation is y = 3x, then y = 3 × 0 = 0, confirming it passes through the origin, and the relationship is proportional. But if the real-world situation has a hidden fee (like a setup charge), the equation could be different. Always verify (0, 0) fits before declaring a relationship proportional.

SUMMARY

Putting It All Together

Summary

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.