Proportional Relationships: Tables and Graphs

Lesson 1 of 2

In this lesson:

• Test whether two quantities are proportional using tables and graphs
• Identify the constant of proportionality across all representations

What You Will Learn in Lesson 1

By the end of this lesson, you should be able to:

1. Determine whether quantities are proportional by testing equivalent ratios
2. Use the graph test: proportional means a straight line through the origin
3. Identify in tables, graphs, equations, verbal descriptions, and diagrams

Which Table Shows a Proportional Relationship?

Both grow steadily — but are both proportional?

The Ratio Test for Proportionality

Compute for every row — not just one.

The constant ratio is called the constant of proportionality.

Table A: Trail Mix at $4.50 per Pound

All ratios equal 4.50 ✓ → Proportional,

Table B: Streaming Plan ($8 Base + $2/Movie)

Ratios differ ✗ → Not proportional

Two Tests: Table vs. Graph

Proportional: straight line through · Not proportional: line misses origin

Proportional Relationships Require Both Tests

Both conditions must hold:

• Table test: equals for all pairs
• Graph test: straight line through

The tests are equivalent — they always agree.

A straight line missing the origin is linear but not proportional.

Third Example: Area vs. Side Length

Ratios increase — not proportional. Graph is a curve, not a line.

Quick Check: Is This Relationship Proportional?

Do all ratios match? Pause before the next slide.

Answer: Row 3 Breaks the Pattern

Row 3: → Not proportional

From "Is It Proportional?" to "What Is ?"

Now that we can test for proportionality, let's name and find the constant.

When a relationship is proportional:

• = the common ratio
• = the unit rate
• = the same number in every representation

One number. Many faces.

Constant of Proportionality:

The constant of proportionality is:

• The common ratio for all pairs in the table
• The unit rate: the value of when
• The number that connects all four representations

Baker Example: Finding in a Table

— proportional, all ratios equal

Baker Data on a Graph — Finding

• Line passes through — confirms proportional
• Point : when , — the unit rate is visible

Same in Every Representation

Baker Equation: Verifying

The equation is . Let's verify with two pairs:

• When : ✓
• When : ✓

is the coefficient of in every proportional equation.

Check In: Find in Three Representations

A car travels at constant speed:

Find , then say where it appears on a graph and in an equation.

Think first, then advance.

Answer: km per Hour

• Table: →
• Graph: line through origin; is the unit rate point
• Equation: — coefficient is

Every 1 hour, the car travels 65 km.

Finding from a Verbal Description

A store sells paint for $28 per gallon.

• (cost per gallon)
• Equation:
• The word "per" always signals the unit rate

No table needed — the verbal description gives directly.

Your Turn: Find from a Graph

Find : locate the point where and read off .

Then write the equation using for cost and for gallons.

Practice: Find from Three Representations

Find the constant of proportionality in each:

1. Table: — what is ?
2. Equation: — what is ?
3. Verbal: "A recipe uses 1.5 cups of oats per serving" — what is ?

Solve all three, then advance for answers.

Answers: Finding in Three Representations

1. — ratio test on any pair
2. — coefficient of
3. cups per serving — the "per" phrase

is always the unit rate: value of when .

Key Takeaways for Lesson 1

✓ Proportional: equals the same for every pair

✓ Graph: straight line through the origin

✓ is the unit rate — same value in all representations

Straight line missing → not proportional

One ratio is never enough — check every row

Lesson 2 Preview: Equations and Graphs

Coming up in Lesson 2:

• Writing and predicting in both directions
• Reading graphs — what and mean in context
• Translating among tables, graphs, equations, and descriptions

Lesson 2 puts your knowledge of to work.