Graphing Rates and Reading Slope

Lesson 1 of 2

In this lesson:

• Graph a proportional relationship as a line through the origin
• See that the unit rate and the slope are the same number
• Find slope as rise over run from any two points

Learning Objectives for This Deck

You should be able to:

1. Graph a proportional relationship, producing a straight line through the origin
2. Explain why the unit rate equals the slope of the graph
3. Calculate slope as rise over run from any two points on the line

A Bakery Sells Cupcakes for $3 Each

A bakery charges $3 per cupcake.

• Let be the number of cupcakes
• Let be the total cost in dollars

If you drew a picture of this on a grid, what would it look like — and where would it start?

Build the Table, Then Plot the Points

Each cupcake adds $3 — the points climb in a straight path from .

Two Features: Straight Line Through Origin

A proportional relationship graph always has both features:

• It is a straight line — the rate never changes
• It passes through the origin — zero cupcakes cost zero dollars

Write the Equation: y = 3x

The cupcake relationship in symbols:

• This is proportional: form
• The constant of proportionality is

A Second Example: A Growing Plant

A plant grows cm per day.

• Equation: — still a line through the origin

A Line That Is Not Proportional

A streaming service charges $5 plus $2 per movie.

Predict: does this line pass through the origin?

Quick Check: Test the Origin

Is a proportional relationship?

Plug in . Do you get ?

How Much Does Cost Rise Each Step?

Back to the cupcakes, .

When goes up by 1, how much does go up?

• You already know this answer — it is

Slope Is Rise Over Run

Steepness measured as the rise divided by the run.

Unit Rate = Slope = k: One Number

For the cupcake relationship:

• Unit rate dollars per cupcake
• Slope
• in

Three names. One number.

Rise Over Run From Two Point Pairs

Cupcake line, :

• From to : rise , run , slope
• From to : rise , run , slope

The Slope Formula From Coordinates

For any two points and :

• Cupcakes:
• Plant:

A Run of One Reveals the Rate

• From , step right — the rise lands at

Slope Carries Units of the Context

Slope is not just a bare number — it has units:

• Cupcakes: slope dollars per cupcake
• Plant: slope cm per day

Units of slope = units of ÷ units of .

Your Turn: Find This Slope

A line passes through and .

What is the rise? The run? The slope?

Solo Check: Write the Slope

A proportional line passes through and .

On your own paper, write the slope before anyone speaks.

Any Two Points Give the Same Slope

• Three different pairs, one slope — slope belongs to the line

Watch Out: Two Common Traps

" is proportional because the slope is 3" — No. It misses the origin, so it is not proportional.

"A flatter line has a bigger slope" — No. Steeper means bigger slope.

Key Takeaways From This Lesson

✓ A proportional relationship graphs as a line through the origin
✓ Unit rate = slope = — one number, three names
✓ Slope from any two points

A straight line that misses the origin is not proportional

Coming Up Next in Lesson Two

In Lesson 2, you'll compare two rates given in different forms — one as a graph, one as an equation, one as a table — and decide which is greater.