Comparing and Interpreting Rates

Lesson 2 of 2

In this lesson:

• Compare two rates given in different representations
• Extract the rate from a graph, table, equation, or words
• Interpret what a slope means in real-world context

Learning Objectives for This Deck

You should be able to:

1. Compare two proportional relationships given in different representations and decide which rate is greater
2. Interpret the slope of a proportional relationship in context, using appropriate units

Who Runs Faster: Graph or Equation?

Two runners are training:

• Runner A: distance shown on a graph through and
• Runner B: distance given by

They're in different formats. How do you even compare them?

Read Each Rate One at a Time

• A: slope mph · B: mph

The Universal Strategy for Comparing

To compare any two proportional relationships:

1. Find each unit rate (slope)
2. Compare the two numbers

A is faster: .

One Rate Hides in Four Disguises

• The rate hides as a ratio, a slope, a coefficient, or a stated value

Comparison 1: Graph vs. Equation

Runner A (graph through ) and Runner B ():

• A: mph
• B: coefficient mph
• A is faster since

Comparison 2: Table vs. Words

Store X rope: 5 ft costs $4.00; 10 ft costs $8.00.
Store Y rope: $0.90 per foot.

Comparison 3: Graph vs. Table

Machine P graphed through and .
Machine Q table: , , .

• P: · Q: — P is faster

Solo Check: Read From a Table

A proportional table: , , .

Which row do you use to find the rate? Write the rate on your own paper.

Your Turn: Compare These Two

Object A: graph through and (miles, hours).
Object B: .

Find each speed. Which is faster?

A Car's Slope: What Does 30 Mean?

A car's graph (gas vs. miles) passes through and .

• Slope

We found the number. But what does actually mean here?

A Template for Interpreting Slope

To say what a slope means:

"For every increase of 1 [x-unit], the [y-quantity] increases by [slope] [y-units]."

Car: "For every 1 gallon, the distance increases by 30 miles" — 30 miles per gallon.

Interpreting the Slopes We Have Met

Read each slope as a sentence:

• Cupcakes, slope : each cupcake costs $3
• Plant, slope : the plant grows cm each day
• Runner A, slope : covers miles each hour

Fuel Efficiency: Is Bigger Better?

• Slope miles per gallon

Here, a bigger slope is better — more miles per gallon.

Does a Bigger Rate Always Win?

It depends on what the slope measures:

• Fuel efficiency (mi/gal): bigger slope is better
• Price (dollars/ft): bigger slope is worse

Always ask: does a bigger rate help or hurt here?

Practice: Compare Two Cars by Efficiency

Car A: 30 miles per gallon. Car B: .

Which is more fuel-efficient — and what does each slope tell you?

Rank Three Cyclists From Fastest

• Cyclist A: graph through and
• Cyclist B:
• Cyclist C: table , ,

Find all three rates. Rank fastest to slowest, on your own.

Watch Out: Three Closing Reminders

Slope and unit rate are the same number — different forms, one value
A line that misses the origin is not proportional — confirm before dividing
Bigger isn't always better — context decides who wins

Key Takeaways From This Lesson

✓ A rate is the common currency — turn any form into "how much per one"
✓ Compare by extract each rate, then compare the numbers
✓ Interpret every slope with units and context

Coming Up Next in This Unit

The next standard asks why slope is the same everywhere on a line — using similar triangles — and what happens when a line does not start at zero: the in .