Lesson 5.6

Rate of Change

Slope isn't just about geometry. It's about speed, cost, growth, and decay. In the real world, Slope has a name: Rate.

Introduction

When we talk about slope in math class, we say "Rise over Run." When we talk about it in physics or economics, we say "Change in Output per Change in Input." It is the same math, but with Units attached.

Past Knowledge

Lesson 5.3 (Slope Formula). You need to know how to calculate .

Today's Goal

Interpret slope in context. Answer the question: "What does the slope represent?"

Future Success

Science classes use Rate of Change constantly. Velocity is the rate of change of Position.

Key Concepts

The "Per" Concept

Slope is a ratio of Units.

Distance / Time

Speed

Miles PER Hour

Cost / Item

Price

Dollars PER Apple

Work / Time

Wage

Dollars PER Hour

The Magic Word is "PER".

Slope = The change in per 1 unit of .

Worked Examples

Example 1: The Hiking Trail

Basic

Time (Hours)	Elevation (Feet)
0	500
2	1500
4	2500

Find and interpret the rate of change.

Step 1: Calculate Slope

Step 2: Add Units

Numerator Unit: Feet.

Denominator Unit: Hours.

Rate = 500 Feet per Hour.

Interpretation: The hiker climbs 500 feet higher every hour.

Example 2: Draining the Pool

Intermediate

Slope is calculated as . Vertical axis is Gallons, horizontal is Minutes.

Interpret the -50

Unit: Gallons per Minute.

Sign: Negative means decreasing.

The pool loses 50 gallons every minute.

Example 3: Comparing Plans

Advanced

Plan A: Pay $50 plus $10/month. Plan B: Pay $0 plus $20/month. Which has a higher Rate of Change?

Plan A

Start: $50

Rate: $10/mo

Plan B

Start: $0

Rate: $20/mo

Plan B has a steeper slope (20 > 10). It grows faster. Even though Plan A starts higher, Plan B will eventually catch up and pass it.

Common Pitfalls

Y per X vs X per Y

Is it "Dollars per Hour" or "Hours per Dollar"? Slope is , so the Y-unit comes first. "Dollars per Hour" is standard.

Negative Rates

Don't say "It decreases by -50 gallons." That's a double negative (decreasing by a negative means increasing). Say "It changes by -50" or "It decreases by 50."

Real-Life Applications

Calories: If you burn 100 calories every 10 minutes, your rate is 10 calories/minute. This helps you calculate how long you need to run to burn off a 250-calorie donut (25 minutes).