Lesson 5.3

The Slope Formula

What if the points are too far apart to count? Or use decimals? We stop counting boxes and start using subtraction.

Introduction

In Lesson 5.2, we counted "Rise" and "Run" visually. But "Counting" is really just finding the Difference between coordinates.

Past Knowledge

Subtraction finds "distance". is the distance between 5 and 2.

Today's Goal

Calculate slope given two points and .

Future Success

This formula is the foundation of the Average Rate of Change in Calculus.

Key Concepts

The Formula

Slope (m)

Numerator

Change in Y (Rise)

Denominator

Change in X (Run)

Crucial Rule

If you start with Point 2 on top, you MUST start with Point 2 on bottom. Order matters.

Worked Examples

Example 1: Positive Slope

Basic

Find the slope between and .

Step 1: Label Points

Step 2: Plug in

Step 3: Solve

Slope = 4

Example 2: Negative Numbers

Intermediate

Find slope between and .

Step 1: Setup

Step 2: Watch Signs!

Numerator:

Denominator: (Minus a negative is plus!)

Step 3: Solve

Example 3: Zero Slope

Advanced

Find slope between and .

Calculation

Rule of Zero

You can divide zero by anything (0 cookies shared by 4 friends = 0 cookies). But you CANNOT divide by zero (Undefined).

Common Pitfalls

X on Top?

Students accidentally write . This calculates . Remember "Rise over Run" means "Y's over X's".

Minus a Negative

often becomes in student work. It should be . Always use parentheses when plugging in negative numbers.

Real-Life Applications

Average Speed: If you are at mile marker 10 at 1:00 PM and mile marker 130 at 3:00 PM, your average speed is the slope.

Points: and .
.
Your "Rate of Change" is 60 miles per hour.