Lesson 5.2

Finding Slope from a Graph

Don't guess. Count. How many steps up? How many steps over?

Introduction

We know slope is "steepness." But exactly how steep? To find the number, we count the vertical change (Rise) and compare it to the horizontal change (Run).

Past Knowledge

Lesson 5.1 (Types of Slope). You know Positive means "Up" and Negative means "Down".

Today's Goal

Calculate exact slope using .

Future Success

This is the visual version of the Slope Formula (), which we learn next lesson.

Key Concepts

Rise Over Run

Slope () is a fraction.

Numerator

Rise

Change in (Vertical)

Up (+) or Down (-)

Denominator

Run

Change in (Horizontal)

Always go Right (+)

Worked Examples

Example 1: Counting Up

Basic

Find the slope between and .

Step 1: Count Rise

Start at (0,1). Count UP to the height of the second point.

We go from to . That is +2 steps up.

Step 2: Count Run

Count RIGHT to the second point.

We go from to . That is +4 steps right.

Step 3: Fraction

. Simplify it.

Example 2: Counting Down

Intermediate

Find the slope between and .

Step 1: Count Rise

Start at the left point (0,2). We have to go DOWN to get to -2.

Count down 4 steps. Rise = -4.

Step 2: Count Run

Count RIGHT 2 steps. Run = +2.

Step 3: Simplify

. Divide it.

Example 3: Integers vs Fractions

Advanced

Why do we leave some as fractions?

Slope = 2

This just means . Up 2, Right 1.

Slope =

Up 2, Right 3.

DO NOT write 0.666!

Graphing logic requires integers for counting. Decimals break the "Rise/Run" method.

Common Pitfalls

Run over Rise?

Slope is ( over ). Students often flip it because comes first in . Remember "Y goes High" (Numerator).

Reducing Wrong

, but . Don't just divide the big number by the small one. Pay attention to which is on top.

Real-Life Applications

Stairs: Building codes are strict about slope. A standard staircase has a rise of 7 inches and a run of 11 inches (). If the specific ratio changes too much, people trip because their muscle memory expects consistent steps.