Slope - Seventh Grade
Rise Over Run, Slope Formula, Graphing & Rate of Change
1. Understanding Slope
What is Slope?
Slope is a measure of how STEEP a line is
It tells you how much a line rises or falls
as you move from left to right
Slope = Rise Over Run
m = rise/run
Where:
m = slope
rise = vertical change (up or down)
run = horizontal change (left to right)
Types of Slope
| Slope Type | Value | Direction |
|---|---|---|
| Positive Slope | m > 0 | Line goes UP ↗ |
| Negative Slope | m < 0 | Line goes DOWN ↘ |
| Zero Slope | m = 0 | Horizontal Line → |
| Undefined Slope | No value | Vertical Line ↑ |
2. Finding Slope from a Graph
Steps
Step 1: Choose two points on the line
Step 2: Count the RISE (vertical change)
• Count UP for positive rise
• Count DOWN for negative rise
Step 3: Count the RUN (horizontal change)
• Always count to the RIGHT (positive)
Step 4: Calculate slope = rise/run
Example
Find the slope of the line through points (1, 2) and (4, 8)
Step 1: Points are (1, 2) and (4, 8)
Step 2: Find the rise
From y = 2 to y = 8
Rise = 8 − 2 = 6 (going UP)
Step 3: Find the run
From x = 1 to x = 4
Run = 4 − 1 = 3 (going RIGHT)
Step 4: Calculate slope
m = rise/run = 6/3 = 2
Slope: m = 2
3. Finding Slope from Two Points
Slope Formula
m = (y₂ − y₁)/(x₂ − x₁)
Where:
(x₁, y₁) = first point
(x₂, y₂) = second point
m = slope
Steps
Step 1: Label the points (x₁, y₁) and (x₂, y₂)
Step 2: Subtract y-coordinates: y₂ − y₁
Step 3: Subtract x-coordinates: x₂ − x₁
Step 4: Divide: (y₂ − y₁)/(x₂ − x₁)
Example 1
Find the slope between (2, 3) and (5, 9)
Step 1: (x₁, y₁) = (2, 3) and (x₂, y₂) = (5, 9)
Step 2: Use formula
m = (y₂ − y₁)/(x₂ − x₁)
m = (9 − 3)/(5 − 2)
m = 6/3
m = 2
Slope: m = 2
Example 2: Negative Slope
Find the slope between (1, 8) and (4, 2)
m = (2 − 8)/(4 − 1)
m = −6/3
m = −2
Slope: m = −2 (negative slope, line goes down)
4. Finding Missing Coordinate Using Slope
Strategy
Use the slope formula and substitute known values
Then solve for the missing coordinate
Example 1: Missing y-coordinate
Find y if the slope between (2, 5) and (6, y) is 3
Step 1: Write the slope formula
m = (y₂ − y₁)/(x₂ − x₁)
Step 2: Substitute known values
3 = (y − 5)/(6 − 2)
3 = (y − 5)/4
Step 3: Solve for y
3 × 4 = y − 5
12 = y − 5
y = 17
Answer: y = 17
Example 2: Missing x-coordinate
Find x if the slope between (3, 4) and (x, 10) is 2
2 = (10 − 4)/(x − 3)
2 = 6/(x − 3)
2(x − 3) = 6
2x − 6 = 6
2x = 12
x = 6
Answer: x = 6
5. Graphing a Line Using Slope
Steps
Step 1: Start with a given point on the line
Step 2: Write slope as a fraction: m = rise/run
Step 3: From the starting point, move UP or DOWN (rise)
Step 4: Then move RIGHT (run)
Step 5: Mark the new point and draw the line
Example
Graph a line with slope m = 3/4 starting from point (1, 2)
Step 1: Plot starting point (1, 2)
Step 2: Slope = 3/4 means rise = 3, run = 4
Step 3: From (1, 2), move UP 3 units
New y = 2 + 3 = 5
Step 4: Move RIGHT 4 units
New x = 1 + 4 = 5
Step 5: New point is (5, 5)
Draw a line through (1, 2) and (5, 5)
Line passes through (1, 2) and (5, 5)
Negative Slope
For negative slope m = −2/3:
Rise = −2 (move DOWN 2)
Run = 3 (move RIGHT 3)
Line goes downward from left to right
6. Rate of Change
What is Rate of Change?
Rate of change is how fast one quantity
changes compared to another quantity
For LINEAR functions, rate of change = SLOPE!
Formula
Rate of Change = Δy/Δx
or
(y₂ − y₁)/(x₂ − x₁)
Constant Rate of Change
Definition: The rate of change is the SAME
between any two points on the line
→ This creates a STRAIGHT LINE
→ The slope is CONSTANT
7. Rate of Change from Tables
Steps
Step 1: Pick any two rows from the table
Step 2: Find the change in y-values
Step 3: Find the change in x-values
Step 4: Divide: Δy/Δx
Example
Find the rate of change:
| x | y |
|---|---|
| 2 | 5 |
| 4 | 11 |
| 6 | 17 |
Solution:
Using first two rows:
Δy = 11 − 5 = 6
Δx = 4 − 2 = 2
Rate of change = 6/2 = 3
Check: Using rows 2 and 3:
Δy = 17 − 11 = 6
Δx = 6 − 4 = 2
Rate of change = 6/2 = 3 ✓
Constant rate of change = 3
8. Rate of Change from Graphs
Key Point
Finding rate of change from a graph is
the SAME as finding slope from a graph!
Steps
1. Pick two points on the graph
2. Count the rise (vertical change)
3. Count the run (horizontal change)
4. Calculate: rise/run
Rate of change = slope = rise/run
Real-World Example
Problem: A car travels 120 miles in 2 hours and 240 miles in 4 hours. What is the rate of change (speed)?
Points: (2, 120) and (4, 240)
Δy = 240 − 120 = 120 miles
Δx = 4 − 2 = 2 hours
Rate = 120/2 = 60
Rate of change = 60 miles per hour
Quick Reference: Slope Formulas
| Method | Formula |
|---|---|
| Basic Slope | m = rise/run |
| Slope from Two Points | m = (y₂ − y₁)/(x₂ − x₁) |
| Rate of Change | Δy/Δx = (y₂ − y₁)/(x₂ − x₁) |
Slope Types
| If m... | Then line... |
|---|---|
| m > 0 (positive) | Goes UP (↗) |
| m < 0 (negative) | Goes DOWN (↘) |
| m = 0 (zero) | Horizontal (→) |
| m = undefined | Vertical (↑) |
💡 Important Tips to Remember
✓ Slope = rise/run: Always vertical change over horizontal change
✓ Formula: m = (y₂ − y₁)/(x₂ − x₁)
✓ Positive slope: Line goes UP from left to right
✓ Negative slope: Line goes DOWN from left to right
✓ Zero slope: Horizontal line (flat)
✓ Undefined slope: Vertical line (straight up and down)
✓ Rate of change = slope: For linear functions, they're the same!
✓ Constant rate of change: Same slope throughout = straight line
✓ From tables: Pick any two rows and use the formula
✓ Graphing with slope: Start at a point, use rise/run to find next point
🧠 Memory Tricks & Strategies
Rise Over Run:
"Rise before you run - vertical before horizontal, get the slope done!"
Slope Formula:
"Y's on top, X's below - that's how the slope formula will go!"
Positive vs Negative:
"Positive slopes climb high, Negative slopes say goodbye (go down)!"
Horizontal vs Vertical:
"Horizontal has zero slope - it's flat, no hope! Vertical is undefined - straight up, you'll find!"
Rate of Change:
"Rate of change and slope are twins - for straight lines, slope always wins!"
Graphing with Slope:
"Start with a dot, rise then run - connect the points and you're done!"
Master Slope! 📈 📊
Remember: Slope = Rise/Run = (y₂ − y₁)/(x₂ − x₁)