What is slope?

Slope tells us how steep a line is and which direction it goes.

Think of slope as answering two questions:

• Does the line go up or down?

• How steep is it?

Real-life examples:

• A hill (steep or gentle)

• A ramp

• Stairs vs. a sidewalk

The big idea (this is the heart of slope)

Slope compares:

how much the line goes up or down
compared to
how much the line goes left or right

This is called:

rise over run

• Rise → up or down

• Run → left or right

Finding slope from a graph

Let’s say you have two points on a line:

Step 1: Find the rise

• Start at y = 2

• Go up to y = 8

Rise = +6

Step 2: Find the run

• Start at x = 1

• Go right to x = 4

Run = +3

Step 3: Write the slope

✅ The slope is 2

That means:

For every 1 unit right, the line goes up 2 units

Positive vs. negative slope

Positive slope 📈

• Line goes up as you move right

• Example:

   

  slope = 3


Negative slope 📉

• Line goes down as you move right

• Example:

   

  slope = −2


Zero slope (flat line)

If a line goes straight across, it has zero slope.

Example:

• No rise

• Only run

Undefined slope (vertical line)

If a line goes straight up and down, the slope is undefined.

Example:

• There is rise

• But no run

You can’t divide by zero, so the slope doesn’t exist.

Slope from an equation

Most lines are written like this:

• m = slope

• b = y-intercept (where the line crosses the y-axis)

Example:

• Slope = 3

• The line goes up 3 for every 1 right

Common beginner mistakes (you’re not alone!)

• ❌ Mixing up rise and run
  ✅ Always do up/down first, then left/right

• ❌ Forgetting negative signs
  ✅ Watch direction carefully

• ❌ Dividing backwards
  ✅ Rise ÷ Run (not the other way)

Why slope matters

Slope helps us:

• Read and draw graphs

• Understand speed (distance over time)

• Compare growth and change

• Prepare for linear equations

It shows up everywhere in algebra.