Slope is just "how fast", written as math.
Every "per" you've ever said (dollars per hour, miles per gallon, feet per second) is a slope. It's how mathematicians measure change.
You drive home for break, 240 miles in 4 hours. You averaged 60 mph.
That number, 60, is the slope of your trip. It's the answer to "how much did one thing change per change in another?" Miles changed by 240. Hours changed by 4. Divide them: 60 miles per hour. Slope is the same idea every time, just dressed up in different units. By the end of this lesson, you'll see them everywhere, and know how to calculate them in 10 seconds flat.
Rise over run. That's the whole game.
Pick two points. Count the rise (vertical change). Count the run (horizontal change). Divide rise by run, that's slope.
"Change in y, divided by change in x." Often written as the letter m.
Slope measures how much y changes for every 1-unit change in x. A slope of 3 means y goes up by 3 every time x goes up by 1.
Four kinds of slope you'll meet
The vocabulary you actually need
- Slope (m) A number that describes how steep a line is and which direction it tilts. The letter m is the standard symbol, there's no deep reason for it, it's just tradition.
- Rise The vertical change between two points. Up is positive, down is negative.
- Run The horizontal change between two points. Right is positive, left is negative.
- Rate of change The plain-English version of slope. "60 mph," "$15 per hour," "3 inches per year", all rates of change.
- Δ (delta) A Greek letter meaning "change in." So Δy / Δx reads as "change in y over change in x", slope, in two characters.
- Ordered pair A point on a graph, written (x, y). Slope needs two of them, that's why the formula has subscripts 1 and 2.
How fast is your savings account growing?
"On day 10, your account had $180. On day 30, it had $480. What's the slope of your savings, your rate of saving in dollars per day?"
Write the two points as ordered pairs.
Time goes on the x-axis (it's the input), money on the y-axis (it depends on time). Translate the sentences into points.
Label them so the formula doesn't trip you up.
Pick either point as (x₁, y₁). The other becomes (x₂, y₂). It doesn't matter which you pick, as long as you're consistent on both top and bottom.
| Label | Value |
|---|---|
| x₁ | 10 |
| y₁ | 180 |
| x₂ | 30 |
| y₂ | 480 |
Plug into the formula.
Slope is rise over run, or (y₂ − y₁) / (x₂ − x₁).
Translate it back to English.
The slope is 15, but 15 what? Look at your axes: y was dollars, x was days. So the slope is 15 dollars per day. You're saving about $15 each day. That's what slope means, never just the number, always the units.
→ slope = rate, with units attachedThree problems. Calculate, read, classify.
Find the slope of the line through the points (2, 5) and (6, 17).
The line below passes through (0, 1) and (4, 3). What's its slope?
Find the slope of the line through (−3, 7) and (5, −1).
Three fast questions before we move on.
Q1. A car drives 180 miles in 3 hours at a steady speed. What's the slope of its distance-vs-time graph?
Q2. What's the slope of the line through (3, 7) and (8, 7)?
Q3. An espresso machine costs $80 to fix per visit, plus the technician's $50 trip fee. If y is the total cost and x is the number of visits, what's the slope?
Slope plus a starting point equals a prediction.
You can now find the slope of any line. But slope alone doesn't tell you where a line is on the graph, just how steep. Pair it with a starting value (like that $50 trip fee in Q3) and you've got a complete linear model. In Lesson 5, we use that model to predict outcomes from data: how much will rent cost in 2030, how far will you drive in 7 hours, how much will you save by graduation.
Continue to Lesson 05Different angle? Need another rep? These are optional — tap any that look helpful.
Slope and rate of change
Explicitly connects slope to rate of change rather than treating it as only a graph skill. Long enough to develop the idea, compact enough to work as a supplement to the lesson.
Slope in the Real World
The real-world interpretation companion: slope as "how much y changes per unit of x." Matches this lesson's dollars-per-hour and miles-per-gallon framing better than a procedural slope-formula video.