Intro to Slope as Rate of Change
Slope is a single number that describes how steep a line is and which way it tilts. More precisely, slope measures how much changes each time increases by . A slope of means the line climbs units for every step right; a slope of means it drops units instead.
You find slope by counting: . The rise is the vertical change between two points on the line, and the run is the horizontal change. Get comfortable counting rise over run and slope stops being a formula to memorize — it becomes something you can see.
Counting rise over run
Pick two points on the line that sit exactly on grid intersections. Count how far you move up or down to get from the first point to the second — that is the rise. Count how far you move right — that is the run. Slope is rise divided by run.
On the line below, the points and both sit on the grid. From the first to the second you rise and run , so the slope is .
The sign tells the direction
A line that goes up as you read it left to right has a positive slope. A line that goes down from left to right has a negative slope — the rise is negative because you are counting downward.
A perfectly flat, horizontal line has a slope of : no rise at all, no matter the run. Before you compute anything, glance at the line's direction and decide the sign. If your arithmetic disagrees with your eyes, recount.
Slope is a rate of change
Off the grid, slope shows up as a rate: miles per hour, dollars per week, feet of climb per foot of distance. If a graph shows distance against time and the line has slope , the object covers miles each hour.
That is why slope matters beyond graphing — any time one quantity changes steadily with another, the slope is the number that says how fast.
Worked examples
Example 1: count from two grid points
A line passes through and . Find its slope.
Answer: slope
Example 2: a negative rise
A line has a rise of for a run of . What is its slope?
Answer: slope
Example 3: slope as a real rate
A mountain road rises meters over a horizontal run of meters. What is the slope of the road?
Answer: slope
Try one yourself
Common questions
Does it matter which two points I pick?
No. Any two points on the same line give the same slope — the rise and run scale together. Pick points that sit exactly on grid intersections so your counts are exact.
What is the slope of a horizontal line?
Zero. There is no rise between any two of its points, and divided by any run is . A vertical line is different — its run is , and its slope is undefined.
What does a bigger slope look like?
A steeper line. A slope of climbs much faster than a slope of . For negative slopes, compare steepness by ignoring the sign: is steeper than .
How do I know if the slope should be negative?
Read the line left to right like a sentence. Going downhill means negative; going uphill means positive. Decide the sign from the picture first, then let the counting confirm it.
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