Mathematics › 📐 Grade 8 › Slope: Measuring Steepness with Rise over Run

Slope: Measuring Steepness with Rise over Run

One number captures how steep a line is and which way it leans — and you read it straight off the graph.

Grade 8Algebra 1

Slope: Measuring Steepness with Rise over Run — illustration

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The big idea: Slope is a single number that answers two questions at once: how steep is a line, and which way does it tilt? You find it by comparing how far the line goes up (the rise) to how far it goes across (the run). Because a straight line has the same steepness everywhere, that ratio is the same no matter which two points you pick.

🎯 By the end, you'll be able to

Define slope as rise divided by run
Read the slope of a line from a graph by building a slope triangle between two points
Tell the sign of a slope (positive, negative, zero, or undefined) from the way a line leans
Compute slope from two coordinate points using the slope formula

📎 You should already know

Plotting points on the coordinate plane
Positive and negative numbers

Steepness needs a number

A ramp, a staircase, a ski slope, a graph of your savings — all of them “go up” at some rate. Saying one is “steeper” than another is a start, but mathematics wants a number so we can compare and calculate. That number is the slope.

The idea is simple: watch how much the line climbs for a given step sideways. A line that climbs a lot while moving only a little across is steep; a line that barely climbs while moving far across is gentle.

🔑 Slope = rise over run

The slope of a line is the rise (how far it goes up) divided by the run (how far it goes across) between any two points on the line. Up and to the right counts as positive; down counts as negative.

\[ \text{slope} = \dfrac{\text{rise}}{\text{run}} = \dfrac{\text{change in } y}{\text{change in } x} \]

Rise is the vertical change; run is the horizontal change. Same idea, two names.

The slope triangle

To measure slope from a graph, pick two points the line passes through and draw a right triangle between them: one horizontal leg (the run) and one vertical leg (the rise). The slope is just rise ÷ run for that triangle.

Here is the powerful part: it does not matter which two points you choose or how big you make the triangle. A straight line has the same steepness everywhere, so every slope triangle you draw on it gives the same ratio.

🎮 Interactive: watch the slope triangle LIVE

Drag the Slope (m) slider and read the orange triangle: for every 1 step right (run = 1), the line rises by m (the rise). Make m bigger to steepen the line, negative to tilt it down, and 0 to lay it flat. The Intercept (b) slider just slides the whole line up or down — it never changes the steepness.

✨ Reading the sign of a slope

Positive slope rises to the right (↗). Negative slope falls to the right (↘). A zero slope is a flat horizontal line — no rise at all. A perfectly vertical line has an undefined slope, because the run is 0 and we cannot divide by zero.

Slope from two points, without a graph

If you know two points, you do not even need to draw. Subtract to get the rise (difference in the y-values) and the run (difference in the x-values), keeping the points in the same order on top and bottom.

\[ m = \dfrac{y_2 - y_1}{x_2 - x_1} \]

The slope formula. Pick a first point (x₁, y₁) and a second (x₂, y₂); order does not matter as long as you are consistent top and bottom.

📝 Worked example: Find the slope of the line through (1, 2) and (4, 8).

Rise = change in y = 8 − 2 = 6.
Run = change in x = 4 − 1 = 3.
Slope = rise ÷ run = 6 ÷ 3.

✓ The slope is <strong>2</strong> — the line rises 2 units for every 1 unit it moves right.

📝 Worked example: Find the slope of the line through (−2, 5) and (2, −3).

Rise = (−3) − 5 = −8.
Run = 2 − (−2) = 4.
Slope = −8 ÷ 4.

✓ The slope is <strong>−2</strong>. The negative sign tells you the line falls as you move right.

⚠️ Keep the order consistent

Whatever point you list first for the rise (top), list first for the run (bottom) too. Mixing the order — y₂ − y₁ over x₁ − x₂ — flips the sign and gives the wrong direction.

Check your understanding

1. A slope triangle on a line has a rise of 4 and a run of 2. What is the slope?

Slope = rise ÷ run = 4 ÷ 2 = 2.

2. A line falls as you move from left to right. Its slope is…

Falling to the right means the rise is negative while the run is positive, so the slope is negative.

3. What is the slope of a flat, horizontal line?

A horizontal line has no vertical change, so rise = 0 and slope = 0 ÷ run = 0.

4. Find the slope of the line through (0, 1) and (5, 1).

Rise = 1 − 1 = 0, run = 5 − 0 = 5, so slope = 0 ÷ 5 = 0. Both points have the same height, so the line is horizontal.

5. Why does every slope triangle on the same straight line give the same slope?

On a straight line the steepness never changes, so a bigger triangle has proportionally bigger rise and run — the ratio stays fixed.

✅ Key takeaways

Slope is a number for steepness: rise (vertical change) divided by run (horizontal change).
Draw a slope triangle between any two points; the ratio is the same wherever you draw it.
Positive slope rises to the right, negative slope falls, zero slope is horizontal, vertical is undefined.
From two points, slope = (y₂ − y₁) / (x₂ − x₁) — keep the order consistent top and bottom.
Slope is exactly the m in y = mx + b, so this is the key that unlocks linear functions.