HighSchoolQ

➕ Algebra 1 · Linear Functions

Slope of a Line

Understand slope as rise over run, find it from two points, and tell positive, negative, zero and undefined slopes apart, with memory tricks, diagrams and games.

Grades 9, 10⏱ 12 min

By the end you can…

  • Describe slope as rise over run and as a rate of change.
  • Calculate the slope of a line from two points with the slope formula.
  • Tell positive, negative, zero and undefined slopes apart.

Lesson (school style)

Slope tells you how steep a line is and which direction it goes. It is the ratio of how much the line rises (up or down) for every step it runs (to the right). Read every line left to right, like a book.

Slope measures steepness

Imagine walking along a line from left to right. Slope compares your vertical change to your horizontal change.

The farther the slope is from 0 (like 5 or −5), the steeper the line. A slope close to 0 (like 1/2 or −1/2) is gentle.

Rise over run

Rise is the vertical change: up is positive, down is negative. Run is the horizontal change: moving right is positive and moving left is negative.

slope = rise ÷ run

Slope is often written with the letter m.

The slope formula

If you know two points (x1, y1) and (x2, y2), then

m = (y2 − y1) ÷ (x2 − x1)

Subtract the y’s on top and the x’s on the bottom, and use the same order for both.

Four kinds of slope

  • Positive: the line goes up from left to right.
  • Negative: the line goes down from left to right.
  • Zero: the line is perfectly flat (horizontal). The rise is 0.
  • Undefined: the line is a vertical wall. The run is 0, and you cannot divide by 0.

Slope is a rate of change

In real life, slope tells you how much one thing changes for each 1 unit of another: miles per hour, dollars per ticket, percentage points of battery per hour.

When you see the word “per,” think slope.

Key words

Slope
A measure of how steep a line is and which way it tilts. It equals rise divided by run (undefined for vertical lines).
Rise
The vertical change between two points.
Run
The horizontal change between two points.
Rate of change
How much one quantity changes for each 1 unit of another quantity.
Undefined slope
The slope of a vertical line, where the run is 0 and division by 0 is not possible.

Worked examples

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

  1. Rise: 4 − 1 = 3.
  2. Run: 5 − 1 = 4.
  3. Slope = rise ÷ run = 3/4.

Answer: Answer: m = 3/4

Example 2. Find the slope of the line through (−2, 5) and (4, −1).

  1. Rise: −1 − 5 = −6.
  2. Run: 4 − (−2) = 6.
  3. Slope = −6 ÷ 6 = −1. The negative sign means the line goes downhill.

Answer: Answer: m = −1

Example 3. Find the slope of the line through (3, 7) and (9, 7), and of the line through (2, 1) and (2, 9).

  1. First line: rise = 7 − 7 = 0 and run = 9 − 3 = 6, so slope = 0 ÷ 6 = 0. It is horizontal.
  2. Second line: rise = 9 − 1 = 8 and run = 2 − 2 = 0, so slope = 8 ÷ 0, which is undefined. It is vertical.

Answer: Answer: The first slope is 0. The second slope is undefined.

Example 4. A phone battery is at 100% at 5 pm and 60% at 9 pm. What is the rate of change?

  1. Treat time as x and battery as y: points (5, 100) and (9, 60).
  2. Slope = (60 − 100) ÷ (9 − 5) = −40 ÷ 4 = −10.
  3. The battery loses 10 percentage points each hour.

Answer: Answer: −10 percentage points per hour

In plain English

Slope is a steepness number. It answers: for every step I take to the right, how many steps do I go up or down?

On a hiking trail that climbs 3 feet for every 6 feet of level (horizontal) distance, the slope is 3 ÷ 6 = 1/2. A steeper climb gives a bigger number. A trail going downhill gives a negative number. A flat trail gives zero. A wall straight up? You never move sideways at all, so you would have to divide by zero. That is not a number, so we call it undefined.

See it

13513 A (1, 1)B (5, 4) run = 4 rise = 3 slope = rise ÷ run= 3 ÷ 4= 3/4up 3 for every 4 right
From A to B the line goes right 4 (run) and up 3 (rise), so the slope is 3/4.
Positiveuphill, m > 0“uphill battle” Negativedownhill, m < 0“all downhill” Zeroflat, m = 0“hit a plateau” Undefinedvertical wallrun = 0 Read every line from left to right
Read left to right: uphill is positive, downhill is negative, flat is zero, and a vertical wall is undefined.

More ways to learn it

Fun ways and memory help

6 fun explanations (movies, sports, everyday life and more) and 6 memory tricks.

Notes and common mistakes

Key ideas, steps, mistakes to avoid and a 3-week memory plan.

Practice it your way

Flashcards

Flip, recall and space your reviews.

Cheat sheet

One page. Everything you need.

True or false

Fast yes/no questions with reasons.

Matching game

Pair them up against the clock.

Practice questions

Step-up questions with explanations.

Solving Two-Step EquationsLaws of Exponents

How this lesson was checked

✔ AI-checked○ Teacher-checked: not yet
About standards and checking

Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Two blind re-solves by a separate AI with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): 56 findings fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-07.

Standards this lesson lines up with:

  • CCSS 8.EE.B.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line.
  • CCSS F-IF.B.6: Calculate and interpret the average rate of change of a function over a specified interval.