Part 4 of 10 · LinesSee the whole course
What is slope in math?
Slope tells you how steep a line is and which way it leans: how far the line goes up (or down) for every 1 step across. It is “rise over run”: slope = rise ÷ run.
Walk along a line in steps
Put your finger on a line and move one step to the right. Does the line go up, down, or stay level? And by how much? That is the slope.
Take the line y = 2x + 1. Each step across, it goes up 2. Another step across, up 2 again. It is always the same, and that steady change is what makes it a straight line. Its slope is 2.
Rise over run
Two words make this precise:
- The run is how far you go across (to the right).
- The rise is how far you go up. If the line goes down, the rise is negative.
The rise goes on top. A line that rises 6 while it goes across 3 has slope 6 ÷ 3 = 2. Slide the slope below and watch the rise and the run change.
Positive, negative and zero slope
Always read a line from left to right, the way you read a sentence.
- Uphill means a positive slope. For y = 2x + 1, the slope is 2.
- Downhill means a negative slope. For y = −x + 4, the line goes down 1 for every 1 across, so the slope is −1.
- Flat means a slope of zero. The line y = 3 never goes up or down: rise 0 ÷ run 1 = 0.
Steeper does not mean a bigger number
Steepness is about the size of the slope, ignoring the sign. A slope of 3 is steeper than a slope of 1. A slope of −3 is exactly as steep as 3, only it goes downhill.
That means −3 is steeper than −1, even though −3 is the smaller number. When you compare steepness, drop the minus sign first. The sign only tells you the direction.
Slope is a rate
Slope is not only for graphs. It is “how much per step”. If you walk 4 kilometers every hour, the slope of your distance is 4 kilometers per hour. If a taxi adds 5 dollars for every kilometer, the slope of the price is 5. A steeper line means something is changing faster.
Watch it explained in under a minute
The five mistakes to avoid
- Dividing across by up. “Run over rise” is backwards: the rise goes on top.
- Dropping the minus sign on a downhill line.
- Reading the line from right to left. Always go left to right.
- Thinking −3 is less steep than −1 because it is the smaller number. Compare sizes, not signs.
- Mixing up a flat line (slope 0) with an upright line. An upright line has no slope at all; see horizontal and vertical lines.
Try it yourself
Tap a question to see the answer.
1. A line rises 8 as it goes across 4. What is its slope?
Rise ÷ run = 8 ÷ 4 = 2.
2. A line falls 6 as it goes across 3. What is its slope?
The rise is −6, the run is 3: −6 ÷ 3 = −2.
3. Which is steeper, a slope of −4 or a slope of 2?
Ignore the signs: 4 is bigger than 2, so −4 is steeper. It goes downhill, and it is steeper than the uphill line.
Where slope fits in
Slope is one of the two numbers that describe any straight line. Next in this course, you will learn how to find the slope when you only know two points: how to find the slope from two points.