Slope 05:00 minutes

Video Transcript

Transcript Slope

On the slopes of the Himalayans, Leonetti the Yeti and his wife Betty are sledding. While Leonetti pushes his wife up the mountain, she nags at him to go faster. It’s hard work, and Leonetti is huffing and puffing with the exertion!

What is Slope?

Why is it such hard work? There’s a mathematical explanation! Slope!

Slope is a number that expresses the steepness of a line. Imagine you draw a line along the incline of the mountainside. Slope is the change in height compared to the change in width.

Formula for Slope

Let’s take a look at the formula for slope. M stands for slope, and the triangle symbol is the Greek letter delta, meaning change or difference.

To determine the slope of a line, select two points on a line then calculate delta y, the change of the y-values, to determine the change of height; divided by delta x, the change of the x-values, to determine the change of width.

When you determine the differences of the x- and y-values, make sure you maintain the order of the coordinates. Alternatively, you can think of this as rise divided by run or simply rise over run.

Slope Calculation Example 1 (Positive Slope)

Let’s figure out the slope of the route where Leonetti is having such a tough time pushing his bride up the mountainside.

Draw a line along the side of the mountain's incline and select any two points on the line. Then determine the ordered pairs for both points, and plug the numbers into the slope formula.

6 − 3 = 3 and 15 − 6 = 9. You can simplify this fraction. So the slope is one over three, or one-third.

Slope Calculation Example 2 (Positive Slope)

What happens if the angle of the incline is different? Imagine Leonetti and Betty take a different route, this time less steep. It’s not such a workout for Leonetti to push Betty up the mountain. How does the slope differ?

Just like before, use the slope formula to divide the change in height by the change in width. One divided by five is equal to one-fifth. FYI, on the same line, regardless of where you select the two points, the slope will always be the same.

On the steeper route, the slope is 1/3 and on the less steep route, the slope is 1/5. So can we conclude, the larger the slope, the steeper the line? That is true for this situation, but the rule is: the larger the absolute value of the slope, the steeper the line.

Calculation example 3 (Negative Slope)

Both times Leonetti has pushed his wife up the mountain. Are you wondering what happens to the slope of the line when the Yetis sled down the mountain? Great question!

To model sledding down the mountain, let's put some new numbers into the slope formula. 2 − 6 = −4, and 15 − 3 = 12. Divide the numbers, and you get a slope equal to negative one third.

So, does that mean we can conclude that a negative slope means a line slanted down? You betcha! And, the greater the absolute value of the negative slope the steeper the line? Right again!

Calculation example 4 (No Slope)

Leonetti and his wife Betty are now on level ground. This time he pulls his wife across the snow. Again, she nags at him to go faster. What happens to the slope when there is no incline or decline?

Since there is no change in height, Y2 − Y1 = 0. X2 − X1 = 13. The slope is zero over 13 which is equal to zero. FYI again, if you draw another line parallel to this line… Guess what? The slopes will be the same!

Slope Tree

I have a fantastic graphic to help you remember all of this. No, it’s not a Christmas tree, it’s a slope tree!

  • If the line slants up, the slope is positive.
  • If the line slants down, the slope is negative.
  • And, if the line is parallel to the x-axis, the slope is zero.

What about the tree trunk? What if the line is parallel to the y-axis? In that case, you would have a zero in the denominator, and we all know you can’t divide a number by zero – it’s not defined, so the slope is not defined as well.

What does all this mean for Leonetti and his nagging wife Betty?

  1. Eugene

    Thank you for your kind comments! We're working hard to continue to bring you videos with easy-to-understand explanations...and fun stories!

    Posted by Eugene L., over 2 years ago
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    Posted by Plano Kellmeyer, almost 3 years ago
  3. Default

    really cool

    Posted by Student033 6, almost 3 years ago