Introduction to slope | Algebra I | Khan Academy

– [Voiceover] As we start to graph lines, we might notice that they’re
differences between lines. For example, this pink or
this magenta line here, it looks steeper than this blue line. And what we’ll see is
this notion of steepness, how steep a line is, how
quickly does it increase or how quickly does it decrease, is a really useful idea in mathematics. So ideally, we’d be
able to assign a number to each of these lines
or to any lines that describes how steep it is, how quickly does it increase or decrease? So what’s a reasonable way to do that? What’s a reasonable way to assign a number to these lines that
describe their steepness? Well one way to think
about it, could say well, how much does a line increase in the vertical direction for a given increase in
the horizontal direction? So let’s write this down. So let’s say if we an increase increase, in vertical, in vertical, for a given increase in horizontal for a given increase a given increase in horizontal. So, how can this give us a value? Well let’s look at that
magenta line again. Now let’s just start at an arbituary point in that magenta line. But I’ll start at a point where it’s going to be easy for me to figure out what point we’re at. So if we were to start right here, and if I were to increase
in the horizontal direction by one. So I move one to the right. To get back on the line, how much do I have to increase in
the vertical direction? Well I have to increase in
the vertical direction by two. By two. So at least for this magenta line, it looks like our increase in vertical is two, whenever we have an increase in one in the horizontal direction. Let’s see, does that
still work if I were to start here, instead of
increasing the horizontal direction by one, if I were increase in the horizontal direction… So let’s increase by three. So now, I’ve gone plus three in the horizontal direction, then to get back on the line, how much do I have to increase
in the vertical direction? I have to increase by one,
two, three, four, five, six I have to increase by six. So plus six. So when I increase by three in the horizontal direction, I increase by six in the vertical. We were just saying,
hey, let’s just measure how much to we increase in vertical for a given increase in the horizontal? Well two over one is just two and that’s the same
thing as six over three. So no matter where I start on this line, no matter where I start on this line, if I take and if I increase
in the horizontal direction by a given amount, I’m going to increase twice as much twice as much in the vertical direction. Twice as much in the vertical direction. So this notion of this
increase in vertical divided by increase in horizontal, this is what mathematicians
use to describe the steepness of lines. And this is called the slope. So this is called the slope of a line. And you’re probably
familiar with the notion of the word slope being
used for a ski slope, and that’s because a ski slope
has a certain inclination. It could have a steep
slope or a shallow slope. So slope is a measure for
how steep something is. And the convention is, is
we measure the increase in vertical for a given
in increase in horizontal. So six two over one is
equal to six over three is equal to two, this
is equal to the slope of this magenta line. So let me write this down. So this slope right over
here, the slope of that line, is going to be equal to two. And one way to interpret that, for whatever amount you increase in the horizontal direction,
you’re going to increase twice as much in the vertical direction. Now what about this blue line here? What would be the slope of the blue line? Well, let me rewrite another
way that you’ll typically see the definition of slope. And this is just the
convention that mathematicians have defined for slope but it’s a valuable one. What is are is our change in vertical for a given change in horizontal? And I’ll introduce a new notation for you. So, change in vertical, and in this coordinate, the vertical is our Y coordinate. divided by our change in horizontal. And X is our horizontal coordinate in this coordinate plane right over here. So wait, you said change in but then you drew this triangle. Well this is the Greek letter delta. This is the Greek letter delta. And it’s a math symbol used
to represent change in. So that’s delta, delta. And it literally means, change in Y, change in Y, divided by change in X, change in X. So if we want to find the
slope of the blue line, we just have to say, well
how much does Y change for a given change in X? So, the slope of the blue line. So let’s see, let me do it this way. Let’s just start at some point here. And let’s say my X changes by two so my delta X is equal to positive two. What’s my delta Y going to be? What’s going to be my change in Y? Well, if I go by the right by two, to get back on the line, I’ll have to increase my Y by two. So my change in Y is also
going to be plus two. So the slope of this blue line, the slope of the blue line, which is change in Y over change in X. We just saw that when our
change in X is positive two, our change in Y is also positive two. So our slope is two divided by two, which is equal to one. Which tells us however much we increase in X, we’re going to increase
the same amount in Y. We see that, we increase
one in X, we increase one in Y. Increase one in X, increase one in Y.>From any point on the line,
that’s going to be true. You increase three in X, you’re going to increase three in Y. It’s actually true the other way. If you decrease one in X, you’re going to decrease one in Y. If you decrease two in X, you’re going to decrease two in Y. And that makes sense from
the math of it as well Because if you’re change
in X is negative two, that’s what we did right over here, our change is X is negative two, we went two back, then your change in Y is going
to be negative two as well. Your change in Y is
going to be negative two, and negative two divided by negative two, is positive one, which

56 thoughts on “Introduction to slope | Algebra I | Khan Academy”

• April 21, 2015 at 8:34 pm

This is great you have several series on integral and differential calculus and now you post a video introducing slope :L

• April 21, 2015 at 8:55 pm

Could have put this up a few weeks ago when we were doing slope in class…

• April 21, 2015 at 9:00 pm

An easier equation for the gradient is RISE/RUN for the graph.try it out,it's really helpful.

• April 21, 2015 at 9:09 pm

Oh! This video is really on time for me. Its our topic for maths

• April 21, 2015 at 9:15 pm

My teacher really likes your channel so keep up the work because this helps me.

• April 21, 2015 at 9:25 pm

The other week my teacher explained this to us it took her 2 classes

• April 21, 2015 at 11:26 pm

Ski slopes are always negative 😉 otherwise you're not skiing lol the thing you ride on is where you experience the positive slope.

• April 22, 2015 at 12:57 am

Great Work !!! I really like your method of teaching !!!

• April 22, 2015 at 1:24 am

Yay i understand it now! Our class finally is learning it and i know how to do it! Thx in the GEM program we are doing this in the 7th grade book i am in 6th grade

• April 24, 2015 at 8:57 pm

i love ur videos! pls keep posting them. im actually doing this stuff right now!!!!

• October 1, 2015 at 1:13 am

wasn't helpful enough for what my school was teaching

• May 15, 2016 at 4:39 pm

Khan Academy is great. It makes stuff I have no idea about in my Algebra class be taught in a more simple way.

• August 4, 2016 at 12:08 am

Whio would

• November 20, 2016 at 8:36 pm

Thanks so much you help me to understand well

• February 9, 2017 at 12:08 am

Thank you sooo much!!!! I didn't understand this in math class and now everything my teacher said finally makes sense!!! You are a true life saver. You just literally turned my c into an A! 🙂

• March 23, 2017 at 4:58 pm

great lesson.. I have an eighth grader struggling in math.. I pray watching your videos will bring her up to speed..the kids laugh at her when she gets something wrong and she gets turned off from school really.. I try to tell her everyone makes mistakes sometime.. I tell her I'm happy when you at least try..thanks again for your video

• March 26, 2017 at 12:15 pm

thanks man i learned every thing

• April 3, 2017 at 2:49 am

This is awesome!!! Now I get it!!! Really wonderful way to explain this!!!

• June 7, 2017 at 1:29 am

How come when my teacher explains it I don't get it, but when you explain it makes much more sense! Thank you Khan Academy!!!!!

• September 8, 2017 at 2:50 pm

You did a decent job explaining the concepts in simple terms, change in y over change in x (rise over run). But it would have made the lesson whole if you would have presented the equation in slope- intercept form for each line, plugged the numbers in from the each of the lines and solved the equation in slope- intercept for both lines (showing the difference of the two slopes). That way we can see how the numbers on the line are represented in the equation and the slope of each line. Thank you for uploading and showing the basics

• September 13, 2017 at 12:20 am

Thanks! I understand now

• September 25, 2017 at 2:18 am

This didn't help. How do I find the line I'm supposed to draw? I get rise over run and the fractions. But how exactly do I draw a line from an equation?

• September 26, 2017 at 1:33 pm

What type of uses does this have in life?

• November 1, 2017 at 12:49 am

Thank you good sir.

• November 2, 2017 at 10:42 am

Please put this into "Algebra" playlist at appropriate index.

• November 20, 2017 at 5:51 pm

i will starg to pass math again thznks

• December 4, 2017 at 1:13 am

I like that there's a link to extra practice after watching the video. It solidifies concepts that were covered in the video

• December 4, 2017 at 5:50 pm

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• December 4, 2017 at 10:18 pm

Thank you so much khan academy you helped me so much so far my math teacher shows us many things like we already know it and doesn't explain his work you have saved my grade thanks

• December 14, 2017 at 11:58 am

I thought it was y2 – y1 / x2 – x1

• December 15, 2017 at 1:37 am

??????????????????????????????? i don't understand

• February 3, 2018 at 1:45 pm

If the definition of slope that told us that " Slope is the numerical measure of steepness" So can we change "m = y2-y1/x2-x1 to m= x2-x1/y2-y1"?

• March 9, 2018 at 2:43 am

nicer expalanation

• April 12, 2018 at 5:59 am

Thank you so much

• May 16, 2018 at 7:15 pm

I'm in algebra 1 and I was in Math 1 in 5th grade for real

• May 19, 2018 at 6:25 pm

great thanks know i understand my homework

• May 22, 2018 at 2:23 am

T pose Fellas

• May 27, 2018 at 6:54 pm

can't thank enough. i'm just soooooooooooooooooo grateful!

• June 3, 2018 at 5:55 pm

thanks for the lesson …. that was really helpful

• September 18, 2018 at 1:12 pm

this helps me understand more

• September 29, 2018 at 3:34 pm

This guys a Genius
Thanks

• October 16, 2018 at 12:27 am

I missed a few days of school due to sickness and came back to algebra class and it felt as if they were speaking a different language. This is so helpful thank you!

• October 17, 2018 at 8:13 pm

I am still confused. Can you be more clear?

• November 20, 2018 at 6:02 am

Great session

• November 28, 2018 at 4:17 pm

heh 600 like

• January 11, 2019 at 8:13 am

Thanks. That helped

• January 15, 2019 at 3:25 pm

Hey! What are you doing here? Do your work

• January 25, 2019 at 1:14 am

I thought it was y=mx+b

• April 10, 2019 at 5:40 pm

Amazing! Thank you!

• May 13, 2019 at 11:31 am

I still don't understand what is this

• June 16, 2019 at 12:51 am

I know this is an old video to comment but I really want somebody to explain to me and answer as to how steepness as Sal puts in the video be defined as "how quickly a line increases or decreases"?? I do understand that slope is the magnitude of how much a line inclines or declines from the horizontal axis or the ratio of change in vertical axis witg respect to change in horizontal axis but the definition mentioned above as by Sal here confuses me. I fail to conceptualise and understand it..can't make sense mathematically. I'd really appreciate and be thankul if someone could /would offer some clarity and explain in mathematical context!:)

• July 12, 2019 at 5:29 pm

What device do you use to write?

• November 1, 2019 at 2:33 pm

Best explanation I heared!

• December 3, 2019 at 3:28 am

How do u keep making everything so clear if you were a teacher everyone would get a hundred