– [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

is your slope again.

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

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

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

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

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

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

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

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

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

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

wasn't helpful enough for what my school was teaching

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

Whio would

Thanks so much you help me to understand well

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! 🙂

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

thanks man i learned every thing

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

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!!!!!

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

Thanks! I understand now

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?

What type of uses does this have in life?

Thank you good sir.

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

i will starg to pass math again thznks

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

ɪᴍ ɪɴ ᴛʜᴇ ᴍɪᴅᴅʟᴇ ᴏғ ᴄʟᴀss ᴀɴᴅ ɪ ᴅᴏɴᴛ ᴋɴᴏᴡ ᴡʜᴀᴛ ᴀ sʟᴏᴘᴇ ɪs

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

I thought it was y2 – y1 / x2 – x1

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

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"?

nicer expalanation

Thank you so much

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

great thanks know i understand my homework

T pose Fellas

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

thanks for the lesson …. that was really helpful

this helps me understand more

This guys a Genius

Thanks

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!

I am still confused. Can you be more clear?

Great session

heh 600 like

Thanks. That helped

Hey! What are you doing here? Do your work

I thought it was y=mx+b

Amazing! Thank you!

I still don't understand what is this

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!:)

What device do you use to write?

Best explanation I heared!

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

wow this video is better than my teacher’s teachings

I'm going to need to watch this video a couple more times.