IB · Mathematics Applications & Interpretation (AI)
AISL 2.1.1 Gradient of a line
Learn AISL 2.1.1 Gradient of a line in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 2.10—Scaling large numbers, log-log graphs.
Learn AISL 2.1.1 Gradient of a line in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 2.10—Scaling large numbers, log-log graphs.
Learn AISL 2.1.1 Gradient of a line in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 2.10—Scaling large numbers, log-log graphs.
The video lesson focuses on the concept of gradient in mathematics, particularly in the context of straight lines. The gradient, denoted as M, is defined as the change in y over the change in x, represented mathematically as .
Key points include:
Overall, the video provides a comprehensive overview of how to calculate and interpret gradients in various contexts.
00:00This lesson we're going to
00:00look at gradient of a
00:02line. Now, in the formulable
00:03plate, you're given a formula
00:04for the gradient. And these
00:07are the formula for equation
00:09of a straight line. The
00:10M, so you're probably familiar
00:12with the letter M being
00:14used for gradient. So M
00:16in mathematics is the gradient.
00:20And almost always, the formula
00:23y equals Mx plus C,
00:25the M tells you the
00:26
00:28y minus y minus y
00:28minus y minus y minus
00:29y minus y minus y
00:29minus y minus y minus
00:29y minus y minus y
00:29minus y minus y minus
00:29minus y minus y minus
00:30minus y minus y minus
00:30y minus y minus y
00:30minus y minus y minus
00:31y minus y minus y
00:31minus y minus y minus
00:37minus y minus y minus
00:38y minus y minus y
00:38minus y minus y minus
00:38minus y minus y minus
00:38y minus y minus y
00:38minus y minus y minus
00:56minus
00:56The change in y over
00:59the change in x and
01:00that's y2 minus y1. It's
01:02the change in y over
01:03x2 minus x1, the change
01:05in x. So imagine I,
01:06let's just draw, let me
01:09just draw, let me just
01:14draw line here. Now, what
01:16is the gradient of this
01:17function? Well, how much has
01:20it changed going up? And
01:23how much has it changed
01:23going up?
01:24across. That's basically what it's
01:26saying. Well, it's changed two
01:27of these and three of
01:29these. Now, these are just
01:30boxes. Our numbers are, they
01:33could actually be meters or
01:34they could be kilometers that
01:35could be anything. It doesn't
01:36really matter. As long as
01:37it's relative, this is two
01:39and this is three. So
01:41the m, the gradient of
01:42this line, is two over
01:45three. The gradient of, let's
01:50say,
01:52this line, well, how much
01:59of a change in the
02:01Y has there been? Well,
02:02let's count these boxes, one,
02:04two, three, four, five, six,
02:05seven, eight. There's a change
02:07of eight, so M is
02:09equal to eight over two.
02:12However, because it's going down,
02:17it's negative, so it's negative
02:19eight over two, and yes,
02:20So you do want to
02:21simplify that eight to the
02:22value of two is four.
02:23So the gradient of this
02:24is negative four. I'll do
02:27two or three more examples
02:28of this. But let me
02:32just give you an example
02:35of gradient in real life.
02:37I like to use the
02:39cycling analogy, because I like
02:40cycling. If you cycle or
02:43if you watch the Tour
02:44de France, you'll know about,
02:46well, if you do road
02:47cycling,
02:48You'll know about gradients of
02:50hills because if you come
02:52if you're cycling along nice
02:53thing a flat road and
02:55then you come across a
02:5610 % Gradient hill and
03:02then you know that goes
03:03on for maybe I don't
03:04know maybe three or four
03:05or sometimes sometimes up to
03:07up to 20 kilometers I
03:10actually cycled up. There's a
03:12famous mountain in France called
03:13long vontu It's very famous
03:15from the Tour de France
03:16It's like a volcano in
03:21the middle of nowhere and
03:24it has like an average
03:25of 10 % for 20
03:26kilometers. It's really, really, really
03:29difficult to cycle up. I
03:30had to stop twice just
03:34to catch my breath. So
03:35what 10 % means is
03:38when it says 10 %
03:39it actually means for every
03:4210 %
03:44is a tenth, so it's
03:461 over 10. So what
03:48that means is for every,
03:50maybe I should do 11%.
03:54Let's say it's 11%. This
03:56is the same as 11
03:57over 100. That means for
03:59every 100, I go across,
04:02I go up 11. Sorry,
04:04I don't know, it might
04:04be something like that. That
04:06might be a gradient of
04:0711%. So if you have
04:09a gradient of, well, in
04:11cycling, like,
04:1216, 17 % is brutal.
04:15It's like even for professionals
04:16that would be very, very
04:17challenging just to cycle up
04:1820%. I think there's hills
04:21like 20, 20, 25 %
04:24and even the professionals complain
04:25that it shouldn't be in
04:26the race. And you'll rarely
04:27see that in a Tour
04:28de France. Skiing is another
04:30good example. I think if
04:34you have 40 % plus,
04:37so 40 % steepness,
04:40which would be 40 %
04:44is obviously 4 out of
04:4610 or 2 fifths. So
04:49if you have a gradient
04:50of 2 fifths, let me
04:51show you what that looks
04:51like here. That means 5
04:53along, 1, 2, 3, 4,
04:555 and 2 up there.
04:59That is a gradient of
05:0040 % or 2 fifths.
05:02Now if I drew this
05:03on for a number of
05:05kilometers and I turned it
05:08into
05:08a ski slope and ports
05:11no all over it that
05:12would be a black slope
05:13or when I say 40
05:15% plus like anything more
05:16than 40 % is considered
05:17a black slope. I think
05:19I think 25 to 40
05:23% is a blue slope.
05:25So if you've got like
05:26a 25 % is a
05:28quarter 25 % is a
05:30quarter. So if you've agreed
05:31in one over four, which
05:33would look like this for
05:35and one one over four
05:36that
05:36would be considered a blue
05:40slope. So I think it
05:41is interesting to think about
05:43where we see a gradient
05:45in real life and what
05:46they actually mean. But in
05:48mathematics, when you see gradients,
05:50this is what they're talking
05:50about. It's the rise over
05:53the run. If they're going
05:55up like this, it's positive,
05:56and if they're going down,
05:57it's negative. It's simple as
05:59that. Let me just do,
06:00let me do one more
06:01of these. This would be,
06:04This would have a gradient
06:06of 1 fourth. So here
06:08m equals 1 fourth. Interestingly,
06:12what would a gradient, what
06:14would a flat, to say
06:16you're cycling along a flat
06:18road, are just a straight
06:19line horizontal line. What gradient
06:21does it have? Well, it's
06:23rise over run. How much
06:24does it rise? Well, it
06:26doesn't rise at all. So
06:27the rise is zero. So
06:28it doesn't matter what the
06:29run is, the gradient. This
06:30is a gradient of zero.
06:33If you really want to
06:33get complicated, let's look at
06:36a vertical line. A vertical
06:38line, what gradient does this
06:40have? Well, this guy's rise
06:43is 4, but it's run
06:44is 0. So here we're
06:46left with 4 over 0.
06:50What happens when you divide
06:51by 0? Well, you can't
06:54do that. So we actually
06:55say, let me get rid
06:57of that, undefined.
07:01So gradient, a vertical line
07:03has an undefined gradient. Okay,
07:07hopefully that clears up what
07:10gradient is. I just want
07:11to apply this formula to
07:13one example to show you
07:14how it works. So I
07:15have the formula nice in
07:17the formula booklet, y2 minus
07:18y1 over x2 minus x1.
07:22I've written x1, y1, x2,
07:24y2 to make it clear
07:25for me. So y2 is
07:272, 2.
07:29minus be careful, it's minus
07:31negative two, so that is
07:32obviously the place people are
07:34gonna make mistakes. Two minus
07:36negative two, which is gonna
07:37become a plus. X two
07:38is five minus negative three,
07:43very, very careful. That's two
07:44plus two over five plus
07:48three equals four over eight
07:52equals a half simplified. So
07:54if the gradient of this
07:55line is a half, and
07:56let me just
07:57to show you if I
07:58was to draw negative three,
07:59negative two here, to five,
08:03two there. Does this have
08:07a gradient of a half?
08:08Well, yes, it does, because
08:10if I go along two
08:12and up one there are
08:15along two and up one.
08:17Or you can go along
08:18eight and up four. It
08:21doesn't make a difference. It's
08:22still going to give you,
08:23this is four rate, this
08:25is one
08:25over two same thing. Okay,
08:27that's the lesson I'm gradient.
08:30Hopefully everything is clear and
08:33understood.