Educational videos for Students in k-12 | Lumos Learning

Educational videos for Students in k-12




Transcript
00:03 Uh huh . Hi , I'm rob . Welcome Math
00:07 antics in this lesson . We're going to learn something
00:09 that's an important foundation for tons of math problems including
00:13 those you'll encounter . We're learning basic algebra . We're
00:17 going to learn about graphing which basically means taking mathematical
00:20 relationships and turning them into pictures . Hey friends ,
00:24 Welcome back to the joy of graphic . We're going
00:26 to pick up right where we left off . We
00:28 already have this nice beautiful function right here . But
00:32 it needs a friend and we're going to do that
00:34 by adding some points . So let's put the next
00:37 point right here . Now all we need to do
00:40 is connect those points because they're all friends and what
00:43 do friends do they stay connected ? Oh and look
00:47 at that , That's beautiful . Well not the kind
00:52 of picture that you'd hang on your wall graphing just
00:55 means making a visual representation of an equation or data
00:59 set so you can understand it better . It's a
01:01 way of helping you literally see how math works when
01:05 math is just a bunch of numbers and symbols on
01:07 a page . It can be pretty abstract and hard
01:09 to relate to . But graphing is like a window
01:12 into the abstract world of math that helps us see
01:15 it more clearly . In fact the focus of our
01:17 lesson today actually looks a bit like a window and
01:20 it's called the coordinate plain . The coordinate plain is
01:23 the platform or stage that are graphing will take place
01:26 on . But to understand how it works , we
01:29 first need to start with its closest relative the number
01:31 line . You remember how a number line works right
01:35 ? A number line starts at zero and represents positive
01:37 numbers as you move to the right and negative numbers
01:40 as you move to the left and there's usually marks
01:43 showing where each imager is along the way . Now
01:45 imagine cloning that number line and rotating the copy counterclockwise
01:50 by 90° so that the second number line is perpendicular
01:53 to the first and they intersect at their zero points
01:56 . What we have now is a number of plain
01:58 . It's basically like a two dimensional version of a
02:01 number line . But that second dimension makes it much
02:04 more useful . With a simple one dimensional number line
02:07 , we could show where various numbers were located along
02:10 that line by drawing or plotting points . But no
02:13 matter how many points we plot , they're always on
02:16 the same line . But with the two dimensional number
02:19 plane , we can plot points anywhere in that to
02:22 the area and that opens up a whole new world
02:24 of possibilities . With the one dimensional number , line
02:28 plotting points was easy . You just needed one number
02:30 to tell you where to plot a point . But
02:32 with the two dimensional number plane , you actually need
02:35 to numbers to plot each point . These two numbers
02:38 are called coordinates because they're the same rank or order
02:41 and they work together to specify the location of a
02:44 point on the number plane . In fact , that's
02:47 why the number plane is often referred to as the
02:50 coordinate plain . It's the stage for plotting coordinates ,
02:54 coordinates . Use a special format to help you recognize
02:56 them . The two numbers are put inside parentheses with
02:59 a comma between them as a separator . So when
03:01 you see two comma five or negative seven comma three
03:05 or zero comma 1.5 . You know you're dealing with
03:08 coordinates . Okay to understand how coordinates work , Remember
03:12 that Our number of plane is formed by combining two
03:14 perpendicular number lines from now on , we're going to
03:17 refer to each one of these number lines as an
03:19 axis . One of the axis is horizontal , like
03:22 the horizon , which means the other axis is vertical
03:25 or straight up and down . And they're often called
03:28 the horizontal and vertical axes because of that . But
03:31 even more often , the axes are referred to by
03:33 variable based names . The horizontal axis is called the
03:37 X axis and the vertical axis is called the Y
03:41 axis . Why use variable names ? Well , there's
03:44 two good reasons . The first is that variable names
03:47 are more flexible than horizontal and vertical , which relate
03:50 to specific orientations in space . That may not always
03:54 be relevant . And the second reason is that each
03:57 of the two coordinate numbers is actually a variable that
04:00 relates to a specific position along one of the two
04:03 axes of the coordinate plain . And since those variables
04:06 are usually called X and Y , it makes sense
04:09 to name the two axes . The same way the
04:12 first coordinate number listed will be called X . And
04:14 the second coordinate number listed will be called Why ?
04:17 And we're always going to list the numbers in that
04:18 same order X first and then Why ? So that
04:21 we never get confused about which is which in fact
04:25 coordinates are often called ordered pairs . Because they're a
04:28 pair of numbers that are always listed in the same
04:30 order X . Value first . Why value second ?
04:33 So if you have the coordinates three comma five ,
04:36 that means X equals three , and Y equals five
04:40 . Pretty easy . Right . But now how do
04:42 we actually plot these coordinates or ordered pairs on the
04:46 coordinate plain ? Well , the first number in the
04:48 ordered pair tells you where along the X axis the
04:51 point is located . And the second number in the
04:54 ordered pair tells you where along the Y axis the
04:57 point is located . The two numbers in an ordered
04:59 pair worked together to define a single point , and
05:03 each one of the numbers only gives you half of
05:05 the information about where that point is . To see
05:07 how this works . Let's plot the coordinates 3:02 .
05:11 First we locate the X . Value along the X
05:13 . Axis , which is at three in this case
05:15 . But instead of putting a point there , we
05:17 draw or just imagine a line perpendicular to the X
05:21 axis that goes through the three . We do that
05:24 because the first number in the ordered pair only tells
05:26 us where along the X axis the point is but
05:29 it could be anywhere along the y axis . We
05:31 won't know that until we plot the second number .
05:34 So temporarily we just draw a line there to represent
05:37 every possible point that could have an X . Value
05:39 of three . Next we locate the y value along
05:42 the Y axis , which is at two in this
05:44 case . But again , instead of putting a point
05:47 there , we draw or just imagine a line perpendicular
05:50 to the y axis that goes through the two .
05:53 We do that because the second number in the order
05:55 pair only tells us where along the y axis the
05:57 point is but it could be anywhere along the X
05:59 axis . So we just draw a line there to
06:02 represent every possible point that could have a Y .
06:04 Value of two . Ah But look what we've got
06:07 now , the first line represents all the possible locations
06:11 were X equals three . And the second line represents
06:14 all the possible locations where Y equals two . And
06:17 the exact point where the two lines intersect represents the
06:20 only point in the entire coordinate plane where both X
06:24 equals three and Y equals two . That intersection is
06:28 the location of our point . Pretty cool . Huh
06:32 ? And that's a really good way to understand how
06:34 the coordinate plain works . But I want to show
06:36 you an even easier and more intuitive way to actually
06:39 plot points . This more intuitive way involves starting with
06:42 the point at the origin of the coordinate plain and
06:45 then treating the coordinates like a set of simple instructions
06:48 that tell you how far to move your point in
06:50 the X and Y directions , for example , to
06:53 plot the coordinates three comma to like before we start
06:56 by imagining a point at the origin zero comma zero
07:00 . Then we look at the first number in our
07:02 ordered pair to see how far we need to move
07:04 our point in the X direction . Since X is
07:07 positive three , we move our point a distance of
07:10 three units in the positive X . Direction . And
07:13 then from there , since why is positive two ?
07:16 We move our point a distance of two units in
07:18 the positive Y direction . So that's a pretty easy
07:21 method for plotting points . Let's try it a few
07:24 more times . So you get the idea , let's
07:26 plot the coordinates negative four comma three . Again we
07:30 start by imagining a point at the origin and then
07:32 let the coordinates tell us how far to move it
07:34 along the X and Y axes . Since X is
07:38 negative four , we move the point a distance of
07:40 four units . But this time in the negative X
07:42 direction , which is to the left . And then
07:45 since why is positive three , we move the point
07:48 a distance of three units and the positive Y direction
07:51 . Now let's plot the coordinates negative three comma negative
07:54 three . In this case X and Y are both
07:57 negative . So starting with the point at zero comma
08:00 zero , we move it three units in the negative
08:02 X . Direction and then three units in the negative
08:04 Y . Direction . And last let's plot the coordinates
08:08 four common negative 2.5 . Starting at zero comma zero
08:12 , we move the 00.4 units and the positive X
08:15 . Direction . And then 2.5 units in the negative
08:17 Y . Direction . Okay , so we've plotted four
08:21 ordered pairs . The easy way . And did you
08:23 notice that each of these points is located in a
08:25 different region of the coordinate plain ? These four regions
08:28 are called quadrants and their boundaries are defined by the
08:31 two axes of the coordinate plain . The quadrants are
08:34 named one through four , so we can easily refer
08:37 to them in conversations if we need to quadrant ,
08:39 one is the upper right quadrant and it contains all
08:42 of the points where both the X and Y values
08:46 are positive Quadrant two is the upper left and it
08:50 contains all of the points that have a negative x
08:52 . value and a positive y value . Quadrant three
08:56 is the lower left and it contains all of the
08:58 points that have both a negative x . And a
09:00 negative y value . And quadrant four is the lower
09:04 right . And it contains all of the points that
09:06 have a positive x value and a negative y value
09:09 . Roman numerals are usually used to label the four
09:12 quadrants and they're in that order . Because that's order
09:15 , you would encounter the quadrants if you started with
09:17 a line segment from the origin to the coordinate one
09:20 comma zero and then rotated that line counter clockwise around
09:24 the origin . Doing this sweeps out of shape called
09:27 a unit circle , which is divided into four quadrants
09:30 , just like the coordinate plain . All right .
09:33 So now you know what the coordinate plain is and
09:36 you know how to plot points on it . But
09:38 you might be wondering what has this got to do
09:40 with basic algebra ? Well , algebra involves many different
09:44 types of equations and functions that are a lot easier
09:47 to understand if we graph their solutions on the coordinate
09:50 plain , as you know . The way to really
09:53 get good at Math is to practice what you've learned
09:55 by doing some exercise problems . Thanks for watching Math
09:58 Antics and I'll see you next time . Oh ,
10:03 okay . It's ah exactly what I wanted , learn
10:11 more at Math Antics dot com .
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