Graphing Points on a Line
Here is a line:
The arrows at each end show that the line really goes on forever.
Each place on the line is called a point. A few of the points on this line are marked
with red dots:
We can number some of the points to make them easier to find. The numbers get
bigger from left to right:
Right now, Sam is sitting on point 4:
On this line, only the even numbers are labeled. The other numbers are marked like
this: I. This mark is called a tick mark.
If Sam wants to find point 5, what should he do?
Sam starts at 0,
and crawls forward. Sam knows that 5 is 1 more than 4, so he counts one
tick mark after 4.
Now Sam is at point 5.
Sam makes a big green dot to show where he has been. He labels the dot
too, so you can tell what it is.
Sam just graphed point 5.
Negative Numbers on a Line
So far, when Sam wanted to graph a point, he started at zero and went forward. What would
happen if he wanted to go the other way? After all, there are lots of points before the one we
labelled zero.
Let's label some more points, going backwards from zero. We'll use the "-" symbol to show that
these numbers are less than zero.
The numbers before zero on the number line are called negative numbers. We read a number like
-4 as "negative four."
Let's ask Sam to graph negative two.
Sam always starts at zero.
Sam knows he has to graph a negative number, so he turns around.
He moves two units away from zero, because negative two is two less than zero.
Finally, Sam marks the point he found with a big green dot.
Sam found -2.
The Plane
Here is a picture of a plane. Two lines are drawn inside the plane. Each of these lines is an axis.
(Together they are called axes.) The axes are like landmarks that we can use to find different
places in the plane.
We can label the axes to make them easier to tell apart. The axis that goes from side to side is the
x-axis, and the axis that goes straight up and down is the y-axis.
Let's zoom in on one corner of the plane. (This corner is called the first quadrant.)
We have marked some of the points on each axis to make them easier to find. The point where
the two axes cross has a special name: it is called the origin.
The gray lines will help us find points. When you make your own graphs, you can use the lines
on your graph paper to help you.
Finding Points in the Plane
We can find every point in the plane using two numbers. These numbers are called coordinates.
We write a point's coordinates inside parentheses, separated by a comma, like this: (5, 6).
Sometimes coordinates written this way are called an ordered pair.
The first number in an ordered pair is called the x-coordinate. The x-coordinate tells
us how far the point is along the x-axis.
The second number is called the y-coordinate. The y-coordinate tells us how far the
point is along the y-axis.
Let's try an example.
A fly is sitting in the plane.
Sam knows that the fly is at point (4, 3). What should he do?
Sam starts at the origin. So far, he has not moved along the x-axis or the y-axis, so he is at point
(0, 0).
Because he wants to find (4, 3), Sam moves four units along the x-axis.
Next, Sam turns around and shoots his tongue three units. Sam's tongue goes straight up, in the
same direction that the y-axis travels.
Sam has found point (4, 3). He eats the fly happily.
Graphing Points in the Plane
You can graph points the same way that Sam found the fly. Let's practice graphing different
points in the plane.
Try the Chameleon Graphing Java Applet
We'll begin by graphing point (0, 0).
Sam starts at the origin and moves 0 units along the x-axis, then 0 units up. He has found (0,0)
without going anywhere!
Sam marks the point with a green dot, and labels it with its coordinates.
Sam has finished graphing point (0, 0).
Next, let's graph point (0, 3).
Sam starts at the origin, just like always. He moves 0 units along the x-axis, because the xcoordinate of the point he is trying to graph is 0.
Sam uses his tongue to move a green dot 3 units straight up.
The final step is labeling the point.
Notice that point (0, 3) is on the y-axis and its x-coordinate is 0. Every point on the y-axis has an
x-coordinate of 0, because you don't need to move sideways to reach these points. Similarly,
every point on the x-axis has a y-coordinate of 0.
Let's end with a more complicated example: graphing point (2, -2).
Sam begins at point (0, 0).
He moves 2 units along the x-axis.
The y-coordinate of the point Sam wants to graph is -2. Because the number is negative, Sam
sticks his tongue down two units. This makes sense, because negative numbers are the opposite
of positive numbers, and down is the opposite of up.
Before he leaves, Sam labels the point he graphed.
Try the Chameleon Graphing Java Applet
Scale
How would you graph the point (60, 70)?
We could start with this graph,
make the x and y axes much longer, and then graph our point. If we tried that, though, the graph
would never fit on this screen.
We could try shrinking the axes, and then graphing the point:
This graph is so small that it is hard to understand.
Instead of trying to mark every whole number on the axes, let's count by tens. When we change
the distance between points on our graph like this, we say that we are changing the scale of the
graph.
Now, let's watch Sam graph the point (60, 70) on this graph. Sam always starts at the origin.
The x-coordinate of the point is 60, so Sam counts to 60 by tens.
Since the point's y-coordinate is 70, Sam must use his tongue to count to 70 by tens, moving
straight up.
Before he leaves, Sam labels the point he graphed.
Estimating Points
Sometimes, the point you want to graph is in between points that are marked on the axes. When
this happens, you must estimate where to put your point.
For example, let's help Sam graph (5, 13) using these axes:
Sam always starts graphing at the origin. (Sam is very small in this picture, so that you can tell
where he is more easily.)
The x-coordinate of the point is 5, so Sam needs to find 5 on the x-axis. 5 is exactly halfway
between 0 and 10, so Sam moves between 0 and 10.
Next, Sam must find the y-coordinate, 13. He knows that 15 is halfway between 10 and 20. 13 is
a little bit less than 15, so Sam tries to put his point a little below the halfway point.
Sam labels the point so we can tell exactly where it is.
When you draw your own graphs, you can pick the scale to make estimating easier. Would (5,
13) have been easier to find if our scale counted by hundreds? What about fives?
Tricky Graphs
Sometimes, graphs can be confusing. For instance, the x-axis of this graph counts by ones, but
the y-axis counts by tens.
Why would somebody want axes with different scales? They might want to graph a bunch of
points with big y-coordinates, but small x-coordinates. The different scales make the graph easier
to read.
Here is a graph of the points (1, 20), (2, 30), and (3, 10):
Changing scales makes the value of a point look bigger or smaller. Suppose that the value of the
y-coordinate stood for your allowance. Then you might want to make the point look very small,
so that you could persuade someone to raise your allowance.
Here is the point (1, 2) graphed using two different scales:
Which scale would you choose for your allowance? What about the distance from your home to
school?
Sometimes, people make graphs that skip several points. Don't let these graphs trick you! Here's
a graph where the y-axis starts at 100:
A fly is sitting at point (2, 102).
Sam wants to eat the fly, so he crawls out to 2 on the x-axis and sticks his tongue straight up.
Unfortunately, his tongue gets tangled in the warp zone before 100. (We told you about this kind
of graph to make it harder for other people to trick you. Don't make graphs that skip points, or
you could get tangled, too.)
Sam will haAxis
An axis is a line that we use to find or draw points and shapes. If we want to talk about more
than one axis, we say "axes."
Here is a picture of two axes in a plane:
Cartesian
A man named René Descartes invented the system that we use to graph points in the plane. (You
can see some pictures of René Descartes at the MacTutor Math History Archive.) The word
Cartesian means "from Descartes," so people often talk about the Cartesian plane when they are
using this system.
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Chameleon
A chameleon is a kind of lizard. Chameleons have long, sticky tongues for catching bugs. They
can change color to warm up or cool off. Chameleons also change color when they are angry or
scared.
Coordinate
A coordinate is a number that tells you where to find a point. Sometimes you need several
coordinates to find exactly where a point is. You can think of a point's coordinates as its address.
Some people talk about "the coordinate plane" to show that you can use coordinates to find every
point in the plane.
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Dimension
An object's dimension tells you how many directions a creature living inside it could move.
Forward and backward count as the same direction.
The word "dimensional" is an adjective that explains how many dimensions something has. For
example, a line is one-dimensional, because a creature that lived on a line could only move
forward or backward. A plane is two-dimensional, because a creature living in a plane could
move forward or backward and up or down.
Our universe is three-dimensional, because we can move forward or backward, up or down, and
side to side. Of course, we can move diagonally or follow a curved path, too. However, these are
really combinations of the three basic directions. You can reach any point just moving front or
back, up or down, and side to side. Try it!
Estimate
Estimate means "make a smart guess." For example, if you saw a bunch of kids in a playground,
you might estimate that there were fifteen children playing. You wouldn't guess 500 children,
unless it was a huge playground, and you wouldn't guess 0 children, because you know there are
some kids there. Whenever you estimate, you should check to see that your answer makes sense.
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Line
In math, a line is always straight. Lines go on forever in both directions.
Here is a picture of a line. A real line is only one-dimensional, but the picture is bigger, so you
can see it more easily.
Negative
A negative number is a number less than 0. We use this symbol for negative numbers: "-". For
example, -10 means negative ten. You can imagine this as a temperature ten degrees below zero.
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Ordered Pair
An ordered pair is a list of two numbers, where the order of the numbers is important. We write
the coordinates of a point as an ordered pair, inside parentheses. Some examples of ordered pairs
are (1, 2) and (6, 77).
Origin
The origin is the point where all of the axes cross. In the plane, the origin has coordinates (0, 0).
In this picture of a plane, the origin has been marked with a red dot:
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Plane
A plane is two-dimensional. It seems flat, like a thin piece of paper. However, unlike pieces of
paper, planes have no edges. A bug could crawl along a plane forever without falling off.
Here is a picture of part of a plane. It has been colored green to make it easier to see.
Point
A point is one place in space. Points are really zero-dimensional. Of course, we can't draw
something that has no dimensions. When we want to show where a point is, we usually draw a
dot, like this: . You can imagine that the real point is in the exact center of the dot.
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Quadrant
The two axes divide the plane into four quarters. These quarters are called quadrants. They are
numbered from one to four, starting in the upper right-hand corner and moving
counterclockwise.
Scale
The distance between marked points on a graph or numberline is called its scale. These two
numberlines use different scales:
A chameleon's skin is covered with flat, hard objects. These are also called scales. The word
scale has several other meanings. How many can you find?
ve to find another fly to eat. Luckily, he is good at finding things.