Mentor: Please, draw a straight horizontal line at the center of your graphing paper. As we count:
"Zero, one, two, three.." we put the numbers on the line, one number per line of the graph
paper. When we count backwards, we distinguish the numbers that come before zero by placing a
"-" sign in front of them, so it goes: "Two, one, zero..." Make sure that you evenly space the
numbers, since the distance from 1 to 2 should be the same as the distance from 2 to 3.

Student: Minus one, minus two, minus three...

Mentor: What we have now is called a "number line" or "coordinate line." It can be used to describe
where a point is on the line. To give the exact "address" of a point, we just look at how far
the point is from zero, using a minus symbol for numbers to the left of zero. Except we don't
call it a minus sign, we refer to these numbers as "negative."

Student: So the address of this point (Student highlights 4) is 4, and the address of this point
(Student highlights -5) is
negative 5.

Mentor: Excellent. Now we want to get more freedom of movement. We will let our points be anywhere on
the paper, not only on the line. To give an address for the points that are not on the number
line we will need to make a vertical number line. Draw a vertical line through the zero of the
horizontal number line. Now label it with positive numbers above the horizontal number line
and the negative numbers below the horizontal number line. Instead of saying horizontal number
line and vertical number line all the time let's call them by their mathematical names. The
horizontal number line is called the x-axis and the vertical number line is y-axis.

Student: Well, I would go up three blocks and then right two blocks.

Mentor: Sure. How else can we get there?

Student: We can first go two blocks to the right, and then three blocks up.

Student: Or we can go one right, three up, and one more right.

Mentor: There are many ways to get from one point to another point (how many ways, by the way?). To
create a standard way of referring to points, mathematicians came to an agreement that they
will always name the point after one special way of walking. Starting from zero, we go all the
way to the right or to the left, counting steps: one, two. Then we go up or down: one, two
three steps up. Then we write the number of steps like that: (2,3). Again, the first number is
"left-right," the second "up-down." A negative sign means either left or down. So, if our
point is (-2, -3), we go two steps to the left, and then three steps down. Do you remember the
names of our number lines?

Student: Yes, the horizontal line is called the x-axis and the vertical line is called the y-axis.

Mentor: Can any one think of a better way to describe the address of a point instead of (left-right,
up-down)?

Student: Could we call the address by the names of the lines?

Mentor: Yes, so the address of a point would be described as (x,y) instead of (left-right, up-down).
The mathematical term for the address of a point is called
coordinates. Now does everyone see how the x-axis and y-axis divide our paper into four sections?

Student: Yes and I bet they have names too!

Mentor: You are right! These sections are called quadrants.

Student: Are they called quadrants because there are four of them and there are four sides to a
quadralateral?

Mentor: Good observation! We get our prefix "quad" from the Latin word "quattuor" which means four.
Each of these quadrants are referred to by a roman numeral.

The first quadrant contains all the points with positive x and positive y coordinates and is represented by
the roman numeral I.

The second quadrant contains all the points with negative x and positive y coordinates and is represented by
the roman numeral II.

The third quadrant contains all the points with negative x and negative y coordinates and is represented by
the roman numeral III.

The fourth quadrant contains all the points with positive x and negative y coordinates and is represented by
the roman numeral IV.

Student: Will any point that has an address of (positive, negative) be in the fourth quadrant?

Mentor: Yes, but let's use the correct vocabulary. Any point that has coordinates of (positive,
negative) is in the fourth quadrant.