This page starts from the beginning. It assumes you know what a negative number is, and nothing else about graphs. Each part uses only what the parts before it have explained, so it is best read in order.
Keep paper and a pencil beside you. Graph paper helps, but any paper will do: you can draw your own grid. When a worked example asks a question, try it yourself before you read the answer.
It is a long page. Two or three sittings is a good pace. There are places to practice along the way, and a check at the end with the reasoning for every answer.
Where we start: the number line
A number line is a straight line with the numbers marked on it in order, evenly spaced. You have seen one on a ruler, and on a thermometer.
In the middle of the number line is zero. To the right of zero are the positive numbers: 1, 2, 3, and so on, getting bigger as you go. To the left of zero are the negative numbers: −1, −2, −3, and so on. The minus sign in front of a negative number tells you it is on the left side of zero. The farther left a number is, the smaller it is.
The arrows at the two ends mean that the line does not stop. It keeps going in both directions without end. We can only ever draw a piece of it.
Every number has one place on the number line, and every place on the line has one number. So one number is enough to say where a point is on a line. Point A in the picture is at −3: start at zero and go 3 steps to the left. Point B is at 4: start at zero and go 4 steps to the right.
How far apart are A and B?
Count the steps from A to B: from −3 to 0 is 3 steps, and from 0 to 4 is 4 more steps. That is 7 steps in all.
You can also subtract: 4 − (−3) = 4 + 3 = 7. Subtracting a negative number is the same as adding.
A distance is always counted as a positive number, the same way a trip of 7 blocks is 7 blocks whichever way you walk.
A number line does not have to lie flat. A thermometer is a number line standing up. Zero is in the middle, the positive temperatures are above it, and the negative temperatures are below it.
The buttons in an elevator are another standing number line: the ground floor, the floors above it, and in some buildings the basement levels below it.
Keep both pictures in mind: a number line lying down, with positive to the right, and a number line standing up, with positive going up. The coordinate plane is made of one of each.
A flat surface needs two numbers
A plane is a perfectly flat surface. Think of the top of a table, the floor of a room, or a sheet of paper. In mathematics, a plane has two more features that no real table has: it has no thickness at all, and it goes on forever in every direction, left and right, up and down, never ending. As with the number line, we can only ever draw a piece of it, and the rest is understood to be there.
On a line, one number was enough to say where a point is. On a plane it is not. Suppose someone tells you, “The keys are 3 steps from the door.” On a narrow hallway, that is enough. In the middle of a large room, it is not: 3 steps which way? To say exactly where something is on a flat surface, you need two numbers: how far across, and how far up or down.
The same idea in Manhattan
Manhattan already works this way. Most of it is laid out as a grid: streets run across the island from east to west, and avenues run along it from south to north.
If someone says, “I’m at 23rd Street and 7th Avenue,” you know exactly which corner they mean. The street number tells you how far uptown they are. The avenue number tells you how far across. Two numbers name one place. One number alone, “I’m on 23rd Street,” would leave you walking along 23rd Street looking for them.
In Manhattan the words “Street” and “Avenue” tell you which number is which, so people can say them in either order. The coordinate plane has no such words, only the two numbers. There, the order is what tells you which number is which. Part 4 comes back to this.
The Manhattan grid is a good picture of the main idea, two numbers for one place, but it is not the same as the grid in mathematics in three ways.
First, Manhattan has no negative streets and no street numbered zero; the grid in mathematics has both. Second, the avenue numbers get bigger as you go west, to the left on a map, while on the mathematics grid numbers get bigger as you go right. Third, the real streets do not run exactly east and west. The grid in mathematics is perfectly even and perfectly square, and it has none of these exceptions.
The two axes
Now we build the grid that mathematics uses. It is called the coordinate plane. It is made from two number lines, one lying down and one standing up.
Step 1: a number line lying down
Draw a number line across the page, with zero in the middle, positive numbers to the right and negative numbers to the left. This line is called the x-axis. (“Axis” means a line that something is measured along. The plural, for two of them, is “axes,” said AK-seez.)
Step 2: a number line standing up
Now draw a second number line straight up and down, so that it crosses the first one exactly at zero. On this line the positive numbers go up and the negative numbers go down, like a thermometer. This line is called the y-axis.
The two axes cross at a right angle. A right angle is a square corner, like the corner of a sheet of paper or the corner where a wall meets the floor. Neither line leans toward the other.
Step 3: the point where they cross
The point where the two axes cross is called the origin. It is zero on the x-axis and zero on the y-axis at the same time. The origin is the starting place: every point on the plane is found by starting at the origin and counting from there.
Finally, lines are drawn through every whole number on both axes, straight across and straight up and down. These are the grid lines. They are there only to help you count; they are not part of any answer.
The arrows at the ends of both axes mean the same thing they meant on the number line: the axes, and the whole plane with them, go on forever.
Words so far
Coordinate plane: the flat surface with the two axes drawn on it, used to name and find points.
x-axis: the number line that lies across, left to right.
y-axis: the number line that stands up, bottom to top.
Origin: the point where the two axes cross; zero on both.
Naming a point: the ordered pair
A point on the coordinate plane is named with two numbers, written inside parentheses with a comma between them, like this: (4, 3).
The first number says how far to go across, left or right, along the x-axis. The second number says how far to go up or down, along the y-axis. Both are counted from the origin.
The two numbers are called the point’s coordinates. The first is the x-coordinate and the second is the y-coordinate. Because the order of the two numbers matters, the pair is called an ordered pair. In general it is written (x, y).
x comes before y in the alphabet. In the same way, the across number comes before the up-and-down number.
Why the order matters
The points (2, 4) and (4, 2) use the same two numbers, but they are different points.
For (2, 4), go 2 across to the right, then 4 up.
For (4, 2), go 4 across to the right, then 2 up.
They land in different places, as the picture shows. Mixing up the order is the most common mistake on the whole topic, so it is worth checking every time.
Plotting a point, step by step
To plot a point means to find its place on the coordinate plane and mark it with a dot. There are three steps, and they are the same every time.
- Start at the origin. Put your pencil on (0, 0).
- Read the first number, and move across. If it is positive, move that many units to the right. If it is negative, move that many units to the left. If it is zero, do not move across at all.
- Read the second number, and move up or down. If it is positive, move that many units up. If it is negative, move down. If it is zero, do not move up or down. Then make the dot.
Plot the point (4, 3).
Start at the origin. The first number is 4, which is positive: move 4 units to the right.
The second number is 3, which is positive: from there, move 3 units up.
Make the dot. In the picture the path is drawn with arrows: first across, then up.
Plot the point (−3, 2).
Start at the origin. The first number is −3, which is negative: move 3 units to the left.
The second number is 2, which is positive: move 2 units up.
The minus sign changed the direction, left instead of right. It did not change the order: across first, then up.
Plot the point (−2, −4).
The first number is −2: move 2 units to the left.
The second number is −4: move 4 units down.
Both numbers are negative, so the point is below and to the left of the origin.
Plot the point (5, −1).
The first number is 5: move 5 units to the right.
The second number is −1: move 1 unit down.
When one of the numbers is zero
A zero means “do not move in that direction.”
For (0, 4): do not move across at all. Move 4 units up. The point sits right on the y-axis.
For (−3, 0): move 3 units to the left, and do not move up or down. The point sits right on the x-axis.
For (0, 0): do not move at all. That point is the origin itself.
So a point with a zero first number always lies on the y-axis, and a point with a zero second number always lies on the x-axis.
Try it: tap the grid
Tap anywhere on this grid. It will draw the path from the origin to the point you tapped, and tell you the point’s name and how to get there.
Practice: plot these points
Plot each point in turn. Tap where you think it goes. If the dot lands in the wrong place, the page tells you which point you plotted instead, so you can see what went wrong and try again.
Reading a point off the grid
Reading a point is plotting in reverse. You see a dot and you name it.
- Find the first number. From the dot, look straight up or straight down to the x-axis. The number there is the x-coordinate. If the dot is to the left of the y-axis, that number is negative.
- Find the second number. From the dot, look straight across to the y-axis. The number there is the y-coordinate. If the dot is below the x-axis, that number is negative.
- Write them in order, across first, inside parentheses: (x, y).
Name the points P, Q and R.
P: straight down from P, the x-axis reads 3. Straight across from P, the y-axis reads 4. P is (3, 4).
Q: straight up from Q, the x-axis reads −5. Straight across, the y-axis reads −2. Q is (−5, −2).
R: R is on the y-axis itself, so there is no distance across: the x-coordinate is 0. It is 5 below the origin. R is (0, −5).
Practice: name these points
Type the coordinates of each point, then press Check. Type a minus sign with the minus key on your keyboard.
The four quadrants
The two axes cut the plane into four regions. Each region is called a quadrant, from a Latin word for a quarter: each is one quarter of the plane.
The quadrants are numbered with Roman numerals, the numbering system of ancient Rome, which you may have seen on clocks and in movie credits: I is 1, II is 2, III is 3, and IV is 4.
The numbering begins in the upper right and goes counterclockwise, which means the opposite way to the hands of a clock. Start in the upper right (I), move up and over to the upper left (II), down to the lower left (III), and across to the lower right (IV).
| Quadrant | Where it is | x is | y is | Example |
|---|---|---|---|---|
| I | upper right | positive | positive | (3, 2) |
| II | upper left | negative | positive | (−3, 2) |
| III | lower left | negative | negative | (−3, −2) |
| IV | lower right | positive | negative | (3, −2) |
You do not need to memorize the table. It follows from how points are plotted: a negative first number sends you left, and a negative second number sends you down.
A point that lies on one of the axes, such as (0, 4) or (−3, 0), is on the border between quadrants, and it belongs to no quadrant. The origin belongs to none either.
Practice: which quadrant?
For each point, choose its quadrant, or “on an axis.” You can work it out from the signs, without drawing.
Distance along a grid line
The distance between two points is how far apart they are, measured in units of the grid. When two points are lined up, straight across from each other or straight up and down from each other, the distance between them is easy to find.
Points straight across from each other
Points that are straight across from each other have the same second number, the same y. Only the first numbers are different.
Take (−3, 2) and (4, 2). Count the units from one to the other along the grid line: from −3 to 0 is 3 units, and from 0 to 4 is 4 more. The distance is 7.
Or subtract the first numbers, the larger minus the smaller: 4 − (−3) = 4 + 3 = 7. This is the same subtraction we did on the number line in Part 1, because it is the same thing: the two points lie on a line.
Points straight up and down from each other
Points that are straight up and down from each other have the same first number, the same x. Only the second numbers are different.
Take (2, 3) and (2, −4). Subtract the second numbers: 3 − (−4) = 3 + 4 = 7. Counting on the grid gives the same: 3 units down to the x-axis, then 4 more.
Shapes on the grid
A rectangle has corners at (−4, 3), (2, 3), (2, −1) and (−4, −1). How long are its sides? What are its perimeter and its area?
The top side runs from (−4, 3) to (2, 3). Same y, so subtract the x values: 2 − (−4) = 6. The rectangle is 6 units wide.
The right side runs from (2, 3) to (2, −1). Same x, so subtract the y values: 3 − (−1) = 4. The rectangle is 4 units tall.
The perimeter is the distance all the way around: 6 + 4 + 6 + 4 = 20 units. The area is the number of unit squares inside: 6 × 4 = 24 square units. You can count the squares in the picture to check.
Three corners of a rectangle are (1, 5), (6, 5) and (6, 2). Where is the fourth corner?
The sides of a rectangle on the grid run straight across and straight up and down. The fourth corner must be straight below (1, 5), so its first number is 1, and straight across from (6, 2), so its second number is 2.
The fourth corner is (1, 2).
Distance when points are not lined up
Most pairs of points are not straight across or straight up and down from each other. The straight line between them is slanted. To find its length, we make it the long side of a right triangle.
Take the points (1, 1) and (4, 5). From (1, 1), draw a line straight across until you are under (4, 5), and then straight up to it. The two lines you drew make a square corner, so together with the slanted line they form a right triangle: a triangle with one right angle.
The across side is 4 − 1 = 3 units. The up side is 5 − 1 = 4 units. These two sides are called the legs. The slanted side, across from the square corner, is called the hypotenuse (hy-POT-en-oos). It is always the longest side.
The Pythagorean theorem connects the three sides. If the legs are a and b and the hypotenuse is c, then a2 + b2 = c2. The test gives you this on the formula sheet.
How far is it from (1, 1) to (4, 5)?
The legs are 3 and 4. So 32 + 42 = c2, which is 9 + 16 = 25.
c2 = 25, so c is the number that times itself makes 25. That number is 5. The distance is 5 units.
Check that the answer makes sense: the slanted distance should be longer than either leg, 3 or 4, but shorter than the two legs added, 7. 5 fits.
You will not be told to draw the triangle. When a question asks for the distance between two points that are not lined up, draw it yourself on your whiteboard: across, then up, then the Pythagorean theorem. The page The Third Side has practice with the theorem itself.
The midpoint
A line segment is the piece of a straight line between two points; the two points are its endpoints. The midpoint of a line segment is the point exactly halfway between its endpoints.
Start on the number line. The point halfway between 2 and 8 is 5: it is 3 units from each. You can find it by adding the two numbers and dividing by 2: (2 + 8) ÷ 2 = 5. Adding and dividing by 2 is finding the average of the two numbers.
On the coordinate plane, the midpoint is halfway across and halfway up at the same time. So you find the average twice: once for the first numbers and once for the second numbers.
What is the midpoint of the segment from (−4, 1) to (2, 5)?
Average the first numbers: (−4 + 2) ÷ 2 = −2 ÷ 2 = −1.
Average the second numbers: (1 + 5) ÷ 2 = 6 ÷ 2 = 3.
The midpoint is (−1, 3). In the picture it sits halfway along the segment. You can check by counting: from each end it is 3 units across and 2 units up or down.
From points to a straight line
So far we have plotted one point at a time. The rest of coordinate geometry is about what happens when many points follow the same rule.
Here is a rule: the second number is always 2 more than the first. Some pairs that follow it are (0, 2), (1, 3) and (−3, −1). A table is a tidy way to list several of them: pick first numbers, and work out each second number from the rule.
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y = x + 2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 |
Now plot every pair in the table. Something happens that you could not have seen from the table alone: the points all lie on one straight line.
Draw the line through them, and extend it in both directions with arrows. Every point on this line follows the rule, including points between the grid marks, such as (0.5, 2.5). And every pair that follows the rule is a point on this line. The line is a picture of the rule.
In algebra the rule is written as an equation: y = x + 2. Here x stands for the first number of a point and y for the second. The equation and the line say the same thing in two ways.
Different rules give different lines. The rule “the second number is twice the first,” written y = 2x, gives a line that climbs more steeply. The rule y = −x gives a line that goes down as you move to the right. How steeply a line climbs or falls is called its slope, and it is where the next page begins.
Lines on the Grid goes on from here: slope, the places where a line crosses the axes, the equation of a line in its three forms, and two lines at once. Its first part is a short review of this page, so you can go straight to its Part 2.
The words on this page, in one place
Number line: a straight line with the numbers marked on it in order, evenly spaced, zero in the middle.
Plane: a perfectly flat surface with no thickness that goes on forever in every direction.
Coordinate plane: a plane with two number lines drawn on it, crossing at zero, used to name and find points.
x-axis: the number line that runs across; positive to the right.
y-axis: the number line that runs up and down; positive going up.
Axes: the plural of axis; both lines together.
Right angle: a square corner.
Origin: the point where the axes cross, (0, 0).
Ordered pair: two numbers in parentheses, (x, y), naming one point; the order matters.
Coordinates: the two numbers of an ordered pair. The x-coordinate comes first, the y-coordinate second.
Plot: to find a point’s place on the plane and mark it.
Quadrant: one of the four regions the axes make, numbered I to IV counterclockwise from the upper right.
Distance: how far apart two points are, always positive.
Right triangle, legs, hypotenuse: a triangle with a square corner; the two sides that make the corner; the long side across from it.
Line segment, endpoints, midpoint: the piece of a line between two points; those two points; the point halfway between them.
Slope: how steeply a line climbs or falls. The next page explains it.
Twelve questions on the whole page
Answer each one, then press Check. Each answer comes with its reasoning. For a point, type both numbers, like (−3, 4), or just −3, 4.
- 1.
What are the coordinates of the origin?
The origin is where the two axes cross, at zero on both. Its coordinates are (0, 0).
- 2.
Starting at the origin, you go 5 units to the left and then 2 units up. What point are you at?
Left makes the first number negative: −5. Up makes the second number positive: 2. The point is (−5, 2).
- 3.
To plot the point (−1, −6), which way do you move first, and how far?
The first number, −1, tells you how far to go across. It is negative, so you move 1 unit to the left. Then you move 6 units down.
- 4.
Are (3, −2) and (−2, 3) the same point?
No. (3, −2) is 3 to the right and 2 down, in quadrant IV. (−2, 3) is 2 to the left and 3 up, in quadrant II. The order of the numbers matters.
- 5.
In which quadrant is the point (−7, −4)?
Both numbers are negative: left and down. That is the lower left, quadrant III.
- 6.
Where is the point (0, −8)?
A first number of 0 means no movement across, so the point is on the y-axis, 8 units below the origin. It is in no quadrant.
- 7.
What is the distance between (−6, 1) and (3, 1)?
The second numbers are the same, so the points are straight across from each other. Subtract the first numbers: 3 − (−6) = 9.
- 8.
What is the distance between (4, 7) and (4, −2)?
The first numbers are the same, so the points are straight up and down from each other. Subtract the second numbers: 7 − (−2) = 9.
- 9.
A rectangle has corners at (1, 1), (6, 1), (6, 4) and (1, 4). What is its area, in square units?
Width: 6 − 1 = 5. Height: 4 − 1 = 3. Area: 5 × 3 = 15 square units.
- 10.
What is the distance between (0, 0) and (6, 8)?
The legs of the right triangle are 6 across and 8 up. 62 + 82 = 36 + 64 = 100, and the number that times itself makes 100 is 10.
- 11.
What is the midpoint of the segment from (2, −3) to (8, 5)?
Average the first numbers: (2 + 8) ÷ 2 = 5. Average the second numbers: (−3 + 5) ÷ 2 = 2 ÷ 2 = 1. The midpoint is (5, 1).
- 12.
The rule is y = x + 2. Which of these points is on its line?
Put the first number in for x and see whether you get the second number. For (3, 5): 3 + 2 = 5. It works. For (5, 3), 5 + 2 is 7, not 3, which shows again that the order matters.