Ordered pairs and the first quadrant
One number says where something is on a line. A map or a floor spreads out in two directions. How do two numbers pin down one spot, and why does it matter which number comes first?
Where you are 4 in The coordinate plane
Why this exists
A The number line gives every number a place, and one number is enough to say where something is on a line: the mark at 7. A flat place is different. A museum floor, a school campus or a sheet of paper spreads out in two directions, across and up. Saying 7 no longer tells you where to go: 7 steps which way?
The fix is to use two number lines at once, one running across and one running up, and to name a spot with two numbers: how far across, and how far up. Ruolan’s museum map works this way. So does the campus map at Yamini and Chukwuma’s school. Once every spot has a pair of numbers, you can find a place from its numbers, write down the numbers of a place you see, and work out how far apart two places are.
The two numbers must always be read in the same order, or one pair could send two people to two different places. That is why they are called an ordered pair. This lesson takes three days. Part 1 builds the grid and the pair. Part 2 plots and names points, including points on the axes and grids that count by 2s, 5s or 10s. Part 3 uses grid maps to describe places and find distances, with coordinates that can be halves.
Definition
The coordinate plane here is a flat grid made from two number lines that are perpendicular: they meet at a right angle, at the 0 of each.
- The x-axis is the number line running across, from left to right.
- The y-axis is the number line running up.
- The origin is the point where they meet. It is 0 on both.
An ordered pair (x, y) is two numbers written in a fixed order inside parentheses, with a comma between them. It names exactly one point:
- the first number, the x-coordinate, says how far to move across, to the right of the origin;
- the second number, the y-coordinate, says how far to move up.
The two numbers together are the point’s coordinates. The origin is (0, 0).
To plot a point is to find it from its coordinates and mark it with a dot. To name a point is the reverse: read its coordinates off the grid. A capital letter in front of the pair names the point itself: P(4, 3) is the point P with coordinates (4, 3).
In this lesson both coordinates are always 0 or more, so no point is ever to the left of the origin or below it. Every point is in the part of the plane called the first quadrant.
Intuition
Part 1: two number lines and an ordered pair
Start with a number line lying flat across the page. That is the x-axis. Now stand a second number line straight up from its 0. That is the y-axis. The two zeros are the same point, the origin, in the lower left corner of the grid. Grid lines run through every whole number on each axis, so the page is covered in squares.
To reach P(4, 3), start at the origin, move 4 units right along the x-axis, then 3 units straight up. The dashed path in the figure does exactly that. Another way to see the same point: P is where the vertical grid line through 4 crosses the horizontal grid line through 3. Those two lines cross at only one point, so the pair names only one point.
The order is a rule everyone agrees on: across first, then up, the same order the axes are named, x then y. One way to remember it: you walk along the hall before you ride the elevator up.
This figure shows why the rule matters. A(2, 5) is 2 across and 5 up. B(5, 2) is 5 across and 2 up. They use the same two numbers, but they are different points: A is high and close to the y-axis, and B is low and farther right. So (2, 5) \ne (5, 2). Swapping the numbers gives the same point only when the two numbers are equal, as in (4, 4).
Part 2: plotting and naming points
Plotting goes from numbers to a dot. For (x, y), find x on the x-axis, follow that vertical grid line up until you are level with y on the y-axis, and put a dot there. Naming goes from a dot to numbers. From the dot, look straight down to the x-axis to read x, then straight across to the y-axis to read y.
A coordinate can be 0. The point (5, 0) is 5 across and 0 up, so it never leaves the x-axis. The point (0, 5) is 0 across and 5 up, so it sits on the y-axis. Every point on the x-axis has y-coordinate 0, and every point on the y-axis has x-coordinate 0. The origin is on both axes.
A grid does not have to count by 1s. When the numbers are large, each square can stand for 2, 5, 10 or more. Each axis must still count in equal steps, like any number line, but the two axes may use different steps. On a grid that counts by 5s, the point (15, 10) is 15 \div 5 = 3 squares across and 10 \div 5 = 2 squares up. So read the numbers on the axes before counting squares.
Part 3: grid maps, distances and halves
A map drawn on a grid turns places into ordered pairs. If the library is at (2, 7), anyone with the map can find it. Each unit on the grid stands for some real length, such as 10 meters, and the map says which.
Two points with the same y-coordinate sit on the same horizontal grid line. Going from one to the other along that line, you move only across, so the distance between them is the difference of their x-coordinates. Two points with the same x-coordinate sit on the same vertical grid line, and the distance is the difference of their y-coordinates. This is length on a number line again, because each grid line is marked like a number line.
Coordinates do not have to be whole numbers. The number \displaystyle 3\frac{1}{2} is halfway between 3 and 4 on a number line, so a point with x-coordinate \displaystyle 3\frac{1}{2} is halfway between the vertical grid lines for 3 and 4. A point with two halves, such as \displaystyle \left(3\frac{1}{2}, 6\frac{1}{2}\right), sits in the very middle of a grid square.
Results
One pair, one point. Every ordered pair names exactly one point, and every point of the grid has exactly one ordered pair.
Order matters. (a, b) and (b, a) are the same point only when a = b.
Points on the axes. A point is on the x-axis exactly when its y-coordinate is 0, and on the y-axis exactly when its x-coordinate is 0. The origin (0, 0) is on both.
Scale. On an axis where each square stands for s, a coordinate c is c \div s squares from the origin.
Distance along a grid line. If two points have the same y-coordinate, the distance between them is the larger x-coordinate minus the smaller. If they have the same x-coordinate, it is the larger y-coordinate minus the smaller.
Why it is true
One pair, one point. The pair (x, y) picks out one vertical line, the one through x on the x-axis, and one horizontal line, the one through y on the y-axis. A vertical line and a horizontal line always cross, at exactly one point. Going the other way, from any point there is one way to look straight down to the x-axis and one way to look straight across to the y-axis, so each point gets one x and one y.
Order matters. (a, b) is on the vertical line through a, and (b, a) is on the vertical line through b. If a and b are different numbers, these are two different vertical lines, which never meet, so the points are different. If a = b, the two pairs are written the same way and name the same point.
Points on the axes. A y-coordinate of 0 means 0 units up, so the point stays on the x-axis; and every point of the x-axis is reached by moving across only. The same reasoning works for the y-axis with an x-coordinate of 0.
Scale. If each square stands for s, then n squares stand for n \times s. A coordinate c is reached after n squares exactly when n \times s = c, that is, when n = c \div s.
Distance along a grid line. Two points with the same y-coordinate sit on one horizontal grid line, and that line is marked just like the x-axis. On a number line the distance between two numbers is the larger minus the smaller: from 2 to 8 you count 6 units. A vertical grid line works the same way with the y-coordinates.
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 to 5 with Part 2, and Examples 6 and 7 with Part 3.
Swapping the two numbers
Ruolan writes that the fossil room is at (6, 1). Yamini copies it as (1, 6). Describe the moves from the origin for each pair. Do the two pairs name the same place?
SolutionIn (6, 1) the first number is the x-coordinate, so it is the move across. The second number is the move up. So (6, 1) is 6 units right and 1 unit up: a spot low down and far to the right.
In (1, 6) the moves are 1 unit right and 6 units up: a spot near the y-axis and high up.
The first point is on the vertical grid line through 6, and the second is on the vertical grid line through 1. Different vertical lines never meet, so the pairs name different places. Yamini’s copy would send a visitor to the wrong room. The same two numbers in the other order name another point; that is what ordered means.
Up first, then across
Chukwuma starts at the origin, walks 3 units right and then 7 units up. Write the ordered pair for where he ends. Yamini also starts at the origin, but walks 7 units up first and then 3 units right. Where does she end?
SolutionChukwuma moved 3 across and 7 up. The across number is written first, so he ends at (3, 7).
Yamini moved the same distances in a different order. Going up first, she reaches (0, 7) on the y-axis. Then 3 units right takes her to the point that is 3 across and 7 up, which is again (3, 7). They end at the same point.
So the order in which you walk the two moves does not change where you end, but the order in which you write the two numbers does. “Across first” is a rule about writing: the first number is always the across number, whichever way someone walked. Writing Yamini’s spot as (7, 3) because she went up 7 first would name a different point.
Reading the museum map
Ruolan’s museum map is drawn on a grid. Give the coordinates of the entrance E, the café C, the gift shop G, the dinosaur hall D and the mineral room M. Which of them lie on an axis?
For each point, look straight down to the x-axis for the first number, then straight across to the y-axis for the second.
- D is above 3 and level with 6, so D(3, 6).
- M is above 7 and level with 4, so M(7, 4).
- C is on the y-axis itself, so it is 0 across. It is level with 5, so C(0, 5).
- G is on the x-axis itself, above 5 and 0 up, so G(5, 0).
- E is at the corner where the axes meet, the origin, so E(0, 0).
C is on the y-axis, G on the x-axis, and E is on both. Notice that C(0, 5) and G(5, 0) use the same two numbers and are, once again, two different points, one on each axis.
Plotting on a grid that counts by 5s
On a grid whose axes both count by 5s, plot F(15, 10), H(5, 35) and K(30, 0). How many squares across and up is each point from the origin?
SolutionEach square stands for 5, so a coordinate divided by 5 gives its number of squares.
- F(15, 10): 15 \div 5 = 3 squares across and 10 \div 5 = 2 squares up.
- H(5, 35): 5 \div 5 = 1 square across and 35 \div 5 = 7 squares up.
- K(30, 0): 30 \div 5 = 6 squares across and 0 squares up, so K is on the x-axis.
Check by reading the axis numbers instead of counting: F sits above the mark 15 and level with the mark 10, as it should. Counting 15 squares across would run off this grid, which has only 8; on a scaled grid the axis numbers, not the squares, are the coordinates.
Choosing a scale
Yamini wants to plot (12, 4), (6, 18) and (0, 14) on a grid that has 10 squares along each axis. Should each square stand for 1, 2, 5 or 10? Use the same scale on both axes.
SolutionTwo things must work. The largest coordinate must fit on the grid, and every coordinate should land on a grid line, so each point can be plotted exactly.
- With 1 per square the axes reach only 10 \times 1 = 10, and 12, 18 and 14 do not fit.
- With 2 per square the axes reach 10 \times 2 = 20, so 18 fits. Every coordinate, 12, 4, 6, 18, 0 and 14, is even, so each one is a whole number of squares: (12, 4) is 6 squares across and 2 up, (6, 18) is 3 across and 9 up, and (0, 14) is on the y-axis, 7 squares up.
- With 5 per square the points would fit, but 12 is not a multiple of 5, so (12, 4) would fall between grid lines.
Counting by 2s is the choice: everything fits and every point sits on grid lines. A scale of 10 would fit too, but the points would crowd into the lower left corner and most of them would fall between lines.
Walking the school campus
On the school campus map each unit stands for 10 meters. The school office is at S(2, 3), the gym at G(8, 3) and the library at L(2, 7). Paths run along the grid lines. How long is the walk from the office to the gym, and from the office to the library? Chukwuma walks from the gym to the library along the paths, through the office. How far does he walk?
S(2, 3) and G(8, 3) have the same y-coordinate, 3, so they lie on the same horizontal grid line. The walk is only across, from 2 to 8:
S(2, 3) and L(2, 7) have the same x-coordinate, 2, so they lie on the same vertical grid line. The walk is only up, from 3 to 7:
Chukwuma walks from the gym to the office and then from the office to the library: 6 + 4 = 10 units, which is 100 meters. The gym and the library share neither coordinate, so no single grid line joins them, and the walk along the paths has to turn a corner at the office.
Halfway between the lines
On a second museum map, a bench is at \displaystyle B\left(3\frac{1}{2}, 2\right) and a statue at \displaystyle T\left(3\frac{1}{2}, 6\frac{1}{2}\right). Describe where each one sits on the grid. How far is the bench from the statue?
Both have x-coordinate \displaystyle 3\frac{1}{2}, halfway between 3 and 4, so both sit halfway between the vertical grid lines for 3 and 4. The bench is 2 up, so it is on the horizontal grid line for 2, at the middle of the side of a square. The statue is \displaystyle 6\frac{1}{2} up, halfway between the lines for 6 and 7, so it sits in the middle of a grid square.
The two share an x-coordinate, so one vertical line joins them and the distance is the difference of the y-coordinates, found by subtracting the mixed numbers:
Check by counting on the figure: from 2 up to 6 is 4 squares, and half a square more reaches \displaystyle 6\frac{1}{2}. That is \displaystyle 4\frac{1}{2} units.
Boundaries
Every coordinate here is 0 or more, so every point is in the first quadrant or on one of the two axes that border it. A point to the left of the y-axis or below the x-axis needs negative numbers; Plotting in all four quadrants extends the grid to all four quadrants.
The subtraction rule for distance works only when two points share a coordinate, so that one grid line joins them. For two points such as (1, 2) and (4, 6), which share neither coordinate, the straight slanted distance is not 4 - 1, not 6 - 2, and not their sum either. The sum is the length of a walk along grid lines that turns a corner. The straight distance between such points needs the The distance formula, a later idea.
Coordinates are numbers on the axes, not counts of squares. They match the count of squares only when each square stands for 1. The point (30, 20) is 3 squares across on a grid counting by 10s, and 30 squares across on a grid counting by 1s.
A point marks a location and has no size. A room or a building on a map is bigger than a point, so its ordered pair names one chosen spot, such as its door.
Where this goes next
Pairs of numbers put a relationship into a picture. In Two number patterns side by side and graphing their pairs, two number patterns are paired term by term and the pairs are plotted as points, so you can see how the patterns grow together. Plotting in all four quadrants lets the axes run past 0 in both directions, and Horizontal and vertical distance finds distances along grid lines anywhere on the plane. Later the same grid carries shapes, the graphs of equations and lines with a slope. The The coordinate plane is where most of the pictures in algebra are drawn.
Exercises
Part 1: the ordered pair
- In the ordered pair (7, 2), which number is the x-coordinate and which is the y-coordinate? Describe the moves from the origin.
- Write the ordered pair for the point 4 units right and 9 units up from the origin.
- Ruolan describes a point as “8 units up and 1 unit right” of the origin. Write it as an ordered pair.
- Do (3, 8) and (8, 3) name the same point? Explain using the moves.
- Which of these pairs name the same point when their two numbers are swapped: (6, 6), (2, 7), (0, 0), (5, 1)?
Part 2: plotting and naming points
Exercises 6 and 7 use this grid.
- Give the coordinates of P, Q and R.
- Give the coordinates of S and T. Which axis is each one on?
- Without a drawing, say where each point is: (0, 7), (9, 0), (0, 0).
- A grid counts by 2s on both axes. A point is 4 squares right and 3 squares up from the origin. What are its coordinates?
- A grid counts by 5s on both axes. How many squares right and up from the origin is (35, 20)?
- A grid counts by 10s across and by 2s up. How many squares right and up from the origin is (60, 14)?
- Chukwuma wants to plot (40, 15), (25, 45) and (0, 30) on a grid with 10 squares along each axis, using one scale on both. Should each square stand for 1, 2, 5 or 10?
- On a grid that counts by 2s on both axes, where does the point (7, 4) go?
Part 3: maps, distances and halves
- How far apart are (1, 6) and (9, 6)?
- How far apart are (4, 2) and (4, 11)?
- On a larger campus map each unit is 10 meters. Yamini’s classroom is at (3, 5) and the cafeteria at (3, 12). How long is the walk along the grid line between them?
- Chukwuma walks along grid lines from the art room at (2, 1) to the music room at (7, 4), going across first and then up. Where does he turn, and how many units does he walk?
- Describe where \displaystyle \left(2\frac{1}{2}, 4\right) and \displaystyle \left(2\frac{1}{2}, 1\right) sit on the grid, and find the distance between them.
- How far apart are \displaystyle \left(1\frac{1}{2}, 3\right) and (6, 3)?
- On Ruolan’s museum map each unit is 4 meters. The stairs are at \displaystyle \left(5, 1\frac{1}{2}\right) and the elevator at (5, 7). How many units apart are they, and how many meters?
- What point is halfway between (3, 2) and (3, 7) on the grid line that joins them?
Reasoning
- Explain why one number is enough to name a point on a number line but not a point on a grid.
- Explain why every point on the x-axis has a y-coordinate of 0.
- Yamini says the distance from (1, 2) to (4, 6) is 4 - 1 = 3. Explain what is wrong, and what the subtraction rule needs.
- A grid counts by 10s across and by 1s up. Explain why this is allowed, and why an axis marked 0, 1, 2, 5, 10 on equal-sized squares is not.
Answers
- x-coordinate 7, y-coordinate 2. The first number is the move across: 7 units right, then 2 units up.
- (4, 9). The across number is written first.
- (1, 8). The across number goes first even when it is said second.
- No. (3, 8) is 3 units right and 8 units up; (8, 3) is 8 units right and 3 units up. They are on different vertical grid lines, through 3 and through 8, so they are different points.
- (6, 6) and (0, 0). Swapping gives the same pair only when the two numbers are equal; (2, 7) becomes (7, 2) and (5, 1) becomes (1, 5), which are other points.
- P(1, 4), Q(5, 7), R(8, 2). Look down to the x-axis for the first number and across to the y-axis for the second.
- S(0, 6) on the y-axis, and T(3, 0) on the x-axis. S is 0 across and T is 0 up.
- (0, 7) is on the y-axis, 7 units up. (9, 0) is on the x-axis, 9 units right. (0, 0) is the origin.
- (8, 6). Each square is 2, so 4 \times 2 = 8 across and 3 \times 2 = 6 up.
- 7 squares right and 4 squares up: 35 \div 5 = 7 and 20 \div 5 = 4.
- 6 squares right and 7 squares up: 60 \div 10 = 6 and 14 \div 2 = 7.
- 5 per square. All six numbers are multiples of 5, and the largest, 45, fits under 10 \times 5 = 50. A scale of 1 or 2 reaches only 10 or 20; with 10, the numbers 15, 25 and 45 fall between grid lines.
- \displaystyle 3\frac{1}{2} squares right and 2 squares up, since \displaystyle 7 \div 2 = 3\frac{1}{2} and 4 \div 2 = 2. It sits on the horizontal line marked 4, halfway between the vertical lines marked 6 and 8.
- 8 units. Same y-coordinate, so subtract the x-coordinates: 9 - 1 = 8.
- 9 units. Same x-coordinate, so subtract the y-coordinates: 11 - 2 = 9.
- 70 meters. Same x-coordinate: 12 - 5 = 7 units, and 7 \times 10 = 70.
- He turns at (7, 1) and walks 8 units: 7 - 2 = 5 across, then 4 - 1 = 3 up, and 5 + 3 = 8.
- Both are halfway between the vertical lines for 2 and 3, one on the horizontal line for 4 and one on the line for 1. Distance 3 units, since they share an x-coordinate and 4 - 1 = 3.
- \displaystyle 4\frac{1}{2} units. Same y-coordinate: \displaystyle 6 - 1\frac{1}{2} = 4\frac{1}{2}.
- \displaystyle 5\frac{1}{2} units, which is 22 meters. \displaystyle 7 - 1\frac{1}{2} = 5\frac{1}{2}, and \displaystyle 5\frac{1}{2} \times 4 = 5 \times 4 + \frac{1}{2} \times 4 = 20 + 2 = 22.
- \displaystyle \left(3, 4\frac{1}{2}\right). The points are 7 - 2 = 5 units apart; half of that is \displaystyle 2\frac{1}{2}, and \displaystyle 2 + 2\frac{1}{2} = 4\frac{1}{2}. The x-coordinate stays 3.
- A number line runs in one direction, so one number says how far along it. A grid spreads in two directions, and one number cannot say both how far across and how far up: (3, 5) and (3, 1) share the number 3 but are different points. Two numbers in a fixed order are needed.
- The y-coordinate says how far up from the x-axis a point is. A point on the x-axis is not up at all, so that number is 0. Every point of the axis is reached by moving across only.
- The two points share neither coordinate, so no grid line joins them, and 3 is only how far across they are. The subtraction rule needs both points on one grid line: the same x-coordinate or the same y-coordinate. A walk along grid lines from (1, 2) to (4, 6) is 3 + 4 = 7 units, and the straight slanted distance is a different number again, found with a later idea.
- Each axis is its own number line, so the two axes may count by different steps: here one square across is 10 and one square up is 1. But along a single axis every square must stand for the same amount. Marks 0, 1, 2, 5, 10 on equal squares would make the steps 1, 1, 3, 5, so equal lengths on that axis would stand for different amounts.