math.wiki
Geometry › Coordinate geometry › The coordinate plane › Plotting in all four quadrants
Open on the map
Math 6 · 6.G.A.3 · 6.NS.C.6B

Plotting in all four quadrants

A map has places west and south of its starting point as well as east and north. How can a pair of numbers name a point on every side?

You need
Ordered pairs and the first quadrantGrade 5
Unlocks
Points that differ only in sign: reflections across the axesMath 6Area of a polygon from coordinatesMath 6Horizontal and vertical distanceMath 6The distance formulaMath 8+1 more
Where you are 4 in The coordinate plane
Geometry11
Coordinate geometry9
The coordinate plane4
Ordered pairs and the first quadrant
Plotting in all four quadrants
Points that differ only in sign: reflections across the axesTwo number patterns side by side and graphing their pairs

When you reach for it

An ordered pair names a point with two numbers: the first says how far across from the origin, the second how far up. With only positive numbers and zero, that reaches just one corner of the plane, above and to the right of the origin. Plenty of pictures need more. A treasure map measured from a well in the middle of an island has places on every side of the well. A robot vacuum that starts from a dock in the middle of a hall wanders behind the dock as well as in front of it. A town measured from its town hall has streets to the west and to the south.

Negative numbers on a number line already point the other way from zero. This lesson puts them on both axes, so every point of the plane gets a pair of numbers, names the four regions the axes make, and chooses how much each grid square stands for.

The procedure

The skill is taught over two days, one part each.

Part 1: plotting and reading points anywhere

The full grid. Take a horizontal number line, with negative numbers to the left of zero, and a vertical number line, with negative numbers below zero, and cross them at right angles at their zeros. These are the axes: the horizontal axis (often called the x-axis) and the vertical axis (the y-axis). They cross at the origin, (0, 0). Each axis is a whole number line, so it keeps equal steps for equal amounts on both sides of zero.

The two numbers mean what they meant before. In a pair such as (-5, 3) the first number says how far to move from the origin across, along the horizontal axis, and the second says how far to move straight up or down. What is new is that each number now carries a direction in its sign:

positivezeronegative
first numbermove rightno move acrossmove left
second numbermove upno move up or downmove down

The size of the number, its distance from zero, says how many units to move. So (-5, 3) means 5 units left and 3 units up. The order of the two moves does not matter, since 3 up and then 5 left ends at the same grid crossing. The order of the two numbers matters completely, because the first is always the across number.

Why a pair names exactly one point: every point on the vertical grid line through -5 is straight above or below the mark for -5, so every one of them has first number -5. Every point on the horizontal grid line through 3 has second number 3. The point (-5, 3) must lie on both lines, and two such lines cross in exactly one place.

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 P
Figure 1The point (-5, 3) sits where the vertical grid line through -5 meets the horizontal grid line through 3.

Reading a point runs the same reasoning backwards. From the point, follow its vertical grid line to the horizontal axis and read the number there: that is the first coordinate. Follow its horizontal grid line to the vertical axis: that is the second. Write them in that order, in brackets, separated by a comma.

Points on an axis. A point on the vertical axis has made no move across, so its first number is 0, as in (0, -4). A point on the horizontal axis has made no move up or down, so its second number is 0, as in (-3, 0).

Numbers between the grid lines. Coordinates need not be integers. The point (2.5, -0.5) is halfway between the vertical grid lines for 2 and 3, and half a unit below the horizontal axis. The grid lines are only a help for counting; every pair of numbers names a point, on a line or not.

Part 2: the four quadrants, and choosing a scale

The quadrants. The two axes cut the plane into four regions called quadrants. They are numbered with Roman numerals, starting at the upper right and going round counterclockwise, the opposite way to a clock’s hands: Quadrant I is upper right, II upper left, III lower left and IV lower right. The first-quadrant grid of Ordered pairs and the first quadrant is Quadrant I.

\text{I}
(+,\,+)
\text{II}
(-,\,+)
\text{III}
(-,\,-)
\text{IV}
(+,\,-)
Figure 2The four quadrants, numbered counterclockwise from the upper right, with the signs of the two coordinates in each.

The quadrant of a point can be told from the signs of its coordinates, with no plotting. Right of the vertical axis means a positive first number and left means a negative one; above the horizontal axis means a positive second number and below means a negative one. Each quadrant is one combination:

Quadrantfirst numbersecond number
Ipositivepositive
IInegativepositive
IIInegativenegative
IVpositivenegative

A point on an axis is in no quadrant. The quadrants are the regions between the axes, and the axes are their borders. A point with a zero coordinate lies on an axis, and since zero is neither positive nor negative it fits no row of the table. The origin, on both axes at once, is in no quadrant either.

Choosing a scale. So far every grid square has been one unit wide and one unit tall. When the numbers are large, one unit per square would need more squares than a page has, so each square is made to stand for more: 10, 50, 150, whatever suits. That amount is the scale of the axis. Three things guide the choice.

  • It must fit. The coordinate farthest from zero along an axis has to land on the paper, so the scale must be at least that number shared out over the squares available.
  • It should be easy to count in. If every coordinate is a whole number of steps, every point lands on a grid line and can be plotted and read exactly.
  • It should use the room. A scale far larger than needed crowds every point into a few squares near the origin, where they are hard to tell apart.

Each axis keeps one scale all the way along, on both sides of zero, because a number line needs equal steps for equal amounts. The two axes may have different scales, which is often the right choice when they measure different things or ranges of very different sizes. The numbers in the margins say which is which.

Worked examples

Examples 1 to 3 go with Part 1. Examples 4 to 7 go with Part 2.

Example 1

Plotting four points

Plot P(3, -5), Q(-4, 1), R(-6, -2) and S(0, -3) on one grid.

Solution

Writing a letter in front of a pair, as in P(3, -5), is the usual way to say “the point P, at (3, -5)”. Each move starts from the origin, and each sign gives a direction.

  • P(3, -5): the first number is positive, so move 3 units right; the second is negative, so move 5 units down.
  • Q(-4, 1): move 4 units left, then 1 unit up.
  • R(-6, -2): both numbers are negative, so move 6 units left and 2 units down.
  • S(0, -3): the first number is zero, so there is no move across; move 3 units down. S lies on the vertical axis.
−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 P Q R S
Figure 3The four points of Example 1. S has first number 0, so it lies on the vertical axis.

To check, read each point back. P sits on the vertical grid line through 3 and the horizontal grid line through -5, which gives (3, -5) again.

Example 2

Where the robot vacuum turned

Veda’s robot vacuum cleans a large hall, and each grid square is one floor tile. Its charging dock is at the origin. The figure shows the path it took: from the dock to A, then to B, C, D and E. Write the coordinates of each turning point.

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 A B C D E
Figure 4The robot vacuum’s path in Example 2, starting from its dock at the origin.
Solution

For each point, read down or up to the horizontal axis for the first number, and across to the vertical axis for the second.

  • A is on the vertical grid line through -4 and the horizontal grid line through 3: A(-4, 3).
  • B is straight below A, so it keeps the first number -4; it is level with -3: B(-4, -3).
  • C is straight across from B, so it keeps the second number -3; its vertical line is through 3: C(3, -3).
  • D is straight above C and sits on the horizontal axis itself, so its second number is 0: D(3, 0).
  • E is on the vertical line through 5 and the horizontal line through 4: E(5, 4).

A move straight up or down, like A to B, never changes the first number, and a move straight across, like B to C, never changes the second.

Example 3

Halfway points on a treasure map

Tove draws a treasure map of an island with the old well at the origin. The first number counts squares east of the well (negative means west) and the second counts squares north (negative means south). Plot the chest at (-2.5, 3), the cave at (4, -1.5), the wreck at (-3.5, -4.5) and the palm tree at (0, 2.5).

Solution

East is right and north is up. A half in a coordinate is half a square, so the point lands halfway between two grid lines.

  • The chest C: 2.5 units left, halfway between the lines for -2 and -3, and 3 units up, on a grid line.
  • The cave V: 4 units right, on a grid line, and 1.5 units down, halfway between the lines for -1 and -2.
  • The wreck W: 3.5 units left and 4.5 units down. Both numbers end in a half, so the point is in the middle of a grid square.
  • The palm tree T: no move across, so it is on the vertical axis, 2.5 units up.
−5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 C V W T
Figure 5Tove’s treasure map in Example 3. A half in a coordinate puts the point halfway between two grid lines.
Example 4

Naming the quadrant from the signs

Without plotting, say where each point lies: (-7, 12), (15, -2), (-0.5, -20), (0, -9) and (6, 0).

Solution

Only the signs decide the quadrant, so read the two signs and match them to the table.

  • (-7, 12): negative then positive, left and up, so Quadrant II.
  • (15, -2): positive then negative, right and down, so Quadrant IV.
  • (-0.5, -20): both negative, so Quadrant III. The first number is tiny next to the second, so the point hugs the vertical axis, but it is still to the left of it, and that is all that counts.
  • (0, -9): the first number is zero, so the point is on the vertical axis, below the origin, and in no quadrant.
  • (6, 0): the second number is zero, so the point is on the horizontal axis, right of the origin, and in no quadrant.
Example 5

What one sign tells you

(a) Remy knows only that a point’s first number is negative. Where can the point be?

(b) Where can a point be if its second number is 0?

Solution

(a) A negative first number puts the point left of the vertical axis. The second number is unknown, so try each sign it could have. A positive second number gives Quadrant II, a negative one gives Quadrant III, and zero puts the point on the horizontal axis, left of the origin. So the point is in Quadrant II, in Quadrant III or on the horizontal axis.

(b) A second number of 0 means no move up or down, so the point is on the horizontal axis. If its first number is also 0, it is the origin. Either way it is in no quadrant.

One sign narrows the choice to half the plane; it takes both signs to pin down one quadrant.

Example 6

Choosing one scale for a town map

Nina maps her town, which is laid out on a street grid with the town hall at the origin. The first number is meters east of the town hall (negative means west) and the second is meters north (negative means south). The library is at (300, 450), the school at (-600, 150), the park at (-450, -750) and the station at (900, -300). Her grid paper has 6 squares on each side of each axis. Choose one scale for both axes that puts every place on a grid crossing, and plot the places.

Solution

The coordinate farthest from zero is the station’s 900. With 6 squares to reach it, each square must stand for at least 900 \div 6 = 150 meters. A scale of 100 is too small: 900 would need 9 squares.

Now test 150 against the other numbers. Its multiples are 150, 300, 450, 600, 750 and 900, and every coordinate here is one of those or its negative, so every place lands on a grid crossing. A larger scale such as 200 would also fit, but 450 is not a whole number of 200s, so the library would fall between grid lines, and the map would use less of the page. So the scale is 150 meters per square.

Count squares from the origin: the library 2 squares right and 3 squares up, the school 4 squares left and 1 square up, the park 3 squares left and 5 squares down, and the station 6 squares right and 2 squares down.

−900 −600 −300 0 300 600 900 −900 −600 −300 0 300 600 900 H L S P T
Figure 6Nina’s town map in Example 6, at 150 meters per square. The margin numbers are written on every second line.
Example 7

Two different scales

Yoko draws a side view of a treasure dive. The first number is meters along the surface from a buoy (negative means toward the shore) and the second is meters above the water (negative means below). The ship’s bow lies at (-30, -6), the anchor at (20, -8), the chest at (40, -5) and the top of a sea cliff at (60, 3). Her grid has room for 6 squares on each side of the vertical axis and 8 squares above and below the horizontal axis. Choose a scale for each axis and plot the points.

Solution

Across, the numbers reach 60 with 6 squares available, so each square must stand for at least 10 meters; and 30, 20, 40 and 60 are all multiples of 10. Up and down, the numbers reach only 8 (the anchor’s -8) with 8 squares available, and every value is a whole number. So the scales are 10 meters per square across and 1 meter per square up.

One scale for both axes would go badly. At 10 meters per square up as well, the bow, the anchor and the chest, between 5 and 8 meters down, would all sit less than one square below the axis, almost on top of one another. At 1 meter per square across, the cliff top would need 60 squares. The axes measure different things, so their scales can differ.

Counting squares: the bow 3 squares left and 6 squares down, the anchor 2 squares right and 8 squares down, the chest 4 squares right and 5 squares down, and the cliff top 6 squares right and 3 squares up.

−40 −30 −20 −10 0 10 20 30 40 50 60 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 W A C T
Figure 7Yoko’s side view in Example 7. Each square is 10 meters across but only 1 meter up.

Boundaries

  • A pair names a point only once the origin, the directions of the axes and their scales are fixed. The same pair on two maps with different origins or scales names two different places. Right and up being positive is a shared agreement; a picture that chooses otherwise has to say so.
  • The order is part of the name. Swapping the two numbers, as in (2, -7) and (-7, 2), gives a different point unless the two numbers are equal.
  • Quadrants hold only points with no zero coordinate. A point on an axis, and the origin, belong to no quadrant.
  • A grid shows only a window. The plane goes on without end in every direction, and a point outside the window still has its place; it needs a larger scale or more paper to be drawn.
  • Each axis needs one evenly spaced scale. If the steps change partway along, the axis is no longer a number line.
  • A point between grid lines can be read off a drawing only approximately.

Where this goes next

Once every point has a pair, the numbers start to describe how points relate. Two pairs that differ only in sign name mirror-image points across an axis, which reflections across the axes takes up. Two points that share a first or a second number lie on one grid line, and horizontal and vertical distance measures how far apart they are. From there the length of any segment comes from the The distance formula, and the area of a shape drawn from its corners from Area of a polygon from coordinates.

Exercises

Part 1: plotting points

Plot each point on one grid and describe the moves from the origin that reach it.

  1. (2, -6)
  2. (-1, 5)
  3. (-5, -4)
  4. (-6, 0)
  5. (-1.5, 4)
  6. (3.5, -2.5)

Part 1: reading points

Write the coordinates of each point in the figure.

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 K L M N
Figure 8The points for Exercises 7 to 10.
  1. Point K
  2. Point L
  3. Point M
  4. Point N

Part 1: maps, floors and streets

  1. Nina’s treasure map has the big rock at the origin; the first number counts squares east and the second counts squares north. The chest is buried 3 squares west and 2 squares south of the rock. Write its coordinates.
  2. Remy’s robot vacuum reports its stops as pairs: the first number is meters east of its dock (negative means west) and the second is meters north (negative means south). The stops are (-2, -1), (-2, 6) and (1, 6). Describe where each stop is from the dock.
  3. In Tove’s town the first number counts blocks east of Main Square and the second counts blocks north. The bakery is at (-7, 0). Where is it?

Part 2: naming the quadrant

  1. Name the quadrant of each point without plotting it: (a) (-8, -1) (b) (9, -14) (c) (-0.3, 6) (d) (11, 0.2)
  2. Say where each point lies: (a) (0, 13) (b) (-2, 0) (c) (0, 0)
  3. Where can a point be if (a) its second number is negative; (b) its first number is positive; (c) both numbers are negative?
  4. Write three different points in Quadrant II, one of them with a fraction as a coordinate.

Part 2: choosing a scale

  1. Veda’s robot vacuum stops at (-240, 80), (160, -200), (-80, -120) and (200, 240), measured in centimeters from its dock. Her grid has 6 squares on each side of each axis. Choose one scale for both axes that puts every stop on a grid crossing and uses as much of the grid as possible.
  2. Yoko maps places in meters from her town hall: (1200, -800), (-2000, 400), (-600, -1600) and (0, 1400). Her grid has 10 squares on each side of each axis. Choose one scale for both axes that puts every place on a grid crossing and uses as much of the grid as possible.
  3. Nina draws a side view of a beach dive. The first number is meters along the beach and the second is meters above sea level (negative means below): (-300, 4), (500, -12), (200, 6) and (-100, -8). Her grid has 6 squares on each side of each axis. Choose an easy scale for each axis so that everything fits on grid crossings.
  4. A grid has 25 per square on the horizontal axis and 5 per square on the vertical axis. A point is 3 squares left of the vertical axis and 4 squares below the horizontal axis. Write its coordinates.

Reasoning

  1. Explain why (4, -1) and (-1, 4) are different points, and name the quadrant of each.
  2. Explain why a point with a zero coordinate is in no quadrant.
  3. A grid uses 20 per square across and 5 per square up. Explain why that is allowed, and why the squares to the left of the vertical axis must stand for 20 each as well.
  4. Name the quadrant of (-37.5, -0.01) and explain how you know without plotting it.
Answers
  1. Start at the origin and go 2 units right and 6 units down.
  2. Go 1 unit left and 5 units up.
  3. Go 5 units left and 4 units down.
  4. Go 6 units left, with no move up or down. The point is on the horizontal axis.
  5. Go 1.5 units left, halfway between the lines for -1 and -2, and 4 units up.
  6. Go 3.5 units right and 2.5 units down. Both numbers end in a half, so the point is in the middle of a grid square.
  7. (-2, -5). K is on the vertical grid line through -2 and the horizontal one through -5.
  8. (5, -1). L is right of the vertical axis and just below the horizontal one.
  9. (-6, 3). M is on the left edge of the grid, level with 3.
  10. (0, 4). N is on the vertical axis, so its first number is 0.
  11. (-3, -2). West is the negative direction for the first number and south for the second.
  12. (-2, -1) is 2 meters west and 1 meter south of the dock. (-2, 6) is 2 meters west and 6 meters north. (1, 6) is 1 meter east and 6 meters north. The first two stops share their first number, so the second stop is straight north of the first.
  13. 7 blocks west of Main Square, on the east-west street through the square. Its second number is 0, so it lies on the horizontal axis.
  14. (a) Quadrant III, both negative. (b) Quadrant IV, positive then negative. (c) Quadrant II, negative then positive. (d) Quadrant I, both positive; 0.2 is small but still positive.
  15. (a) On the vertical axis, above the origin. (b) On the horizontal axis, left of the origin. (c) It is the origin. None of the three is in a quadrant.
  16. (a) Quadrant III or Quadrant IV, or the vertical axis below the origin, since the first number may be negative, positive or zero. (b) Quadrant I or Quadrant IV, or the horizontal axis right of the origin. (c) Quadrant III only.
  17. For example (-1, 1), (-20, 7) and \displaystyle \left(-\frac{1}{2}, 3\right). Each has a negative first number and a positive second number, so each is in Quadrant II.
  18. 40 centimeters per square. The largest number is 240, and 240 \div 6 = 40. Every coordinate (80, 120, 160, 200, 240 and their negatives) is a multiple of 40, so every stop lands on a crossing. A scale of 20 would need 12 squares to reach 240.
  19. 200 meters per square. The largest number is 2000, and 2000 \div 10 = 200. Every coordinate (400, 600, 800, 1200, 1400, 1600, 2000, their negatives, and 0) is a multiple of 200.
  20. 100 meters per square across and 2 meters per square up. Across, 500 \div 6 is a little over 83, and every number is a multiple of 100, so 100 fits on crossings. Up, 12 \div 6 = 2, and 4, 6, 8 and 12 are all even.
  21. (-75, -20). Three squares of 25 is 75, to the left, so the first number is -75; four squares of 5 is 20, downward, so the second is -20.
  22. The first number is always the move across. (4, -1) is 4 right and 1 down, in Quadrant IV; (-1, 4) is 1 left and 4 up, in Quadrant II. Different moves end at different points.
  23. A zero first number means no move across, so the point is on the vertical axis; a zero second number puts it on the horizontal axis. The axes are the borders between quadrants, not part of any. In the sign table every quadrant needs each number positive or negative, and zero is neither.
  24. The two axes are separate number lines, so each may have its own unit, as long as the margins say what it is. But each axis needs equal steps for equal amounts all along. If the squares left of zero stood for a different amount, points left of the vertical axis would be drawn in the wrong places.
  25. Quadrant III. Both numbers are negative, so the point is left of the vertical axis and below the horizontal one. The sizes do not matter: -0.01 puts it only a hair below the axis, but below is below.
kg1128 · updated 2026-10-12 · CC-BY-SA-4.0