6th Grade · 6.NS.C.8

The coordinate plane with four quadrants in 6th grade

In 6th grade, the coordinate plane has four quadrants instead of one. The grid now goes to the left of $(0, 0)$ and below it, so a point such as $(-3, 5)$ has a place. Your child learns to plot points in all four quadrants and to find the distance between two points on the same row or column of the grid.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

What it means, in plain words

The coordinate plane is made of two number lines that cross at 0. The horizontal one is the x-axis and the vertical one is the y-axis. The point where they cross is the origin, $(0, 0)$ . Each point has an address, called its coordinates, written $(x, y)$ . The first number says how far to go left or right, and the second says how far to go up or down. In 5th grade, children used only the top-right part of the grid, where both numbers are positive.

The two axes split the coordinate plane into four quadrants, numbered I to IV. Quadrant I is the top right. The numbers then go around the opposite way to a clock: II is the top left, III the bottom left and IV the bottom right. The signs of the coordinates tell the quadrant. In Quadrant II, $x$ is negative and $y$ is positive, as in $(-3, 5)$ . In Quadrant III both numbers are negative, and in Quadrant IV only $y$ is negative.

Problems often use maps or plans, where each unit stands for a block or a foot. When two points have the same x-coordinate, they sit on one vertical line. If both points are on the same side of the x-axis, subtract to find the distance. If they are on opposite sides, add their distances from the axis using absolute value (the distance from 0): from $(1, 4)$ to $(1, -3)$ is $|4| + |-3| = 7$ units. Points with the same y-coordinate work the same way, across instead of up and down.

Your child should be able to:

Name the four quadrants and tell which signs the coordinates have in each one
Plot and read points with negative coordinates in all four quadrants
Find the distance between two points with the same x-coordinate or the same y-coordinate
Solve map and plan problems on a coordinate grid

Example

In which quadrant is the point $(-6, -2)$ , and how do you plot it?

Start at the origin, $(0, 0)$ .
The x-coordinate is $-6$ , so move 6 units to the left.
The y-coordinate is $-2$ , so move 2 units down and mark the point.
Both coordinates are negative, so the point is in Quadrant III.

Example

On a garden plan, the tomatoes are at $(-4, 5)$ and the beans are at $(-4, -3)$ . Each unit is 1 foot. How far apart are the tomatoes and the beans?

Both points have the x-coordinate $-4$ , so they sit on the same vertical line.
The tomatoes are 5 feet above the x-axis. The beans are $|-3| = 3$ feet below it.
The points are on opposite sides of the axis, so add: $5 + 3 = 8$ .
The tomatoes and the beans are 8 feet apart.

Where children usually get stuck

Your child plots $(2, -5)$ at the spot for $(-5, 2)$ , 5 units left and 2 units up.
The two numbers look alike, and nothing in $(2, -5)$ says which one means left or right. The order of the coordinates was never tied to a meaning.
What helps. Use a phrase such as "walk along the hall, then take the stairs". The first number moves left or right, and the second moves up or down.
For the points $(-4, 5)$ and $(-4, -3)$ , your child writes $5 - 3 = 2$ feet.
The minus sign on $-3$ gets dropped, so the child treats both points as if they were above the x-axis.
What helps. Count the grid squares between the two points together. There are 8, which shows that the distances on the two sides of the axis add up.
Your child calls the top-left quadrant Quadrant IV, numbering the quadrants the way you read a page.
Quadrants are numbered the opposite way to a clock, starting at the top right. That is not the order the child reads in or the way a clock turns.
What helps. Draw a large letter C over the grid. The pen starts at the top right and passes through I, II, III and IV in order.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
The Library is shown on the map below. Marcus's claim: "The Library is at (2, 4)."
Select the statement that best describes Marcus's mistake.
A
The x- and y-coordinates were reversed.
B
The x-coordinate has the wrong sign.
C
The y-coordinate has the wrong sign.
D
The student read a different location on the map.
Show the answer and how to solve it
The correct answer is that the x- and y-coordinates were reversed because the Library is at (4, 2), not (2, 4).
1. 1Find the Library on the map. It is in Quadrant I, shown by the red star.
2. 2The Library is 4 units right of the origin and 2 units up. The correct ordered pair is (4, 2).
3. 3Marcus wrote (2, 4). Both digits appear, but they have traded positions—the x-value 4 and the y-value 2 were listed in reverse order.
Common mistake. A common mistake is picking a sign-error choice because Marcus's pair looks different. But the Library is in Quadrant I with positive coordinates—Marcus simply swapped 4 and 2.
The idea underneath. In an ordered pair, x tells left or right of the origin and y tells up or down, so changing one sign moves the point to a different quadrant even when the digits stay the same.
2On level
Points A and B are shown on the coordinate plane below.
Enter the horizontal distance between A and B.
Distance:units
Show the answer and how to solve it
Answer. 6
The answer is 6 because points A and B are 6 units apart horizontally.
1. 1Point A is at (-9, 1) and point B is at (-3, 1). Since both points share the same y-coordinate of 1, the distance between them is purely horizontal.
2. 2Find how far apart the x-coordinates are. Subtracting -9 is the same as adding 9.
3. 3So the horizontal distance between A and B is 6 units.
Common mistake. A common mistake is writing 12, found by adding |-9| + |-3| $= 9 + 3$ . Point A is at $x = -9$ , 9 units left of zero. Point B is at $x = -3$ , 3 units left of zero. Both points are left of zero, so their horizontal gap is $9 - 3 = 6$ .
The idea underneath. When two points share the same y-coordinate, their horizontal distance is the gap between their x-coordinates. Different points have a positive distance.
3On level
Points A and B are shown in the graph below. A student tried to find the horizontal distance between the points.
The student's work is shown below.
$\displaystyle 3 - 2 = 1$
Select the statement that best explains the mistake.
A
The student subtracted the distances from 0 instead of combining them across the y-axis.
B
The student used the y-values instead of the x-values.
C
The student forgot that distance must be positive.
D
The student added the absolute x-values even though the points are on the same side of the y-axis.
Show the answer and how to solve it
The correct statement is that the student subtracted the distances from 0 instead of combining them across the y-axis. The horizontal distance between the points is 5 units.
1. 1Both points share y = 1, so the distance between them is purely horizontal.
2. 2Point A is at x = -3, which is 3 units left of 0. Point B is at x = 2, which is 2 units right of 0.
3. 3The student correctly identified 3 and 2 as the distances from 0, but then subtracted them. Because the points are on opposite sides of the y-axis, those two parts should be added:
4. 4The mistake is subtracting the distances from 0 instead of combining them across the y-axis. The correct horizontal distance is 5 units.
Common mistake. A common mistake is to subtract the distances from 0 — here 3 and 2 — as if one cancels part of the other. When the points are on opposite sides of the y-axis, those two parts must be added to find the full horizontal distance of 5 units.
The idea underneath. Horizontal distance depends on how far apart the x-values are, and when points lie on opposite sides of 0, the left part and right part add to make the whole distance.
4Stretch
Point A and point B are shown in the table below.
Point Ordered pair
A (3, 7.5)
B (3, -2.5)
Enter the positive vertical distance between the points.
Distance:units
Show the answer and how to solve it
Answer. 10
The answer is 10 because the y-values 7.5 and -2.5 are 10 units apart.
1. 1Both points have x = 3, so the points are stacked vertically. Only the y-values, 7.5 and -2.5, determine the distance.
2. 2Use the absolute value of the difference between the y-values.
3. 3The positive vertical distance is 10 units.
Common mistake. A common mistake is computing $-2.5 - 7.5$ and entering -10. Distance cannot be negative, so the correct answer is 10.
The idea underneath. When two points share the same x-coordinate, they are directly above and below each other, so vertical distance comes only from the y-values.

Point	Ordered pair
A	(3, 7.5)
B	(3, -2.5)