Back to Solve real-world and mathematical problems by graphing points in all four quadrants

Graphing in All Four Quadrants and Coordinate Distances

Work through each section in order. For graphing problems, use the coordinate grid provided or sketch one yourself. Show your work for distance problems using absolute value.

Grade 6·22 problems·~35 min·Common Core Math - Grade 6·standard·6-ns-c-8

Work through problems with immediate feedback

Recall / Warm-Up

The point $(-3, 5)$ is located in which quadrant of the coordinate plane?

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

Which of the following points has a negative $x$ -coordinate and a negative $y$ -coordinate?

$(-2, 4)$

$(3, -1)$

$(-5, -3)$

$(0, -6)$

What is $|-7|$ ?

Fluency Practice

$Coordinate plane with points P at (-4, 3), Q at (2, -5), R at (-1, -2), and S at (5, 1)$

The coordinate plane below shows four points. What are the coordinates of point $P$ ?

$(3, -4)$

$(-4, -3)$

$(-4, 3)$

$(4, 3)$

Point $A$ is at $(5, -3)$ and point $B$ is at $(-2, -3)$ . What is the distance between them?

$Coordinate plane showing points C at (-3, 6) and D at (-3, -1) connected by a vertical dashed segment$

Point $C$ is at $(-3, 6)$ and point $D$ is at $(-3, -1)$ . These two points share the same $x$ -coordinate. What is the distance between them?

Point $E$ is at $(4, 3)$ and point $F$ is at $(4, -4)$ . What is the distance from $E$ to $F$ ?

Points $G(-6, 2)$ and $H(1, 2)$ share the same $y$ -coordinate. Which expression gives the distance between them?

$|2 - 2|$

$|{-6} - 1|$

$|{-6} - 2|$

$|1 + 2|$

Varied Practice

$Coordinate plane with points W at (3,-4), X at (-2,4), Y at (-2,-4), Z at (2,-4)$

The coordinate plane below shows four points. Which point has coordinates $(-2, -4)$ ?

Point $W$

Point $X$

Point $Y$

Point $Z$

$Coordinate plane showing points J at (-5, 1) and K at (4, 1) connected by a horizontal segment$

Points $J(-5, 1)$ and $K(4, 1)$ share the same $y$ -coordinate. They are connected by a horizontal segment on the coordinate plane below. What is the length of segment $JK$ ?

To find the distance between $(-2, 7)$ and $(-2, -3)$ , first notice the points share the same ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . That means they lie on the same ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ line. Subtract the ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ coordinates and take the absolute value: $|7 - (\underline{\hspace{5em}})| = \underline{\hspace{5em}}$ .

shared coordinate axis:

line orientation:

coordinate type:

missing value:

distance:

Point $M$ is at $(1, -2)$ and point $N$ is at $(1, 5)$ . What is the distance from $M$ to $N$ ?

$-7$

A rectangle has three of its four vertices at $(-3, 4)$ , $(5, 4)$ , and $(5, -2)$ . What are the coordinates of the fourth vertex? Explain how you know.

Word Problems

$Coordinate map showing City Hall at (2,5), Library at (2,-3), and Community Center at (-4,-3) with dashed segments connecting them$

A city park uses a coordinate grid where each unit represents 1 block. City Hall is at $(2, 5)$ , the library is at $(2, -3)$ , and the community center is at $(-4, -3)$ .

How many blocks is it from City Hall to the library?

How many blocks is it from the library to the community center?

If you walk from City Hall to the library to the community center, how many total blocks do you walk?

$Coordinate map showing Ranger Station at (-5,3), Waterfall at (3,3), and Campsite at (3,-4) with dashed segments connecting them$

A hiking trail map uses a coordinate grid where each unit is 1 kilometer. The ranger station is at $(-5, 3)$ , the waterfall is at $(3, 3)$ , and the campsite is at $(3, -4)$ .

What is the distance from the ranger station to the waterfall?

What is the distance from the waterfall to the campsite?

Error Analysis

$Contrast card showing that without absolute value the subtraction order matters, but with absolute value both orders give 6$

Priya is finding the distance from $(3, 4)$ to $(3, -2)$ .

She writes: distance $= 4 - (-2) = 4 + 2 = 6$ .

Then she wonders: 'But if I go from $(-2)$ to $4$ , I get $-2 - 4 = -6$ . That gives a negative distance — something must be wrong!'

What should Priya do to make sure she always gets a correct, non-negative distance?

Always subtract the first $y$ -value from the second: distance $= y_2 - y_1$

Use absolute value: distance $= |y_1 - y_2|$

Ignore negative signs and just add the magnitudes

The distance is 0 because both points are on the same vertical line

$Contrast card showing Marcus subtracting the shared y-coordinates to get 0 versus the correct subtraction of x-coordinates to get 6$

Marcus is finding the distance between points $(-4, 1)$ and $(2, 1)$ .

He writes: distance $= |1 - 1| = 0$ .

"These points are 0 apart," he concludes.

What mistake did Marcus make? Choose the best description.

Marcus subtracted the coordinates that are the same, instead of the ones that differ

Marcus should have added the coordinates instead of subtracting

Marcus forgot to write the absolute value bars

Marcus used the wrong formula — he should have added the $x$ and $y$ values

Challenge / Extension

A rectangle has vertices at $(-3, 5)$ , $(4, 5)$ , $(4, -2)$ , and $(-3, -2)$ . What is the perimeter of the rectangle? Show your work by finding the length and width separately.

Two hikers both start at the origin $(0, 0)$ . Hiker A walks to point $(5, 2)$ and Hiker B walks to point $(5, -3)$ . Which hiker is farther north or south from the other? By how many units? Explain how you use coordinates and absolute value to find your answer.

0 of 22 answered