Map Distances and Why Absolute Value Works | Grade 6 Math | LevelKite

A coordinate plane is a grid built from two number lines that cross: the horizontal $x$ -axis and the vertical $y$ -axis. They cross at the origin, $(0, 0)$ . Every point has an ordered pair $(x, y)$ as its address. The $x$ -coordinate tells how far left or right the point is from the $y$ -axis. The $y$ -coordinate tells how far up or down it is from the $x$ -axis.

You have already found distances between points on a coordinate plane. Now you will use that skill on a map. Each place is a point, and each grid unit, the gap between two neighboring grid lines, stands for a real distance. You will also explain why absolute value, a number's distance from $0$ , gives the distance between two places on the same grid line. Then you will learn to spot mistakes in distance work.

The figure below is a map of a small town. North is at the top, so the $x$ -axis runs west to east and the $y$ -axis runs south to north. The grid lines are 1 unit apart, and each grid unit stands for 1 block. Streets run along the grid lines, and every place sits where two grid lines cross.

Point A is the library at $(-5, 4)$ .
Point B is the school at $(3, 4)$ .
Point C is the park at $(3, -1)$ .
Point D is the pool at $(3, -6)$ .
Point E is the museum at $(-5, -3)$ .
Point F is the bakery at $(-1, -3)$ .

The town map, where 1 grid unit stands for 1 block. Point A is at

(-5, 4)

, B at

(3, 4)

, C at

(3, -1)

, D at

(3, -6)

, E at

(-5, -3)

and F at

(-1, -3)

.

To read a place on the map, start at the origin. The library is at $(-5, 4)$ : go 5 units west, then 4 units north. It works the other way too. The place at $(3, -1)$ is 3 units east and 1 unit south of the origin, so it is the park. A negative $x$ -coordinate means west of the $y$ -axis, and a positive one means east. A negative $y$ -coordinate means south of the $x$ -axis, and a positive one means north.

On this map, two places are on the same east-west street when they have the same $y$ -coordinate. The library and the school both have $y = 4$ . Two places are on the same north-south street when they have the same $x$ -coordinate. The school, the park and the pool all have $x = 3$ .

Get every distance from the coordinates, not from how long a gap looks in the picture.

Worked example

From the library to the school

You walk along the street from the library, Point A at $(-5, 4)$ , to the school, Point B at $(3, 4)$ . Each grid unit stands for 1 block. How many blocks do you walk?

Compare the two ordered pairs. Both places have $y$ -coordinate $4$ , so they are on the same horizontal grid line. The walk goes straight east, and only the $x$ -coordinate changes.
$(-5, 4) \to (3, 4)$
Walk from the library to the $y$ -axis. The $x$ -coordinate goes from $-5$ to $0$ . The grid lines are 1 unit apart, so this part is 5 grid units long.
$\text{library to axis: } 5 \text{ units}$
Keep walking from the $y$ -axis to the school. The $x$ -coordinate goes from $0$ to $3$ , so this part is 3 grid units long.
$\text{axis to school: } 3 \text{ units}$
Add the two parts, because together they make the whole walk.
$5 + 3 = 8$
Change grid units to blocks. Each grid unit stands for 1 block, so 8 grid units stand for 8 blocks.
$8 \times 1 \text{ block} = 8 \text{ blocks}$

Answer

You walk $8$ blocks east from the library to the school.

Look again at the $5$ and the $3$ in that walk. They are absolute values. The absolute value of a number is its distance from $0$ on a number line. You write it with bars: $|-5| = 5$ and $|3| = 3$ .

On a horizontal grid line, the $x$ -coordinates work like a number line, and its $0$ is on the $y$ -axis. So each coordinate holds two facts. Its sign tells which side of the axis the place is on. Its absolute value tells how far the place is from the axis. A coordinate of $0$ puts the place right on the axis. Distance asks only how far, so absolute value is the part you need.

In the figure below, which shows that street from the town map, the point labeled $(0, 4)$ is where the street crosses the $y$ -axis. The library at $(-5, 4)$ and the school at $(3, 4)$ are on opposite sides of that axis. So the walk has two parts: library to axis, $|-5| = 5$ units, and axis to school, $|3| = 3$ units. That is why you add: $|-5| + |3| = 5 + 3 = 8$ .

Grid lines are 1 unit apart. The library at

(-5, 4)

is

5

units from

(0, 4)

on the

y

-axis, and the school at

(3, 4)

is

3

units from it on the other side. Together:

5 + 3 = 8

units.

Worked example

Two places on the same side of an axis

The park is Point C at $(3, -1)$ and the pool is Point D at $(3, -6)$ . Use absolute value to find how many blocks apart they are. Explain why your calculation gives the distance.

Compare the ordered pairs. Both places have $x$ -coordinate $3$ , so they are on the same vertical grid line. Only the $y$ -coordinate changes, so work with $-1$ and $-6$ .
$(3, -1) \quad \text{and} \quad (3, -6)$
Find each place's distance from the $x$ -axis. On a vertical grid line, the $y$ -coordinates work like a number line with its $0$ on the $x$ -axis. In the figure after this example, that $0$ is the point labeled $(3, 0)$ . Take the absolute values.
$|-1| = 1 \qquad |-6| = 6$
Decide the sides. Both $y$ -coordinates are negative, so both places are south of the $x$ -axis, on the same side. The park is between the axis and the pool.
The pool's 6 units from the axis are made of the park's 1 unit plus the gap, in grid units, from the park to the pool.
$1 + \text{gap} = 6$
Subtraction undoes that addition: taking the park's 1 unit away from the pool's 6 units leaves only the gap. So subtract the smaller absolute value from the larger.
$\text{gap} = |-6| - |-1| = 6 - 1 = 5$
Change grid units to blocks. Each grid unit stands for 1 block, so 5 grid units stand for 5 blocks.
$5 \times 1 \text{ block} = 5 \text{ blocks}$

Answer

The park and the pool are $5$ blocks apart. Subtracting gives the distance because both places are on the same side of the $x$ -axis, so the park's distance from the axis is part of the pool's distance.

Grid lines are 1 unit apart. The park at

(3, -1)

is

1

unit below

(3, 0)

on the

x

-axis, and the pool at

(3, -6)

is

6

units below it. The gap between them is

6 - 1 = 5

units.

Key concept

Why absolute value gives distance

Two points on the same horizontal or vertical line share one coordinate. Only the other coordinate changes. Call the two changing coordinates $a$ and $b$ . Then $|a|$ and $|b|$ are the points' distances from the axis that the line crosses.

Opposite sides of the axis ( $a$ and $b$ have different signs): the path crosses the axis, so the distance is $|a| + |b|$ .
Same side of the axis ( $a$ and $b$ have the same sign): the nearer point's distance is part of the farther point's distance, so subtract the smaller absolute value from the larger.
One point on the axis ( $a$ or $b$ is $0$ ): $|0| = 0$ , so the distance is the absolute value of the other coordinate.

This works only when the two points share a coordinate.

Worked example

Find the mistake in Kai's route

Kai walks along the streets from the library, Point A at $(-5, 4)$ , south to the museum, Point E at $(-5, -3)$ , then east to the bakery, Point F at $(-1, -3)$ . Kai writes: Leg 1 is $|4| + |-3| = 7$ blocks. Leg 2 is $|-5| + |-1| = 6$ blocks. The total is $13$ blocks. Find and fix Kai's mistake.

Check that each leg stays on one grid line and uses the coordinate that changes. Leg 1: both points have $x = -5$ , so $y$ changes from $4$ to $-3$ . Leg 2: both points have $y = -3$ , so $x$ changes from $-5$ to $-1$ . Kai chose the right coordinates for both legs.
Check the sides for leg 1. The $y$ -coordinates $4$ and $-3$ have different signs, so the library and the museum are on opposite sides of the $x$ -axis. The leg crosses the axis, so adding is correct.
$|4| + |-3| = 4 + 3 = 7$
Check the sides for leg 2. The $x$ -coordinates $-5$ and $-1$ are both negative, so the museum and the bakery are both west of the $y$ -axis, on the same side. Adding is the mistake. Kai's $6$ measures a walk from the museum to the $y$ -axis and then back to the bakery.
Fix leg 2. The bakery's 1 unit from the axis is part of the museum's 5 units, so subtract the smaller absolute value from the larger. The map agrees: from $x = -5$ to $x = -1$ is 4 grid units, because the grid lines are 1 unit apart.
$|-5| - |-1| = 5 - 1 = 4$
Add the two correct legs to get the length of the whole route. Each grid unit stands for 1 block, so the route is 11 blocks long.
$7 + 4 = 11$

Answer

Kai added on leg 2, but the museum and the bakery are on the same side of the $y$ -axis, so that leg is $5 - 1 = 4$ blocks. The whole route is $7 + 4 = 11$ blocks, not $13$ blocks.

Five questions for checking distance work

Ask yourself	Mistake it catches	Example from the town map
Do the two points share a coordinate?	Using the absolute value rule when the points are not on one grid line	Library $(-5, 4)$ and bakery $(-1, -3)$ share none. Add the legs of a street route instead.
Did the work use the coordinate that changes?	Using the coordinate that is the same for both points	School $(3, 4)$ and park $(3, -1)$ : $x$ is the same, so use $y$ . They are $4 + 1 = 5$ blocks apart.
Are the points on opposite sides of the axis?	Subtracting the absolute values when you should add	Library and school: $5 + 3 = 8$ blocks, not $5 - 3 = 2$ blocks.
Are the points on the same side of the axis?	Adding the absolute values when you should subtract	Museum and bakery: $5 - 1 = 4$ blocks, not $5 + 1 = 6$ blocks.
Is the answer not negative, and does it have a unit?	Writing a negative distance or leaving off the unit	Library to school: write $8$ blocks, not $-8$ and not just $8$ .

The big idea: on a coordinate map, two places on the same horizontal or vertical grid line share one coordinate, and the other coordinate does the work. Its sign tells which side of the axis a place is on, and its absolute value tells how far the place is from that axis. So you add the absolute values when the places are on opposite sides of the axis, and you subtract the smaller from the larger when they are on the same side. Then you turn grid units into the map's real unit, such as blocks. To catch mistakes, ask the five questions in the table every time you check distance work.