Absolute Value: Magnitude, Order, and Common Mistakes | Grade 6 Math | LevelKite

You already know that the absolute value of a number is its distance from $0$ on the number line. In this lesson you use that distance to describe real quantities.

In a real situation, a positive or negative number answers two questions:

Which side of zero? The sign tells you the direction, such as above or below, or gain or loss.
How far from zero? The absolute value tells you the size.

The size of a quantity, without its direction, is its magnitude. A magnitude is a distance, so it is never negative.

Start with elevation, which tells you how high something is compared with sea level. Here you measure it in feet. Sea level is $0$ , a positive elevation is above sea level, and a negative elevation is below it. A diver is at an elevation of $-40$ feet.

On the number line below, every tick (each small mark on the line) has a printed number, and the numbers go up by $10$ from one tick to the next. So each step from one tick to the next is $10$ feet. The marker sits exactly on the tick at $-40$ . Start at $0$ and count equal steps to the marker: $10$ , $20$ , $30$ , $40$ . That is $4$ steps of $10$ .

So the diver's distance from $0$ is $4 \times 10 = 40$ , and $|-40| = 40$ . The negative sign tells you the diver is below sea level. The absolute value tells you how far: $40$ feet.

The marker at

-40

is

4

equal steps of

10

from

0

, so

|-40| = 40

.

More situations: the sign gives the direction from 0 and the absolute value gives the magnitude.

Situation	What 0 means	Absolute value	What the magnitude tells you
Account balance of $-30$ dollars	No money and no debt	$\|-30\| = 30$	A debt of $30$ dollars
Elevation of $25$ feet	Sea level	$\|25\| = 25$	$25$ feet above sea level
Change in a game score of $-12$ points	No change in the score	$\|-12\| = 12$	A loss of $12$ points
River level of $-1.5$ feet compared with normal	The river's normal level	$\|-1.5\| = 1.5$	$1.5$ feet below the normal level

In every row, the sign gives the direction from $0$ and the absolute value gives the magnitude. The elevation is positive, so that number and its absolute value are equal. Words such as debt, loss, above and below already give the direction, so the number that goes with them is the absolute value.

Some number lines print a number at only some of the ticks. The ticks are still equally spaced, so before you count steps, find the size of one step. Take two printed numbers that are next to each other, find how far apart they are, and divide by the number of equal steps it takes to get from one to the other. The next example uses a line like this.

Only the whole numbers are printed. Count the equal steps between

-1

and

0

to find the size of one step.

Worked example

Reading a temperature below zero

Point A on the number line above marks the temperature in a town at dawn, in degrees Celsius (°C). On this scale, $0$ °C is the freezing point of water, and negative temperatures are below it. What is the temperature, and how far is it from the freezing point?

Find the size of one step. Not every tick has a printed number. The printed numbers −1 and 0 are next to each other and are 1 apart. It takes 2 equal steps to get from one to the other, so divide 1 by 2.
$1 \div 2 = 0.5$
Count equal steps from 0 to Point A. Point A sits exactly on the fifth tick to the left of 0, so it is 5 steps away. Each step is 0.5, so multiply to find the distance from 0.
$5 \times 0.5 = 2.5$
Use the direction to name the number. Point A is to the left of 0, so the number is negative. The temperature is −2.5 °C.
Write the distance you counted as an absolute value. An absolute value is a distance from 0, so it has no negative sign.
$|-2.5| = 2.5$
Read both parts in the situation. The negative sign means below the freezing point. The absolute value, 2.5, is the magnitude: how many degrees below.

Answer

The temperature is $-2.5$ °C. Because $|-2.5| = 2.5$ , it is $2.5$ degrees below the freezing point of water.

Now compare two quantities. You can ask two different questions, and they do not always have the same answer.

Order asks which number is less. On a horizontal number line, the lesser number is the one farther to the left.
Comparing absolute values asks which number is farther from zero, in either direction. When both numbers sit on ticks of the same line, you find out by counting equal steps from $0$ to each one.

A bank account shows the difference. An account balance is an amount of money in dollars, and $0$ means no money and no debt. A negative balance means the owner owes money, and the size of what the owner owes is the debt. On the number line below, every tick has a printed number and each step is $10$ dollars. Maya's balance is the marker at $-50$ . Leo's balance is the marker at $-30$ .

-50

is to the left of

-30

. It is also more steps from

0

:

5

steps of

10

compared with

3

steps of

10

.

Worked example

Lower balance, greater debt

Maya's account balance is $-50$ dollars and Leo's is $-30$ dollars, as marked on the number line above. Whose balance is less? Who has the greater debt?

Answer the order question by position. The marker at −50 is to the left of the marker at −30, and on a number line the number farther left is the lesser one. So Maya's balance is less than Leo's.
$-50 < -30$
Find each distance from 0 by counting equal steps of 10. The marker at −50 sits 5 steps from 0. The marker at −30 sits 3 steps from 0.
$5 \times 10 = 50 \quad\text{and}\quad 3 \times 10 = 30$
Write the distances as absolute values, because the absolute value of a number is its distance from 0. Here each one is the size of a debt in dollars.
$|-50| = 50 \quad\text{and}\quad |-30| = 30$
Answer the debt question by comparing the absolute values. 50 is greater than 30, so Maya's debt is greater than Leo's.
$|-50| > |-30|$
Check that both statements can be true together. They can, because they answer different questions. The first compares positions on the line. The second compares distances from 0.

Answer

Maya's balance is less, because $-50 < -30$ . Maya also has the greater debt, because $|-50| > |-30|$ : she owes $50$ dollars and Leo owes $30$ dollars.

Key concept

Position and distance from zero are different

The sign of a number tells you which side of $0$ a quantity is on. Its absolute value tells you the quantity's magnitude: how far it is from $0$ . When a number sits exactly on a tick of a line with equal steps, that distance is the number of steps from $0$ to the tick, times the size of one step.

So there are two kinds of comparison. An order statement such as $-50 < -30$ compares positions: farther left is less. An absolute value comparison such as $|-50| > |-30|$ compares distances from $0$ . Both can be true together, even when one says less than and the other says greater than.

When do the two comparisons point the same way? It depends on the signs. Take two different numbers, neither of them $0$ .

Both positive, such as $20$ and $45$ : the lesser number is also closer to $0$ . So $20 < 45$ and $|20| < |45|$ .
Both negative, such as $-50$ and $-30$ : the lesser number is farther from $0$ . So $-50 < -30$ but $|-50| > |-30|$ .
One negative, one positive: the negative number is always the lesser one, but you must count steps to compare the absolute values. For example, $-50 < 20$ and $|-50| > |20|$ , while $-30 < 45$ and $|-30| < |45|$ .

This is why a balance less than $-30$ dollars means a debt greater than $30$ dollars. Any balance less than $-30$ lies to the left of $-30$ . So it is negative and farther from $0$ than $-30$ is. On a line where both numbers sit on ticks, you count more equal steps from $0$ to reach it.

You can use these ideas to find mistakes in someone's work. Test each statement with three checks.

Is an absolute value written as a negative number? It cannot be. When a number sits on a tick, its absolute value is the number of equal steps from $0$ to that tick, times the size of one step. So it is $0$ or positive.
Is it an order statement or an absolute value comparison? Check order by position: farther left is less. Check absolute values by counting equal steps from $0$ on the same line.
Does the reason match the statement? A fact about distances from $0$ does not settle an order statement by itself, and a fact about order does not settle an absolute value comparison by itself. You also need the signs.

Worked example

Find the mistakes

Diver A is at an elevation of $-15$ feet and Diver B is at $-9$ feet, where $0$ is sea level. Riley writes three statements. Statement 1: $|-15| = -15$ . Statement 2: $-15 > -9$ , because $15 > 9$ . Statement 3: Diver A is deeper than Diver B. Which statements have a mistake, and what are the correct statements?

Check statement 1. On a line with steps of 1 foot, −15 sits 15 equal steps from 0. A count of steps is never negative, so the absolute value is 15. Riley kept the negative sign, so statement 1 has a mistake.
$|-15| = 15$
Check statement 2. It is an order statement, so use position. On the line, −15 is to the left of −9, so −15 is the lesser number. Statement 2 has a mistake.
$-15 < -9$
Name the mistake in statement 2. Riley's reason, that 15 is greater than 9, compares the two distances from 0. That comparison of absolute values is true, but it does not give the order. For two negative numbers, the one farther from 0 is the lesser number.
$|-15| > |-9|$
Check statement 3. Both divers are below sea level, so the deeper diver is the one more steps from 0. Diver A is 15 feet below and Diver B is 9 feet below, so statement 3 is correct.
$15 > 9$

Answer

Statements 1 and 2 have mistakes. The correct statements are $|-15| = 15$ and $-15 < -9$ . Statement 3 is correct: Diver A's depth of $15$ feet is greater than Diver B's depth of $9$ feet.

Takeaway. A positive or negative number carries two facts. Its sign gives a direction from $0$ . Its absolute value gives its magnitude, which you find on a number line by counting equal steps from $0$ to its tick and multiplying by the size of one step. Order compares positions on the line, and absolute value compares distances from $0$ , so the two comparisons can point in opposite directions. A balance of $-50$ dollars is less than a balance of $-30$ dollars, yet it is the greater debt. When you check work, ask which question each statement answers: position, or distance from $0$ .