Absolute Value and Two Questions

In this lesson:

• Define as the distance from on the number line
• Read as the size, or magnitude, of a signed quantity
• Hold two readings of one pair: order and magnitude

What You Will Be Able to Do

By the end, you will:

1. Compute as the distance from on the line
2. Interpret as the size of a signed quantity
3. Hold two facts about a pair: order and magnitude

Recall the Puzzle From Last Lesson

We wrote . And: the debt of dollars is bigger than the debt of dollars.

How can the same pair of numbers give two opposite-sounding answers?

Plot Five and Negative Five

Plot and on a horizontal number line. Count the steps from to each.

• to : 5 steps right
• to : 5 steps left

Same distance, opposite directions.

The Definition: Distance From Zero

The absolute value of a rational number , written , is the distance from to on the number line.

Three Facts From the Definition

From "distance from ":

• for every — distance is never negative
• — opposites sit the same distance away
• — zero is its own distance from itself

Reference Line — Several Distances

Each arrow's length is that number's absolute value.

Compute By Measuring on the Line

Compute each by reading the distance off a number line:

Distance is positive; the input's sign does not matter.

The Shortcut Is a Consequence, Not the Rule

A common shortcut: " removes the negative sign."

This works for most cases — because and are the same distance from .

But: the shortcut is a consequence of the definition, not the definition itself.

Diagnostic — What Is ?

The shortcut: "drop the negative sign." But has no sign to drop. What does the shortcut say?

The definition: is the distance from to — which is .

. Zero is the only number with absolute value .

Check-In: Compute Two Absolute Values

Compute:

Read off the line — distance from in each case.

Answers: ; .

Magnitude in Three Real-World Contexts

Each signed value has a direction and a size:

• Balance dollars: the size of the debt is dollars
• Submarine at ft: the depth measurement is ft
• Temperature C: the cold severity is C

Two Questions Now — Where, and How Far

For any pair of numbers, we can ask two different questions:

• Order: Where are they on the line? (left-right)
• Absolute value: How far is each from ? (distance)

The answers can disagree. That is the point.

Order and Distance Read Off One Plot

For two numbers and on a line:

• Order reads left versus right — which side
• Absolute value reads distance from — how far each segment is

Same plot, two readings.

Side By Side: Two Negatives Compared

Both readings, same plot.

Both Inequalities Are True at Once

For the pair and :

Order points one way; absolute value points the other. Both are right.

Worked Example — The Standard's Debt Sentence

Two account balances: dollars and dollars.

• Order: — the balance is lower
• Magnitude: — the debt of is bigger

A balance less than represents a debt greater than .

Worked Example — Two Elevations Below Sea Level

A scuba diver at ft. A submarine at ft.

• Order: — submarine sits at a lower elevation
• Magnitude: — the submarine's depth is greater

Lower elevation, deeper dive — both true together.

Two Questions Give Two Inequalities

The standard's verbatim language:

An account balance less than dollars represents a debt greater than dollars.

The "less than" answers the order question. The "greater than" answers the absolute-value question.

Mixed Practice — Two Questions Each

For each pair of values, answer both questions:

1. Temperatures F and F
2. Balances and dollars
3. Elevations ft and ft

Which is less? Which has the larger absolute value?

Your Turn — Two Cold Cities

Anchorage hits F. Fairbanks hits F.

Tasks:

1. Draw a number line and plot both
2. Write both inequalities — order and absolute value
3. Interpret each in one sentence

Practice Answers and Common Errors

Mixed:

1. and
2. and
3. and

Solo: and . Fairbanks: lower reading, bigger cold severity.

Three Traps to Watch Out For

Trap 1: is not "drop the sign." It is distance from .

Trap 2: Order and absolute value are different questions. They can disagree.

Trap 3: "Less than" answers position. "Bigger debt" answers magnitude.

Where vs. How Far — and What's Next

Order asks: Where on the scale is each number?
Absolute value asks: How far from is each?

Next: in 6.NS.C.8, becomes the distance to an axis on the coordinate plane.