Ordering and Absolute Value: Rational Numbers

Grade 6 | 6.NS.C.7

In this lesson:

• Order rational numbers using the number line
• Write and interpret real-world inequalities
• Define and evaluate absolute value

Learning Objectives for This Lesson

By the end of this lesson, you will:

1. Interpret as a position statement on the number line
2. Write inequalities for rational numbers in real-world contexts
3. Define absolute value as distance from 0; interpret
4. Distinguish absolute value comparisons from ordering comparisons

Which City Is Colder in January?

Minneapolis: −13°C | Moscow: −10°C

The Ordering Rule: Further Left Is Less

On a number line (left to right):

• The number to the left is always less than the number to the right
• This holds for all rational numbers — positive, negative, zero
• "Further left" means less — not "further from 0"

Ordering Pairs on the Number Line

• is right of : ✓
• is right of : ✓
• is left of : ✓

Comparing Four Pairs Using the Number Line

Real-World Inequalities: Context to Symbols

Backward: means Dead Sea is lower.

Check-In: Compare Elevations Using Ordering

Death Valley: −86 m | Dead Sea: −430 m

Which location is lower? Write an inequality comparing these elevations.

Think first, then advance for the answer.

Check-In Answer: Dead Sea Is Lower

The Dead Sea (−430 m) is lower — it has the smaller number.
−430 is further left on the number line than −86.

Absolute Value: Distance from Zero

How far is each number from 0?

Both 7 and −7 are 7 units from 0 — same distance, opposite directions.

Absolute Value Notation and Key Facts

• — 7 units from 0
• — also 7 units from 0
• — zero distance
• always — never negative

Evaluating Absolute Values: Five Examples

Absolute Value Describes Real-World Magnitude

Absolute value strips away direction — keeps only size.

• Balance : debt of
• describes the size of the debt
• Sign → direction; absolute value → amount

Check-In: Evaluate and Reflect on Absolute Value

What is ?

Can the absolute value of any number be negative?

Think, then advance.

Check-In Answer: Distance Is Never Negative

No — absolute value is never negative.

−8 is 8 steps from 0, so , not .

Common error: . The result is always .

Two Different Questions for Any Pair

For any two numbers, we can ask two separate questions:

When Ordering and Magnitude Disagree

Compare and :

Ordering (position):

Magnitude (distance from 0):

Both statements are true simultaneously — they answer different questions.

The Standard's Debt and Balance Example

"Balance less than −$30 → debt greater than $30."

• Balance less than −30: more negative →
• Debt greater than $30: larger magnitude →

Guided Practice: Two Questions Each

For each pair, answer: (a) which is greater? (b) which has greater absolute value?

1. and
2. and
3. and

Work each pair, then advance for answers.

Guided Practice Answers: Two Questions Compared

| Pair | Greater | Greater |
|------|---------|----------------|
| | | ← disagree |
| | | ← disagree |
| | equal | equal |

Two negatives: ordering and magnitude usually disagree.

Key Takeaways from Today's Lesson

✓ Further left = less — always, for all rational numbers

✓ Absolute value = distance from 0; always

✓ Ordering and magnitude are separate questions

— further from 0 does NOT mean greater

, not — distance is never negative

AND can both be true

Coming Up Next: Coordinate Plane Distances

Next: 6.NS.C.8 — Distances in the coordinate plane

• Points on the same line: distance = absolute value of the difference
• From to :
• Today's foundation carries directly into that work