Ordering Rational Numbers on the Line

In this lesson:

• Read as " is to the left of " on a number line
• Order pairs of negative numbers without trusting digit-size
• Translate signed real-world quantities through the line, not the words

What You Will Be Able to Do

By the end, you will:

1. Compare two rational numbers using a number line
2. Read off the line, not from a sign rule
3. Translate verbal comparisons into inequalities
4. Avoid the "more negative is greater" trap

Three Thermometers — Which Is Warmest?

A morning weather report shows three readings:

• City A:
• City B:
• City C:

Which city is warmest? Coldest?

Plot Two Numbers and Look

Draw a horizontal number line and plot and .

• sits to the right of
• The number further right is greater

So: .

The Geometric Definition of Less and Greater

For any two rational numbers and on a horizontal number line:

• means is to the left of
• means is to the right of

This is the definition. The symbolic comparisons follow from the picture.

Reading Off the Number Line

is to the right of , so .

The Line Can Stand Up Too

On a vertical number line:

• means is below
• means is above

Example: on a thermometer, is below — and .

Read These Three Pairs Off the Line

In each row, the smaller number sits to the left.

Quick Predict — Which Is Greater?

Question: Which is greater, or ?

Pause. Pick one. Then explain.

A common wrong answer says — because "7 is bigger than 3." But the line says is to the right of , so .

Positives Are Always Greater Than Negatives

Compare and :

• sits to the right of
• sits to the left of

Any positive number is greater than any negative number — the line makes it visible.

Check-In: Identify the Greater Negative

Question: Which is greater, or ? Justify in one sentence using the number line.

Write your answer before continuing.

Answer: , because is to the right of on the number line.

A Thermometer Is Still a Number Line

A temperature scale is a number line:

• Warmer = further to the right (or up)
• Colder = further to the left (or down)

So every warmth comparison is a number-line comparison.

Three Real-World Scales — All Are Number Lines

Warmer, higher, less indebted — all map to "greater" on the signed scale.

Verbal and Symbolic Say the Same Thing

The standard's example, verbatim:

means " is warmer than ."

The inequality and the verbal phrasing say the same thing — read off the same line.

Worked Example — Two Cities, Two Temperatures

City A is at . City B is at . Which is colder?

Plot: sits to the left of on the temperature line.
Read: .
Translate: City A is colder than City B.

Worked Example — Two Famously Low Places

Death Valley sits at about . The Dead Sea shore sits at about . Which is lower?

Plot: is far to the left of .
Read: .
Translate: The Dead Sea is lower than Death Valley.

Plot First, Then Translate to Words

A common error: writing for " is colder than ."

The error: colder sounds like more. But on the line, colder = to the left = less than.

Rule: Plot first. Read the inequality off the line. Then translate to words.

Mixed Practice — Three Pairs

For each pair, write the inequality and one verbal sentence:

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

Plot each pair on a line first.

Your Turn — Submarine and Diver

A submarine sits at . A scuba diver is at .

Tasks:

1. Draw a vertical number line and plot both depths
2. Write an inequality comparing the two
3. Write one sentence saying which is deeper

Answers and Common Errors to Avoid

Mixed practice:

1. . The first reading is colder.
2. . Negative two-hundred feet is lower.
3. . Negative seventy-five dollars is the lower balance — bigger debt.

Solo: . The submarine is deeper.

Watch Out — Two Traps to Avoid

Trap 1: "More negative = greater." is less than , not greater. Plot first.

Trap 2: "Colder means more, so write ." Colder corresponds to to the left, which is less than. Translate from the line, not from the word.

Order Is Position on the Line

Key takeaways:

• means is left of (below, on a vertical line)
• For two negatives, the one closer to is greater
• Verbal comparisons translate through the line

One more question is coming about the same pair.

Coming Up — Distance From Zero

In Lesson 2, we'll meet absolute value:

• is the distance from to on the line
• For two negatives, larger means smaller on the line
• A balance less than is a debt greater than