B. Number · Number Sense: Big Numbers, Square Roots and Rational Numbers

Comparing and Ordering Rational Numbers

B1.3 A1About 40 minutes

Learning goals

I can compare two rational numbers using >, < and =.
I can explain why −0.3 is greater than −0.8.
I can compare negative fractions using equivalent fractions, decimals or benchmarks.
I can order positive and negative fractions and decimals from least to greatest.

Words to know

greater than (>)
Farther to the right on a horizontal number line.

less than (<)
Farther to the left on a horizontal number line.

benchmark
A friendly number used to compare, like −1, −1/2, 0, 1/2 or 1.

common denominator
A denominator two fractions can share, so they are easy to compare.

Remember from Grade 6

In Grade 6 you compared integers (−5 < −2) and compared fractions and decimals by changing them to the same form. Today you will compare negative fractions and decimals too.

The coldest night

Here are the overnight low temperatures for five nights in one week in February in a small Ontario town:

Day	Low temperature (°C)
Monday	−6.5
Tuesday	−2 1/4
Wednesday	0.5
Thursday	−8.0
Friday	−2.3

Which night was the coldest? Which was the warmest?

How to compare rational numbers

On a horizontal number line, numbers get greater as you move right.
Every negative number is less than 0, and every positive number is greater than 0.
For two negative numbers, the one closer to 0 is greater. −2 is warmer than −8, so −2 > −8. In the same way, −0.3 > −0.8.

So for negative numbers, the number that looks bigger (8 or 0.8) is actually less. Think of temperature: −8°C is colder than −2°C.

The five temperatures on a number line. −2.3 and −2 1/4 (−2.25) are very close together. Let's zoom in.

Zoomed in from −2.5 to −2. −2.3 is to the left of −2.25, so −2.3 < −2.25.

Worked example

Example 1: The coldest to the warmest

Order the five temperatures from coldest to warmest.

Change −2 1/4 to a decimal: −2.25.
The only positive number, 0.5, is the warmest.
Among the negatives, the one farthest from 0 is the coldest: −8.0, then −6.5.
Compare −2.3 and −2.25: write them with the same number of places, −2.30 and −2.25. 2.30 is farther from 0 than 2.25, so −2.30 < −2.25.

−8.0 < −6.5 < −2.3 < −2 1/4 < 0.5. Thursday was the coldest night and Wednesday was the warmest.

Worked example

Example 2: Compare negative fractions

Which is greater: −2/3 or −3/4?

Compare the positive fractions first. Use a common denominator of 12: 2/3 = 8/12 and 3/4 = 9/12. So 2/3 < 3/4.
For negatives, the order flips: −8/12 is closer to 0 than −9/12.

−2/3 > −3/4

Worked example

Example 3: Mixed forms

Order from least to greatest: −1.25, 3/8, −1 1/3, 0.4, −0.05.

Sort by sign. Negatives: −1.25, −1 1/3, −0.05. Positives: 3/8, 0.4.
Change to decimals: −1 1/3 = −1.333…, 3/8 = 0.375.
Negatives: −1.333… is farthest from 0, then −1.25, then −0.05.
Positives: 0.375 < 0.4.

−1 1/3 < −1.25 < −0.05 < 3/8 < 0.4

Use more than one strategy

Change to the same form (decimals or common denominators), use benchmarks (Is it less than −1? Between −1 and −1/2?), or sketch a number line. If one strategy gets messy, switch. Then check with a second strategy. Explaining why your answer makes sense is as important as getting it.

Hands-on activity

Coldest card wins

You need: 16 small cards or paper slips, a marker, and a partner.

Write one number on each card: −0.4, −0.04, −1/2, −1/3, −2.25, −2.5, 0.6, −0.6, −1 3/4, −1.8, 3/8, −3/8, −5/8, −0.62, 0, −1.
Shuffle and deal 8 cards to each player, face down.
Each player flips one card. The lower number (the "colder" temperature) wins both cards. The winner must explain why their number is lower. If the partner disagrees, check on a sketched number line.
Play until the cards run out. Then play again where the greater number wins.

What to look for: explanations like "−5/8 is −0.625, which is farther from 0 than −0.62, so it is lower."

Just for fun

Your toes already know how to compare negative numbers. On a winter morning they will happily tell you that −2°C is greater than −8°C!

Your turn

Use >, < or =: −0.4 ___ −0.04

Use >, < or =: −1/2 ___ −1/3

Use >, < or =: 0.25 ___ −2.5

Use >, < or =: −3.15 ___ −3.105

Order from least to greatest: 0.6, −0.6, −1.6, 1/6

Which is the greatest: −2/5, −0.35 or −1/2?

Which temperature is the coldest: −12.5°C, −12.05°C or −12.55°C?

More practice: warm-up

Use >, < or =: −3 ___ 2

Use >, < or =: −5 ___ −7

Use >, < or =: −0.9 ___ 0.1

Use >, < or =: 0 ___ −1/4

Which is greater: −1.5 or −2.5?

More practice: on your own

Order from greatest to least: −0.75, −3/5, −0.7, −2/3

Use >, < or =: −5/8 ___ −0.6

Which is greater: −1 1/4 or −1.3?

Order from least to greatest: 2.1, −2.01, −2.1, 2.01, −2

A pond's water level changed by −0.45 cm, +0.3 cm, −0.5 cm and −0.05 cm on four days. Which day had the biggest drop?

Challenge

Find a rational number between −3/4 and −2/3.

True or false: "If a > b, then −a > −b." Give an example.

Ana says −4.5 > −4.25 because 4.5 > 4.25. Explain her mistake.

Check your understanding

Quiz

Which number is the greatest?

Which statement is true?

Which list is in order from least to greatest?

Which temperature is the coldest?

Which sign makes this true? −2/3 ___ −0.6

Which statement is correct?

For parents and teachers

Big idea: This lesson finishes B1.3: compare and order rational numbers, including negative fractions and decimals. The key idea is that for negative numbers, the one closer to zero is greater.

Common difficulties:

Ignoring the sign (saying −4.5 > −4.25 because 4.5 > 4.25). Temperature talk helps: "Which is colder?"
Comparing decimals by length (thinking −3.105 is less than −3.15 because it has more digits). Add zeros so they have the same number of places.
Comparing fractions by the denominator only. Use common denominators or decimals.

How to help: Play "coldest card wins" and ask for a reason every time. This lesson builds A1 skills: choosing a strategy, explaining and justifying, and respectfully checking a partner's reasoning.