B. Number · Number Sense: Numbers to One Million, Integers, Decimals and Fractions

Comparing and Ordering Integers, Decimals and Fractions Together

B1.3 B1.6 A1About 40 minutes

Learning goals

I can compare a fraction and a decimal.
I can compare fractions using equivalent fractions, decimals or benchmarks.
I can put integers, decimals and fractions in order on one number line.
I can choose a strategy and explain why it works.

Words to know

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

common denominator
The same denominator for two or more fractions, so they can be compared, like 25/40 and 24/40.

ascending order
From least to greatest.

descending order
From greatest to least.

Remember

You can now compare whole numbers, integers and decimals to thousandths. In Grade 5 you compared fractions using benchmarks (0, 1/2, 1) and equivalent fractions. In the last lesson you changed fractions into decimals. Now we'll put everything together on one number line.

The class number line

Ms. Okafor's Grade 6 class hangs a string across the classroom as a number line. Each student gets a card: −3, 0.7, 3/4, 0, 1 1/5, −1, 0.725. They have to stand in order from least to greatest. Where should everyone go?

Strategies

Sort by sign first. Negative numbers are less than 0, and 0 is less than all positive numbers.
Change to the same form. Decimals are often easiest: 3/4 = 0.75 and 1 1/5 = 1.2. (Or change decimals to fractions with a common denominator.)
Use benchmarks. Is it less than 1/2, between 1/2 and 1, or more than 1?
Line up the places to compare decimals.

Some of the cards on a number line from −3 to 2: −3, −1, 0.7, 3/4 and 1 1/5. (0 is on the line too, and 0.725 is just between 0.7 and 3/4, too close to draw at this size.)

Worked example

Example 1: Order the class cards

Order from least to greatest: −3, 0.7, 3/4, 0, 1 1/5, −1, 0.725

Negatives first: −3 is farther left than −1, so −3, −1.
Then 0.
Change the rest to decimals to thousandths: 0.7 = 0.700, 3/4 = 0.750, 0.725, 1 1/5 = 1 2/10 = 1.200.
Order them: 0.700, 0.725, 0.750, 1.200.

−3, −1, 0, 0.7, 0.725, 3/4, 1 1/5

Worked example

Example 2: Compare two fractions

Which is greater: 5/8 or 3/5? Use two strategies.

Strategy 1, decimals: 5/8 = 0.625 and 3/5 = 0.6 = 0.600. 0.625 > 0.600.
Strategy 2, common denominator: 8 × 5 = 40. 5/8 = 25/40 and 3/5 = 24/40. 25 > 24.

5/8 > 3/5. Both strategies agree, so we can be confident.

Worked example

Example 3: Use benchmarks

Order from least to greatest: 3/7, 0.9, 1/10, 5/9

Compare each to 1/2: 3/7 is less than 1/2 (3.5 would be half of 7). 5/9 is more than 1/2 (4.5 would be half of 9).
1/10 = 0.1 is close to 0. 0.9 is close to 1.
So 1/10 is least, then 3/7 (just under 1/2), then 5/9 (just over 1/2), then 0.9.

1/10, 3/7, 5/9, 0.9

Placed to scale: 1/10 is near 0, 3/7 (about 0.43) is just under 1/2, 5/9 (about 0.56) is just over 1/2, and 0.9 is near 1.

Choose your strategy

There is often more than one good way to compare. Pick the one that feels easiest for the numbers you have, then explain why you chose it. If your first strategy gets messy, it's okay to switch to another one.

Just for fun

If the cards on the class number line could talk, −3 would say: "Don't mind me, I'll just hang out here at the cold end of the string."

Hands-on activity

Clothesline number line

You need: a piece of string or yarn about 2 m long, 10 clothespins (or tape), 10 index cards or paper scraps, and a marker.

Tie or tape the string between two chairs.
Write these on the cards: −2, −1/2, 0, 0.3, 1/3, 0.33, 5/8, 0.6, 1, 1.25.
Clip 0 and 1 on the string first, leaving room on both sides.
Shuffle the other cards. Take turns with your grown-up placing one card at a time. Explain your choice each time: "I put 5/8 here because it's 0.625, which is a bit more than 0.6."
When all the cards are up, check the order together.

Grade 7 peek: one card, −1/2, is a negative fraction. Negative fractions are not part of Grade 6 (they come in Grade 7), but you can still place it: it goes halfway between −1 and 0. Ask: is −1/2 more or less than −2?

Correct order: −2, −1/2, 0, 0.3, 0.33, 1/3, 0.6, 5/8, 1, 1.25. (1/3 = 0.333… is just a little more than 0.33.)

Your turn

Use >, < or =: 0.4 ___ 2/5

Use >, < or =: 7/8 ___ 0.87

Use >, < or =: −2 ___ 1/4

Use >, < or =: 2/3 ___ 0.6

Order from least to greatest: 0.5, 3/8, −1, 0.45

Order from greatest to least: 1 1/2, 1.05, 1 1/4, 1.5

More practice: warm-up

Use >, < or =: 1/2 ___ 0.5

Use >, < or =: 0.3 ___ 3/4

Which is greater: −5 or 0.1?

Is 4/9 more or less than 1/2?

Order from least to greatest: 1, 0, −1

More practice: on your own

Use >, < or =: 3/4 ___ 7/10. Use two strategies.

Order from least to greatest: 0.65, 5/8, 2/3, 0.6

Recipe A uses 0.75 L of milk. Recipe B uses 4/5 L. Which uses more?

Write a fraction that is between 0.2 and 0.3.

Order from least to greatest: −4, 2.05, 2 1/10, 0, −10, 2

Challenge

Order from least to greatest: 5/6, 0.83, 4/5, 0.8333

Find a decimal and a fraction that are both between 3/8 and 1/2.

Three hikers measure their snack bags. Ali: 0.25 kg. Bea: 3/10 kg. Cal: 275 g. Order the masses from least to greatest. Explain how you made them the same form.

Check your understanding

Quiz

Which is greater: 3/5 or 0.65?

Which list is in order from least to greatest?

Which is equal to 0.125?

Which fraction is less than 1/2?

Where does 2 1/4 go on a number line?

Which is the least?

For parents and teachers

Big idea: B1.3 asks Grade 6 students to compare and order integers, decimal numbers and fractions, separately and in combination, in various contexts. B1.6 (fraction–decimal equivalences) gives them the tools to do it.

Common difficulties:

Comparing digits instead of amounts (for example, thinking 3/5 is greater than 0.65 "because 5 is bigger than 0.65"). Change both to the same form first.
Forgetting negatives are always less than positives.
Larger denominator = larger fraction. 1/8 is less than 1/5. Benchmarks and decimals help.
Repeating decimals. 5/6 = 0.8333… is greater than 0.8333, even though they look the same at first.

How to help: The clothesline number line is a great family activity. Ask "How do you know?" and "Can you check it another way?" Choosing and explaining strategies is part of Strand A (mathematical processes). The −1/2 card gives a peek at negative fractions, which your child will study more in Grade 7.