Grade by Grade
B. Number · Number Sense: Numbers to 100 000, Fractions, Decimals and Percents

Comparing and Ordering Fractions

B1.4 A1About 40 minutes
Learning goals
  • I can compare fractions with the same denominator or the same numerator.
  • I can use 1/2 and 1 as benchmarks to compare fractions.
  • I can use equivalent fractions to compare fractions with different denominators.
  • I can order fractions from least to greatest and greatest to least.

Words to know

benchmark
A friendly number you compare with, like 0, 1/2 or 1.
common denominator
The same denominator. 2/3 and 3/4 can both be written in twelfths: 8/12 and 9/12.
unit fraction
A fraction with 1 as its numerator, like 1/5.

Remember from Grade 4

In Grade 4 you compared fair shares. You learned that when a whole is shared among more people, each piece is smaller: 1/8 of a pizza is less than 1/6 of the same pizza. This year you'll compare any fractions from halves to twelfths.

Let's start with a question

Ava and Ben each have a sub sandwich of the same size. Ava eats 3/8 of hers. Ben eats 3/4 of his. Who ate more?

Both ate 3 pieces. But Ben's pieces are fourths, which are bigger than eighths. So Ben ate more: 3/4 > 3/8.

Five smart strategies

  1. Same denominator? Compare the numerators. 5/8 > 3/8 (5 pieces > 3 pieces of the same size).
  2. Same numerator? The smaller denominator means bigger pieces. 2/3 > 2/5.
  3. Benchmark 1/2. Is each fraction less than or more than half? 3/8 < 1/2 (half of 8 is 4), but 4/6 > 1/2 (half of 6 is 3).
  4. Make the denominators the same. Use equivalent fractions: 2/3 = 8/12 and 3/4 = 9/12, so 2/3 < 3/4.
  5. How far from 1? 7/8 is one eighth away from 1. 5/6 is one sixth away. Eighths are smaller, so 7/8 is closer to 1 and greater.

Remember: only compare fractions of the same-size whole.

2/38/123/49/12
2/3 = 8/12 and 3/4 = 9/12. The shaded part of 3/4 is a little longer, so 3/4 > 2/3.
Worked example

Example 1: Same denominator

Compare 5/12 and 7/12.

  1. The pieces are the same size (twelfths).
  2. 5 pieces is less than 7 pieces.

5/12 < 7/12

Worked example

Example 2: Same numerator

Compare 3/5 and 3/8.

  1. Both have 3 pieces.
  2. Fifths are bigger than eighths, because the whole is cut into fewer parts.

3/5 > 3/8

Worked example

Example 3: Use the benchmark 1/2

Compare 3/8 and 5/6.

  1. Half of 8 is 4. 3/8 is less than 1/2.
  2. Half of 6 is 3. 5/6 is more than 1/2.

3/8 < 5/6

Worked example

Example 4: Make the denominators the same

Compare 2/3 and 3/4.

  1. Find a denominator both can become: 3 × 4 = 12.
  2. 2/3 = 8/12 (× 4) and 3/4 = 9/12 (× 3).
  3. 8/12 < 9/12.

2/3 < 3/4

011/41/22/35/6
A number line from 0 to 1 marked in twelfths. 1/4 = 3/12, 1/2 = 6/12, 2/3 = 8/12 and 5/6 = 10/12.
Worked example

Example 5: Order a list

Order from least to greatest: 1/2, 2/3, 1/4, 5/6.

  1. Write them all in twelfths: 1/2 = 6/12, 2/3 = 8/12, 1/4 = 3/12, 5/6 = 10/12.
  2. Order the twelfths: 3, 6, 8, 10.

1/4, 1/2, 2/3, 5/6

Check with a second strategy

There's often more than one way to compare. Try one strategy, then check with another. If your two answers don't match, that's a great clue to look again!

Just for fun

Funny but true: years ago, a restaurant chain sold a 1/3-pound burger to compete with a 1/4-pound burger. Many customers skipped it because they thought 1/4 was bigger, since 4 is bigger than 3! You know better: thirds are bigger pieces than fourths.

Hands-on activity

Fold a fraction wall

You need: 6 strips of paper all the same length (cut from the long side of a sheet; ask a grown-up to help with scissors), a pencil and a ruler.

  1. Leave one strip whole. Label it 1.
  2. Fold one strip in half. Label each part 1/2.
  3. Fold one in half, then in half again: fourths. Fold another in half three times: eighths.
  4. Fold one into thirds (like folding a letter). It may take a few tries! Fold the last strip into thirds the same way, then fold it in half: sixths.
  5. Line up the strips one above the other. Use them to compare: 3/4 or 5/8? 2/3 or 3/4? 5/6 or 7/8? Write each answer with > or <.

Check: 3/4 > 5/8, 2/3 < 3/4, 5/6 < 7/8.

Your turn

1

Use >, < or =: 4/9 ___ 7/9

2

Use >, < or =: 2/5 ___ 2/7

3

Use >, < or =: 1/3 ___ 4/12

4

Use >, < or =: 5/8 ___ 3/10

5

Use >, < or =: 3/4 ___ 5/6

6

Order from least to greatest: 3/8, 1/8, 7/8

7

Order from least to greatest: 1/2, 1/6, 1/3

More practice: warm-up

1

Use >, < or =: 1/4 ___ 3/4

2

Use >, < or =: 1/2 ___ 1/10

3

Which is bigger: 1/5 or 1/9?

4

Is 4/8 equal to 1/2?

5

Which is closer to 1: 9/10 or 1/10?

More practice: on your own

1

Compare 7/12 and 5/12.

2

Compare 2/3 and 7/12.

3

Compare 4/10 and 3/5.

4

Order from greatest to least: 5/6, 1/4, 1/2

5

Lin ate 2/3 of a sandwich. Raj ate 3/6 of a sandwich the same size. Who ate more?

Challenge

1

Compare 6/7 and 8/9 without drawing.

2

Order from least to greatest: 3/4, 5/6, 7/12, 2/3

3

Ali says 1/4 of his pizza could be more than 1/2 of Bea's pizza. Could he be right? Explain.

Check your understanding

Quiz

1.

Which is greater?

2.

Which is greater?

3.

Which fraction is less than 1/2?

4.

Compare 2/3 and 5/6.

5.

Which list is in order from least to greatest?

6.

Why is 1/8 less than 1/6 (of the same whole)?

For parents and teachers

Big idea: B1.4 asks Grade 5 students to compare and order fractions from halves to twelfths, including improper fractions and mixed numbers (next lesson), in various contexts. Children should have several strategies, not just one rule.

Common difficulties:

  • Thinking a bigger denominator means a bigger fraction (1/8 > 1/6). Fold paper strips to show it.
  • Comparing only numerators when denominators differ (3/8 > 2/3 "because 3 > 2").
  • Forgetting that wholes must be the same size.

How to help: At snack time ask, "Would you rather have 2/3 or 3/4 of this granola bar?" Ask your child to explain two ways. Praise checking with a second strategy, and patience while folding thirds. Supervise scissors when cutting strips.