Ordering fractions

Q: What are benchmark fractions?

Easy reference points, usually 0, 1/2 and 1. Deciding whether each fraction is above or below 1/2 often orders them quickly.

Q: Can I use decimals to order fractions?

Yes. Divide each numerator by its denominator and compare the decimals: 5/8 = 0.625 and 2/3 ≈ 0.667.

Q: What grade learns ordering fractions?

Grade 3 orders simple fractions, and grade 4 orders any fractions (4.NF.A.2).

To put fractions in order, compare them two at a time, or rewrite them all with the same denominator and order the numerators.

Same denominator: order the numerators

When the denominators match, the pieces are the same size, so just count the pieces. 2/7, 5/7, 3/7 in order from least to greatest is 2/7, 3/7, 5/7.

This is the easiest case, and every other method turns the problem into this one.

See it: pictures to draw with your child

Copy these onto paper or a whiteboard. A quick sketch makes the idea concrete before any written method.

Bars of the same length: 5/8, 2/3 and 3/4 in order.

Same numerator: the bigger the denominator, the smaller the fraction.

Benchmarks: 3/8 is below 1/2 (4/8), and 7/8 is close to 1.

Same numerator: the bigger denominator is smaller

When the numerators match, there are the same number of pieces, but the pieces are different sizes. Bigger denominators make smaller pieces. So 3/4, 3/8, 3/5 in order from least to greatest is 3/8, 3/5, 3/4.

Children often get this backwards at first. A quick picture of a pizza cut into 4 and one cut into 8 fixes it.

Use benchmarks: 0, 1/2 and 1

Many fractions can be ordered by asking “is it more or less than a half?” A fraction is more than 1/2 when the numerator is more than half the denominator: 5/8 is more than 4/8. It is close to 1 when the numerator is nearly the denominator: 9/10.

Order 2/5, 7/8, 1/2: 2/5 is less than a half, 7/8 is nearly 1, so the order is 2/5, 1/2, 7/8. No calculation needed.

When all else fails: a common denominator

For fractions like 2/3, 5/8 and 3/4, rewrite them all with the same denominator, the least common multiple of the bottoms. Here that is 24:

2/3 = 16/24
5/8 = 15/24
3/4 = 18/24

Now order the numerators: 15/24, 16/24, 18/24, so 5/8, 2/3, 3/4. See finding a common denominator. Decimals work too: 0.625, 0.667, 0.75.

Order fractions on the fraction wall

Shade fractions on different rows. The one reaching furthest right is the greatest.

Ascending and descending order

Ascending means from least to greatest, smallest first. Descending means from greatest to least. Read the question carefully: many mistakes are correct orders written the wrong way round. A quick check is that the first fraction in ascending order should be the one closest to 0.

Quiz: order and compare

Find the smallest fraction, and choose <, > or =.

Where this fits: fractions step by step

What to say to your child

A short script for teaching it the first time. Pause after each line and let your child do the next step.

“Which is bigger, 3/4 or 3/8?” (Hmm…)
“Picture two pizzas. One cut in 4, one in 8. Which slices are bigger?” (The quarters.)
“So 3 quarters is more than 3 eighths.”
“Now 2/5, 1/2 and 7/8. Which is less than a half?” (2/5.) “Which is nearly 1?” (7/8.)
“So the order is 2/5, 1/2, 7/8.”

More worked examples

Example 1: same denominator

4/9, 1/9, 7/9 → 1/9, 4/9, 7/9.

Example 2: same numerator

2/3, 2/9, 2/5 → 2/9, 2/5, 2/3.

Example 3: benchmarks

1/6, 5/6, 1/2 → 1/6 is small, 5/6 is nearly 1: 1/6, 1/2, 5/6.

Example 4: common denominator

3/4, 5/6, 2/3 → 9/12, 10/12, 8/12 → 2/3, 3/4, 5/6.

Practice questions (answers below)

Ask these out loud or write them down. Check each answer, and ask “how did you get that?” for one or two.

Order from least: 5/8, 1/8, 3/8
Order from least: 1/2, 1/5, 1/3
Order from least: 4/7, 4/5, 4/9
Which is greater: 3/5 or 1/2?
Order from least: 1/2, 3/10, 9/10
Order from least: 2/3, 3/4, 7/12
Order from greatest: 1/4, 5/8, 1/2
Which is closest to 1: 4/5, 7/8, 2/3?
Is 5/12 more or less than 1/2?
Stretch: order 3/5, 0.5, 2/3

Show the answers

1/8, 3/8, 5/8
1/5, 1/3, 1/2
4/9, 4/7, 4/5
3/5
3/10, 1/2, 9/10
7/12, 2/3, 3/4
5/8, 1/2, 1/4
7/8
Less
0.5, 3/5, 2/3

Common mistakes

Ordering by denominators aloneSays 1/8 > 1/4 because 8 > 4.Fix: with the same numerator, the bigger denominator is smaller.

Comparing tops and bottoms separatelyThinks 3/8 > 2/3 because 3 > 2 and 8 > 3.Fix: compare the whole fraction: use a common denominator or benchmarks.

Writing the order backwardsGives greatest to least when asked for least to greatest.Fix: check the first answer is the one closest to 0.

How to teach it at home

Fraction line-up

Write fractions on cards and peg them on a string from 0 to 1.

Pizza pick

Ask which pizza share is bigger: 3/8 or 1/2?

Half or not

Call out fractions and say “more” or “less” than a half.

By grade

Grade 3 (3.NF.A.3): compare fractions with the same numerator or denominator.
Grade 4 (4.NF.A.2): compare any two fractions using benchmarks or common denominators.
Grade 5: order fractions, decimals and mixed numbers together.
Next: improper fractions and mixed numbers.

More for each grade: Grade 3 math and the CBSE class 3 maths guide; Grade 4 math and the CBSE class 4 maths guide; Grade 5 math and the CBSE class 5 maths guide.