4th Grade · 4.NF.A.2

Comparing fractions in 4th grade

Comparing fractions means deciding which of two fractions is greater, such as $\frac{4}{5}$ and $\frac{7}{8}$ , and writing the answer with >, < or =. In 4th grade, the two fractions can have different top numbers and different bottom numbers. Your child learns several ways to decide and explains each answer with a drawing or a reason.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as $\frac{1}{2}$ . Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

What it means, in plain words

A fraction has two parts. The bottom number, the denominator, tells how many equal pieces the whole is cut into. The top number, the numerator, tells how many of those pieces you have. In 3rd grade, your child compared fractions that share one of these numbers, such as $\frac{2}{8}$ and $\frac{6}{8}$ .

Comparing fractions in 4th grade means comparing fractions with different denominators, where neither number matches. One way is to compare each fraction with a landmark such as $\frac{1}{2}$ or 1. Another is to rewrite both fractions with the same denominator, called a common denominator, or with the same numerator. Any common denominator works, so your child does not have to find the smallest one. A third way looks at how far each fraction is from 1 whole: $\frac{4}{5}$ is $\frac{1}{5}$ away from 1 and $\frac{7}{8}$ is only $\frac{1}{8}$ away, so $\frac{7}{8}$ is greater.

Two fractions can only be compared when they are parts of the same whole. Half of a big sandwich is more food than half of a small one, even though both are $\frac{1}{2}$ . Your child records the result with > (is greater than), < (is less than) or = (is equal to) and shows why with a drawing, such as two bars of the same length.

Your child should be able to:

Compare a fraction with $\frac{1}{2}$ or 1 to decide which of two fractions is greater
Rewrite two fractions with a common denominator and compare the numerators
Record a comparison with >, < or =
Explain the answer with a drawing of two equal wholes

Example

Which is greater, $\frac{2}{5}$ or $\frac{7}{10}$ ?

Half of 5 pieces is $2\frac{1}{2}$ pieces. 2 pieces is less than that, so $\frac{2}{5}$ is less than $\frac{1}{2}$ .
Half of 10 pieces is 5 pieces. 7 pieces is more than that, so $\frac{7}{10}$ is more than $\frac{1}{2}$ .
One fraction is below one half and the other is above it, so $\frac{7}{10} > \frac{2}{5}$ . $\frac{7}{10}$ is greater.

Example

Which is greater, $\frac{1}{3}$ or $\frac{5}{12}$ ?

Both fractions are less than $\frac{1}{2}$ , so the landmark does not decide it.
Cut each third into 4 equal pieces. Then $\frac{1}{3} = \frac{4}{12}$ .
Now both fractions count twelfths. 5 twelfths is more than 4 twelfths, so $\frac{5}{12} > \frac{1}{3}$ . $\frac{5}{12}$ is greater.

Where children usually get stuck

Your child says $\frac{3}{5}$ and $\frac{4}{6}$ are equal, because each one is 2 pieces short of a whole.
This is real reasoning: the child compares how much is missing instead of looking at one number at a time. The missing pieces are different sizes, though. Sixths are smaller than fifths, so a gap of 2 sixths is smaller than a gap of 2 fifths.
What helps. Draw two bars of the same length, cut one into fifths and the other into sixths, and shade both fractions. The empty part of the sixths bar is shorter, so $\frac{4}{6}$ is greater.
Your child says $\frac{1}{8}$ is greater than $\frac{1}{3}$ , because 8 is greater than 3.
With whole numbers, a bigger number always means more. In a fraction, a bigger denominator means the whole is cut into more pieces, so each piece is smaller.
What helps. Cut two strips of paper of the same length. Fold one into 3 equal parts and the other into 8, then hold one part of each side by side.
To get a common denominator, your child changes only the bottom number and writes $\frac{3}{5}$ as $\frac{3}{10}$ .
The child treats the denominator as a label to swap. Cutting every fifth into 2 pieces also doubles the number of shaded pieces, so the top number has to change too.
What helps. Draw $\frac{3}{5}$ as a bar, then split each fifth in half with a pencil line. Count the shaded pieces together. There are 6, so $\frac{3}{5} = \frac{6}{10}$ .

Try four practice questions

Real 4th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up

Find the missing denominators to make both fractions have a numerator of 2. Then select the symbol that compares the two fractions.

Original fraction	New numerator	New denominator
$\displaystyle \frac{1}{3}$	2
$\displaystyle \frac{2}{5}$	2
Now compare the fractions:
$\displaystyle \frac{1}{3}$		$\displaystyle \frac{2}{5}$

Show the answer and how to solve it

The missing denominators are $6$ and $5$ , and $\frac{1}{3} < \frac{2}{5}$ .

1Look at the two fractions, $\frac{1}{3}$ and $\frac{2}{5}$ , and the target common numerator of $2$ .
2To rewrite $\frac{1}{3}$ with a numerator of $2$ , multiply both the numerator and denominator by $2$ (because $1 \times 2 = 2$ ):
3The fraction $\frac{2}{5}$ already has a numerator of $2$ , so its denominator stays $5$ . Now compare $\frac{2}{6}$ and $\frac{2}{5}$ : they share the same numerator. Since $6 > 5$ , the pieces in $\frac{2}{6}$ are smaller, so $\frac{2}{6} < \frac{2}{5}$ , which means $\frac{1}{3} < \frac{2}{5}$ .

Common mistake. A common mistake is to think that the fraction with the larger denominator is the larger fraction. When two fractions have the same numerator, the OPPOSITE is true: the one with the smaller denominator is larger, because the same number of pieces is taken from larger pieces.

The idea underneath. When two fractions share a numerator, the one with the smaller denominator is larger — the pieces are bigger, so the same number of pieces gives a larger amount.

2On level
Maya compared two fractions using 0 and 1 as benchmarks. Maya's work is shown below.
Fraction Where it falls
$\displaystyle \frac{3}{4}$
less than 1
$\displaystyle \frac{4}{4}$
equal to 1
$\displaystyle \frac{3}{4} < \frac{4}{4}$
Select the statement that best describes Maya's work.
A
Maya should have marked $\frac{4}{4}$ as greater than $1$ .
B
Maya compared only the numerators of $\frac{3}{4}$ and $\frac{4}{4}$ .
C
The comparison is correct.
D
Maya identified each benchmark correctly but reversed the comparison symbol.
Show the answer and how to solve it
Maya's benchmark notes are both correct and the comparison $\frac{3}{4} < \frac{4}{4}$ is correct, so the comparison is correct.
1. 1Look at Maya's work. Maya wrote two benchmark notes and one comparison statement.
2. 2Check the first benchmark note. The fraction $\frac{3}{4}$ has a numerator smaller than its denominator ( $3 < 4$ ), so it is less than $1$ . Maya's note 'less than $1$ ' is correct.
3. 3Check the second benchmark note. The fraction $\frac{4}{4}$ has a numerator equal to its denominator, so it equals $1$ . Maya's note 'equal to $1$ ' is correct.
4. 4Use the benchmarks to compare. Since $\frac{3}{4}$ is less than $1$ and $\frac{4}{4}$ equals $1$ , the left fraction is smaller, so the correct symbol is $<$ . Maya wrote $<$ , which matches.
5. 5Both benchmark notes are correct and the comparison symbol is correct, so Maya's work has no error.
Common mistake. A common mistake is assuming a comparison must contain an error. When every benchmark note and the symbol are correct, the right choice is that the comparison is correct.
The idea underneath. To judge a benchmark comparison you must check two things: each fraction's benchmark note and the comparison symbol. Here both are right.
3On level
These fractions name parts of the same whole. Select the symbol that completes the comparison.
$\displaystyle \frac{2}{5}\;\text{?}\;\frac{3}{7}$
A
$\displaystyle <$
B
$\displaystyle =$
C
$\displaystyle >$
Show the answer and how to solve it
$\frac{2}{5}$ is less than $\frac{3}{7}$ . After rewriting both with numerator $6$ , you get $\frac{6}{15}$ and $\frac{6}{14}$ . Because $15 > 14$ , the fraction $\frac{6}{14}$ has bigger pieces, so $\frac{2}{5} < \frac{3}{7}$ .
1. 1Look at the numerators of $\frac{2}{5}$ and $\frac{3}{7}$ . The numerators are $2$ and $3$ . The smallest number that both $2$ and $3$ divide into is $6$ , so rewrite both fractions with numerator $6$ .
2. 2Rewrite $\frac{2}{5}$ with numerator $6$ by multiplying the top and bottom by $3$ :
3. 3Rewrite $\frac{3}{7}$ with numerator $6$ by multiplying the top and bottom by $2$ :
4. 4Now compare $\frac{6}{15}$ and $\frac{6}{14}$ . With the same numerator, the fraction with the smaller denominator is larger because the whole is split into fewer, bigger pieces. Since $15 > 14$ :
5. 5So $\frac{2}{5} < \frac{3}{7}$ . The correct symbol is $<$ .
Common mistake. A common mistake is thinking the fraction with the larger denominator is greater after rewriting. With the SAME numerator, the SMALLER denominator gives the LARGER fraction, because the whole is split into fewer (and therefore bigger) pieces.
The idea underneath. When two fractions have the same numerator, the one with the SMALLER denominator is LARGER, because the whole is divided into fewer, bigger pieces. Making common numerators is often easier than making common denominators when the numerators have a small common multiple.

Fraction	Where it falls
$\displaystyle \frac{3}{4}$	less than 1
$\displaystyle \frac{4}{4}$	equal to 1

4Stretch

Liam wants to compare

\frac{5}{8}

and

\frac{7}{12}

by giving them a common denominator.

Step 1	$\displaystyle \frac{5}{8} = \frac{15}{24}$
Step 2	$\displaystyle \frac{7}{12} = \frac{14}{24}$
Step 3	$\displaystyle \frac{15}{24} < \frac{14}{24}$
Step 4	$\displaystyle \frac{5}{8} < \frac{7}{12}$

Select the statement that best describes the steps shown.

After converting the fractions, Liam put the wrong symbol between them.

Liam's work is correct.

Liam changed a denominator without multiplying the matching numerator.

Liam multiplied the numerator by the wrong number when converting one fraction to the common denominator.

Show the answer and how to solve it

Both of Liam's conversions are correct, but he used the wrong symbol. Since $\frac{15}{24} > \frac{14}{24}$ , the correct conclusion is $\frac{5}{8} > \frac{7}{12}$ .

1Look at Liam's two conversions in Steps $1$ and $2$ . He is changing $\frac{5}{8}$ and $\frac{7}{12}$ to a common denominator of $24$ .
2Check each conversion. To go from denominator $8$ to $24$ , multiply by $3$ ; to go from denominator $12$ to $24$ , multiply by $2$ .
3Both conversions are correct. Now compare the new numerators: $15 > 14$ , so $\frac{15}{24} > \frac{14}{24}$ . Liam wrote $<$ in Step $3$ instead of $>$ .
4The conversions are fine. The only error is the comparison symbol: Liam used $<$ when the correct symbol is $>$ . This also makes Step $4$ wrong.

Common mistake. A common mistake is to check only the conversions and then accept the symbol without verifying it. Always compare the converted numerators directly: $15 > 14$ means the symbol must be $>$ , not $<$ .

The idea underneath. After converting to common denominators, the symbol comparing the original fractions must follow the comparison of the new numerators. Here $15 > 14$ means the symbol must be $>$ , not $<$ .