3rd Grade · 4th Grade · 3.NF.A.3 · 4.NF.A.1

Equivalent fractions in 3rd and 4th grade

Equivalent fractions are fractions that name the same amount, such as $\frac{1}{4}$ and $\frac{2}{8}$ . In 3rd grade, your child sees this with pictures and number lines and compares fractions that share a top or bottom number. In 4th grade, your child learns how to find equivalent fractions by multiplying the top and bottom by the same number. Your child also explains why that works.

What the standards say

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

3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.A.3.a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
3.NF.A.3.b Recognize and generate simple equivalent fractions, (e.g., $\frac{1}{2}$ = $\frac{2}{4}$ , $\frac{4}{6}$ = $\frac{2}{3}$ ). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
3.NF.A.3.c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
3.NF.A.3.d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

4.NF.A.1 Explain why a fraction $\frac{a}{b}$ is equivalent to a fraction $\frac{n \times a}{n \times b}$ by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

What it means, in plain words

3rd grade: the same amount, and comparing fractions 3.NF.A.3

Two fractions are equivalent when they are the same size. Fold a paper strip into 4 equal parts and shade 1 part. Fold the same strip once more so it has 8 parts. The shaded piece now covers 2 parts, so $\frac{1}{4} = \frac{2}{8}$ . On a number line, the two fractions sit on the same point.

Equivalent fractions in 3rd grade also include whole numbers. Four fourths fill the whole strip, so $\frac{4}{4} = 1$ . A whole number can be written as a fraction too: 3 whole pies cut into halves make 6 halves, so $3 = \frac{6}{2}$ .

Your child also compares fractions by their size. When comparing fractions with the same numerator, look at the bottom number. $\frac{2}{3}$ is more than $\frac{2}{8}$ , because thirds are bigger pieces than eighths. When the bottom numbers match, the fraction with more pieces is bigger. A comparison only works when both fractions are of the same whole. Half of a large pizza is more than half of a small one.

Your child should be able to:

Show with a picture or a number line that two fractions are equivalent
Write a whole number as a fraction, such as $2 = \frac{4}{2}$
Compare two fractions with the same top or bottom number and explain why
Record a comparison with >, = or <

Example

Which is more of the same pan of brownies, $\frac{2}{3}$ of the pan or $\frac{2}{8}$ of the pan?

Both fractions have 2 pieces, so look at the size of the pieces.
A pan cut into 3 parts has bigger pieces than a pan cut into 8 parts.
2 big pieces are more than 2 small pieces, so $\frac{2}{3} > \frac{2}{8}$ .
$\frac{2}{3}$ of the pan is more brownie.

4th grade: why multiplying the top and bottom works 4.NF.A.1

Here is how to find equivalent fractions in 4th grade: multiply the top and the bottom by the same number. Multiply both numbers in $\frac{2}{5}$ by 2 and you get $\frac{4}{10}$ . Multiply both by 20 and you get $\frac{40}{100}$ . From one fraction, your child can make a whole list of equivalent fractions.

Equivalent fractions in 4th grade come with a reason, shown with a picture. Draw $\frac{2}{5}$ on a strip, then cut each fifth in half. Now there are twice as many parts, 10 instead of 5, and each part is half as big. The shaded amount has not changed, and it now covers 4 of the 10 parts. The number of parts and their size both change, but the fraction stays the same size.

This builds on 3rd grade, where a fraction became a number with its own place on the number line. $\frac{2}{5}$ and $\frac{4}{10}$ are one point on the line with two names. Your child does not have to write every fraction in its smallest form. Simplifying is a handy habit, not part of this standard.

Your child should be able to:

Multiply the top and bottom of a fraction by the same number to make an equivalent fraction
Explain with a picture why the parts get smaller as the number of parts grows
List several fractions equal to one fraction, such as $\frac{1}{4} = \frac{2}{8} = \frac{3}{12}$
Decide whether two fractions are equivalent

Example

Find a fraction equal to $\frac{3}{4}$ that has 12 as its bottom number.

$4 \times 3 = 12$ , so the bottom number was multiplied by 3.
Multiply the top number by 3 as well: $3 \times 3 = 9$ .
In a picture, each fourth is cut into 3 smaller parts, so 3 shaded fourths become 9 shaded twelfths.
The equivalent fraction is $\frac{9}{12}$ .

Where children usually get stuck

Your child says $\frac{3}{8}$ is bigger than $\frac{3}{4}$ , because 8 is bigger than 4.
Your child reads the bottom number as a count, where more is more. It tells the size of the pieces, and more pieces means smaller pieces. Years of whole numbers make this one of the most common fraction mistakes.
What helps. Fold one paper strip into fourths and another into eighths. Shade 3 parts of each and lay the strips side by side.
To find a fraction equal to $\frac{1}{3}$ , your child adds 1 to the top and the bottom and writes $\frac{2}{4}$ .
Your child knows both numbers must change in the same way, but adding does not keep the size. Only multiplying cuts every part into the same number of smaller parts.
What helps. Draw $\frac{1}{3}$ and $\frac{2}{4}$ on two strips of the same length. The shaded lengths do not match, and that settles it.
Your child says $\frac{4}{8}$ is more than $\frac{1}{2}$ , because 4 pieces are shaded and only 1 is.
Your child counts the shaded pieces and ignores their size. Each eighth is much smaller than a half, and 4 eighths cover exactly the same space as 1 half.
What helps. Break a chocolate bar with 8 squares into two halves. Count the squares in one half together.

Try four practice questions

Real 3rd 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
Match each fraction to the number line that shows the same point.
Connect each item on the left to its match on the right.
Fraction
Number Line
$\displaystyle \frac{1}{3}$
$\displaystyle \frac{1}{2}$
Use Tab to navigate between items. Click or tap a card to select it, then click or tap a card on the opposite side to create a match. You can also drag a card or its connector dot to a card on the opposite side.
Show the answer and how to solve it
$\frac{1}{3}$ pairs with the line marked at $\frac{2}{6}$ , and $\frac{1}{2}$ pairs with the line marked at $\frac{3}{6}$ . The line marked at $\frac{1}{6}$ has no equivalent in the fraction column.
1. 1Look at the first fraction, $\frac{1}{3}$ . All number lines show sixths, so find an equivalent fraction with a denominator of $6$ .
2. 2Do the same for $\frac{1}{2}$ . Multiply both the top and bottom by $3$ to get an equivalent number of sixths.
3. 3Now match each fraction to the number line whose marked point is at the equivalent position. $\frac{1}{3}$ matches the line with point at $\frac{2}{6}$ . $\frac{1}{2}$ matches the line with point at $\frac{3}{6}$ . The remaining number line shows $\frac{1}{6}$ , which is not equivalent to either fraction, so it stays unmatched.
Common mistake. A common mistake is matching by a shared digit instead of by equivalence. For example, a student might match $\frac{1}{2}$ to the line with the point at $\frac{2}{6}$ because both contain a $2$ , but $\frac{1}{2}$ actually equals $\frac{3}{6}$ , not $\frac{2}{6}$ .
The idea underneath. Two fractions name the same point on a number line when one is a multiplied form of the other. Multiplying both the top and bottom of $\frac{1}{3}$ by $2$ gives $\frac{2}{6}$ -- same point, just split into more equal parts.
2On level
Divide the numerator and denominator of the fraction below by 2. Which fraction is the result?
$\displaystyle \frac{6}{10}$
A
$\displaystyle \frac{3}{10}$
B
$\displaystyle \frac{2}{5}$
C
$\displaystyle \frac{3}{4}$
D
$\displaystyle \frac{3}{5}$
Show the answer and how to solve it
$\frac{6}{10}$ divided by $2$ gives $\frac{3}{5}$ , because $6 \div 2 = 3$ and $10 \div 2 = 5$ .
1. 1Look at the fraction $\frac{6}{10}$ . To divide it by $2$ , divide both the numerator and the denominator by $2$ .
2. 2Divide the numerator by $2$ :
3. 3Divide the denominator by $2$ :
4. 4Put the new numerator over the new denominator to get the result:
Common mistake. A common mistake is dividing only the numerator (getting $\frac{3}{10}$ ) and leaving the denominator unchanged. Both the numerator AND the denominator must be divided by the same number for the new fraction to name the same amount as the original.
The idea underneath. When you divide both the numerator and the denominator by the same number, the new fraction names the same amount as the original — that is what makes the two fractions equivalent.
3On level
Look at the number line. Read Maya's claim. Select the statement that best describes Maya's claim.
$\displaystyle \text{Maya says the marked point is }\frac{2}{6}\text{ and }\frac{1}{3}\text{.}$
A
Both $\frac{2}{6}$ and $\frac{1}{3}$ name the marked point.
B
Neither $\frac{2}{6}$ nor $\frac{1}{3}$ names the marked point.
C
Only $\frac{1}{3}$ names the marked point.
D
Only $\frac{2}{6}$ names the marked point.
Show the answer and how to solve it
Both $\frac{2}{6}$ and $\frac{1}{3}$ name the marked point. The marked tick is $\frac{2}{6}$ .
1. 1Look at the number line. The marked point sits exactly on the tick labeled $\frac{2}{6}$ .
2. 2 $\frac{2}{6}$ is the label at the marked tick, so it names the marked point.
3. 3Split each third into $2$ equal parts to make sixths. $\frac{1}{3}$ covers $2$ sixths, and $\frac{2}{6}$ covers $2$ sixths. They cover the same distance, so $\frac{1}{3}$ names the marked point.
4. 4Both $\frac{2}{6}$ and $\frac{1}{3}$ name the marked point.
Common mistake. Check each fraction against the marked tick $\frac{2}{6}$ . Different-looking fractions can name the same point; nearby points can still be different.
The idea underneath. Fractions name the same point when they cover the same distance from zero on the same whole. Check each fraction in the claim.
4Stretch
Maya is checking whether she can find a whole number $n$ such that multiplying the numerator and denominator of $\frac{2}{4}$ by $n$ gives $\frac{3}{6}$ . Her work is shown below.
Numerator
$\displaystyle 2 \times 1 = 2$
Denominator
$\displaystyle 4 \times 1 = 4$
Maya's claim
$\displaystyle n = 1$
Select the statement that best describes Maya's work.
A
The fractions are equivalent, but no whole-number n multiplies the first fraction to make the second.
B
The student used the numerator change but did not check the denominator change.
C
The computation is correct.
D
The student used different multipliers for the numerator and denominator.
Show the answer and how to solve it
The fractions $\frac{2}{4}$ and $\frac{3}{6}$ are equivalent, but $2 \times n = 3$ has no whole-number solution. The correct statement is: 'The fractions are equivalent, but no whole-number n multiplies the first fraction to make the second.'
1. 1Look at Maya's work below.
2. 2The fractions $\frac{2}{4}$ and $\frac{3}{6}$ ARE equivalent (they both equal $\frac{1}{2}$ ), but no whole-number $n$ satisfies $2 \times n = 3$ — only a non-whole multiplier would work.
3. 3Check whether any whole number connects the numerators:
4. 4Maya's work has the error: the fractions are equivalent but no whole-number $n$ can multiply $\frac{2}{4}$ to give $\frac{3}{6}$ . The correct statement is: 'The fractions are equivalent, but no whole-number n multiplies the first fraction to make the second.'
Common mistake. A common mistake is assuming Maya's work is complete because she showed a whole-number attempt ( $n = 1$ ). However, $2 \times 1 = 2 \neq 3$ , so $n = 1$ does not connect $\frac{2}{4}$ to $\frac{3}{6}$ .
The idea underneath. Equivalent fractions may share the same value, but multiplying to reach one from the other requires a whole-number multiplier. If no whole n satisfies a × n = c, the task cannot be completed by multiplication.

Numerator	$\displaystyle 2 \times 1 = 2$
Denominator	$\displaystyle 4 \times 1 = 4$
Maya's claim	$\displaystyle n = 1$