4th Grade · 4.NF.B.4

Multiplying fractions by whole numbers in 4th grade

Multiplying fractions by whole numbers means finding several equal groups of the same fraction, such as 3 groups of $\frac{2}{5}$ . In 4th grade, your child learns that $3 \times \frac{2}{5}$ is $\frac{2}{5}$ added 3 times. Your child finds the answer by counting pieces and uses it to solve word problems.

What the standard says

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

4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
4.NF.B.4.a Understand a fraction $\frac{a}{b}$ as a multiple of $\frac{1}{b}$ .
4.NF.B.4.b Understand a multiple of $\frac{a}{b}$ as a multiple of $\frac{1}{b}$ , and use this understanding to multiply a fraction by a whole number.
4.NF.B.4.c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.

What it means, in plain words

This topic builds on what your child already knows: $4 \times 3$ means 4 groups of 3. A unit fraction is one equal piece of a whole, such as $\frac{1}{3}$ . In 4th grade, your child first sees any fraction as a number of unit fractions. $\frac{5}{3}$ is 5 one-thirds, or $5 \times \frac{1}{3}$ , in the same way that 50 is 5 tens.

Multiplying fractions by whole numbers then works the same way. $3 \times \frac{2}{5}$ means 3 groups of 2 fifths, which is 6 fifths, so $3 \times \frac{2}{5} = \frac{6}{5}$ . The whole number multiplies the top number. The bottom number stays the same, because the pieces are still fifths. Some worksheets call this topic fractions times whole numbers.

Multiplying fractions by whole numbers in 4th grade also shows up in word problems. The whole number is usually how many times something happens, and the fraction is the amount each time. Your child also says which two whole numbers the answer falls between, which is a quick way to check that it makes sense.

Your child should be able to:

Write a fraction as a whole number times a unit fraction, such as $\frac{7}{4} = 7 \times \frac{1}{4}$
Multiply a whole number by a fraction by counting groups of pieces
Write an answer greater than 1 as a mixed number, such as $\frac{8}{3} = 2\frac{2}{3}$
Solve word problems and say which two whole numbers the answer falls between

Example

What is $4 \times \frac{2}{3}$ , and which two whole numbers is it between?

$4 \times \frac{2}{3}$ means 4 groups of $\frac{2}{3}$ : $\frac{2}{3} + \frac{2}{3} + \frac{2}{3} + \frac{2}{3}$ .
Each group is 2 thirds, so that is $4 \times 2 = 8$ thirds, or $\frac{8}{3}$ .
6 thirds make 2 wholes, so $\frac{8}{3} = 2\frac{2}{3}$ . The answer is $\frac{8}{3}$ , which is between 2 and 3.

Example

A recipe uses $\frac{3}{4}$ cup of milk. How much milk do you need to make the recipe 5 times?

Multiply: $5 \times \frac{3}{4}$ .
That is 5 groups of 3 fourths, or $5 \times 3 = 15$ fourths: $\frac{15}{4}$ .
12 fourths make 3 wholes, so $\frac{15}{4} = 3\frac{3}{4}$ .
You need $3\frac{3}{4}$ cups of milk, which is between 3 and 4 cups.

Where children usually get stuck

Your child writes $2 \times \frac{3}{8} = \frac{6}{16}$ , multiplying both the top and the bottom number.
The child multiplies every number in sight. The bottom number names the size of the pieces, and 2 groups of eighths are still eighths.
What helps. Say it in words: 2 groups of 3 eighths is 6 eighths. Draw the two groups if needed and count the pieces, which gives $\frac{6}{8}$ .
Your child says $\frac{5}{3}$ is not a real fraction, because the top number is bigger than the bottom.
Fractions were first learned as parts of one whole, so having more pieces than one whole holds seems impossible.
What helps. Count thirds out loud on a number line: 1 third, 2 thirds, 3 thirds (that is 1 whole), 4 thirds, 5 thirds. The count simply keeps going past 1.
For "5 friends each eat $\frac{2}{3}$ of a sandwich", your child writes $5 + \frac{2}{3} = 5\frac{2}{3}$ sandwiches.
The child sees a whole number and a fraction and puts them together, without asking what the 5 means in the story.
What helps. Draw 5 plates with $\frac{2}{3}$ of a sandwich on each and count the thirds. There are 10 thirds, or $3\frac{1}{3}$ sandwiches.

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
Drag unit fraction tiles into the slots to build the fraction shown. Leave any extra slots empty.
$\displaystyle \frac{2}{3}$
Tap or click a tile to drop it into the highlighted blank, or tap an empty blank first to choose a different one. Tap a filled blank to send its tile back.
$\displaystyle \frac{1}{2}$
$\displaystyle \frac{1}{3}$
$\displaystyle \frac{1}{4}$
Tap an item to place it in the next empty slot, or drag it to a specific slot. With a keyboard: select an item with Enter or Space, then select a slot to place it. Press Delete or Backspace on a slot to remove an item.
Show the answer and how to solve it
Place $2$ copies of the unit fraction $\frac{1}{3}$ into the slots. The extra slots remain empty because $\frac{2}{3}$ is exactly $2$ copies of $\frac{1}{3}$ .
1. 1Look at the target fraction.
2. 2Look at the parts of $\frac{2}{3}$ . The denominator $3$ tells you the unit piece is $\frac{1}{3}$ . The numerator $2$ tells you to use $2$ of those pieces.
3. 3Drag $2$ copies of the $\frac{1}{3}$ tile into the slots. The remaining slots stay empty.
Common mistake. A common mistake is to use a unit fraction whose denominator does not match the target. For $\frac{2}{3}$ the unit fraction must be $\frac{1}{3}$ , not $\frac{1}{2}$ (using the numerator as the denominator) or $\frac{1}{4}$ (a neighboring denominator).
The idea underneath. $\frac{2}{3}$ is the same as $\frac{1}{3}$ added $2$ times. The numerator counts how many unit pieces; the denominator names which unit piece.
2On level
Drag tiles into the blanks to complete the sentence.
Tap or click a tile to drop it into the highlighted blank, or tap an empty blank first to choose a different one. Tap a filled blank to send its tile back.
$\displaystyle \frac{3}{6} \text{ means }$
copies of
.
3
6
$\displaystyle \frac{1}{6}$
$\displaystyle \frac{3}{6}$
$\displaystyle \frac{1}{3}$
Tap an item to place it in the next empty slot, or drag it to a specific slot. With a keyboard: select an item with Enter or Space, then select a slot to place it. Press Delete or Backspace on a slot to remove an item.
Show the answer and how to solve it
The fraction $\frac{3}{6}$ means $3$ copies of $\frac{1}{6}$ . The numerator $3$ counts how many copies; the denominator $6$ names the unit fraction.
1. 1Look at the fraction $\frac{3}{6}$ shown in the sentence frame.
2. 2The denominator $6$ tells the size of each equal part. Each part is $\frac{1}{6}$ of a whole.
3. 3The numerator $3$ tells how many of those $\frac{1}{6}$ parts you have. So $\frac{3}{6}$ means $3$ copies of $\frac{1}{6}$ .
4. 4Drag the tile $3$ into the first slot (the count) and the tile $\frac{1}{6}$ into the second slot (the unit fraction).
Common mistake. A common mistake is dragging $\frac{1}{3}$ into the second slot. The unit fraction is $\frac{1}{6}$ — its denominator must match the displayed denominator $6$ , not the numerator $3$ .
The idea underneath. Every fraction is built by counting copies of a unit fraction. For $\frac{3}{6}$ , the unit fraction is $\frac{1}{6}$ and there are $3$ copies. The bottom number names the unit; the top number counts copies.
3On level
Diego was asked to write an explanation of what $\frac{3}{8}$ means. Here is what he wrote:
$\displaystyle \frac{3}{8} \text{ is } 3 \text{ copies of } \frac{1}{6}$
Select the statement that best describes Diego's explanation.
A
Both the number of copies and the unit fraction are correct.
B
The number of copies is wrong, but the unit fraction is correct.
C
The number of copies is correct, but the unit fraction is wrong.
D
Both the number of copies and the unit fraction are wrong.
Show the answer and how to solve it
The correct choice is "The number of copies is correct, but the unit fraction is wrong." Diego's count of $3$ matches the numerator of $\frac{3}{8}$ , but his unit fraction $\frac{1}{6}$ should be $\frac{1}{8}$ to match the denominator $8$ .
1. 1First, figure out what $\frac{3}{8}$ actually means in copies-of language. The numerator $3$ tells how many copies, and the denominator $8$ names the unit fraction $\frac{1}{8}$ .
2. 2Now look at Diego's claim again.
3. 3Compare the two parts of the claim. Diego's count of $3$ matches the numerator $3$ of $\frac{3}{8}$ , so the number of copies is correct. But Diego's unit fraction $\frac{1}{6}$ does NOT match $\frac{1}{8}$ , so the unit fraction is wrong.
4. 4So the correct description is: "The number of copies is correct, but the unit fraction is wrong."
Common mistake. A common mistake is mixing up which number tells the count and which names the unit fraction. In $\frac{3}{8}$ , the numerator $3$ tells how many copies, and the denominator $8$ names the unit fraction $\frac{1}{8}$ .
The idea underneath. $\frac{3}{8}$ is $3$ copies of the unit fraction $\frac{1}{8}$ . The top number counts the copies; the bottom number names the size of each copy. Every fraction works the same way — check both parts independently when judging an explanation.
4Stretch
Click the tick on the number line where the fraction belongs.
$\displaystyle \frac{5}{4}$
Tap or click the number line to place your point. Drag it to move, or tap it again to remove.
Show the answer and how to solve it
$\frac{5}{4}$ is the point you reach after $5$ equal jumps of $\frac{1}{4}$ from $0$ . The correct tick is the fifth one after $0$ . You will pass $1$ on the way and end up between $1$ and $2$ .
1. 1The number line goes from $0$ to $2$ . It is divided into $4$ equal parts per whole-number interval, so each tick mark is one $\frac{1}{4}$ .
2. 2Starting at $0$ , count $5$ equal jumps of $\frac{1}{4}$ .
3. 3After $5$ jumps you land at $\frac{5}{4}$ . Click the fifth tick after $0$ to show your answer.
Common mistake. A common mistake is counting the starting point ( $0$ ) as the first jump. Each jump moves $\frac{1}{4}$ , so $5$ jumps starting from $0$ land at $\frac{5}{4}$ , not $\frac{4}{4}$ . A second common mistake is stopping at $1$ instead of continuing past it.
The idea underneath. On a number line, $\frac{5}{4}$ is just $5$ copies of $\frac{1}{4}$ — the numerator counts the jumps, and the denominator tells you the size of each jump.