3rd Grade · 3.OA.A.4

Finding the unknown factor in 3rd grade

An unknown factor is a missing number in a multiplication or division equation, as in $4 \times ? = 36$ . A factor is one of the numbers you multiply. In 3rd grade, your child finds the missing number by using how multiplication and division fit together.

What the standard says

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

3.OA.A.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers.

What it means, in plain words

In $4 \times 9 = 36$ , the numbers 4 and 9 are the factors, and 36 is the product, the answer to a multiplication. When one factor is hidden behind a box or a question mark, it is called the unknown factor or the missing factor. Your child reads $4 \times ? = 36$ as a question: 4 times what makes 36?

Multiplication and division are the same fact seen from two sides. The numbers 4, 9 and 36 make a fact family, a set of facts that use the same three numbers: $4 \times 9 = 36$ , $9 \times 4 = 36$ , $36 \div 4 = 9$ and $36 \div 9 = 4$ . So a missing number multiplication problem can be solved with division, and a division problem can be solved with multiplication.

The missing number can sit anywhere. It can be the total, as in $? \div 3 = 7$ , or the number you divide by, as in $40 \div ? = 8$ . It can even be on the left of the equal sign, as in $24 = ? \times 6$ . The equal sign means "is the same as", so both sides must have the same value.

Your child should be able to:

Find the missing factor in an equation such as $? \times 6 = 54$
Find a missing number in a division equation such as $40 \div ? = 8$
Use a fact family to turn a division into a multiplication
Solve an equation where the unknown is on the left of the equal sign

Example

What number makes $4 \times ? = 36$ true?

Read it as a question: 4 times what makes 36?
Count by 4s up to 36: 4, 8, 12, 16, 20, 24, 28, 32, 36. That is 9 jumps.
Check: $4 \times 9 = 36$ .
The missing factor is 9.

Example

What number makes $? \div 5 = 6$ true?

Read it as: what number, split into 5 equal groups, gives 6 in each group?
That number is 5 groups of 6, so multiply: $5 \times 6 = 30$ .
Check: $30 \div 5 = 6$ .
The missing number is 30.

Where children usually get stuck

For $4 \times ? = 36$ , your child multiplies the two numbers shown and writes 144.
Your child sees a times sign and multiplies the numbers that are there. The equation was read as an instruction, not as a question with a missing piece.
What helps. Read the equation out loud together as a question: "4 times what makes 36?"
Your child sees $24 = ? \times 6$ and says the problem is written backward.
Most equations your child has seen put the answer after the equal sign, so the sign has come to mean "the answer is". It means "is the same as", and it works in both directions.
What helps. Read the equal sign as "is the same as" every time: "24 is the same as what times 6?"
Your child knows $36 \div 4 = 9$ but cannot answer $4 \times ? = 36$ .
Your child learned multiplication facts and division facts as two separate lists, so the two equations look unrelated. They use the same three numbers.
What helps. Make a triangle card with 4, 9 and 36 at the corners. Cover one corner with a thumb and ask which number is hiding.

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
Find the missing number that makes the equation true.
$\displaystyle ? \div 2 = 4$
Show the answer and how to solve it
Answer. 8
The missing number is $8$ . To find a missing dividend, multiply the divisor by the quotient: $2 \times 4 = 8$ .
1. 1The ? marks the missing dividend — the number being divided. You need to find the number that, when divided by $2$ , gives $4$ .
2. 2Division and multiplication are inverse operations. If $? \div 2 = 4$ , then the missing number must equal $2 \times 4$ .
3. 3Multiply the divisor by the quotient:
4. 4Check by substituting $8$ back into the equation:
Common mistake. A common mistake is computing $4 \div 2$ — treating the equation as ordinary division. The ? is the dividend (the number being divided), so the missing number is found by multiplying $2 \times 4$ , not by dividing $4$ by $2$ .
The idea underneath. When the dividend is missing in a division equation, multiply the divisor and the quotient. Division undoes multiplication, so $2 \times 4$ rebuilds the original number that was divided.
2On level
What number goes in the blank?
$\displaystyle \underline{\phantom{00}} \times 7 = 21$
Show the answer and how to solve it
Answer. 3
The missing number is $3$ because $3 \times 7 = 21$ .
1. 1Look at the equation. We need to find the number that, when multiplied by $7$ , equals $21$ .
2. 2The equation shows groups of $7$ with $21$ in all. The blank tells how many groups there are. Divide $21$ into groups of $7$ to count the groups:
3. 3Check by multiplying $3$ and $7$ to make sure the product is $21$ :
Common mistake. A common mistake is using a neighboring fact, like thinking $4 \times 7 = 21$ (which actually equals $28$ ). Always check by multiplying your answer back to make sure it equals the product shown.
The idea underneath. The blank counts groups of $7$ that make $21$ in all. Dividing $21$ into groups of $7$ finds that count: $21 \div 7 = 3$ .
3On level
What number is missing from the equation below?
$\displaystyle 9 \div \underline{\phantom{00}} = 3$
Show the answer and how to solve it
Answer. 3
The missing number is $3$ because $3 \times 3 = 9$ , which means $9 \div 3 = 3$ .
1. 1Look at the division equation. We need to find what number divides $9$ to give a result of $3$ .
2. 2Think of a multiplication fact instead. The equation $9 \div ? = 3$ means: what number times $3$ equals $9$ ?
3. 3Recall the multiplication fact $3 \times 3 = 9$ . So the missing number is $3$ .
4. 4Check by dividing: $9 \div 3 = 3$ . The equation is true.
Common mistake. A common mistake is to use the wrong operation, like subtracting ( $9 - 3 = 6$ ) instead of dividing. Remember that division and multiplication are inverse operations: think about which number, multiplied by $3$ , gives $9$ .
The idea underneath. Division and multiplication are inverse operations. To find a missing divisor, ask: what number times the quotient equals the dividend?
4Stretch
Maya is solving an equation. Her work is shown below.
$\displaystyle \begin{array}{l} ? \div 6 = 8 \\ \text{She wrote: } ? = 8 \div 6 \end{array}$
Select the statement that best describes Maya's work.
A
Maya divided 8 by 6 instead of multiplying to find the correct value 48.
B
Maya used the divisor 6 as the dividend instead of the correct value 48.
C
Maya found the correct value 48.
D
Maya added 8 and 6, getting 14 instead of the correct value 48.
Show the answer and how to solve it
Maya divided $8$ by $6$ instead of multiplying. To find the missing number being divided, multiply: $6 \times 8 = 48$ .
1. 1Look at the equation Maya is solving and what she wrote.
2. 2The equation $? \div 6 = 8$ asks: what number, when divided by $6$ , gives $8$ ? To find the missing number being divided (the dividend), multiply the divisor by the quotient.
3. 3Maya wrote $? = 8 \div 6$ . She divided instead of multiplying. The missing number is $6 \times 8 = 48$ .
4. 4The matching statement says Maya divided $8$ by $6$ instead of multiplying to find the correct value $48$ .
Common mistake. A common mistake is dividing whenever a division equation has a missing number. When the missing number is the dividend (the number being divided), you multiply the divisor by the quotient. For $? \div 6 = 8$ , the missing dividend is $6 \times 8 = 48$ .
The idea underneath. Multiplication and division are inverse operations. To find a missing dividend, multiply the divisor by the quotient. In this problem, the divisor is $6$ and the quotient is $8$ , so the missing number is $6 \times 8 = 48$ .