3rd Grade · 3.OA.B.6

Fact families in 3rd grade

Fact families are groups of math facts that use the same three numbers, such as 4, 9 and 36. In 3rd grade, your child learns that division is multiplication with a missing number, so every multiplication fact also gives two division facts.

What the standard says

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

3.OA.B.6 Understand division as an unknown-factor problem.

What it means, in plain words

A factor is a number you multiply. The product is the answer. In $4 \times 9 = 36$ , the factors are 4 and 9 and the product is 36. The fact family for these numbers has four facts: $4 \times 9 = 36$ , $9 \times 4 = 36$ , $36 \div 4 = 9$ and $36 \div 9 = 4$ .

Relating multiplication and division is the big idea of this topic. A division question asks for a missing factor. To find $45 \div 5$ , your child asks: 5 times what number makes 45? The answer, 9, is the missing factor. This way, division does not need a new list of facts to memorize.

Fact families in 3rd grade are often shown as a triangle. The product sits at the top and the two factors sit at the bottom corners. Cover the top number and you multiply to find it. Cover a bottom number and you divide.

Your child should be able to:

Write all four facts in a fact family from three numbers
Turn a division question into a multiplication question with a missing factor
Use a multiplication fact to answer a division fact, such as $40 \div 8$ from $5 \times 8 = 40$
Name the factors and the product in a multiplication fact

Example

Write the fact family for 3, 7 and 21.

21 is the biggest number, so it is the product. 3 and 7 are the factors.
Multiplication facts: $3 \times 7 = 21$ and $7 \times 3 = 21$ .
Division facts: $21 \div 3 = 7$ and $21 \div 7 = 3$ .
These four facts make the fact family for 3, 7 and 21.

Example

What is $48 \div 6$ ? Solve it by finding the missing factor.

Write it as multiplication: $6 \times ? = 48$ .
Use a known fact: $6 \times 8 = 48$ .
The missing factor is 8, so $48 \div 6 = 8$ .

Where children usually get stuck

Your child knows $5 \times 9 = 45$ but draws 45 tally marks and circles groups of 9 to find $45 \div 9$ .
Division was taught as its own topic, with sharing pictures, and never got tied to the times facts already known.
What helps. Ask every division fact as a multiplication question: 9 times what makes 45? Keep asking it that way until your child starts to ask it first.
For $30 \div 6$ , your child writes $30 \times 6 = ?$ .
Your child knows multiplication is involved but not where each number goes. In a fact family, the biggest number is always the product, never a factor.
What helps. Draw a fact triangle with 30 at the top and 6 in one bottom corner. Ask what number goes in the empty corner.
Your child writes the fact family for 2, 8 and 16 as $2 \times 8 = 16$ , $8 \times 2 = 16$ , $16 - 8 = 8$ and $16 - 2 = 14$ .
In 1st and 2nd grade, fact families joined addition and subtraction. The old pattern carries over: two facts forward, then two facts with a minus sign.
What helps. Say that subtraction undoes adding and division undoes multiplying. Then ask which sign the last two facts need.

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
Liam was asked to write an unknown-factor multiplication equation that matches the division equation. The first line below shows the division equation. The second line shows what Liam wrote.
$\displaystyle \begin{array}{l} 10 \div 2 = ? \\ 2 \times ? = 10 \end{array}$
Select the statement that best describes Liam's work.
A
The question mark is a factor, but the dividend and divisor are switched.
B
The question mark is in the product position.
C
The equation correctly matches the division with a missing factor.
D
The multiplication fact is true, but there is no question mark.
Show the answer and how to solve it
Liam's equation $2 \times ? = 10$ has the same missing factor and product as $2 \times ? = 10$ . It is a correct match.
1. 1In $10 \div 2 = ?$ , $10$ is the dividend, the number being divided. The divisor is $2$ , the number we divide by.
2. 2The matching multiplication is $2 \times ? = 10$ . The divisor $2$ and the missing number are factors. The dividend $10$ is the product.
3. 3Liam's equation $2 \times ? = 10$ has the same missing factor and product as $2 \times ? = 10$ . It is a correct match.
4. 4The statement that describes Liam's work is: "The equation correctly matches the division with a missing factor."
Common mistake. A matching equation keeps the dividend as the product and uses the $?$ for a missing factor. $2 \times ? = 10$ does both.
The idea underneath. For $10 \div 2 = ?$ , the matching multiplication is $2 \times ? = 10$ . The same missing number makes both equations true.
2On level
Which multiplication equation with a missing factor matches the division equation below?
$\displaystyle 32 \div 8 = ?$
A
$\displaystyle 8 \times ? = {}$
$\displaystyle 32$
B
$\displaystyle 8 \times 32 = {}$
$\displaystyle ?$
C
$\displaystyle 8 \times 4 = {}$
$\displaystyle 32$
D
$\displaystyle 32 \times ? = {}$
$\displaystyle 8$
Show the answer and how to solve it
The matching equation is $8 \times ? = 32$ . The divisor goes in the known-factor position, the ? remains as the unknown factor, and the dividend is the product.
1. 1Look at the division equation. The number being divided is 32 (the dividend), and we are dividing by 8 (the divisor). The ? stands for the missing answer.
2. 2Division and multiplication are related. The question 'how many 8s make 32?' is the same as 'what number times 8 equals 32?' So we can rewrite the division using a multiplication equation with a missing factor.
3. 3The divisor (8) goes first as the known factor, the ? stays as the missing factor, and the dividend (32) is the product. This matches the equation $8 \times ? = 32$ .
Common mistake. The equation $8 \times 4 = 32$ is a true multiplication fact. But this question asks for a missing factor, so choose $8 \times ? = 32$ .
The idea underneath. Every division equation has a matching unknown-factor multiplication. For example, $32 \div 8 = ?$ can be rewritten as $8 \times ? = 32$ : the divisor becomes the known factor, the unknown quotient becomes the missing factor, and the dividend becomes the product.
3On level
Maya wants to solve $24 \div 4$ . Maya's work is shown below.
$\displaystyle \begin{array}{l} \text{Step 1: } 4 \times 6 = 24 \\ \text{Step 2: So } 24 \div 4 = 6 \end{array}$
Select the statement that best describes Maya's work.
A
Maya used a true multiplication fact, but it does not equal the dividend.
B
Maya's true fact matches the dividend, but the quotient is the divisor instead of the missing factor.
C
Maya used a multiplication fact that is not true.
D
Maya's reasoning is correct.
Show the answer and how to solve it
Maya's multiplication fact ( $4 \times 6 = 24$ ) is true and the product matches the dividend $24$ . Since $4 \times 6 = 24$ , the answer is $24 \div 4 = 6$ . Maya's reasoning is correct.
1. 1Maya's work has two parts: a multiplication fact (Step $1$ ) and a division conclusion (Step $2$ ). To check Maya, look at Step $1$ first.
2. 2Is $4 \times 6 = 24$ true? Yes — that is a true multiplication fact.
3. 3Now check whether the product matches the dividend. Maya is solving $24 \div 4$ , so the dividend is $24$ . The product in Step $1$ is also $24$ .
4. 4Since the product equals the dividend, the multiplication fact is exactly what is needed to solve $24 \div 4$ . Maya's division conclusion in Step $2$ is also correct: because $4 \times 6 = 24$ , the answer $24 \div 4 = 6$ follows directly.
Common mistake. A common mistake is looking only at whether the multiplication fact is true, without checking whether the product equals the dividend. Here, both conditions are met: $4 \times 6 = 24$ is true and $24$ is the dividend.
The idea underneath. To solve a division problem using multiplication, the multiplication fact must end with the dividend. To solve $24 \div 4$ , find $4 \times \,? = 24$ , and the missing factor ( $6$ ) is the quotient.
4Stretch
Riley used the array shown to write a multiplication equation. Use the first factor for the number of rows and the second factor for the number of items in each row.
$\displaystyle \text{Riley's equation: } 7 \times 6 = 42$
Select the statement that best describes Riley's equation.
A
Riley's equation matches all three counts in the array.
B
Riley's second factor does not match the number in each row.
C
Riley's first factor does not match the number of rows.
D
Riley's stated total does not match the number of items in the array.
Show the answer and how to solve it
Riley's equation matches all three counts in the array.
1. 1Count the rows, the items in each row, and all the items in the array.
2. 2The array has $7$ rows with $6$ items in each row. There are $7 \times 6 = 42$ items in all.
3. 3The first factor $7$ matches the $7$ rows. The second factor $6$ matches the $6$ items in each row. The stated total $42$ matches the $42$ items in the array.
4. 4All three counts match. The equation $7 \times 6 = 42$ is correct.
Common mistake. Do not assume there is an error. All three numbers match the array: $7$ rows, $6$ in each row, and $42$ items in all.
The idea underneath. Compare the first factor with the number of rows, the second factor with the number in each row, and the stated total with the number of items in the array.