Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Every division fact hides a multiplication fact. To find 32 ÷ 8, a third grader can ask, "8 times what number makes 32?" The answer, 4, is the missing factor. Seeing division this way means students do not have to learn a whole separate set of division facts; they lean on the multiplication facts they are already building.
The idea depends on understanding that multiplication and division are inverse operations: one undoes the other. If 8 groups of 4 make 32, then splitting 32 into groups of 8 gives 4 groups, and splitting it into 8 groups gives 4 in each. Fact families such as 4, 8 and 32 make the connection concrete, as do arrays that can be read as a multiplication or a division. Students should be comfortable rewriting a division question as a multiplication question with a blank, then using a known fact, skip counting or an array to fill it in.
Students who memorize 42 ÷ 6 = 7 separately from 6 × 7 = 42 carry twice the load. Practise each division fact alongside the multiplication fact it comes from.
For 32 ÷ 8 some children write 8 × 32. Model the sentence "8 times what makes 32?" until it becomes automatic.
Because multiplication makes numbers bigger, some students expect division answers to be big too. Compare 8 × 4 = 32 with 32 ÷ 8 = 4.
A family such as 3, 6, 18 should give 3 × 6, 6 × 3, 18 ÷ 3 and 18 ÷ 6. Writing 6 ÷ 18 shows the total is not being placed first.
Find 56 ÷ 7 by thinking about multiplication.
Answer: 56 ÷ 7 = 8, because 7 × 8 = 56.
Triangle flash cards with the product in the top corner are a simple tool: cover any corner and students name the missing number using multiplication thinking. Writing division as "7 × _ = 56" on the board every time for a few weeks helps make the switch automatic.
This standard is often assessed by asking which multiplication fact helps solve a division problem, or by asking students to complete a fact family. It feeds straight into fluency with division in 3.OA.C.7.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: D) 9 × 4 = 36
36 ÷ 9 asks "9 times what makes 36?" Since 9 × 4 = 36, the answer is 4.
Answer: 7
6 × 7 = 42, so 42 ÷ 6 = 7.
Answer: 8
Think 3 × ? = 24. Since 3 × 8 = 24, the quotient is 8.
Answer: C) 9 ÷ 45 = 5
In a fact family the product (45) is divided by one factor to give the other. 9 ÷ 45 is not 5.
Answer: 9
Think 7 × ? = 63. Since 7 × 9 = 63, the answer is 9.
Answer: A) 48 ÷ 8 = 6
If 8 × 6 = 48, then 48 shared into 8 groups gives 6, so 48 ÷ 8 = 6.
A full lesson with slides, activities and an exit ticket on division as an unknown-factor problem, pitched to grade 3 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 3.OA.B.6 with an answer key, ready in about a minute.
Make a worksheet →Turn division as an unknown-factor problem into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Because it lets students use their multiplication facts to divide. Instead of a separate memory task, division becomes "what times this number gives that number?"
Seeing division as the inverse of multiplication is the basis for long division, dividing fractions and solving equations such as 7x = 56 in middle school.