What is this skill?
Missing factor problems are multiplication equations with a blank, like 6 × ? = 42 or ? × 8 = 32. Third graders find the unknown number by thinking about multiplication facts they know, or by dividing: 42 ÷ 6 = 7, so the missing factor is 7. A factor is one of the numbers being multiplied, and the product is the answer.
These problems help children see division as a missing factor problem. "What is 42 ÷ 6?" and "6 times what is 42?" are the same question. Third graders also learn that a symbol or blank can stand for an unknown number, an early taste of the letters that appear in algebra.
Why it matters
Missing factor problems are where multiplication and division come together. Children who can solve 6 × ? = 42 quickly have flexible fact knowledge: they can use what they know about multiplication to solve division, and they see the two operations as connected rather than separate lists to memorize.
The skill is used constantly later. Long division asks "How many times does 7 go into 45?", and finding equivalent fractions or common denominators asks "What do I multiply 3 by to get 12?" Solving simple equations in middle school starts with exactly this kind of thinking.
They also build understanding of the equals sign as meaning "the same value as," which helps children work with equations written in different orders.
Common mistakes
- Dividing or multiplying the two numbers that are shown without thinking about what the problem asks. For 6 × ? = 42, multiplying 6 × 42 is a common slip. Ask "What number goes in the box so the sentence is true?"
- Subtracting the numbers, such as answering 36 for 6 × ? = 42, because it looks like a missing addend problem from earlier grades. Point to the × sign and ask what it means.
- Being confused when the blank comes first, as in ? × 8 = 32. The turnaround rule helps: it is the same as 8 × ? = 32.
- Not checking. Putting the answer back in the equation (6 × 7 = 42) is a quick way to confirm it.
How to help at home
- Play "I am thinking of a number": "I multiplied my number by 4 and got 28. What is my number?"
- Use objects: put 30 buttons into 5 equal piles and ask how many in each, then write the matching 5 × ? = 30.
- Write fact family triangles with one corner blank, which is exactly a missing factor problem.
- Ask your child to explain how they found the answer. Some will recall the fact, some will skip count, and some will divide, and all are good strategies to talk about.
Worked examples
These are real questions from the Tutorsaurus missing factor problems practice, with the answers worked out.
Example 1
? × 5 = 10
Answer: 2. 5 × 2 = 10, so the missing factor is 2. You can also divide: 10 ÷ 5 = 2.
Example 2
2 × ? = 14
Answer: 7. 2 × 7 = 14, so the missing factor is 7. You can also divide: 14 ÷ 2 = 7.
Example 3
6 × ? = 42
Answer: 7. 6 × 7 = 42, so the missing factor is 7. You can also divide: 42 ÷ 6 = 7.