Missing numbers in multiplication & division sentences
A box in a number sentence stands for a missing number, as in \(\square \times 7 = 35\). In Tasmanian Year 4 maths this subtopic uses multiplication facts and the link between multiplying and dividing to find it (Australian Curriculum AC9M4A02, AC9M4N06).
Students turn a missing-number multiplication into a division and back again, use arrays and fact-family triangles, and decide whether number sentences are true or false. They balance both sides, as in \(7 \times 4 = 14 \times \square\), and write a number sentence with a box to match a story, such as sharing balls into equal bags or finding the cost of one item. It is part of Year 4 Patterns and Number Sequences.
Theory
Missing numbers in multiplication and division sentences is a Year 4 maths number and algebra skill: finding the number that fits in a box, like \(7 \times \square = 28\) or \(36 \div \square = 4\). This page explains the inverse relationship between multiplication and division, how to use fact families and arrays, and what the equals sign really means. This page uses Year 4 multiplication facts.
A number sentence uses numbers, an operation and an equals sign, like \(2 \times 7 = 14\). Sometimes one number is hidden in a box, and your job is to find it: \(2 \times \square = 14\).
The equals sign means "the same as". Both sides of the sentence must have the same value, so a missing-number sentence asks which number makes the two sides balance.
Multiplication and division are inverse operations — they undo each other. This is the key to finding the box. The factors are the numbers you multiply, and the product is the answer. In \(8 \times 4 = 32\), the factors are 8 and 4 and the product is 32.
Missing-number sentences do not need formulas, but these facts help you fill in the box. One multiplication fact gives you a whole fact family.
How to find a missing number
- Read the sentence. Find where the box sits and remember the equals sign means both sides are the same.
- Choose the inverse. If a factor is missing, divide the product by the factor you know. If the product is missing, multiply the two factors.
- Check. Put your number back into the box and work it out to make sure both sides balance.
| \(\square\) | \(=\) | \(48 \div 8\) |
| \(\square\) | \(=\) | \(6\) |
| \(\text{check: } 8 \times 6\) | \(=\) | \(48\) |
A factor is missing, so divide. The box is \(6\).
| \(\square\) | \(=\) | \(35 \div 5\) |
| \(\square\) | \(=\) | \(7\) |
| \(\text{check: } 35 \div 7\) | \(=\) | \(5\) |
Split \(35\) into groups of \(5\). There are \(7\) groups.
| \(4 \times \square\) | \(=\) | \(28\) |
| \(\square\) | \(=\) | \(28 \div 4 = 7\) |
There are \(7\) rolls in each row.
| \(3 \times 8\) | \(=\) | \(24\) |
| \(\square \times 2\) | \(=\) | \(24\) |
| \(\square\) | \(=\) | \(24 \div 2 = 12\) |
Work out one side first, then balance the other. The box is \(12\).
Common pitfalls
Frequently asked questions
How do you find a missing number in a multiplication sentence?
Divide the product by the factor you already know. For \(9 \times \square = 72\), work out \(72 \div 9 = 8\). Then check by multiplying \(9 \times 8 = 72\).
What is a fact family?
A fact family is a group of number sentences made from the same three numbers. From \(8\), \(4\) and \(32\) you get \(8 \times 4 = 32\), \(4 \times 8 = 32\), \(32 \div 8 = 4\) and \(32 \div 4 = 8\).
What does the equals sign really mean?
It means "the same as". Both sides must have the same value, so \(3 \times \square = 21\) asks which number makes both sides balance. It does not mean "write the answer here".
How do you find a missing number in a division sentence?
It depends where the box is. For \(\square \div 5 = 6\) you multiply, \(5 \times 6 = 30\). For \(32 \div \square = 8\) you divide, \(32 \div 8 = 4\).
Are multiplication and division opposites?
Yes. They are inverse operations, which means they undo each other. If \(8 \times 4 = 32\) then \(32 \div 8 = 4\).
How can I check a missing number answer?
Put your number back into the sentence and work it out. If you found \(\square = 8\) in \(9 \times \square = 72\), check by multiplying \(9 \times 8 = 72\).