ACARA v9 CONTENT DESCRIPTION “find unknown values in numerical equations involving multiplication and division using the properties of numbers and operations”
Builds on Inverse Operations and Fact Families (AC9M5A01). Year 5 puts the inverse relationship between multiplication and division to work, using it to find an unknown number in an equation. When a multiplication or division statement has a missing value, the matching inverse operation, together with the properties of the numbers involved, reveals it. Knowing where the unknown sits and which operation undoes the one in the equation turns finding it into a reliable, repeatable step, and it lays the groundwork for the algebra of later years.
Finding an unknown in an equation
A numerical equation can have a missing number, shown by a symbol or a box, and the task is to find the value that makes the equation true. Six multiplied by some number equals forty-two asks which number fills the gap. Rather than guessing, the missing value can be found by using what is known about multiplication and division. The key idea is that an equation is a balance: whatever the unknown is, both sides must come out equal, so the number that balances the statement is the answer.
Finding an unknown in an equation
The missing factor is the product divided by the known factor.
Which number makes the equation true?
Multiplication and division undo each other
Multiplication and division are inverse operations, meaning each undoes the other. If eight multiplied by five gives forty, then forty divided by five gives back eight, and forty divided by eight gives back five. This inverse relationship is the main tool for finding unknowns. When a number is multiplied by something to reach a result, dividing the result by that something recovers the original. Recognising which operation will undo the one in the equation is the first step in solving it without trial and error.
Multiplication and division undo each other
Dividing the product recovers a factor.
Multiplication and division are inverse operations.
Where the unknown can sit
The unknown can sit in different places, and each position is handled the same way. In a number multiplied by four equals twenty-eight, the unknown is a factor, found by dividing twenty-eight by four. In thirty-six divided by some number equals nine, the unknown is the divisor, found by dividing thirty-six by nine. In some number divided by five equals six, the unknown is the starting number, found by multiplying five by six. Reading the equation carefully to see what the unknown represents tells which inverse step to take.
Where the unknown can sit
Factor, divisor or dividend: the same method works.
What is the missing number?
Using fact families to find unknowns
A multiplication and division fact family ties three numbers together with four related facts. The numbers seven, six and forty-two give seven multiplied by six, six multiplied by seven, forty-two divided by six, and forty-two divided by seven. Because these four facts describe the same relationship, knowing any two of the numbers lets the third be found. Fact families turn finding an unknown into recalling a known relationship: if two numbers of a family are given, the missing one is already part of the same set of facts.
Using fact families to find unknowns
Knowing two numbers of a family finds the third.
A fact family ties three numbers together.
Solving with the inverse operation
Putting it together, solving for an unknown means applying the inverse operation to the numbers that are known. To find the value in some number divided by three equals eight, multiply eight by three to get twenty-four, because multiplication undoes the division. To find the value in five multiplied by some number equals forty-five, divide forty-five by five to get nine. The properties of the operations guarantee this works every time: the inverse step rearranges the equation so the unknown stands alone, and the arithmetic then gives its value.
Solving with the inverse operation
Undo the operation to leave the unknown alone.
Solve ? ÷ 3 = 8 with the inverse operation.
Finding unknowns with confidence
Finding unknowns in multiplication and division equations comes down to a clear routine: see where the unknown sits, choose the operation that undoes the one in the equation, and use fact families and the properties of numbers to carry it out. Checking the answer by putting it back into the equation confirms the balance holds. With these habits a child can solve for a missing factor, divisor or dividend with confidence, and is ready for the more general equation-solving of later years.
Quick self-check
1. In 6 × ? = 42, the missing number is...
2. Division undoes multiplication, so if 8 × 5 = 40 then...
3. In ? ÷ 4 = 5, the missing number is...
4. Which step finds the unknown in ? × 3 = 21?
5. A multiplication and division fact family links...