This lesson shows how multiplication and division are connected as inverse operations, using fact families and arrays with numbers up to 999. Students build a fact family from one statement, use multiplication to check a division answer, and solve missing-number problems by switching between the two operations.
Why multiplication and division are connected
Multiplication and division are inverse operations: one undoes the other. If you know that \( 6 \times 7 = 42 \), you already know two division facts for free: \( 42 \div 6 = 7 \) and \( 42 \div 7 = 6 \). This connection is what lets you use a multiplication fact you already know to solve or check a division problem, even when the numbers get as large as 999.
Instead of memorizing multiplication and division as two separate skills, it helps to think of them as one family of related facts built from the same three numbers.
Fact families: one grouping, four facts
A fact family is a set of related multiplication and division statements that all use the same three numbers, two factors and their product. For the numbers 6, 7, and 42, the fact family looks like this:
\( 6 \times 7 = 42 \)
\( 7 \times 6 = 42 \)
\( 42 \div 6 = 7 \)
\( 42 \div 7 = 6 \)
Notice the product, 42, always ends up on the left of a division and on the right of a multiplication. The two smaller numbers, 6 and 7, swap places but the family stays the same. The diagram below shows this relationship as a fact triangle, a quick way to see all four facts at once.
Fact triangle for the numbers 6, 7, and 42: multiplying the bottom two numbers gives the top; dividing the top by either bottom number gives the other.
This same idea works for larger numbers. If \( 40 \times 9 = 360 \), then the fact family also gives you \( 360 \div 40 = 9 \) and \( 360 \div 9 = 40 \), without any new work.
Seeing the relationship in an array
An array of rows and columns is a great way to see multiplication and division as two directions of the same picture. If you already looked at dividing with arrays, this will feel familiar: the same grid answers both "how many in total" and "how many in each group."
The same array of 3 rows of 4 shows both \( 3 \times 4 = 12 \) and \( 12 \div 4 = 3 \).
Using multiplication to check division
Because multiplication reverses division, it is a fast way to check that a division answer is correct, even with three-digit numbers. Multiply the quotient by the divisor and see if you land back on the original number.
Example: Find \( 963 \div 3 \), then check the answer.
Dividing gives \( 963 \div 3 = 321 \). To check, multiply the quotient by the divisor: \( 321 \times 3 = 963 \). Since this matches the original number, the division is correct.
This checking method works no matter how the division was solved, whether by sharing into equal groups as in sharing and grouping up to 999, or by working through a written division statement.
Solving missing-number problems
The multiplication and division relationship is especially useful when one number in a statement is missing. Instead of guessing, switch to the related operation.
Example: Solve for the missing factor: \( 4 \times \square = 348 \).
Since multiplication and division undo each other, rewrite the problem as a division: \( \square = 348 \div 4 \). Dividing gives \( \square = 87 \), so \( 4 \times 87 = 348 \).
Example: Solve for the missing divisor: \( 720 \div \square = 90 \).
Rewrite this as multiplication instead: \( 90 \times \square = 720 \), so \( \square = 720 \div 90 = 8 \).
Key idea to remember
Every multiplication fact has a matching division fact, and every division fact has a matching multiplication fact. Once you spot the fact family behind a problem, you can move freely between \( a \times b = c \) and \( c \div b = a \) or \( c \div a = b \). This flexibility makes solving, checking, and finding missing numbers up to 999 much faster than working each fact from scratch.