Lesson 47: Relating Multiplication and Division, Part 2

New Concept

Together, all four facts form a multiplication and division fact family.

What’s next

Next, you'll use this relationship to solve division problems and write your own fact families.

Property

The phrase 'fifteen divided by three' can be shown in three different ways, using a division box, a division sign, or a division bar. All three formats represent the same calculation.

Example

1. 'Thirty-five divided by seven' can be written as: $7\overline{)35}$, $35 \div 7$, or $\frac{35}{7}$. 2. 'Fifty-four divided by nine' can be written as: $9\overline{)54}$, $54 \div 9$, or $\frac{54}{9}$.

Explanation

Think of these as different outfits for the same math operation! Whether you see a box, a sign, or a bar, the mission is always to divide a number into equal groups. Learning to recognize all three styles means you'll never be confused by a problem's format. It’s like knowing a superhero's identity even when they change costumes.

Property

A multiplication fact has three numbers. We can form one other multiplication fact and two division facts with these three numbers. For example:

Example

1. Using the numbers 4, 5, and 20, the fact family is: $4 \times 5 = 20$, $5 \times 4 = 20$, $20 \div 5 = 4$, and $20 \div 4 = 5$. 2. Using 3, 7, and 21, the fact family is: $3 \times 7 = 21$, $7 \times 3 = 21$, $21 \div 7 = 3$, and $21 \div 3 = 7$.

Explanation

Think of numbers like 4, 5, and 20 as a close-knit family. They are always related! If you know that $4 \times 5 = 20$, you automatically know its twin $5 \times 4 = 20$. This family connection also gives you the division facts for free: $20 \div 5 = 4$ and $20 \div 4 = 5$. It’s a four-for-one deal!

Property

There are special rules for dividing by 1, dividing a number by itself, and dividing zero. 1. A number divided by one is itself ($8 \div 1 = 8$). 2. A non-zero number divided by itself is one ($\frac{9}{9} = 1$). 3. Zero divided by a non-zero number is zero ($4\overline{)0} = 0$).

Example

1. 'Twelve divided by one' is solved as $12 \div 1 = 12$. 2. 'Fifteen divided by fifteen' is solved as $\frac{15}{15} = 1$. 3. 'Zero divided by five' is solved as $5\overline{)0} = 0$.

Explanation

These are division's easiest cheat codes! Any number divided by 1 is just itself, because you're just making one big group. Any number divided by itself is 1, like sharing ten cookies with ten people—everyone gets one. And if you have zero cookies to share among four friends, how many does each person get? Zero, of course!