Zeros and Division
Frank wrote, “18 ÷ 3 = 7.”
Tanya wrote, “18 ÷ 3 = 6.”
She said, “One of us must be wrong. There can't be two different answers to the same division problem.”
Mrs. Dewey said, “That's right, Tanya. Each division problem has a unique solution. That means that each division problem has only one correct answer. Work together to find the correct answer. Try using fact families.
- Write the fact family for 18 ÷ 3. Who is correct, Tanya or Frank?
Tanya wrote: 18 ÷ 3 = ? 3 ? = 18
3 6 = 18, so 18 ÷ 3 = 6
“That's good thinking,” said Mrs. Dewey. “Let's use your reasoning to think about division and zero. Find 0 ÷ 24.”
Tanya replied, “To find 0 ÷ 24, I find the only number that you can multiply by 24 and get 0. Since any number times zero is zero, 24 0 = 0 and 0 ÷ 24 = 0.”
Tanya wrote: 0 ÷ 24 = ? 24 ? = 0
0 24 = 0, so 0 ÷ 24 = 0
- Use Tanya's reasoning to find 0 ÷ 5.
Tanya began, “To find 24 ÷ 0, I find the number that you can multiply by 0 and get 24. But, no number makes the number sentence 0 ? = 24 true. What do I do?
“Since there is no solution for 0 ? = 24, we say that 24 ÷ 0 is undefined. In fact, if you use your reasoning with any number divided by zero, you will find the same thing. So, mathematicians say that division by zero is undefined.”
- Use Tanya's reasoning to find 5 ÷ 0.