Zero in division
This is a complete lesson for third grade, with explanations and exercises, about dividing by zero. We study the idea of sharing bananas between zero persons, which is absurd. Even saying that division by zero equals zero doesn't work, because then the check by multiplication fails.
We could think of sharing 6 bananas between 0 persons. But that doesn't make sense. We can't even talk about how many each one gets, because there is no one around. We could think of making groups of 0. How many groups would you get? Again, you would not get anywhere, you would never get those 6 bananas put into groups of 0. You might think that maybe 6 ÷ 0 = 0 or that each person gets zero bananas. Check it with multiplication! You would get 0 × 0 = 6, which is not true! So 6 ÷ 0 = 0 does not work either. 

Dividing six by zero (6 ÷ 0) is “undefined.” Basically, you can't do it. 

What about 0 ÷
0? Couldn't we say 0 ÷
0 = 0? 0 ÷ 0 is hard. The answer could be zero, but actually the answer could be any number : Let's say that 0 ÷
0 = 2. Check by multiplying: 2 × 0 = 0; OK. So 2 would work. So, we cannot find just ONE answer. We say that the answer cannot be determined. 

Dividing a number by zero doesn't work. 
What about zero divided by something? That is perfectly fine. 0 ÷ 5 = 0 “If there are zero bananas and five people, each person gets 0 bananas.” 
1. Divide. CROSS OUT all the problems that are impossible. Think about sharing bananas.
a. 4 ÷ 1 = ______ 4 ÷ 0 = ______ 
b. 14 ÷ 14 = ____ 0 ÷ 0 = _____ 
c. 1 ÷ 1 = _____ 7 ÷ 0 = _____ 
d. 0 ÷ 5 = _____ 5 ÷ 5 = _____ 
e. 0 ÷ 1 = _____ 0 ÷ 4 = _____ 
f. 0 ÷ 14 = _____ 14 ÷ 0 = _____ 
g. 0 ÷ 3 = _____ 0 ÷ 1 = _____ 
h. 10 ÷ 10 = ____ 1 ÷ 1 = _____ 
In multiplication, zero works just fine! Multiplication means you have many groups of the
same size. 



2.
Multiply. Then for each multiplication, make a division sentence if possible.
Some divisions are not possible!
a. 6 × 1 = ______ ______ ÷ ____ = ____ 
b. 0 × 8 = ______ ______ ÷ ____ = _____ 
c. 5 × 7 = ______ _______ ÷ ____ = _____ 
d. 10 × 11 = ______ ________ ÷ ____ = ____ 
e. 1 × 1 = ______
______ ÷ ____ = ____ 
f. 1 × 8 = ______ ______ ÷ ____ = _____ 
g. 0 × 0 = ______ ______ ÷ ____ = ____ 
h. 5 × 9 = ______ ______ ÷____ = _____ 
i. 9 × 0 = ______ ______ ÷ ____ = ____ 
3. Solve, and write a number sentence for each word problem.
a.
Sally had 30 kg of rice. She put equal amounts into six bags. How much was in each bag? _______ _______ = _______ 
b. There are six minivan taxis at the
airport, and each can hold seven passengers. How many passengers can they take in total? _______ _______ = _______ 
c. Greg bought
three cartons of eggs, with 12 eggs in each carton. How many eggs did he get? _______ _______ = _______ 
d. The airplane had 56 passengers.
Each minivan taxi can hold seven passengers. How many minivan taxes are needed to take these passengers to a hotel? _______ _______ = _______ 
e. There are five tables, and each has four legs. How many legs are there in total? _______ _______ = _______ 
f. Kelly poured
a total of four cups of milk into four glasses. How much milk was in each glass? _______ _______ = _______ 
4. Divide. CROSS OUT all the problems that are impossible.
a. 9 ÷ 1 = ______ 9 ÷ 0 = ______ 
b. 0 ÷ 20 = _____ 20 ÷ 0 = _____ 
c. 11 ÷ 1 = ______ 8 ÷ 0 = ______ 
d. 0 ÷ 0 = ______ 0 ÷ 10 = ______ 
5.
Make a QUESTION for each situation. (Think what you can find out using what the
problem tells you.) Then solve your question.
a.
Mark, Jack, and Joe decided to share their toy cars evenly in a game. Mark had 18 cars, Jack had 7, and Joe had 11.

b.
Mrs. Elliott hired six children to do yard work. She paid one of them $15, and the rest of them $10 each.

c. The books on Alice's
reading list have: 320, 129, 120, and 235 pages.

d. Jeremy wants to read
two books that have 32 and 40 pages. He reads 12 pages a day.

e.
Kelly
had 80 cm of red material and 40 cm of blue material. She cut it all into 20cm pieces.


f. A child
arranged toy cars in rows of six cars. He made seven rows like that. The eighth row had three cars.


This lesson is taken from Maria Miller's book Math Mammoth Division 1, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.
Math Mammoth Division 1
A selfteaching worktext for 3rd grade that covers division concept, division & multiplication fact families, word problems, division facts, remainder, zero and one in division, and more.Download ($3.70). Also available as a printed copy.