# Zero in Division

 What do you think 6 ÷ 0 would be?

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.
Let's say that 0 ÷ 0 = 0.  Check by multiplying: 0 × 0 = 0; OK. So 0 would work.
Let's say that 0 ÷ 0 = 11.  Check by multiplying: 11 × 0 = 0; OK. So 11 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.
You can find the total by adding. Therefore:

 5 × 0  = 0 + 0 + 0 + 0 + 0 =  0 (five groups of zero items)
 0 × 3  =  0 (zero groups of three items)

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

 Division and multiplication involving 0 leads to some funny or interesting situations. Can you solve what stands for?

 a.  0 ÷ = 4 b.  0 × = 0 c.    ÷ 0 = 6 d.   0 × = 3

This lesson is taken from my book Math Mammoth Division 1.

#### Math Mammoth Division 1

A self-teaching 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.