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When Dividing, Zero Is No Hero

Have you ever wondered why we can't divide by zero?  I remembering asking that long ago in a math class, and the teacher's response was, "Because we just can't!"  I just love it when things are so clearly explained to me. So instead of a rote answer, let's investigate the question step-by-step.

The first question we need to answer is what does a does division mean?  Let's use the example problem on the right.

  1. The 6 inside the box means we have six items such as balls. (dividend) 
  2. The number 2 outside the box (divisor) tells us we want to put or separate the six balls into two groups. 
  3. The question is, “How many balls will be in each group?” 
  4. The answer is, “Three balls will be in each of the two groups.” (quotient)

Using the sequence above, let's look at another problem, only this time let's divide by zero.
  1. The 6 inside the box means we have six items like balls. (dividend) 
  2. The number 0 outside the box (divisor) tells us we want to put or separate the balls into groups into no groups. 
  3. The question is, “How many balls can we put into no groups?” 
  4. The answer is, “If there are no groups, we cannot put the balls into a group.” 
  5. Therefore, we cannot divide by zero because we will always have zero groups (or nothing) in which to put things. You can’t put something into nothing.
Let’s look at dividing by zero a different way. We know that division is the inverse (opposite) of multiplication; so………..
  1. In the problem 12 ÷ 3 = 4.  This means we can divide 12 into three equal groups with four in each group.
  2. Accordingly, 4 × 3 = 12.  Four groups with three in each group equals 12 things.
So returning to our problem of six divided by zero..... 
  1. If 6 ÷ 0 = 0....... 
  2. Then 0 × 0 should equal 6, but it doesn’t; it equals 0. So in this situation, we cannot divide by zero and get the answer of six.

We also know that multiplication is repeated addition; so in the first problem of 12 ÷ 3, if we add three groups of 4 together, we should get a sum of 12. 4 + 4 + 4 = 12

As a result, in the second example of 6 ÷ 0, if six zeros are added together, we should get the answer of 6. 0 + 0 + 0 + 0 + 0 + 0 = 0 However we don’t. We get 0 as the answer; so, again our answer is wrong.

It is apparent that how many groups of zero we have is not important because they will never add up to
equal the right answer. We could have as many as one billion groups of zero, and the sum would still equal zero. So, it doesn't make sense to divide by zero since there will never be a good answer. As a result, in the Algebraic world, we say that when we divide by zero, the answer is undefined. I guess that is the same as saying, "You can't divide by zero," but now at least you know why.
Free - Why We Can't Divide by Zero

If you would like a free resource about this very topic, just click under the resource title page on your right.

Put a LID on It!

There are so many things we consider to be trash, when in reality, they are treasures for the classroom. One that I often use is plastic lids from things like peanut canisters, Pringles, coffee cans, margarine tubs, etc.  These lids can be made into stencils to use when completing a picture graph.

Students must first of all understand what a picture graph is.  A pictorial or picture graph uses pictures to represent numerical facts. Sometimes it is referred to as a representational graph. Each symbol or picture used on the graph represents a unit decided by the student or teacher. Each symbol could represent one, two, or whatever number you want.  This type of graph is used when the data being gathered is small or approximate figures are being used, and you want to make simple comparisons.

Here is what you do to make ready-made picture graph stencils.
  • Choose the size of lid that you want and turn it over. Then trace a pattern on the plastic lid.  Make sure you are using the bottom of the lid so the rim does not interfere when the children use it to trace. 
  • To make the stencil, cut out the pattern using an Exacto knife. You might choose to do zoo animals:  a zebra, a lion, a bear, an elephant or a giraffe. 
  • Have a large sheet of paper ready with a question on it such as: “What is your favorite zoo animal?”
  • The students then select the stencil (picture) that is their favorite animal and trace it in the correct row on the graph.
Below is a sample of this type of graph. It is entitled, What is Your Favorite Season?  A leaf is used for fall; a snowflake represents winter; a flower denotes spring, and the sun is for summer. Notice at the bottom of the graph that each tracing will represent one student.

You could craft stencils for modes of transportation, geometric shapes, pets, weather, etc. The list is infinite.  But what if you don't want to or don't have time to make all of those stencils? Then save the strips that are left when you punch out shapes using a die press. They are instant stencils!

If you are interested in additional graphing ideas, check out the resource entitled: Graphing Without Paper or Pencil.   You might also like Milk Lid Math.  This four page handout contains numerous math activities that utilize a free manipulative.