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When Dividing, Zero Is No Hero - Why We Can't Divide by Zero

Have you ever wondered why we can't divide by zero?  I remember 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 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.

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

A Go Figure Debut for a South Wales Teacher Who's New!

Meet Walton 
Walton started teaching 20 years ago, in the Peace Corps in Vanuatu (Vanuatu is a double chain of 13 principal and many smaller islands in the south-western Pacific Ocean.) a few months after September 11th! He was teaching English to Francophone (French-speaking),
students trying to get their GED-equivalent so they could go to school in France. It was a great little program to fill the gap between the local educational system and the requirements of French universities. So he fell in love with teaching with these hard working students in a ni-Van (abbreviation for Ni-Vanuatu) high school!

Since then, he has taught and tutored in public and private schools throughout the world including Kazakhstan and Russia. One year, he did an orientation for Afghan students who were planning to come and study in the U.S. in high school which was very rewarding.

Walton stopped teaching full-time in 2012, when he was teaching at the University of New Haven and since then have been doing private tutoring online as well as materials writing. So his classroom is his desk! The good news is that adjusting to COVID wasn't a big problem as he was already teaching online-although However, he feels for anyone who had to adjust on the spot and especially his son's poor teachers who had to teach hybrid! Walton has had materials published by OUP, Compass Publishing, and Macmillian, as well as his own materials which makes him a Freelance Writer and Editor.

His favorite part of teaching is when a student is working to grasp something, and they just can't get it. Then you figure out how to present it just the right way or something just clicks in their heads and they get it! It just makes it worth all the hard times and struggle. And you can see their excitement and enthusiasm. It's especially fun when they start overusing something because they are so proud of themselves for getting it!

For hobbies, Walton likes to read mystery novels and play Minecraft by himself but also with his nine year old son. In the summer, he and his son go fishing almost every day. It's been fun for him to watch his son learn problem solving and experimentation as they try to figure out what the fish are biting at, where they are, and so on.

FREE
His featured  free product, The Candy Thief, is a critical thinking mystery activity where students use clues to solve a mystery. This one has a Halloween theme, as students figure out who stole someone's Halloween candy and is designed for younger learners. There are three witness, but one of them is lying. Can your students ready their statements and figure out who the liar is? It is a great warm-up, critical thinking game, conversation starter, and filler of a time killer for early finishers.

$3.00
Walton has chosen to highlight as his paid product, Scary Story Writing Lesson Plan. This complete Halloween writing lesson plan uses the genre-based approach to writing to teach students how to write a scary store In the Halloween Scary Story Lesson Plan, students read several scary stories, including “Movie of Death,” They also talk about horror movies they are familiar with. They are then guided through an analysis of key features of a scary story. These include typical settings for a scary story, building tension, the concept of a twist and endings. Finally, students write their own scary story with those genre-specific elements. It's a fun lesson on how to write a scary story by giving students examples of scary stories and tools to analyze the elements of a scary story.

I find Walton's resources very unique and engaging. With Halloween just around the corner, check out what he has to offer. There is probably something in his store that you can use this month!

"BOO" to Fractions? Recognizing Equivalent Fractions


Here is a Halloween riddle: Which building does Dracula like to visit in New York City? Give up? It's the Vampire State Building!! (Ha! Ha!) Here is another riddle. What do ghosts eat for breakfast?  Scream of Wheat and Ghost Toasties!








Okay, so what do these riddles have to do with teaching math? I have been attempting to come up with ways for my students to recognize fractional parts in lowest terms. As you know from this blog, I have used Pattern Sticks, the Divisibility Rules, and finding Digital Root. These are all strategies my students like and use, but to be a good mathematician requires practice - something most of my students dread doing. I can find many "drill and kill" activities, but they tend to do just that, drill those who don't need it and kill those who already know how to do it. So to drill and "thrill", I created fractional word puzzles for specific times of the year.

The one for October is Halloween Fraction Riddles. It contains eight riddles that the students must discover by correctly identifying fractional parts of words. For instance, my first clue might be:

$3.00
$3.00
The first 2/3's of WILLOW. The word WILLOW contains six letters. It takes two letters to make 1/3; therefore, the first 2/3's would be the word WILL. This causes the students to group the letters (in this case 4/6), and then to reduce the fraction to lowest terms. The letters are a visual aid for those students who are still having difficulty, and I observe many actually drawing lines between the letters to create groups of two. 

At first, I thought my students would breeze through the activities, but to my surprise, they proved to be challenging as well as somewhat tricky - just perfect for a Trick or Treat holiday. Maybe this is an activity you would like to try with your intermediate or middle school students. Just click on this link: Halloween Fraction Riddles.


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Conducting Effective Parent/Teacher Conferences

If you are like most teachers, you are preparing for your first round of parent/teacher conferences. Now that I teach on the college level, this is one activity I currently don't have to do, but when I did, I really did enjoy them. Why? Because I was prepared with more than just the student's grades. Here are some of the ways I got ready.

First, in preparing for parent/teacher conferences, what can you do on a daily basis? Is the conference based on simply talking about grades or are there additional items that need discussing? How can an observation be specific without offending the parent or guardian? How is it possible to remember everything?

I kept a clipboard in my classroom on which were taped five 6” x 8” file cards so they overlapped - something like you see in the two pictures above. Each week, I tired to evaluate five students, writing at least two observations for each child on the cards. At the end of the week, the file cards were removed and placed into the children's folders. The next week, four different students were chosen to be evaluated. In this way, I did not feel overwhelmed, and had time to really concentrate on a small group of children. By the end of 4-5 weeks, each child in the class had been observed at least twice. By the end of the year, every child had been observed at least eight different times.

Below are sample observations which might appear on the cards.

Student
Date
             Observation
IEP
ESL

Mary Kay
  8/20


  8/28
Likes to work alone; shy and withdrawn;  wears a great deal of make-up.

She has a good self concept and is friendly. Her preferred learning style is  visual based on the modality survey.


X


    Donald
  9/19


  9/21
Leader, at times domineering, likes to  play games where money is involved.

His preferred learning style is auditory  (from the modality survey). He can be a  “bully,” especially in competitive games. He tends to use aggressive language with  those who are not considered athletic.




By the time the first parent/teacher conferences rolled around, I had at least two observations for each child. This allowed me to share specific things (besides grades) with the parents/guardians. As the year progressed, more observations were added; so, that a parent/guardian as well as myself could readily see progress in not only grades, but in a student's behavior and social skills. The cards were also an easy reference for filling out the paperwork for a 504 plan or an IEP (Individual Education Plan). As a result of utilizing the cards, I learned pertinent and important facts related to the whole child which in turn created an effective and relevant parent/teacher conference.
$1.95

To keep the conference on the right track, I also created a checklist to use during parent/teacher conferences.  It featured nine characteristics listed in a brief, succinct checklist form. During conferences, this guide allowed me to have specific items to talk about besides grades. Some of the characteristics included were study skills and organization, response to assignments, class attitude, inquiry skills, etc. Since other teachers at my school were always asking to use it, I rewrote it and placed it in my TPT store. It is available for only $1.95, and I guarantee it will keep your conferences flowing and your parents focused! When you have time, check it out!