### The Mysterious Case of Zero, the Exponent - Why any Number to the Zero Power Equals One

Sometimes my college students like to ask me what seems to be a difficult question. (In reality, they want to play Stump the Teacher.)  I decided to find out what sort of answers other mathematicians give; so, I went to the Internet and typed in the infamous question, "Why is any number to the zero power one?"  It was no surprise to find numerous mathematically correct answers, most written in what I call "Mathteese" - the language of intelligent, often gifted math people, who have no idea how to explain their thinking to others.  I thought, "Wow!  Why is math always presented in such complicated ways?"  I don't have a response to that, but I do know how I introduce this topic to my students.

Since all math, and I mean all math, is based on patterns and not opinions or random findings, let's start with the pattern you see on the right.  Notice in this sequence, the base number is always 3.  The exponent is the small number to the right and written above the base number, and it shows how many times the base number, in this case 3, is to be multiplied by itself.

(Side note: Sometimes I refer to the exponent as the one giving the marching orders similar to a military commander. It tells the base number how many times it must multiply itself by itself. For those students who still seem to be in a math fog and are in danger of making the grave error of multiplying the base number by the exponent, have them write down the base number as many times as the exponent says, and insert the multiplication sign (×) between the numbers. Since this is pretty straight forward, it usually works!)

Notice our sequence starts with 31 which means 3 used one time; so, this equals three; 32 means 3 × 3 = 9, 33 = 3 × 3 × 3 = 27, and so forth. As we move down the column, notice the base number of 3 remains constant, but the exponent increases by one. Therefore, we are multiplying the base number of three by three one additional time.

Now let's reverse this pattern and move up the column. How do we get from 243 to 81? That's right! We divide by three because division is the inverse operation (the opposite) of multiplication. How do we get from 243 to 81? That's right! We divide by three because division is the inverse operation (the opposite) of multiplication. Notice as we divide each time, the exponent by the base number of 3 is reduced by one. Let's continue to divide by three as we move up the column. 27÷ 3 = 9; 9 ÷ 3 = 3. Now we are at 31 = 3 which means we must divide 3 by 3 which gives us the quotient of 1. Notice, to fit our pattern, the 3 in the left hand column would have to be 30 ; so, 30 must equal one!

This works for any number you wish to put in the left hand column. Try substituting the base number of three with two. Work your way up the sequence dividing by two each time. You will discover that two divided by two equals one (20 = 1). Therefore, we can conclude that any natural number with an exponent of 0 is equal to one. (Zero is not included; it's another mystery to solve.)

What happens if we continue to divide up the column past 30 ?   (Refer back to the sequence on the left hand side.)  Based on the pattern, the exponent of zero will be one less than 0 which gives us the base number of 3 with a negative exponent of one or 3-1 .    To maintain the pattern on the right hand side, we must divide 1 by 3 which looks like what you see on the left. Continuing up the column and keeping with our pattern, 3 must now have a negative exponent of 2 or 3-2 and we must divide 1/3 by 3 which looks like what is written on the right.

Each time, the exponent by the base number of 3 is reduced by one. Let's continue to divide by three as we move up the column. 27÷ 3 = 9; 9 ÷ 3 = 3. Now we are at 31 = 3 which means we must divide 3 by 3 which gives us the quotient of 1. Notice, to fit our pattern, the 3 in the left hand column would have to be 30 ; so, 30 must equal one! This works for any number you wish to put in the left hand column. Try substituting the base number of three with two. Work your way up the sequence dividing by two each time. You will discover that two divided by two equals one (20 = 1). Therefore, we can conclude that any natural number with an exponent of 0 is equal to one. (Zero is not included; it's another mystery to solve.)

As a result, the next two numbers in our pattern are..............

Isn't it amazing how a pattern not only answers the question: "Why is any number to the zero power one?" But it also demonstrates why a negative exponent gives you a fraction as the answer. (By the way math detectives, do you see a pattern with the denominators?)

Mystery Solved!   Case Closed!

This lesson is available on a video entitled:  Why Does "X" to the Power of 0 Equal 1?

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Want simple, visual answers to other difficult math questions? Try this resource entitled Six Challengig Math Questions with Illusratrated Answers. Many of the answers feature a supplementary video for a more detailed explanation.

### A Go Figure Debut for a Digital Teacher Who is New!

Ian has been a classroom teacher 12 years in two states (Virginia and Massachusetts) and has worked with students in a general education and advanced academic setting in grades 3-5. What he loves most about teaching is inspiring awe and wonderment.  Ian pushes his students to think beyond the standards, to question the world around them, and to seek out answers to their wonderings. He encourages his students to utilize technology, to demonstrate their understanding and to showcase their findings. His classroom has a certain buzz of excitement while students work on their latest problem-based learning activity, such as delivering the news for the southeast region, or researching a signature dish for their Midwest Cooking Star episode. His students enjoy many healthy, competitive digital games to review and reinforce concepts.

Ian is a soon-to-be father of four, who loves to spend the summers on the beaches of Massachusetts, the falls hiking and exploring the outdoors, and the winter months building snowmen and sledding. His family is a running family, with a combined 15 marathons between him and his wife.

Ian's Teachers Pay Teachers store, Mr. Kidders Keys To Online Learning, contains 112 resources with 12 of them offered as free. The purpose of his store is to help facilitate a digital friendly classroom and is intended to make the hectic life of a teacher more manageable. It features a variety of digital morning meeting, virtual, or brain break games, some of which are also content based. His three main lines of games are Zoomed In! - Reveal! - and Dash and Discover! In addition to the digital games, he has a growing library of digital math resources, mainly for grades 3-5: digital escape rooms, self-graded Google Forms with embedded video tutorials, interactive skip counting activities.

 FREE Resource
One of those digital math resources is entitled Grade 4 Multiplication: One Digit by Multi-Digit Multiplication Word Problems Google Form. Enjoy this FREE grade 4 Google Form multiplication quiz that is a self-graded word problem assessment involving one digit by multi-digit questions. It gives instant feedback for teachers! Just download it by clicking the link under the resource cover on your right.

 Only \$3.00
Ian's featured paid resource is a mystery image game that will boost your students' social-emotional health and get them engaged with online learning. Animal Adaptations Edition of Reveal! The Mystery Picture Game is ideal as a review or pre-assessment of animal adaptations. The Google Slideshow can be used during distance learning video calls or in the classroom. This involves NO PREP and will leave your students smiling.

Finally, Ian has a few interactive, narrated virtual field trips that explore American history through famous works of art. All the digital activities in his store are ready to use, no preparation required, no need to print or copy anything (some even come with optional printouts). Take some time to check them out!

### Recycled Butterflies - Earth Day Ideas

When one of my grandchildren was in kindergarten, she came home one day with egg carton caterpillars.  I know most of us have made one of these in our lifetime, but to these two, they were the best craft ever!

She told me that her teacher was raising butterflies in her classroom, and soon the butterflies would hatch.  Anticipation and excitement reigned until the day she came out of school telling everyone that one of the butterflies had hatched.  However, much to her chagrin, the teacher was going to let it go.  My granddaughter just couldn't understand why or how her teacher could do that!

But, here is the good part!  She got to make a cocoon out of a toilet paper cylinder.  She covered it by gluing on white cotton balls.  Then she made a butterfly out of tissue paper and a small plastic bag tie.  She put the butterfly inside the cocoon and then pretended to have the butterfly hatch!  This was done over and over and over until the cocoon was no more.  Luckily, I was able to get pictures before both were literally destroyed!

Now, what does all of this have to do with math?  I contemplated all the ways to use recycled products to make items for the classroom.  Thus
 \$7.50
Trash to Treasure
was created. It is 34 pages of art ideas, fun and engaging mini-lessons as well as cute and easy-to-construct crafts all made from recycled or common, everyday items.

Find out more than 14 ways to use milk lids for math. Did you know that you can practice math facts using clear plastic containers? Learn how to take two plastic plates and turn them into angle makers. How about using two plastic beverage lids to make card holders for kindergartners or for those whose hands are disabled? Discover ten ways to use carpet squares as well as nine ways to use old calendars. How about playing hop scotch on old carpet squares? Were you aware that butter tubs can become an indoor recess game to practice addition or multiplication facts? These are just a few of the fun and exciting activities that use recycled items found in this 34 page resource entitled Trash to Treasure.

Because these numerous activities vary in difficulty and complexity, they are appropriate for any PreK - 3rd classroom, and the visual and/or kinesthetic learners will love them.