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The Mysterious Case of Zero, the Exponent


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!


A "Go Figure" Debut for Someone Who's New!

We have many talented teachers who become sellers on Teachers Pay Teachers.  Presently TPT is over 61,000 strong; so, sometimes it is hard for a new store to get noticed. Every once in awhile, I will be introducing one of those new sellers to my blog readers.  I choose these particular stores because they contain high quality items, resources that are out of the ordinary and something that my husband or I can download and use in the classroom.

Acorn's Store
Today I would like you to meet Acorn, a teacher from Dublin, Ireland who is "nuts" about science.  Barry has been a science and ICT (Information Communications Technology - a fancy term for computers!) teacher for 20 years. He has also been a career guidance teacher, and he is the author of two books. Barry describes his teaching style as "friendly", and adds that he is a passionate believer that if you can't make something easy to understand then you don't understand it.

I first found his store because I was intrigued with his free resource. I downloaded it for my husband who teaches science, and he was "hooked". Acorn's products are for grades 4-11 although my husband thinks they are perfect for middle schoolers. They are engaging and hold the students attention. Acorn uses humor interspersed with necessary science knowledge that students are required to know. Check out his freebie entitled: Ionic & Covalent Bonding. It's an animated journey into chemistry and is the love story that was never told. In other words, it is a Chemical Romance. A short "fast draw" animation including cheesy science jokes demonstrates how ionic bonding occurs between sodium and chlorine.

Welcome to the world of Teachers Pay Teachers, Acorn. We look forward to reviewing more of your unique products!



Ben Franklin - A Math-a-Magician!

Available on Amazon
for about $3.99


Did you know that Benjamin Franklin created many inventions, including the Magic Square? (A magic square is a box of numbers arranged so that any line of numbers adds up to be the same number, including the diagonals!) Richard Walz has written a historical fiction book about this. It is fun for the students to read while at the same time it gives them a great deal of historical information. It also contains many activities that can be used along with the book.

History shows that Franklin served as clerk for the Pennsylvania Assembly. Uninterested in the meetings, Ben would doodle on a piece of paper to pass the time. In 1771, he stated, "I was at length tired with sitting there to hear debates, in which, as clerk, I could take no part, and which were often so un-entertaining that I was induc'd to amuse myself with making magic squares or circles" (Franklin 1793). So being bored, Ben wrote down numbers in a box divided into squares, and then pondered how the numbers added up in rows and columns...and thus the Magic Square was born. In fact, he studied and composed some amazing magical squares, even going so far as to declare one square “the most magically magical of any magic square made by any magician.”

I love to use Magic Squares in my classroom. The construction and analysis of magic squares provides practice in mental arithmetic, operations with numbers, geometry, and measurement plus it encourages logical reasoning and creativity, all in a game-like setting.  Furthermore, they are a powerful tool for teaching students basic addition skills since each row, column, and diagonal must add up to be the same sum.

One effective way to use a Magic Square is to omit a few of the numbers from the boxes, then have students try to figure out which numbers are missing. To find these numbers, first the students will have to calculate the magic sum. A Magic Square also provides an engaging way to develop mental math skills. Try using magic squares as a warm-up at the beginning of math class or as a math center activity. In addition, students might also want to create their own Magic Squares and then have their classmates solve them.

Below is a magic square for you to solve. You are to arrange the digits 1-9 in the squares below
so that each column, row and diagonal adds up to 15. Can you do it?


To find a solution to this magic square puzzle, look under
Answers to Problems at the top of the Home page.

Your students can make their own Magic Squares by following these steps.  Begin by using a box divided into nine squares.  (There are larger ones, but as they grow in size so does the difficulty.)

  1. The first numeral is placed in the top row, center column.
  2. An attempt is always make to place the next numeral in the square above and to the right of numeral last placed.  All the rest of the rules tell you what to do when rule #2 cannot be satisfied.
  3. If, in the placement of the next numeral according to rule #2, the numeral falls above the limits of the magic square, place the numeral in the bottom square of the next column to the right of the last placed numeral.
  4. If, in the normal placement of the next numeral, it falls to the right of the limits of the magic square, that numeral is placed in the left-hand square of the row above the last placed numeral.
  5. If the cell above and to the right is filled, place the numeral in the cell immediately below the last square filled.
  6. Using this method, filling the upper right-hand cell completes a sequence of moves. Then this happens, the next numeral is placed in the cell immediately below the upper right hand corner square.
Do these steps sound absolutely confusing?  Maybe the pictures below will help to clarify the rules.




Now have your students try this.  Using the blank nine squared Magic Square seen above, use the numerals 11, 12, 13, 14, 15, 16, 17, 18 and 19 to make each row (horizontal, diagonally & vertical) add up to 45. Ask your students if they see a pattern between this new Magic Square and the first one.  (Ten has simply been added to each digit.)  You might also try making your own at Make Your Own Magic Squares.

This post has only scratched the surface of Magic Squares, but isn't that like most things in math?  I trust your students will give Magic Squares a try while having fun doing it!

Check out my newest product entitled The A, B, C's of Number Tiles.  All of the 26 letter puzzles of this 42 page handout are solved in a similar way that magic squares are solved. The activities vary in levels of difficulty. Because the pages are arranged alphabetically, and are not in any particular order based on difficulty, the students are free to skip around in the book. Since the students do not write in the book, the math-a-magical puzzles can be copied and laminated so that they can be used from year to year.