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It Depends on the Angle

My Basic Algebra Concepts class just started a brief chapter on geometry...my favorite to teach!  We are currently working on angles, and as we went through the definitions, I noticed my students were having difficulty distinguishing complimentary from supplementary angles.  Since most of my students are visual learners, I had to come up with something that would jog their memory.

The definition states that complementary angles are any two angles whose sum is 90°.  (The angles do not have to be next to each other to be complementary.)  As seen in the diagram on the left, a 30° angle + a 60° angle = 90° so they are complementary angles.  Notice that the two angles form a right angle or 1/4 of a circle.

If I write the word complementary and change the first letter  "C" into the number nine and I think of the letter "O" as the number zero, I have a memory trick my mathematical brain can remember. 


Supplementary Angles are two angles whose sum is 180°.  Again, the two angles do not have to be together to be supplementary, just so long as the total is 180 degrees. In the illustration on your right, a 110° angle + a 70° angle = 180°; so, they are supplementary angles.  Together, they form a straight angle or 1/2 of a circle. 
If I write the word supplementary and alter the "S" so it looks like an 8, I can mentally imagine 180°.


 Since there are so many puns for geometric terms. I must close with a bit of geometry humor.




Having Fun With Angles



Want a variety of hands-on ideas on how to introduce angles to your students?
Check out this resource.





Mathematics Tips for Parents


Boy by "My Cute Graphics"
Success in school starts and continues at home, but many parents feel inadequate when it comes to helping their children with math. While parents can usually find time to read a story to their children, thereby instilling a love for books, they are often at a loss as to how to instill a love and appreciation for mathematics.  Like reading, mathematics is a subject that is indeed necessary for functioning adequately in society.  Here are some tips to help you as you work with your child this school year.

Recognize that you make an important difference in your child's education.   Most children develop a sense of numbers way before the "regular" school years.  If you have a young child, take advantage of those early years through activities at home that teach and at the same time are enjoyable.  You might take your child on a counting walk in your neighborhood to count how many trees, shrubs, plants, houses, birds, dogs, etc. you see.  Look for twigs or pine cones or leaves, etc. and have your child count as many as s/he can. Then lay them side by side to compare the length and ask your child, "Which is the longest, which is the shortest? Are there any that are the same length?"

Provide experiences at home that help your child be successful, and seek ways to let children, even very young children, know that they are needed and important.  Cooking is a fun way to do this. Help your child follow the directions on a Kool Aid packet or frozen juice can to make refreshments for the family.  Help your child cut a fruit or vegetable into halves, fourths, thirds, etc. Let them help prepare a meal while asking, "What do you do first? Second? Third?"  or better yet, allow them to measure the ingredients for a recipe.

Children do not need a lot of motivation when it comes to recognizing and learning the value of coins.  You know they are interested when they start bugging you for money.  However, it is not sufficient for children to be able to just recognize coins, they must also know the value of these coins.  The best way to accomplish this is to use real money.  You might show your child two or more coins and have him/her tell you the total value of the coins.  Or hold up a coin.  After your child identifies it, discuss what the coin would buy at the store.  When going to the grocery store, give your child his/her own money to buy something.  Have them select an item that costs less than the money you have given them.  You can also do a similar activity by asking them to determine what are the fewest number of coins it would take to pay for the item. Give your child a practical math experience by estimating how long it takes to prepare a meal from start to finish.

Parents' attitudes toward mathematics have an impact on children's attitudes; so, be patient with your child.  A wrong answer on a math test or a homework assignment is not a time for scolding.  It tells you to look further, to ask questions, and to find out what the wrong answer is saying about your child's understanding.  Ask your child to explain how they solved the problem.  Most importantly, relax!  Know that neither you nor the teacher needs to be perfect for your child to learn math.  Remember, one bad math assignment/test will not destroy your child's ability to learn math.

But what if you need some assistance?  Luckily, in today's world, we can find mathematical help at the click of a button.  Below are some great places to go and find outside help if your child is struggling or if you need more information for yourself.

Study Shack is a great place to find or make flashcards, play hangman, do matching activities or crosswords.  It has activities for grades 1-6 as well as addition, multiplication, algebra and geometry.  Cliff's Notes for Math is site that has notes, examples and quizzes for your older children.  The subject areas include Basic Math through Calculus.  There are many on-line math dictionaries.  My favorite is A Math Dictionary for Kids because it includes animation and interactive activities.  Even You Tube is a great resource for students struggling with a concept and needing an alternative way of seeing it. 

Finally, talk about people who use math in their jobs, including builders, architects, engineers, computer professionals, and scientists.  Point out that even if your child does not plan to pursue a career in which s/he will use math, learning it is still important because math teaches you how to solve problems and how to think logically.  AND we use math everyday!


My Cute Graphics offers FREE clip art and images for teachers, classroom projects, web pages, blogs, scrapbooking, print and more. Check out the website by clicking either under the boy sitting on the equal sign at the beginning of this article or on the purple letters in this paragraph.

Parabola - The Arch Enemy?

I love to relate math to the real world with my students because that is the only way they will see the relevance.  Our family (son + his wife + three grandkids + husband) just returned from a trip to Orlando.  While driving home from Cocoa Beach, my daughter-in-law noticed a purple arch.  Being that my mind is always, always thinking about math, I informed her that it was a parabola with a negative slope.  My son, who is an engineer, starting talking about slope, and the two of us shared some equations such as the one for lines y = mx + b (much to the chagrin of the other travelers stuck in the car with us).

Mathematically speaking, a parabola is a two-dimensional, symmetrical curve or simply, a special curve shaped like an arch.  All parabolas are vaguely “U” shaped, and they have a highest or lowest point called the vertex.  The vertex is the place the parabola makes it sharpest turn.  Any point on a parabola is at an equal distance from a fixed point (the focus), and a fixed straight line (the directrix). Parabolas may open up or down and may or may not have x-intercepts, but they will always have a single y-intercept. Those that open up have a positive slope (they form a smile) and those that open down have a negative slope (they form a frown).  We always get a parabola when we graph a quadratic equation, an equation that contains a variable that is squared such as y2 = 20x or x2 - 9y = 0.

Now that all of this is as clear as mud for many of you, let's look at some parabolas in real life.  Yes, they are out there!  Can you identify the ones below?


Did the last picture stump you?  Well, it would unless you were from Los Angeles.  It is the
Encounter Restaurant, atop Los Angeles International Airport’s landmark Theme Building.

Other real life examples include....

1) Throwing or Kicking a Ball - If you throw a baseball, kick a soccer ball, shoot an arrow, fire a missile, or throw a stone, it will arc up into the air and come down again following the path of a parabola! (Except for how the air affects it.) The next time you watch a football being thrown from the quarterback to a receiver, think of a parabola.

2) Roller Coasters that arc up and down and sometimes around - the one ride I avoid! When a coaster falls from the peak (vertex) of the parabola, it is rejecting air resistance, and all the bodies are falling at the same rate. The only force here is gravity. Most people (I am NOT included) enjoy or get a thrill out of parabolic-shaped coasters because of the intense pull of gravity.

3) Reflectors - Parabolas are also used in satellite dishes and flashlights. In satellite dishes it helps reflect signals that then go to a receiver, which interprets the signals and shows satellite-transmitted channels on your television. In flashlights, car headlights and spotlights, the parabolic shape helps reflect light. Notice the beam of light coming from the flashlight on your right. See how the light appears to be in the shape of a parabola?


4) Suspension Bridges such as the Golden Gate Bridge, the Brooklyn Bride, the Washington Bridge, etc.  Suspension bridges are capable of spanning long distances and actually are the only type of bridge to span the longest distances possible for a bridge. This is because the shape of the suspension bridge is actually one of the most stable structures there is. In the image of above, can you see how the cables form parabolas?

So now you know parabolas are everywhere even when you are playing ports, watching T.V., riding a roller coaster at your favorite theme park or going cross a suspension bridge.  So what kind of parabola will you display on your face today…a negative parabola (a frown?) or a positive parabola (a smile)?




Elvis and PEMDAS



Any math teacher who teaches the Order of Operations is familiar with the phrase, "Please Excuse My Dear Aunt Sally".  For the life of me, I don't know who Aunt Sally is or what she has done, but apparently we are to excuse her for the offense.  In my math classes, I use "Pale Elvis Meets Dracula After School".  Of course both of these examples are mnemonics or acronyms; so, the first letter of each word stands for something.  P = parenthesis, E = exponents, M = multiplication, D = division, A = addition, and S = subtraction

I have always taught the Order of Operations by just listing which procedures should be done first and in the order they were to be done.  But after viewing a different way on Pinterest (originally posted on Math = Love by Sara Hagen), I have changed my approach. Here is a chart with the details and the steps to "success" listed on the right.


Since multiplication and division as well as addition and subtraction equally rank in order, they are written side by side. What I like about this chart is that it clearly indicates to the student what they are to do and when.  To sum it up:

When expressions have more than one operation, follow the rules for the Order of Operations:
  1. First do all operations that lie inside parentheses.
  2. Next, do any work with exponents or radicals.
  3. Working from left to right, do all the multiplication and division.
  4. Finally, working from left to right, do all the addition and subtraction.
Failure to use the Order of Operations can result in a wrong answer to a problem.  This happened to me when I taught 3rd grade.  On the Test That Counts, the following problem was given.

The correct answer is 11 because you multiply the 4 x 2 and then add the 3, but can you guess which answer most of my students chose?  That's right - 14!  From that year on, the Order of Operations became a priority in my classroom.  Is it a priority in yours?  Should it be?







PEMDAS











I have a new product that I recently added to my store entitled: Order of Operations - PEMDAS, A New ApproachThis ten page resource includes a lesson plan outline for introducing PEMDAS, an easy to understand chart for the students, an explanation of PEMDAS for the student as well as ten practice problems.  It is aligned with the fifth grade common core standard of 5.OA.1. Just click on the words under the cover page if it is something you might like.

O-H-I-O and Octagons


An octagon is any eight sided polygon.  We often use a stop sign as an example of an octagon in real life.  But in a actuality, a stop sign is a regular octagon meaning that all of the angles are equal in measure (equiangular) and all of the sides have the same length (equilateral).  For an eight sided shape to be classified as an octagon, it needs to have only eight sides.

I got to thinking about this since fall is just around the corner, and our family are BIG Ohio State football fans.  Being raised in Ohio and having relatives who taught at Ohio State has fueled this obsession, but so has doing graduate work there.  If you aren't familiar with the Ohio State Buckeyes, here is your opportunity to learn something new.

On the right you will see one of the many symbols for THE Ohio State University.  The red "O" is geometric because it is an octagon (just count the sides). Even the beginning of the word Ohio is an octagon. (I just adore mathematics in real life!)

The Ohio Stadium, a unique double-deck horseshoe design, is one of the most recognizable landmarks in all of college athletics. It has a seating capacity of 102,329 and is the fourth largest on-campus facility in the nation. Attending football games in the Ohio Stadium or watching the game on television is a Saturday afternoon ritual for most Ohio State fans.  The stadium is even listed in the National Registry of Historic Places. Anyone (and we have) who has been to a game in the giant horseshoe understands why. There are few experiences more fun or exciting!  In the middle of the football field is the octagonal O as seen in the picture below. (Another example of math in real life!)

And then there is the Ohio State Marching band which I love!  Their half times programs are awesome, and many are posted on You Tube.  The signature formation of the band, which is performed before, during halftime or after home games, is Script Ohio. Each time the formation drill is performed, a different fourth or fifth year band member who plays the sousaphone has the privilege of standing as the dot in the "i" in "Ohio." The crowd always goes wild!  (If you don't believe me, click here to see a video and be sure to check out the "O" on the drums.)  Once, when we were at a game, the band formed a double script Ohio. Even though this "O" is not an octagon since it is written in cursive, I still had to mention this distinctive and unique half time tradition.


Before I continue this posting, I must answer the age old question, "What is a buckeye?"  Since I grew up in Ohio, this question is easy for me to answer, but for everyone else, a buckeye is a nut. (I bet many of you thought it was candy.) Buckeye trees grow in many places in Ohio. The trees drop a "fruit" that comes in a spiked ball with a seam that runs around it. If you crack the seeds open, you can remove the "buckeye." When the nut dries, it is mostly brown in color but it has a light color similar to an over-sized black-eyed pea on one end. This coloration bears a vague resemblance to an eye hence the name, buckeye. 

Then there is Brutus Buckeye, (a student dressed in a costume) the official mascot of THE Ohio State University; so, you might say, since I was born and raised in Ohio, I am a nut!  Brutus (as seen on the left) wears a headpiece resembling a buckeye nut, a block O hat, (another octagon), a scarlet and gray shirt inscribed with the word "Brutus" on the front and the numbers "00" on the back.  Brutus also wears red pants with an Ohio State towel hanging over the front, and high white socks with black shoes. Both male and female students may carry out the duties of Brutus Buckeye as long as they are a committed Ohio State fan.


O-H-I-O
Finally, if you ever are lucky enough to see four people with their hands in the air, forming letters of the alphabet, it is most likely four Ohio state fans spelling out O-H-I-O!  That's how our grandchildren learned how to spell it! (The picture on the right is of our youngest son with his four groomsmen on the day of his wedding.)

And it is so-o-o easy to remember.  Just use this riddle:  What is round on the ends and high in the middle?  You guessed it - OHIO!


Patterns - Even in Sound!

Resonance Sound Experiment
As you know, all math is based on patterns.  In fact, patterns are everywhere.  This year, my husband is going to do an experiment with his students on sound and vibration.  He was looking for some ideas on You Tube, and came across The Amazing Resonance Experiment.

If you click under the picture on the left, you will see an incredible thing happen when salt is put on the surface of a metal plate connected to a tone generator and then vibrated with different sound frequencies. Different patterns emerge in a seemingly graceful dance.  As the plate vibrates at different frequencies, the salt particles fall into different resonant patterns.  It is mind-boggling how the frequencies create such detailed and meticulous patterns of the salt grains. When the video began, I expected all of the patterns to be symmetrical, but many were not (although, as a mathematician, I thought they were trying!).  I do think it is beautiful how one pattern morphs into the next. Notice how the patterns become more complex as the tone increases in frequency.  Also take note of how the individual shapes (circles, squares, etc.) that form the patterns keep getting smaller and smaller as the frequency becomes higher.

According to physics, everything is frequency and vibration. "These salt patterns are a result of micro bendings in the material due to the vibration waves going through it. The salt gathers at stationary points on the plate where there is the energetic most convenient place. In other words, the salt gathering results from vibrating the plate at different frequencies.  However, the patterns are not connected solely to the frequency, but rather to the frequency combined with the shape of the metal plate.  Using the same frequencies on differently shaped plates (round, triangular, etc.) would produce different patterns. Even changing the material would affect the result." The experiment has been tried with flour, sand and sugar with the flour forming clumps (not a desired result). 

So take five minutes to watch the video and see the shape of sound which is beautiful, remarkable as well as fascinating !  Don't you love science?  It's just as amazing as math!! 

Dinner Dilemma and the Roll of a Die

Being a grandparent lets you try some new discipline methods that you never thought of as a parent. My grandchildren don't always like what I serve for dinner (Unbelievable, isn't it?); so, many times food is left on their plates. My children want their children to at least take a bite of everything on their plate which often times feels like a monumental task for our grandchildren. The Solution? I have an oversized sponge die on hand for such occasions. The child who doesn't want to eat rolls the die, and the number that comes up is how many bites they must take before dessert is served. Now, the child must argue with the die and not the parent or me! (It's difficult to argue with an inanimate object.)

Besides taking care of a dinner dilemma, my grandchildren are learning to conserve sets. (Oh, there's the math part of this article!) Since there are no numbers on the die, only dots, the child must count the dots to find out the number. Surprisingly, even the youngest are learning to recognize the dot patterns and can state the number of dots without counting. This indicates they are learning to conserve sets, a necessary prerequisite to memorizing the math facts. If you aren't sure what conserving sets means, go back and read my blog posting entitled Can't Memorize Those Dreaded Math Facts. In the meantime, enjoy a new way to enjoy dinner because it is pretty dicey!


You might like a math game that uses dice. It is called Bug Ya and can be purchased at my store. Three games are included in the four page resource packet. One is for addition and subtraction; the second is for multiplication, and the third game involves the use of money. The second and third games may involve subtraction with renaming and addition with regrouping based on the numbers that are used. All the games have been developed to extend the recall of facts through playful and intelligent practice. Be sure and download the preview.


Getting to Know You...



School is beginning for most of us.  I teach on the college level, but I still feel the most important thing I can do is to make the students feel connected to one another so that they at least know one other person in the class.  I always start each new class by playing a true/false game.  I start off the first class by listing four items about myself, three that are true and one that is false.  The students try to discover the false one.  On a 3” × 5” card, I then have the students write four things about themselves, three true and one false from which we, as a class, try to find the false one.  I then collect and save the cards.
 
At the next class meeting, I will choose 3-4 cards from which to read the true statements. As a class, we try to match the student to the card.  It really helps the students to relax and have fun at the same time plus they get to know each other. I usually do this activity for a couple of weeks until I sense that the students are comfortable being in the group.
 
By the way, here are my four statements.  Can you choose the false one?
  1. I have eight grandchildren, five of whom are adopted.
  2. My husband asked me to marry me on our first date.
  3. I am a big Jayhawk (Kansas University) fan.  (We live in Kansas.)
  4. I have been teaching for over 30 years.
 
Give up?  You can find the answer on the page entitled Answers to Questions.

 





You might be interested in two other back to school items.  First is the Back to School Glyph for grades K-3.



Secondly, for new teachers, there is a Beginning of the Year Checklist. This four page comprehensive checklist walks you through 88 items which should to be established before the first day of school.

Not on the Test

Not on the Test
While watching my granddaughter at her tennis lesson, I was visiting with two teachers.  One was a retired fourth grade teacher and the other currently taught Algebra in middle school.  Both we decrying the fact that each year the students come with knowledge that is more narrow than broad.  They both felt this was because more and more time is now spent on testing or getting ready for testing.  As I stated in my January 25, 2012 posting entitled The Pros and Cons of Testing, "High stakes tests have become the “Big Brother” of education, always there watching, waiting, and demanding our time. As preparing for tests, taking pre-tests, reliably filling in bubbles, and then taking the actual assessments skulk into our classroom, something else of value is replaced since there are only so many hours in a day.  In my opinion, tests are replacing high quality teaching and much needed programs such as music and art."

A long time ago, a friend sent me a song written by Tom Chapin with John Forster called Not on the Test. I saved it, and I listen to it often, especially when I am having a "down" day.  Tom and John wrote the song to express their disappointment in the lack of arts education in many public schools.  Even though the song refers to No Child Left Behind, with Common Core approaching with its own set of tests, I think you might get a much needed laugh from the song.  Just click on the link under the picture, and let me know what you think!

Math or Maths?

I keep seeing the word "maths" on Pinterest. hmm   To me it is confusing and puzzling in addition to being a misspelling or misuse of the word "math".  Where in the world did this word come from, who created it, and why?

Math can be used in the singular form ("math") or in the collective form ("math") without distinction. Math is a field of study which is divided into specific multiple disciplines such as algebra, geometry, etc. which then are again divided into mathematical subgroups.

Likewise, "physics" is single word for a particular field of study, even though it ends in "s". You don't study "a physic"; you study "physics". Just like physics, mathematics is considered singular, but maths? Isn't that like saying deers for deer or sheeps for sheep?

Well, here is the mathematical truth.  Maths is the term commonly used in England, Australia, New Zealand, etc.  It is a shortened form of mathematics.  They pluralize the word, and refer to studying "maths" because mathematics has an "s" on the end.  So the answer to this "maths" question is that it depends on where you live as to what word you use.  Therefore be aware of the geographical differences so you can use the correct form of the word in your writing or speaking.  And when you see the word "maths", don't jump to the conclusion it is a misspelling or a misuse.  Recognize that the writer most likely lives outside of the United States.



Interested in "Math" stuff?  There are 51 resources in my store just for math.  Just go to the category of Math Stuff.

The Mathematics of a Hail Storm

Kansas had an absolutely terrible storm Sunday night.  The sky was a murky green (this is never a good sign), and the hail was the size of golf balls.  It broke three of our four basement windows; so, we had glass everywhere plus hail stones were coming into our family room like crazy.  We just stuffed the windows with pillows and tried to clean up.  Our neighbors have siding damage that looks like someone went through our neighborhood with a machine gun when instead it was the force of the hail.  When the storm let up, everyone was outside and most were thanking God for sparing our homes.  Hail was laying everywhere in piles; so, my husband, being the scientist he is, was examining some of the big pieces.  He showed me how the hail was formed.
 
He explained that hail is created when raindrops are lifted up into the atmosphere during a thunderstorm and then super cooled by temperatures below freezing, turning them into balls of ice. The faster the updraft on these balls of ice, the bigger they can grow. On the hailstones found in our yard, we could actually see several rings inside of them which indicated they were cycled through the thunderstorm more than once. What started out as a minute raindrop became a golf ball sized chunk of ice. According to Dr. Dick Orville of Texas A & M University, large hailstones have been clocked traveling more than 90 miles per hour. I don’t know how fast these golf ball sized ones were traveling, but fast enough to make huge holes in aluminum siding.
 
Let’s look at the math of concentric circles.  Concentric means two or more circles of different sizes that all have the same center -- like a bulls eye target. 
On the right is a picture of one of “OUR” hail stones. If you look closely, you can see four different circles, (a pattern) one inside the other.  That means it was recycled through the thunderstorm at least four times. 
AND it was not only hard, but heavy.  According to the Internet, the largest hailstone ever recorded in the United States occurred in 1970 in Coffeyville, Kansas which is about two hours southeast of Wichita.  It was a stone that weighed 1.6 pounds and measured 5.5 inches when it fell. YIKES!!  I am glad I missed that one.
 

Ice Investigation
 
 
Since ice seems to be the theme of this post, I want to mention my newest Teacher Pay Teachers resource entitled Ice InvestigationThis 20 page resource is a six lesson science investigation for grades 3-4 which uses ice cubes.  The inquiry guides the student through the six steps of the scientific method.  The unit consists of a three page student investigation organizer, a property word list, an optional student checklist, and a four point grading rubric for the teacher.  If you are interested, just click under the cover on your right. 
 

Divide to Conquer?

As presented in the posting of April 10th, there is another way to approach long division.  However, since many of you are required to present it the long way, here are a couple of ideas to make it easier for your students.

First of all, have the students use graph paper.  The squares help to keep the numbers aligned which seems to be a problem for many students.  If you don't have graph paper, you can download free templates at Donna Young's Free Graph Paper and make your own.  I like the idea of separating the problems with lines to ensure there is no cross over from one problem to another.

 

Secondly, try using the mnemonic of Does McDonald's Sell Cheese Burgers.  I've  see this acronym many times on Pinterest, but usually the C is omitted.


Check means that after the student has subtracted, they should check to see if the remainder is smaller than the divisor.  If it is equal to or larger, then they enough was not taken out of the dividend.  This is a step often skipped when long division is taught; yet, if the student doesn't check and make the needed correction, the answer (quotient) will be wrong.

In order to learn division, the student must first have a good understanding of multiplication. Students don’t need to perfectly know all of the times tables, but a majority of the facts or having a reasonably quick strategy to figure out the answer is necessary.

Start by practicing division using the number series the students can easily skip count such as 2 and 5. Then gradually move up to nine.  After that, move to division by double digit numbers using 10 since most students know how to skip count by 10.  Once the concept is understood, teaching division will become more about guided practice to help your child to become comfortable with the division operation which, in reality, is a different kind of multiplication practice.






Is zero even? What an "odd" question!

My daughter and her husband are heading to Las Vegas with his family to celebrate his parent's 50th wedding anniversary.  I guess when you are at the roulette table, (never been there or done that) and you bet "even" and the little ball lands on 0 or 00, you lose. Yep, it's true; zero is not considered an even number on the roulette wheel, something you better know before you bet.  This example is a non-mathematical, real-life situation where zero is neither odd or even.  But in mathematics, by definition, zero is an even number. (An even number is any number that can be exactly divided by 2 with no remainder.)  In other words, an odd number leaves a remainder of 1 when divided by 2 whereas an even number has nothing left over.  Under this definition, zero is definitely an even number since 0 ÷ 2 = 0  has no remainder. 

Zero also fits the pattern when you count which is the same as alternating even (E) and odd (O) numbers.
Most math books include zero as an even number; however, under special circumstances zero may be excluded.  (For example, when defining even numbers to mean even NATURAL numbers.)  Natural numbers are the set of counting numbers beginning with 1 {1, 2, 3, 4, 5....}; so, zero is not included. 
Consider the following simple illustrations.  Let's put some numbers in groups of two and see what happens.



As you can see, even numbers such as 4 have no "odd man out" whereas odd numbers such as 3 always have one left over.  Similarly, when zero is split into two groups, there is not a single star that does not fit into one of the two groups.  Each group contains no stars or exactly the same amount.  Consequently, zero is even.

Algebraically, we can write even numbers as 2n where n is an integer while odd numbers are written as 2n + 1 where "n" is an integer.  If n = 0, then 2n = 2 x 0 = 0 (even) and 2n + 1 = 2 x 0 + 1 = 1 (odd).  All integers are either even or odd. (This is a theorem).  Zero is not odd because it cannot fit the form 2n + 1 where "n" is an integer. Therefore, since it is not odd, it must be even.

I know this seems much ado about nothing, but a great deal of discussion has surrounded this very fact on the college level.  Some instructors feel zero is neither odd or even.  (Yes, we like to debate things that seem obvious to others.)

Consider this multiple choice question.  (It might just appear on some important standardized test.)Which answer would you choose and why?

             Zero is…
                      a.) even
                      b.) odd
                      c.) all of the above
                      d.) none of the above


Mathematically, I see zero as the count of no objects, or in more formal terms, it is the number of objects in the empty set.  Also, since zero is defined as an even number in most math textbooks, and is divisible by 2 with no remainder, then "a" is my answer.