Go Figure!

Is zero an even number? Now, that's 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.

My Students Are Having Difficulty Memorizing Those Dreaded Math Facts!

Many of my college students come to me without knowing their math facts. Some do, but most do not. Since we use calculators in the class, it really isn't an issue. It just takes those students longer to do a test or their homework. One day, the students in my Basic Algebra Concepts class (a remedial math class) were playing a math game to practice adding and subtracting positive and negative numbers. We were using double die (see picture) where a small dice is located inside a larger dice. (I have to keep an eye on these because they tend to "disappear." The students love them!) I noticed one of my students continually counting the dots on the die. He was unable to see the group of dots and know how many were in the set. It was then that I realized he could not subitize sets. (to perceive at a glance the number of items presented)

Subitizing sets means that a person can look at a grouping or a set and identify how many there are without individually counting them. (i.e. three fingers that are held up) When a child is unable to do this, they cannot memorize math facts since memorizing is associating an abstract number with a concrete set. Many teachers as well as parents fail to recognize the root cause of this memorization problem. AND no amount of practicing, bribing, yelling, or pulling out your hair will change the situation. So what can you do?

First of all, the problem must be identified. Use a dice and see if the child must count each dot on each face. Try holding up fingers or laying out sets of candy (M&M's - yummy!) or using dominoes. Put five beans in a container, and ask the child how many are in the box. (They may count them the first few times.) Take them out, and put them back in. Ask the child again how many there are. If, after several times, s/he is unable to recognize the set as a whole, then s/he cannot subitize sets.

How do you help such a child? If you have small children at home, begin subitizing sets by holding up various combinations of fingers. My youngest grandson just turned four; so, we worked on holding up two fingers on one hand and two fingers on the other; then one and three fingers, and of course, four fingers. I also like to use dominoes. They already have set groupings which can be identified, added, subtracted, and even multiplied. A dice is great because the child thinks you are playing a game, not doing math. Roll one dice, and ask the child to identify the set of dots. Try the bean idea, but continue to change the number of beans in the box. My grandchildren love the candy idea because they are allowed to eat them when we are done. (All children need a little sugar now and then even though their parents try to control the intake. I love being a Grandma!)

Gregory Tang has written two wonderful books for older children, The Grapes of Math and Math for All Seasons, which emphasize subitizing sets. At times, I even use them in my college classes! I was fortunate to attend two of his workshops presented by Creative Mathematics. He not only has a sense of humor, but his books can be read again and again without a child becoming bored. Check them out!

The ROOT of the Problem - Finding Digital Root to Reduce Fractions

When students skip count, they can easily say the 2's, 5's, and 10's which translates into easy memorization of those particular multiplication facts. Think what would happen if every primary teacher had their students practice skip counting by 3's, 4's, 6's, 7's, 8's and 9's! We would eradicate the drill and kill of memorizing multiplication and division facts.

Since many of my college students do not know their facts, I gravitate to the Divisibility Rules. Sadly, most have never seen or heard of them. I always begin with dividing by 2 since even numbers are understood by almost everyone. (Never assume a student knows what an even number is as I once had a college student who thought that every digit of a number must be even for the entire number to be even.) We then proceed to the rules for 5 and 10 as most students can skip count by those two numbers.

Finally, we learn about the digital root for 3, 6, and 9. This is a new concept but quickly learned and understood by the majority of my students. (See the definition below which is from A Simple Math Dictionary available on TPT).

Here are several examples of finding Digital Root:

a) 123 = 1 + 2 + 3 = 6. Six is the digital root for the number 123. Since 123 is an odd number, it is not divisible by 6. However, it is still divisible by 3.

b) 132 = 1 + 3 + 2 = 6. Six is the digital root for the number 132. Since 132 is an even number, it is divisible by 6 and by 3.

c) 198 = 1+ 9 + 8 = 18 = 1 + 8 = 9. Nine is the digital root for the number 198; so, 198 is divisible by 9 as well as by 3.

4d201 = 2 + 0 + 1 = 3. Three is the digital root for the number 201; so, 201 is divisible by 3.

The first time I learned about Digital Root was about eight years ago at a workshop. I was beside myself to think I had never learned Digital Root. Oh, the math classes I sat through, and the numbers I tried to divide by are too numerous to mention! It actually gives me a mathematical headache. And to think, not knowing Digital Root was the ROOT of my problem!

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A teacher resource on Using the Divisibility Rules and Digital Root is available at Teachers Pay Teachers. If you are interested, just click under the resource cover on your right.

Different Ideas on How to Teach Long Division

As presented in an earlier posting, 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 acronym (a mnemonic device) of Does McDonald's Sell Cheese Burgers. I have 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 than the divisor, then 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.