Faux Diamonds

In some preschool and kindergarten classes across the country, the geometric shape formerly known as a diamond is now being called a rhombus.  Why?  Does it matter? 

To be honest, a diamond is not technically a mathematical shape whereas a rhombus is.  When someone says the word rhombus, you know they are referring to a quadrilateral that has all four sides the same length; the opposite sides are parallel, and the opposite angles are equal.  (Mathematical Warning: A rhombus is not thinner than a diamond, AND the plural form, rhombi, is not a dance performed on the program Dancing With the Stars.)  

But what comes to mind when you hear the word diamond?  If you are a woman, you might envision a large sparkling gem setting on the ring finger of your left hand.  If you are a guy, you might think of a baseball infield. (The distance between each base is the same, making the shape a diamond.)  If you play cards, the word might bring to mind a suit of playing cards, OR you might recall a line in the song, Twinkle, Twinkle, Little Star.  Calling a rhombus a diamond is similar to calling a child a "kid" (could be a baby goat), or a home your "pad" (might be a notebook).  The first is an accurate term, the second one is not. 

So how does this affect you as a teacher?  It doesn't, unless rhombus is on a local benchmark or state test.  But if you are an elementary grade teacher, please use the correct mathematical language because a middle school math teacher will thank you; a high school geometry teacher will sing your praises, (see song below) and a college math teacher, like me, will absolutely love you for it!

Rhombus, Rhombus, Rhombus

  (sung to the "Conga" tune)
(The song where everyone is in a line with their hands on each other's shoulders)

 Rhombus, rhombus, rhombus;

Rhombus, rhombus, rhombus

Once it was diamond;

Now it's called a rhombus.

Much Ado About Nothing

I have decided to post (this is an updated previous post from 2011) some questions about zero that my college students have asked me in class.  I will say this, "Zero can surely give you a severe headache unless one knows its properties."  

Question #1 - Do you know why zero is an even number?    All mathematics is based on patterns.  Because of this, I know that an even plus an even number will always give me an even answer; an odd number added to an odd also gives me an even answer, and an odd number plus an even gives me an odd answer. In other words:    E + E = E     O + O = E     O + E = O



The numbers 4 and -4 are both even numbers. If we add them together, their sum is zero.  Based on the math pattern of  E + E = E,  then zero has to be even as well.  If we substitute zero in other problems such as 1 + 0 = 1, it fits the O + E = O  rule just as 2 + 0 = 2 fits the E + E = E  rule.

In Algebra, even numbers can be written as 2 x n where n is an integer.  Odd numbers can be written in the form of  2 x n + 1.  If we have n represent 0, then  2 x n = 0 (even) and  2 x n + 1 = 1. (odd)

I say all of this to relate an actual incident that occurred in my classroom.  I wrote the number 934 on the white board, and commented that since it was even it was divisible by 2.  One of my students was perplexed because he did not understand how 934 could be even when it contained two odd numbers and only one even number.  He actually thought that all the digits of a number had to be even for the number to be even.  Funny?  Not really!  Amazingly, he had made it through 12 years of school without understanding Place value as it relates to even numbers. Unfortunately, I had assumed that everyone (especially my college students) knew what an even number was.  I no longer make assumptions about students and their math knowledge!

Question #2 - Is zero positive or negative? The definition for positive numbers is all numbers greater than zero, and the definition for negative numbers is all numbers less than zero. Therefore zero can be neither positive or negative.

Question #3 - Is zero a prime or composite number? To be a prime number, a number must have only two positive divisors, itself and one. Zero has an infinite number of divisors so it is not prime. A composite number can be written as a product of two factors, neither of which is itself. Since zero cannot be written as a product of two factors without including itself, zero, it is not composite.

Question #4 - Why can't you divide by zero? I love this question. Back in the dark ages when I asked it, I was always told, "Because I said so." Being an inquisitive student was not a blessing when I was growing up. Math teachers who knew all did not want to be questioned!!!! Anyway, I don't mind the question, and here is my practical answer.

First, we must understand division. Division means putting or separating a number of items into a number of specific groups or sets. When you divide, such as in the problem 12 divided by 2, you are really putting 12 things into two groups or two sets. Therefore, if you have the problem 8 divided by 0, it is impossible to put eight things into no groups. You cannot put something into nothing!



Hopefully, this clears up a few things about zero.  I leave you with this math cheer.  (I always wanted to be a cheerleader!)

                            Zero, two, four, six, eight,

Who do we appreciate?

Even numbers! Even Numbers! Even Numbers!

        

SMART Goals

My Math Study Skills class has just started chapter #5 on setting goals.  So many times my students will write goals such as "I will study more for math".  Sounds great, but this statement isn't a goal.  It is not specific or measurable, and I have no idea who is doing the goal.  Instead it should read something like this:  "I plan to set aside 15 minutes each Monday through Friday to study math."

Since mnemonic devices are a way to help students remember, I introduce the acronym (a word form created from the first letters of a series of words) SMART.

 

      A well written goal is learner oriented.  It emphasizes what the student is expected to do, not what the instructor will do.  It focuses on the outcome and not the learning activities that will lead to that outcome.  It uses clearly stated verbs that describe a definite action or behavior.  Finally, a well written goal describes an observable and measurable performance or end product.

     

      I keep this stair step visual in front of my students during the five weeks they are tracking their three math goals. It helps them to set-up attainable goals.

      When they accomplish a set goal, I have 

      noticed they feel more confident about math which, in turn, improves their self-esteem and helps the

      student to become a more internally motivated student.

I use a booklet called My Goal Tracker by Laura Candler, a top seller on Teachers Pay Teachers, which is free.  If you are interest in having your students set goals and keep track of how they are doing, I would suggest downloading this well laid out and easy to use booklet.

It's A Square Deal

My math-a-magical powers are back.  In my January 23rd post, I demonstrated how to easily divide by the decimal .25.  Today let's look put on our magician's hat and learn how to square any two digit number that ends in 5.

In the number below, look at the digit in the tens place (3) and count up one more.  In this case when we count up, the 3 becomes a 4.  Now multiply the two numbers together, 4 × 3 which equals 12.  Next do 5² which is 5 × 5 = 25.  Therefore, the answer to 35² is 1,225.

Let's try another one.  Remember, it must be a two digit number that ends in 5 for the magic trick to work. This time let's start with 65 and follow the exact same procedure used above.

Did you get the right answer?  Have your students try this using all the two digits numbers that end in 5, beginning with 15 and finishing with 95.  Since all math is based on patterns, ask hem to carefully look for a pattern and describe it.  I think you will be very surprised at what you find!

 

The Power of Math Tricks

Math tricks will never make you a great mathematician, but in the eyes of some, you can be a fantastic math-a-magician. My college students love it when I show them a trick they can then take home to amaze and impress their peers, parents, children or the best yet - their spouses. 

Remember when I demonstrated how to easily multiply by 11 in the post The Eleventh Hour?  Or how about the trick of multiplying by 12 in Quick Times?   Here is a new one I recently showed my students.

First, let's look at a problem where a number is divided by the decimal .25. 

The above example is really 9 x 4 which is 36, but why is this true?  Hopefully your students know that .25 is equivalent to 1/4; so this problem can be reworded as  9 divided by 1/4.  As seen below, when dividing a whole number by a fraction, we find the reciprocal of 1/4 and then multiply which gives us the answer of 36.

 

Based on the sample above, anytime a problem requires dividing by .25, simply multiply by four to get the correct answer.  Try these without using a calculator or paper and pencil.





Instead of using the reciprocal to divide fractions, I teach my students that this is the "cross" method. Simply look at the original problem and cross multiply as seen in the illustration below. 



Fractions for the
Confused
and Bewildered

First multiply the bottom right denominator with the top left numerator. (4 x 9)  Next multiply the bottom left denominator with the top right numerator, (1 x 1) and you get an answer of 36.  When doing the fractions this way, there is no confusion on the students' part about which fraction to invert. If you would like a more details on how to divide fractions this way, go to the post entitled: Don't Flip!

If you are interested in other alternative ways to teach the four operations of fractions, you can check out the resource on your right.


By the way, the answers to the above problems are  a) 24    b) 80    c) 72    d) 380.    How did you do?