Thursday, June 23, 2011

Let's Go Fly A Kite!

One of the comments on the posting Faux Diamonds was quite intriguing. One teacher wrote, "Would you believe on the NY state 4th grade math test this year, they would not accept "diamond" as an acceptable answer for a rhombus, but they did accept "kite"!!!!! Can you believe this? Since when is kite a shape name? Crazy."

Well, believe it or not, a kite is a geometric shape! The figure on the right is a kite. In fact, since it has four sides, it is classified as a quadrilateral. It has two pairs of adjacent sides that are congruent (the same length). The dashes on the sides of the diagram show which side is equal to which side. The one dashed sides are equal to each other, and the two dashed sides are equal to each other.

A kite has just one pair of equal angles. These congruent angles are a light orange on the illustration at the left. A kite also has one line of symmetry which is represented by the dotted line. (A line of symmetry is an imaginary line that divides a shape in half so that both sides are exactly the same. In other words, when you fold it in half, the sides match. It is like a reflection of yourself in a mirror.)

The diagonals of the kite are perpendicular because they meet and form four right angles. In other words, one of the diagonals bisects or cuts the other diagonal exactly in half. This is shown on the diagram on the right. The diagonals are green, and one of the right angles is represented by the small square where the diagonals intersect.

There you have it! Don't you think a geometric kite is very similar to the kites we use to fly as a kids? Well, maybe you didn't, but I do remember observing Ben Franklin flying one! Anyway, as usual, the wind is blowing strong here in Kansas, so I think I will go fly that kite!

Monday, June 13, 2011

Accentuate the Positive; Eliminate the Negative

Have you ever wondered why a negative number times a negative number equals a positive number?  As my mathphobic daughter would say, "No, Mom.  Math is something I never think about!"  Well, for all of us who tend to be left brained people, the question can be answered by using a pattern.  Have you noticed a reoccurring theme in my articles?  All Math is Based on Patterns!

Let's examine 4 x -2 which means four sets of -2.  Using the number line above, start at zero and move left by twos - four times. Voila!  The answer is -8.  Locate -8 on the number line above.  Now try 3 x -2.  Again, begin at zero on the number line, but this time move left by twos - three times. Ta-dah!  We arrive at -6.  Therefore, 3 x -2 = -6.

Here is what the mathematical sequence looks like.  Moving down the sequence, observe that the farthest left hand column decreases by one each time, while the -2 remains constantSimultaneously, the right hand answer column increases by 2 each time.  Therefore, based on this mathematical pattern, we can conclude that a negative times a negative equals a positive!!!!

I wish to thank my long time friend, Barbara, who teaches mathematics at a University in North Carolina, for sharing this pattern with me.

Wednesday, June 8, 2011

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.  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.

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!

Wednesday, June 1, 2011

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, 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.