What Trapezoid?

I disagree with the definition of trapezoid that was presented in Holt Geometry, the textbook my students are using. In it, trapezoid is defined as "a quadrilateral with exactly one pair of parallel sides." I believe that a trapezoid is "a quadrilateral with at least one pair of parallel sides."

Before I go any further, let me clarify that I am using the word trapezoid as Americans call this shape, which both the British and the Australian call trapezium.

If you Google definition of trapezoid, you will get the impression that mathematicians are starkly divided in their definitions of the term trapezoid. The truth is they are. One group leans toward the exclusive definition that Holt Geometry prefers, while the second group highly favors the inclusive definition. The second group is where I belong.

I found online a University of Washington  course syllabus for Math 444. While it adopted the inclusive definition, it explained that "the advantage of the (exclusive definition) is that it allows a verbal distinction between parallelograms and other quadrilaterals with some parallel sides." The same syllabus also explained that "the advantage of the inclusive definition is that any theorem proved for trapezoids is automatically a theorem for parallelograms." This simply means that if trapezoids have at least one pair of parallel sides, then every parallelogram easily qualifies as a trapezoid.

Proponents from both sides have their reasons. But the proponents for the inclusive definitions have gone as far as using higher level mathematics to explain themselves.

My own explanation as a proponent of inclusive definition is much more simple. Here it is:

Given Parallelograms 1, 2 and 3 as shown in the photo.

Parallelogram 1 is a 2" x 3" rectangle. Parallelogram 2 is a 3' x 3' square. Parallelogram 3 is a nonrectangle parallelogram with a base of 4 cm. and a height of 2 cm.

The universal formula that we use to determine the area of each of these parallelograms is A = bh. Applying this formula, we get areas of 6 square inches, 9 square feet, and 8 square centimeters, respectively.

Now, let us find the areas of the same figures but we will use instead the universal formula to determine the area of a trapezoid, A = (1/2)[b1 + b2]h.

For Parallelogram 1:
A = (1/2)[b1 + b2]h  = (1/2)[3" + 3"]2" = 6 square inches

For Parallelogram 2:
A = (1/2)[b1 + b2]h =  (1/2)[3' + 3']3' = 9 square feet

For Parallelogram 3:
A = (1/2)[b1 + b2]h = (1/2)[4 cm + 4 cm]2 cm = 8 square centimeters

The results are the same!

Clearly, the universal formula for the area of a trapezoid applies perfectly to each of the given parallelograms. This only shows that every parallelogram is a trapezoid and the inclusive definition is justified.

For practical purposes, I still go by our textbook's preferred definition though. But I make sure that my students are aware of the significance of the inclusive definition of trapezoid. This way, when they chance upon a college professor who is an ardent supporter of said definition, they would be okay.



If the inclusive definition of trapezoid is adopted, the resulting 2D classification chart would look like this.

To view an online discussion on this topic, visit The Math Forum's Ask Dr. Math.

Classroom Economics: Saving for Rainy Days

Holt Geometry, the textbook my students use, defines postulate as "a statement that is accepted as true without proof." According to Reader's Digest Oxford Complete Wordfinder, postulates, or axioms, are used as "basis for mathematical reasoning." Geometry, in fact, is built on a strong foundation of postulates that include the following: There is exactly one line that passes through any two points. I am not quite knowledgeable in the science of finance, but I would assume that one of its postulates must be the following: There'll be nothing for one to withdraw if there was nothing deposited in the first place. Postulate or not, I use it to strengthen the foundation of my own teaching.

At the beginning of each school year, making deposits is on top of my priority list. Like a hungry eagle looking for food, I scour for and swoop on every opportunity - big and small- to make a deposit.

Giving generous compliments is making deposits. While I strive to get to know my students, I make sure that I give each one of them appropriate and hopefully nurturing compliments. Lots of them. I compliment anything about the individual that I can safely and appropriately compliment on. I celebrate every little positive thing I see.

Greeting and wishing them well on their birthdays is making deposits. So are grieving with them when someone in their lives perished, listening to them when their bffs break their hearts, and celebrating the genius behind each academic mistake.

Shaking it in the middle of the dance floor during school dances and being a kewl teacher the right way are making deposits.

Calling parents to tell them positive things is an example of making multiple deposits. With one single deposit, I get to  increase my deposits in two accounts instead of just one.

Sharing my own life stories, whenever appropriate, allows me to deposit to a host of accounts, not just two, with one transaction.

Although I am a sucker for making deposits, I am well-aware that my deposits, like bank deposits, are limited to certain currencies, and because I'm just a teacher, I cannot just deposit large amounts whose sources I cannot justify. Certain amounts of deposits are certainly going to raise alarms. I certainly do not want my students to feel uncomfortable with me.

We do not like to withdraw monies we have saved, but rainy days are bound to come. It is for this reason that we save in the first place.

Making withdrawals is getting after my students for a host of reasons, such as misbehaving, failing to turn in homework or project, violating school or classroom rules, and failing to perform in class satisfactorily or according to certain mutually accepted higher expectations. Sometimes, I withdraw before it becomes necessary to withdraw.

Just like in real banking and finance, my withdrawals have limits. But unlike real banks, my banks, which in this case are my students, are not "financially" stable and well-founded. They are kids, and they are volatile. Sometimes, I withdraw as much as I needed. But most of the time, I withdraw according to the conditions that my banks are in.

No matter what the state of economy is, it is deemed wise to always save. In fact, it is a value we are encouraged to teach our youth.

My classroom economics is always a winner. I may fail to help all students of mine achieve academic mastery. But with my classroom economics, I fervently hope to touch their lives with mine.