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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)?

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.

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.

Are Trapezoids "Trapping" Your Students?

In a previous posting (Aliens and Trapezoids, July 7, 2011) I shared with you how I taught my students to remember the word trapezoid.  Today, I would like to talk about the characteristics of a trapezoid. 

As I search on Pinterest, I find quadrilaterals that look like the one on the right classified as "trapezoids".  Indeed they are, but this is a special kind of trapezoid because it has one set of equal sides, one line of symmetry and one set of parallel sides.  It is called an isosceles trapezoid.  Isosceles means "having two equal sides" just as an isosceles triangle has.
BUT to be a trapezoid, the only characteristic needed is one set of parallel sides.  Look at the red quadrilateral on the left.  It is a trapezoid because it has one set of parallel sides.  YET, students rarely see this kind except on those math tests that COUNT!  Why?  Because those trapezoids (and test writers) are out to "get" your students.  So think about it.  Are the trapezoids in your classroom trying to "trap" your students or can your students recognize a trapezoid even if it doesn't have symmetry? 


I love teaching geometry, and therefore I have several geometry products for sale in my Teachers Pay Teachers Store.

Geometry Parodies - The four page handout includes 20 unusual definitions of geometric terms such as “A place where people are sent for committing crimes.” Each definition is a play on words or a parody.

Plane Geometry Test - This 100 point assessment is over
plain geometry concepts and focuses on using and applying geometry. Measuring and categorizing angles, identifying lines, angles, quadrilaterals, etc., solving for circumference, using formulas, recognizing symmetry, and comparing using and applying geometry. Measuring and categorizing angles, identifying lines, angles, quadrilaterals, etc., solving for circumference, using formulas, recognizing symmetry, and comparing two quadrilaterals are included.

Plane Geometry Vocabulary Crossword - This puzzle is designed so that the student will practice and use geometric vocabulary. It is a free form crossword puzzle that features 25 different geometry terms. The 25 clues are in the form of definitions which emphasize points, lines, and angles.

Solid Geometry Test - This 100 point test is a summative assessment given at the end of the solids unit in our math book. It highlights using and applying formulas to find area, perimeter, circumference, surface area, and volume.

Solid Geometry Vocabulary Crossword - This crossword puzzle is designed to practice geometric vocabulary and recognize formulas. It is a free form crossword puzzle that features 23 different geometric terms or formulas. The 23 clues are in the form of definitions or a formula format which give emphasis to polyhedrons, circles, and formulas for area, surface area, and volume.