Photosophics 1

"Even if it means finding myself
naked and exposed,
I would still choose to be right."

I wake up to this view from my apartment window everyday for a year now.
Photographed February 4, 2011. Location: Las Misiones Apartments, Mission, TX USA.

Age Doesn't Matter!

My friend Norma asked me a question last night via Facebook. For practical purposes, I am rephrasing her question as:

Why is the sum of the person's two-digit birth year and that person's age plus one always equal to 111?

Norma, who is now 34, was born in '76. She observed that 76 + 34 + 1 = 111.

I thought of Matty. He was born in '83. Last year, he turned 27.  Observe that 85 + 25 + 1 = 111.

My student Alisha, who was born in '96, celebrated her 14th birthday last year. Well, 96 + 14 + 1 = 111.

"So why does this work?," Norma insisted.

Readers, what do you think?


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SOLUTION:
When you say your birth year is 76, you actually mean 1976, which is 1900 + 76.

So if your birth year is X, then it is actually (1900 + X).

If you were born in the year (1900 + X), then in the year 2010, your age was [2010 - (1900 + X)] years old.

So your birth year plus your age plus one
= X + [2010 - (1900 + X)] + 1
= X + 2010 - 1900 - X + 1
= 111.

It doesn't matter how old the person is, the result will always be 111.


Norma's question reminded me of a similar problem. Last December, in a statewide conference I attended in Dallas, the following question (that I also rephrased) was raised for us to play with:

Add your age by the end of this year to the year you were born. What do you get?

Now your turn. Prove why it works.

Waking Up to a Math Magic on a Lazy Sunday Morning

As soon as I woke up this morning, I lazily reached for my phone. It's a habit. Every morning I wondered who left me a message when I was asleep.

Buried in this pile of e-mail notifications from Facebook and stores such as Kohl's, Target, Express, et cetera, there's this comment left by Akai Avenue, a loyal follower of Teaching Garden, in the preceding blog post. She just wanted to share with me a "mathematics magic".

Now let me share the same with you all, ninos y ninas!

Either by hand or using a calculator, do the following:

Multiply: 259 X your age X 39

What did you get?

STOP. Don't scroll down yet. Give it some more thought. Why did it give you that answer? What do you think would your friend get?

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Interesting, right?

My answer was 434343. I've never been reminded of my age this way. Sort of humiliating! :D

I assumed right away that the product of 259 and 39 must be 10101, and I was right.

Observe:
10101 X 24 = 242424
10101 X 16 = 161616
10101 X 57 = 575757

Also observe that the sum of the digits of 10101 is 3. If you divide the former by the latter, you'll get 3,367, which can be factored into 13 and 259. The latter can be factored further into 7 and 37.

Therefore, 10101 = 3 X 7 X 13 X 37.

Seeing 7 and 13, which are both associated with the word "lucky", and knowing that 3 X 37 = 111, I thought there must be a better way to present this math magic.

Oh, I know! Said math magic could use a makeover. And here it comes:

Multiply your age by 111 and by lucky numbers 7 and 13.

Doesn't this new version sound more appealing?

Now time to get out of bed. :D

Teachership

I've been anticipating some news from the Philippines, so every morning, before I head to work, I'd get online to check my e-mails and to spend a few minutes on Facebook.

Today, there was nothing worth noting, except three of my FB friends are celebrating their birthdays. One of them is Cynthia, a junior student at my school, who was with me in Geometry last year. On her wall, I wrote: "Happy birthday, Cynthia! God bless you." I didn't get online again until lunchtime. When I did, I found this message:   

Cynthia Fregoso: thank u mr. jope i apreciate ur teachership:)

Teachership! Whoa!

I've known teachership to be a legitimate word, but upon reading it in her message, the word hit me in a way that I cannot possibly explain.

I am aware that my students appreciate me as their teacher. Modesty aside, the compliments come my way so often that I do not thirst for them anymore. But there was something in Cynthia's message that touched me so profoundly and I know that it has something to do with her use of the word teachership.

It's past midnight now, but I have not figured it out yet. Maybe I don't have to figure it out, right?

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.

In Search of a Good Teacher

If we, teachers, only listen to our students, they could teach us a thing or two.

Karidme Rocha and Shawn Klesel are both wonderful students of mine in Geometry. Like a host of my former and current students, they're both on my Facebook friends list.

The following is part of a thread that stemmed from Kari's status that she posted on December 30th:

Kari:      In search of a prince charming:)
Shawn:      im a prince and im charming but i aint no prince charmingXD
Kari:      o wow Shawn lol you need to be both at the same time lolXD
Shawn:      hahaha only to certain ppl at certain timeXD

Upon reading this thread, I reflected. Face to face, I looked intently at this teacher that they call Mr. Jope.

You are a teacher, and you are good. But are you a good teacher?
Sometimes? All the time?

Good teacher TO WHOM?

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.