How do you measure two-thirds?

From an article by Mary Ann Bragg which appeared on CapeCodeOnline and was also printed in this month’s College Mathematics Journal:

TRURO — Voters narrowly approved one of four zoning amendments late Tuesday night at the annual town meeting. But town officials were still looking at the exact vote count on that article yesterday.

In a vote of 136 to 70, voters passed a new time limit on how quickly a cottage colony, cabin colony, motel or hotel can be converted to condominiums. The new limit requires that those properties be in operation for three years before being converted to condominiums.

The idea behind the zoning amendment is to slow the pace of condominium development in Truro and preserve more affordable accommodations for tourists, according to citizens proposing the warrant article.

Currently Truro does not allow condominiums complexes to be built outright in its zoning bylaws. Instead, property owners must build a cottage colony, cabins, motel or hotel first and then covert it to condominiums through a special permit.

The exact count of the vote — 136 to 70 —had town officials hitting their calculators yesterday. The zoning measure needed a two-thirds vote to pass. A calculation by town accountant Trudy Brazil indicated that 136 votes are two-thirds of 206 total votes, said Town Clerk Cynthia Slade.

But is it?  Is 136 a sufficient number of votes to be considered two-thirds of the total 206 votes?  Let’s check:

If you use the fact that $$\frac{2}{3} \approx 0.66$$ and then proceed to multiply 206 by 0.66 you get 135.96.  There were 136 votes in favor which is  more than 135.96 so that means it passes, right?  If you think so, then you’d be WRONG!!!

The main problem is the rounding.  In fact, $$\frac{2}{3} = 0.666666\ldots$$ or using repeated decimal notation, $$\frac{2}{3} = 0.\bar{6}$$.  When you round, you are actually creating an error that, in this case, makes a pretty significant difference.

Think of it another way, lets compare 136 / 206 to 2 / 3.  First, just do it by decimal approximation:

$$\frac{136}{206} \approx 0.660194174757 < 0.6666666667 \approx \frac{2}{3}$$

My calculator cannot exactly represent either of these fractions but its accurate to 12 decimal places and I can clearly see that 136/206 < 2/3 so the vote should not pass.

Do you remember another way you can compare fractions?  Find a common denominator and convert each fraction, then compare.

$$\frac{136}{206} \cdot \frac{3}{3} = \frac{408}{618}$$

$$\frac{2}{3} \cdot \frac{206}{206} = \frac{412}{618}$$

So, here we see that, again,

$$\frac{136}{206} = \frac{408}{618} < \frac{412}{618} = \frac{2}{3}$$

This second method of checking is even better than the first because there are no approximations involved.  We’ve confirmed, absolutely, that 136 votes out of a total of 206 does NOT constitute two-thirds.

Fortunately, a good citizen made an anonymous call in Truro, MA, to clear this up.  What perplexes me is that they decided they needed to let the State Attorney General’s office decide on the correct count. The mathematical explanation wasn’t good enough. Can you say quantitative illiteracy?

Mathematical Moment #2

(Continuation of a series from a while back… –> Mathematical Moments)

In the fourth grade, I was first given the opportunity to participate in a UIL competition that I really enjoyed.  In Texas, there are quite a few academic contests which are part of the UIL (University Interscholastic League), and by that time, I had dabbled in only one event and it was called “Story Telling”.  In that event, you and a group of students were read a story and then one by one, you were asked to re-tell to the story to a judge who would score your storytelling ability.  I tried it, and to be honest, I stunk.

Fortunately, in the fourth grade, another event became available to me, namely “Number Sense.”  In this competition, you are given a 10 minute test with 80 math questions, almost entirely arithmetic. The most challenging aspect of the test is that all calculation must be done in your head, no scratch paper or marks on the test are allowed.  Also, no calculators are allowed.  It’s all in your head.

The teacher who coordinated the extra-curriculum program for Number Sense asked each of the fourth grade teachers to recommend students who might be interested and appropriately skilled to participate.  For whatever reason, I was the only fourth grader that ended up being involved.  I can vividly recall the first couple of meetings with the Number Sense team.  I can’t be sure of who the older kids were, but I would hazard to guess that it included the same individuals that I would later compete against in Number Sense throughout high school, namely Nick Hiemstra and Tony Cook.

Three things come back to my mind as I think about those early meetings:

1. Holes in the ceiling tiles.  It was in these first two meetings that I was taught the art of flicking pencils into the ceiling tiles above.  The room we used was an older room that wasn’t being used for anything else and when I arrived, I discovered a group of guys having the best time flicking pencils up into the ceiling.  There were holes everywhere and a least a couple dozen pencils sticking out. I tried my hand at it and succeeded at least a few times before we were caught and strongly urged not to continue this act of destruction. So we continued at a later time.

2. Rolling Chalkboard.  For some reason, I never imagined that a chalkboard would be anywhere but fastened to a wall.  It seems silly now, but I was amazed in fourth grade by a chalkboard on wheels that you would flip over and write on both sides.  Why it amazed me, I have no idea, but I still remember that feeling of awe toward a chalkboard.  I guess it was my destiny to build a career around such a thing: writing on the board…

3. Mental Arithmetic. I was like a dry sponge immersed in the ocean when it came to this first exposure to a vast wealth of tricks for mental calculation.  Tricks for multiplying by 11, by 25, by 125, for squaring numbers ending in 5, for adding fractions whose numerator is 1, for multiplying any two digit numbers together, for adding long sequences of numbers, and on and on.  I fell in love with mathematics for the first time.  I had been fast at the Multiplication tables in 3rd grade but this opened a whole new world to me.

The one tragedy I remember that came out of this new experience was when I misunderstood our sponsor and believed that I was actually on the team for that first contest.  The team was traveling to River Road in Amarillo.  I called the sponsor the night before to find out when we were leaving and learned that I was not going.  oh, the pain.  I still remember how that felt.  I knew I was going to have to get good and I vowed to memorize the little 20-page (or so) red paper booklet with all the number sense tricks.  After only a few weeks, I had done just that.

There are lot more Mathematical Moments that come from my involvement in Number Sense but my fourth grade year was the first and probably set me on the road to where I am today.

Mathematical Moment #1

I am adding a new category to this blog because I’d like to document some of the experiences that I have had over the years that drew me to mathematics.  (Aside: I wonder if I have too many categories as it is.  I seem to have abandoned some previous ideas.  Maybe this summer will give me a chance to get back to them.)

Mathematical moments are those special moments that involve a specific application or perhaps just a specific individual that helped mathematics come alive for me.  Obviously, I have a deep love for the subject matter.  I have found a place where I “fit”, a career in which I can experience a true sense of discovery and excitement.  That connection I feel towards my chosen field has developed over time.  I often point to the fact that my first year in college was spent as a Religion Major.  I had abandoned my first love (with respect to my education).  Something was missing that year and fortunately I found what it was.  Is this over the top?  I don’t think so as long as everyone understands that I speaking within the arena of my education and my career, neither of which compare to my faith or my family. Anyways, I added the math major and the rest, as they say, is history.

My earliest mathematical moment:

I’ll start with the earliest mathematical memory of significance. I am sure there were even earlier indications for my “analytical side” or “mathematical side” in my childhood.  In fact, I vaguely recall a particular time when I just about bugged my mother to death asking her to count to 10 for me, over and over and over.  But beyond that, I remember a time in third grade, in Mrs. Russell’s class, that we were working on Multiplication tables.  We had been memorizing them for some time and we were handed a multiplication test which was to be timed.  Much like the one here: Multiplication Timed Test,

We had taken them before, but it seemed like all of a sudden, something clicked on.  The answers began to flow like I couldn’t believe.  There were 100 questions and, if I am not mistaken, we had 2 minutes to take the test.  I don’t think I had finished one before but on this day, bang boom zip!  I was done and there was time left to spare.  Not only that but I had aced it.  Up to that point in my life, I don’t think there was anything quite like that experience.  I looked around at those around me as they were methodically moving down their pages with their answers.  Some nearing the end, some only halfway done, but only I was looking around.  Once time was up and the tests were graded, I new I had found my favorite subject.