How many taxicabs in NYC? #Fractions — eureca
Or, What on earth is a guess-estimate?
Or, Fractions. What?
Guess-estimate, though the name may sound new, is a kind of problem pretty much widespread in business domain. Guess-estimates involve a mix of educated guesswork and estimation to ascertain the solution/ near-solution to a problem, an example of which most conveniently is the headline of this post.
Let’s sort the NYC taxicab problem first
Population of NYC is around 8.5 million. Given it’s a working district and one of the busiest city in the world, with majority of the population working, it is safe to assume that 90% of the population, on an average is commuting everyday to work. 90% comes to around 7.6 million. In general, for a busy city like the NYC, usual demographic trend dictates the following split of commuting population by the conveyance of commute
Train/ Metro — 40% ~ 3.04 million
Personal Vehicle — 10% ~ 0.76 mn
City Buses — 20% ~1.52 mn
Cab — 30% ~ 2.28 mn
Cab for last mile connectivity — 50% of Train/ Metro population ~ 1.52 mn
Thus, the total population traveling in cabs per day is around (2.28 mn+ 1.52 mn) 3.8 mn.
Assuming, on an average, a taxi takes a 30 mins trip on an average and carries 1.5 people per trip (3 seats — half filled at one time), number of people catered to by 1 cab in an hour is 3. Total number of people catered to by 1 cab in a day, assuming working day is of 12 hours is around 36 people per day.
Hence, number of cabs in NYC should be near 3.8 mn/ 36 ~ 1,00,000 cabs, with definitely a good margin of error. Surprisingly, however we are very close. The actual number is around 1,03,000. Obviously, there are multiple places where I have taken liberty of educated guesswork, but even if it is simplified to a large extent, we will be good with the final number. After all, it is an estimate.
But, this is NOT something I want to call your attention to.
Hence, the headline, is a misnomer.
What are we talking about then
TBH, we do these guess-estimates daily, even multiple times a day without at times even knowing that we are in the middle of one. We do it while estimating calorific consumption in 100 lbs of pulses in our favourite fitness tracker (MyFitnessPal, hands down for me). We do it while watching a movie and estimating how much time before the interval so that we may gorge on our favourite popcorn in between. We do it every time we shop, we chop and we plop.
What though is much more interesting to see is the number of times these estimations involve calculations and/ or estimations based on fractions.
Note that percentages are just gussied up versions of fractions. Hence, they mean the same for the purpose of this post.
Though if you look at several curriculum designs, you’d find, which I did, that kids are underprepared for this type of estimation, even if the very basic ones. As an adult even, for example, if you ask me to sum a half (1/2) and one-fourth (1/4), I will dive headway into calculating Lowest Common Multiple, proceed to find suitable multiples of the numerators and add them to get to the sum. Well, this is how most of us are/ were wired. On the other hand, if I pour half a glass full and add a quarter more, you can easily make out that the glass is three fourth full overall. You most definitely would not go for finding out Lowest Common Multiple, let along some other calculation aspects inside your heads.
Our minds are as brilliant as they come. At the time when we are using our logical side of our heads to find connection between cause and event — in this case, adding one-fourth of the glass’ content in it leads to making the glass three-fourth full — we are also using our pattern recognition abilities to gauge that the resultant indeed is going to be a three-fourth, which is why before even observing the water being poured into the glass, we can make out what the resultant fraction (3/4) will be.
Fractions are highly visual and that’s how they should be taught
Probably, you might be wondering that the pattern sure works for simple fractions but how could it possibly help in finding sums of complicated fractions! And you most probably are right, but that’s how we start learning any concept. We start with simple fractions in a visually presented format which uses our logical and pattern recognition abilities and then we try and extend it to complicated examples, and only then can we build in the concept of Lower Common Multiple and its application to fraction calculation. For an instance, once you know the concept of a half, visually, the size of parent object which is to be halved ceases to matter, because howmuchever is the size, you can visually make out the two child objects which would be created post the halving operation.
Segue into the Cake Problem
Why the segue? Because the cake problem is an important problem in understanding fractions, and is in several ways, a foundational problem in the realm of applied mathematics to test children’s conceptual understanding. It does of course has application even in higher mathematics world including but not limited to algorithms.
There are several variations to the cake problem. Since brevity is the soul of wit, here is how it is:
Divide a square cake into three equal pieces. — The Cake Problem.
Just think about the problem for a second. There is a very high probability that you’d try and draw two parallel lines on the surface with equal width, which is for sure correct. How else though can we do it? The uncanny thing is most kids we spoke to for this problem followed the same procedure. Only a few thought of a way which was unconventional. If the angle at the center of the square cake casted by the two sides of a piece of cake is 60 degrees, our problem is solved without a recourse to parallel lines.
The thought process that go into devising a strategy for this extraordinary partition is resoundingly unconventional and is an ingenious one on top of it. But then how this can be imagined if not for the visual representation of the fractions!
The same problem with importance to proportionality and varied fractions (1/6s and 1/3) is presented in the following figure.
Back to eureca
These are what guided us as motivations towards fractions module within eureca. We wanted a highly visual and intuitive way for kids to learn the concept of fractions from, and not with only conventional representations but with the real life ones juxtaposed with abstract ones.
Why are real life cases important though? Unfortunately, we begin to find the implications and applications of fractions in our daily lives once we are full-blown adults, as pointed out surreptitiously :) by the very first example in the post: a guess-estimate case study from any of the consulting job interviews across MBA campuses globally. This is to say that, the association of fractions in their own right with real life objects and situations forms a mental picture much later than we would like to. Such associations though have a much higher chance to make a subject, particularly mathematics which thrives on the abstract, more interesting and applied. Adding to it, if these associations can be made by the kids in our own environment in a thoroughly active and participatory way rather than only through observational nature of most multimedia, that will be the closest to reality we can get. Hence follows the representation of such cases on fractions in Augmented Reality within eureca.
A quick video says more than words. So, here goes.
Hope you enjoy the module. Holler, for we love to talk.
