Archive of posts filed under the Uncategorized category.

## Calculus Gave Me a Speeding Ticket

Years ago, one sunny Sunday afternoon, I was driving home from visiting friends at college and received a speeding ticket.  I didn’t realize it at the time, but calculus played an important role in my citation.

You see, this was no ordinary speeding ticket, the kind where a police officer paces the offender or uses radar to measure a vehicle’s speed.  My speed was calculated from an airplane high above the road.  And the Mean Value Theorem clinched the case.

Aerial speed enforcement works like this:  large marks painted on the road divide the highway into quarter-mile intervals.  A pilot flying overhead uses a stopwatch to time a suspected speeder from one mark to the next.  Say the pilot records a time of 12 seconds; a simple calculation converts one quarter mile per 12 seconds into 75 miles per hour; this information, the average speed on this interval, is radioed to the police on the ground who then stop and ticket the driver.

What I didn’t realize at the time was how crucial calculus is in all of this.

A fundamental theorem of calculus, the Mean Value Theorem (MVT), relates the average rate of change of a function with the instantaneous rate of change of the function.  Suppose we have some function of time, $f(t)$, and suppose that we know the value of this function at two times, say $f(t_1)$ and $f(t_2)$.  The average rate of change of $f(t)$ between $t_1$ and $t_2$ is

$f_{avg} = \frac{f(t_2) - f(t_1)}{t_2 - t_1}$

The MVT tells us that, as long as $f(t)$ is a differentiable function, then at some time between $t_1$ and $t_2$, say at t = c, the instantaneous rate of change of $f(t)$ must have been equal to the average rate of change of $f(t)$ from $t_1$ and $t_2$.  That is,

$f'(c) = \frac{f(t_2) - f(t_1)}{t_2 - t_1}$

where $f'(x)$ is the derivative of $f(x)$, the instantaneous rate of change of $f(x)$.

What does this have to do with my speeding ticket?  Well, as I’m moving along the highway in my car, the pilot records two values of my position function, $x(t)$, at two different times, $t_1$ and $t_2$.  The pilot then computes my average speed

$x_{avg} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}$

Here’s where calculus comes in.  The Mean Value Theorem says that, at some point between those two times my instantaneous speed must have been equal to my average speed.  If my average speed was above the legal limit, then at some time between $t_1$ and $t_2$, my instantaneous speed must have been above the limit, and at that moment, I was guilty of speeding.

I wonder if it would have helped to argue that my position function wasn’t differentiable!

## How Many Gregarious Mongolians Are There?

When I visited Mongolia 15 years ago, a young man approached us on the streets of Ulan Bator and offered to show us around. His name was Soyoloo, and we spent a few fun days hanging out with him.

I have many fond memories of Mongolia, so this NYT piece on Ulan Bator’s underground homeless population immediately caught my attention. The first picture is of a man named Soyoloo.  I wondered if this was the same Soyoloo I met so long ago.

How likely is that?  A naive estimate looks like this:  half of the one million people in Ulan Bator are men, so there’s a 1 in 500,000 chance that a randomly-selected (that is, randomly photographed) man from Ulan Bator is the man I met 15 years ago.  At 0.0002%, this does not seem very likely.

But here’s where conditional probability comes in.  The man in the photograph isn’t just any randomly-selected man; he’s a randomly-selected man named Soyoloo.  What I really want to know is, given the condition that a randomly selected man from Ulan Bator is named Soyoloo, what is the probability that he’s the man I met 15 years ago?

Naturally the answer depends on how many men in Ulan Bator are named Soyoloo.  ”Soyoloo” doesn’t appear on any list of most popular Mongolian names, so I am going to estimate that at most 0.5% of Mongolian men are named Soyoloo (this would be roughly equivalent to ‘Andrew’ in the US, the 35th most popular male name).  This means that at most 2,500 men (0.5% of 500,000) in Ulan Bator are named Soyoloo.  Therefore, if a man in Ulan Bator named Soyoloo is selected at random, there is a 1 in 2,500 chance (0.04%) it’s the man I met.

This is the power of conditional probability.  The man in the photograph and the man I met share a common characteristic:  being named Soyoloo.  Knowing this condition greatly alters the probability that they are the same person.

In fact there’s another characteristic the two men share:  they are both comfortable striking up relationships with foreign strangers, whether it’s me or a photojournalist.  Let’s call this characteristic gregariousness.  Now the question is, what is the probability that a randomly-selected gregarious male resident of Ulan Bator named Soyoloo is the person I met 15 years ago?

Well, how many gregarious Mongolians are there?  Let’s say 5% of all Mongolians are gregarious.  Then there are roughly 125 gregarious Mongolian men named Soyoloo in Ulan Bator, and thus, a 1 in 125 (0.8%) chance that the man in the photo is the same man I met.  Not a lock, but not a longshot, either!

This probability may seem surprisingly high, given the distance in both time and space,  but I have a feeling it’s him.  His eyes look warmly familiar.

## Regents Recap — January 2013: Question Design

Here is another installment in my series reviewing the NY State Regents exams in mathematics.

One consequence of scrutinizing standardized tests is a heightened sense of the role question design plays in constructing assessments.

Consider number 14 from the Integrated Algebra exam.

In order to correctly answer this question, the student has to do two things:  they need to locate the vertex of a parabola; and they need to correctly name a quadrant.

Suppose a student gets this question wrong.  Is it because they couldn’t find the vertex of a parabola, or because they couldn’t correctly name the quadrant?  We don’t know.

Similarly, consider number 21  from the Geometry exam.

This is textbook geometry question, and there’s nothing inherently wrong with it.  But if a student gets it wrong, we don’t know if they got it wrong because they didn’t understand the geometry of the situation, or because they couldn’t execute the necessary algebra.

Using student data to inform instruction is a big deal nowadays, and collecting student data is one of the justifications for the increasing emphasis on standardized exams.  But is the data we’re collecting meaningful?

If a student gets the wrong answer, all we know is that they got the wrong answer.  We don’t know why; we don’t know what misconceptions need to be corrected.  In order to find out, we need to look at student work and intervene based on what we see.

And what if a student gets the right answer?  Well, there is a non-zero chance they got it by guessing.  In fact, on average, one out of four students who has no idea what the answer is will correctly guess the right answer.  So a right answer doesn’t reliably mean that the student knows how to solve this problem, anyway.

So what then, exactly, is the purpose of these multiple choice questions?

## Fibonacci Flushers

While travelling in Europe, I became fascinated with the variety of toilet-flushing mechanisms I encountered.  The typical toilet had a low-flow / high-flow option (which I imagine saves a great deal of water in the long run) , and a lot of creativity emerged in the way this two-flush system was implemented.

While documenting the many ways to flush, I found this rectangular model oddly familiar and appealing.

And then it hit me:  this looks like the golden rectangle!

The golden ratio has long been used by artists and architects to create aesthetically pleasing work.  It is, after all, the divine proportion.  Could it be that these toilet-makers took their cues from the masters of art and math?  I had to find out.

I dropped my image into Geogebra and took some measurements.

The total length of the rectangle divided by its height is around 1.71.  So, it’s not quite the golden ratio, but it’s pretty close.  This flush-design is about 90% divine, I suppose.

Maybe their next design will be closer to the perfect proportion.

## Differentiating Consumer Quantities

One thing I’d like to see more of in math education is an appreciation for practical consumer math.  For example, suppose your favorite cereal brand unceremoniously starts arriving in a smaller box.

Students should learn to be on the lookout for things like this.  They should develop a quantitative curiosity, exploring the various ways that quantity can be disguised.

And ultimately be able to put together a real quantitative analysis that helps them make good decisions.

This is math that everyone can use!