Acceleration

The following two questions came from a middle-school math textbook. The first is reasonable, while the second is a classic example of an author being overly cute when writing a homework problem.

A car slams on its brakes, coming to a complete stop in 4 seconds. The car was traveling north at 60mph. Calculate the acceleration.
A rocket blasts off. At 10 seconds after blast off, it is at 10,000 feet, traveling at 3600mph. Assuming the direction is up, calculate the acceleration.

For the first question, we’ll assume constant deceleration (after all, this comes from a middle-school textbook). First, let’s convert from miles per hour to feet per second:

$60 ~ \frac{\hbox{mile}}{\hbox{hour}} = 60 ~ \frac{\hbox{mile}}{\hbox{hour}} \times \frac{1 ~ \hbox{hour}}{3600 ~ \hbox{second}} \times \frac{5280 ~ \hbox{feet}}{1 ~ \hbox{mile}} = 88~ \frac{\hbox{feet}}{\hbox{second}}$

The deceleration is therefore equal to the change in velocity over time, or

$\frac{-88 ~ \hbox{feet/second}}{4 ~ \hbox{second}} = -22 ~\hbox{ft/s}^2$

Now notice the word north in the statement of the first question. This bit of information is irrelevant to the problem. I presume that the writer of the problem wants students to practice picking out the important information of a problem from the unimportant… again, a good skill for students to acquire.

Let’s now turn to the second question. At first blush, this also has irrelevant information… it is at 10,000 feet. So I presume that the author wants students to solve this in exactly the same way:

$3600 ~ \frac{\hbox{mile}}{\hbox{hour}} = 3600 ~ \frac{\hbox{mile}}{\hbox{hour}} \times \frac{1 ~ \hbox{hour}}{3600 ~ \hbox{second}} \times \frac{5280 ~ \hbox{feet}}{1 ~ \hbox{mile}} = 5280 ~ \frac{\hbox{feet}}{\hbox{second}}$

for an acceleration of

$\frac{5280 ~ \hbox{feet/second}}{10 ~ \hbox{second}} = 528 ~\hbox{ft/s}^2$

The major flaw with this question is that the acceleration of the rocket completely determines the distance that the rocket travels. While middle-school students would not be expected to know this, we can use calculus to determine the distance. Since the initial position and velocity are zero, we obtain

$x''(t) = 528$

$x'(t) = \int 528 \, dt = 528t + C$

$x'(0) = 528(0) + C$

$0 = C$

$\therefore x'(t) = 528t + 0 = 528t$

$x(t) = \int 528t \, dt = 264t^2 + C$

$x(0) = 264(0)^2 + C$

$\therefore x(t) = 264t^2 + 0 = 264t^2$

Therefore, the rocket travels a distance of $264 ~ \hbox{feet/second}^2 \times (10 ~ \hbox{second})^2 = 26400 ~ \hbox{feet}$ . In other words, not 10,000 feet.

As a mathematician, this is the kind of error that drives me crazy, as I would presume that the author of this textbook should know that he/she just can’t make up a distance in the effort of making a problem more interesting to students.

Teaching base 10

Price (on the day of this writing) on Amazon.com for a base-10 starter kit, featuring 100 small plastic blocks, 30 rods (representing 10), 10 plates (for 100), and one big cube (for 1000): $31.32.

Cash value of 100 pennies, 30 dimes, 10 dollar coins, and a $10 bill: $24.00.

Analog clocks

What time is it?

From the perspective of an elementary student — even a good student who generally can read the hands of a clock correctly — the answer probably is 2:50. After all, the hour hand is pointing much closer to the 2 than to the 1.

To address this misconception, perhaps the best suggestion that I’ve heard is using the analogy of waiting for your birthday. A child doesn’t turn 7 until his (or her) birthday comes. On the day before his birthday, he still has to say that he’s 6. Likewise, if the minute hand is just a few minutes before 7:00 and the hour hand is not all the way at seven, we still have to say that the time is six-something.

Sorting and interpretative dance

Algorithms for sorting a list, like Bubble Sort, Quick Sort, etc., often require some sort of visual interpretation to make dry algorithms more accessible to students. Here are some YouTube videos that brilliant do this via the medium of folk dancing.

Quick Sort

Merge Sort

Insert Sort

Select Sort

Shell Sort

Bubble Sort

Infraction

While I can’t take credit for this one-liner, I’m more than happy to share it.

A colleague was explaining his expectations for simplifying expressions such as

$\displaystyle \frac{\displaystyle ~~~\frac{2x}{x^2+1}~~~}{\displaystyle ~~~\frac{x}{x^2-1}~~~}$

Of course, this isn’t yet simplified, but his students were balking about doing the required work. So, on the spur of the moment, he laid down a simple rule:

Not simplifying a fraction in a fraction is an infraction.

Utterly brilliant.

Ontologically

Case Study #465 of how not to teach a class.

Me: Ontologically, what kind of object is this?

Student: It might help if I knew what “ontologically” meant.

Math education: an international comparison

A quote from Prof Brian Butterworth, an Emeritus professor from the Centre of Educational Neuroscience at the University College London:

The UK is not very good at maths. We are about average looking at all [Organization for Economic Cooperation] countries. So, we are significantly worse than Canada and Australia and much worse than China and Japan although we are a bit better than Germany and significantly better than the United States.

Credit: Learn maths to boost the economy, scientist advises, The Guardian, November 18, 2010.

Independence

Once in my probability class, a student asked a reasonable question — could I intuitively explain the difference between “uncorrelated” and “independent”? My expert answer: it’s like the difference between “mostly dead” and “all dead.”

Greek letters

One evening, I was watching the Philadelphia Eagles play the Chicago Bears on NBC’s Sunday Night Football telecast. The bottom of the screen showed the score: “PHI 14 CHI 7.” As my wife walked by, she innocently asked, “Why are there Greek letters on the screen?”

We’ve been fans of $\phi$ and $\chi$ ever since.

Student misconceptions about PEMDAS

Simplify $6/2*(1+2)$ .

A Common Incorrect Answer. According to PEMDAS, we should handle the parentheses first. So $6/2*(1+2) = 6/2*3$ . Next, there are no exponents, so we should proceed to multiplication. So $6/2*3 = 6/(2*3) = 6/6$ . Finally, we move to division, and we obtain the answer $6/6 =1$ .

The above answer is incorrect and (even worse) arises from a natural but unfortunate misconception of the way that children are commonly taught order of operations. If you don’t see the misconception, please give it some thought before continuing.

The mnemonic PEMDAS, commonly taught in the United States, stands for

Parentheses

Exponents

Multiplication

Division

Addition

Subtraction

I personally never learned this memorization trick when I was in school. What I do remember, from learning BASIC computer programming around 1980, was the mnemonic My Dear Aunt Sally. I’m told that in the United Kingdom (and perhaps elsewhere in the English-speaking world) schoolchildren are taught BIMDAS, where B stands for Brackets and I stands for Indices.

Unfortunately, all of these memorization devices suffer from a common flaw: they do not indicate that multiplication and divison have equal precedence, and that addition and subtraction have equal precedence. In other words, the order of operations really are

Parentheses

Exponents

Multiplication and Divison (left to right)

Addition and Subtraction (left to right)

Therefore, the correct answer to the above problem is

$6/2*(1+2) = 6/2*3 = (6/2)*3 = 3*3 = 9$ .

In brief, though not intended by teachers, PEMDAS and BIMDAS perhaps promote the misconception that multiplication takes precedence over division and addition takes precedence over subtraction. To avoid this misconception, one of my colleagues suggests that PEMDAS be taught more visually as

P
E
MD
AS

so that students will have a better chance of remembering that MD and AS should have equal precedence.