## Posts tagged ‘system’

### Thinking > Symbols

As if having to wear a mask while flying isn’t scary enough, last week I had a harrowing experience.

See Valentin Kirilov’s full video at https://vimeo.com/156775315.

It’s normally a three-hour flight from Yellowknife in the Northwest Territories to Portland, but about 20 minutes into a recent flight, there was an unsettling noise followed by an announcement from the pilot: “Ladies and gentlemen, this is your captain. Please don’t be alarmed by the noise you just heard near the left wing. Let me assure you that everything is okay. We’ve lost an engine, but the other three engines are working just fine. They’ll get us to Portland safely. But instead of taking 3 hours, it’s going to take 4 hours.”

She said this matter-of-factly, as if she were commenting on the color of my shirt or pointing to a squirrel that was enjoying an acorn. It was reassuring that she didn’t seem worried — but still, we had lost an engine! The cabin grew quiet as we waited to see what would happen next. But, nothing. We continued to fly as if nothing had happened, and the din of conversation slowly resumed.

And then there was another noise near the right wing, followed by the pilot: “Ladies and gentlemen, this is your captain again. Unfortunately, we’ve lost another engine. But the two remaining engines are functioning properly. We’re going to get you to Portland safely, but now it’s gonna take about 6 hours.”

Needless to say, you could hear a pin drop as we passengers waited with baited breath. And sure enough, there was another noise, and the pilot again: “Ladies and gentlemen, we’ve lost a third engine, but we’ll make it safely to Portland with the one engine we’ve got left. It’s just gonna take 12 hours.”

That was when the guy in the seat next to me lost it. “I sure hope we don’t lose that last engine,” he yelled, “or we’re gonna be up here all day!”

That story isn’t true. None of it. Not. One. Single. Fact. I mean, come on! Yellowknife? Why would I need to go to Yellowknife? To brush up on my Dogrib?

But the following story is true.

My honorary niece Ivy occasionally calls for help with math. Not surprisingly, the frequency of calls has increased with online learning during the pandemic. This is a modified version of a question about which she recently called:

A plane traveling against the wind traveled 2,460 miles in 6 hours. Traveling with the wind on the return trip, it only took 5 hours. What was the speed of the plane? What was the speed of the wind?

It’s assumed that the plane flew at the same average speed in both directions and that the wind was equally strong throughout. On a standardized test, those assumptions would need to be stated explicitly. But on this blog, I make the rules, and I refuse to add more words to explain reasonable assumptions without which the problem would be impossible to solve.

As presented to Ivy in an online homework assignment, the problem stated that x should represent the plane’s speed and y should represent the speed of the wind. My first question was, “Why? What’s wrong with p and w as the variables for plane and wind, respectively?” But my second, and perhaps more important, question was, “Why are they forcing algebra on a problem that is much easier without it?”

Somewhere, many years ago, someone decided that the methods of substitution and elimination were critically important, if not for success in life then at least for success in high school. Anecdotally, this seems to be true; I was a bad-ass systems-of-equations solver in high school, and I was also the captain of the track team that won the county title, so… yeah. Sadly, that someone seemed to put less stock in a slightly more critical skill: thinking.

Here’s the deal. Against the wind, the plane covered 2,460 miles in 6 hours, so the plane’s speed on the way out was 410 miles per hour. With the wind, the plane covered those same 2,460 miles in just 5 hours, so the plane’s speed on the way back was 492 miles per hour. Since the wind is constant in both directions, the unaided speed of the plane must have been 451 miles per hour, exactly halfway between the against-the-wind and with-the-wind speeds. Visually,

The top portion of that diagram represents the plane flying against the wind; the bottom portion represents the plane flying with the wind. This makes it obvious that the the unaided speed of the plane is equidistant from 410 miles per hour and 492 miles per hour, and that distance is the speed of the wind.

Algebraically, the problem might be solved like this…

$6 \cdot (x - y) = 2460 \rightarrow x - y = 410 \\ 5 \cdot (x + y) = 2460 \rightarrow x + y = 492$

which is no different than the logical approach above: x ‑ y is the speed of the plane against the wind, and x + y is the speed of the plane with the wind, so the numbers are obviously the same. But using algebra just seems so much more… cumbersome.

Students in elementary classrooms often explain their thinking with drawings and diagrams. Unfortunately, some of that great thought is abandoned when students take a course called Algebra, and the mechanical skill of symbolic manipulation replaces the mathematical skill of logical reasoning. That’s a tragedy. As Jim Rubillo, former executive director at NCTM, used to say, “We have to stop being so symbol‑minded.”

### All Systems Go

I noticed the boys having an intense conversation in front of this sign at our local pizza shop:

When I asked what they were doing, they said, “We’re trying to figure out how much one slice and a beer would cost.”

As you read that, there were likely two thoughts that crossed your mind:

• Why can’t these poor boys look at a pizza menu without perceiving it as a system of equations?
• Why are eight-year-olds concerned with the price of beer?

The answer to both, of course, is that I’m a terrible father, and both beer and math are prominent in our daily lives.

But you have to admit that it’s pretty cool that my sons recognized, and then solved, the following system:

$\begin{array}{rcl} 2p + s & = & 6.00 \\ 2p + b & = & 8.00 \\ p + s & = & 3.50 \end{array}$

They didn’t use substitution or elimination because they didn’t have to — and, perhaps, because they don’t know either of those methods yet. But mental math was sufficient. If two slices and a soda cost $6.00, and one slice and a soda cost$3.50, then one slice must be 6.00 – 3.50 = $2.50. Consequently, two slices cost$5.00, so a beer must be 8.00 – 5.00 = $3.00. A beer and a slice will set you back$5.50.

I remember once visiting a classroom in Somerville, MA, and the teacher was reviewing the substitution method. My memory is a bit fuzzy, but the problem she solved on the chalkboard was something like this:

Mrs. Butterworth’s math test has 10 questions and is worth 100 points. The test has some true/false questions worth 8 points each and some multiple-choice questions worth 12 points each. How many multiple-choice questions are on the test?

The teacher then used elimination to solve the resulting system:

$\begin{array}{rcl} t + m & = & 15 \\ 5t + 10m & = & 100 \end{array}$

The math chairperson was standing next to me as I watched. “Why is she doing that?” I asked. “You don’t need elimination. It’s clear there have to be 8 or fewer multiple-choice questions (8 × 12 = 96), so why not just guess-and-check?”

“Because on the MCAS [Massachusetts Comprehensive Assessment System], if they tell you to use elimination but you solve the problem a different way, it’ll be marked wrong.”

So much for CCSS.Math.Practice.MP1. Although most of us would like students to “plan a solution pathway rather than simply jumping into a solution attempt,” apparently students in Massachusetts need to blindly follow algorithms and not think for themselves.

The following is my favorite system of equations problem:

I counted 34 legs after dropping some insects into my spider tank. How many spiders and how many insects?

Why is this my favorite system of equations problem? Because there is a unique solution, even though it results in just one equation with two unknowns. Traditional methods won’t work, and students have to think to solve it. Blind algorithms lead nowhere.

Other things that lead nowhere are spending your leisure hours reading a math jokes blog. But since you’re here…

Why did the student put his homework in a fish bowl?
He was trying to dissolve an equation.

An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn’t care.

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

## MJ4MF (offline version)

Math Jokes 4 Mathy Folks is available from Amazon, Borders, Barnes & Noble, NCTM, Robert D. Reed Publishers, and other purveyors of exceptional literature.