## Archive for April, 2011

### Impossible Final Exams

The end of the semester is approaching. If you still haven’t prepared a final exam for your students, consider using the MJ4MF Final Exam (PDF). Those teaching in online college classes can send this exam to their students.

Alternatively, here are some questions you could use, depending on what course you’re teaching:

**Geometry:** Provide one real-life application of Ceva’s theorem that’s *useful*.

**Algebra:** From the real world, provide one example of a quadratic equation with integer coefficients that has integer solutions.

**Analysis:** Derive the Euler-Cauchy equation using only a straightedge and compass.

**Biology:** Create life.

**Computer Science:** Write a fifth-generation computer language. Using this language, write a computer program to finish the rest of this exam for you.

And here’s a mathy joke about final exams…

A mathematician, who had earned his PhD nearly 30 years ago, returned to the school from which he matriculated. He visited with the faculty in the math department, and they shared their exams with him. “Why, the questions on these tests are the same ones I answered when I was a student here!” he said.

“That’s true,” said one of the professors, “but the answers are all different.”

### On the Road Again

The summer driving season is just around the corner, and a gallon of gas costs a dollar more than it did at this time last year. We can all take solace in the fact that Exxon isn’t feeling any ill effects.

Hopefully the following driving-related math jokes will make you feel a little less depressed.

Which American highway is related to the length of the vector (1, 8, 1)?

Route (root) 66.

Perhaps the next one hits a little too close to home.

A large group of math majors were headed to a party, but they only had one car. Consequently, one person served as the driver and shuttled folks to the party in groups. When the last of them arrived, they realized they didn’t really feel like interacting with any of the other party-goers. So, they repeated the process of returning home one group at a time, in their only car.

In short, they commute but don’t associate.

How is the group described above like the averaging function *x* AVG *y* = ½(*x* + *y*)? The averaging function is commutative:

*x* AVG *y* = *y* AVG *x*

However, it is not associative:

(*x* AVG* y*) AVG *z* ≠ *x* AVG (*y* AVG *z*)

### Wish I Had Said That

The Northern Virginia Math Teachers Circle (NoVaMTC) met for the first time last night. Professor Bob Sachs from George Mason University led us through a fantastic activity about Conway Rational Tangles. A Google search will provide lots of information about them; however, you’ll have more fun by investigating them yourself. Let me give you a quick overview…

Take two pieces of rope, each about 10 feet long. (It’s easier to see what happens if the ropes are different colors, but it’s not a requirement.) Give each end of the rope to a different volunteer. Of course, this implies that you’ll need four volunteers. But to explore, you can just take two small pieces of string and perform the moves yourself. Call the four volunteers A, B, C, and D, and assume they are arranged as shown below.

With the ropes arranged in this way, there are two possible moves:

**Twist:** The two people on the left switch positions; the person in front lifts his rope, and the person in back goes underneath the front person’s rope. It is very important that this move is always done consistently. That is, it must always be the two people on the left who switch positions, and the front person must always lift the rope over the back person.

The results after one and two twists are shown below.

**Rotate:** All four volunteers move a quarter-turn clockwise; from the original position, A would move to B, B would move to C, C would move to D, and D would move to A.

There is also a third move, **Display**, which just means that the volunteers closest to the audience lower their ends of the rope, and the volunteers further away raise their ends of the rope. That way, everyone in the audience gets a good look at the ropes. This move is only important if doing the activity with a group.

And basically, that’s it. Now do a few moves to sufficiently tangle the rope. The question is:

**Is there a series of twists and rotates
that can be used to untangle the ropes?**

You might find it surprising that it is always possible to untangle the rope using only twists and rotates. But even knowing that, figuring out how to do it is far from trivial.

Try it for yourself. Put just one twist in the rope. Now, what series of twists and rotates will untangle the rope?

Try it with two twists. Or maybe twist, rotate, twist. Or maybe twist, rotate, rotate.

Is there a way to codify this system? Given a series of twists and rotates to sufficiently tangle the rope, could you describe the series of twists and rotates that would untangle the rope? That’s the ultimate goal. Play with it for a bit.

Okay, that’s enough of an introduction to Conway Tangles. Take some time to play. If you want to learn more about it, I recommend Tom Davis’s description of how to use this activity in a classroom.

As the NoVaMTC group was exploring tangles, some of the members were hung up on the notion that just untangling the ropes was not sufficient; they also wanted the four volunteers to be returned to their original position. When they asked Professor Sachs about this, he responded that it’s like the Albert Einstein quote:

Make things as simple as possible, but not simpler.

He explained that when solving a problem, he’ll make some assumptions that he might revisit later. But at the beginning, he’ll analyze a simplified model first, hoping that it offers insight for how to solve the original problem. Such is the case with tangles: Let’s explore the problem by first figuring out how to untangle the rope, then later maybe we’ll return to the problem of getting all four volunteers back to their original positions.

This reminded me of the George Polya quote:

If there is a problem you can’t solve, then there is an easier problem you can solve: Find it.

Have fun exploring Conway Tangles!

For a diversion, here are a few other mathy quotes that you might enjoy…

A math student’s best friend is BOB (the Back Of the Book). Just remember that BOB doesn’t come to class on test days.

Tell students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors.

What is now proved was once only imagined.

Those who say it cannot be done should not interrupt those who are doing it.

### Analyzing My Favorite Game

I’ve discussed my favorite game before, which is played as follows:

- On a piece of paper, everyone playing writes down a positive integer.
- Show your number to a neighbor (for verification purposes only).
- The winner is the person who wrote down
*the smallest integer not written by anyone else*.

I recently used this game with a group of 32 people at the end of a presentation. The first round was a sample round only, and folks didn’t know the rule for determining a winner before choosing their number. (People often find the rule confusing, so I often do a first round where I don’t tell folks the rule until *after* everyone has written down a number. Then I give the rule, and we determine the winner to provide an example of the rules in action.)

We played six rounds. I gave the winner of the third and sixth rounds a copy of *Math Jokes 4 Mathy Folks*; the other rounds were just for fun.

I’ve always been curious about the strategy that folks use when playing this game, so I asked folks to record their numbers for each round, and then I collected their choices. Geek that I am, I analyzed the results, and I thought I’d share them with you. (Don’t you feel special?)

In Round 1, choices were all over the charts. This is to be expected, since folks had no idea why they were choosing a random number. Choices ranged from 3 to 99, with a mean of 15.9.

After the rules were revealed, though, things got more interesting. The charts below show the choices during Rounds 2‑6. (Horizontal axis is the number chosen; vertical axis is the number of attendees who chose the number.)

Some observations about these results:

- The maximum number chosen by any player decreased in each round.
- The average value chosen for Rounds 2‑6 was 8.5, 7.5, 5.3, 7.7, and 6.4, respectively. It’s interesting that the average decreased to 5.3, then shot back up to 7.7. This might be explained by the trend in winning values. The winner in Rounds 2 and 3 chose 2. In Round 4, at least three players chose each of the numbers 1‑4, and the winner chose 5. Players may have assumed that too many players were tending to choose low numbers, so they chose slightly higher numbers in Round 5.
- No fewer than five players chose the number 7 in every round.
- Interestingly, the winner who chose 1 in Round 5 was also the winner who chose 1 in Round 6. She went to the well twice — and it paid off!

A few days later, I ran the same experiment with a different group of 35 people. The results were slightly different.

- The average value chosen decreased in every round, as follows: Round 1, 11.96; Round 2, 7.54; Round 3, 7.53; Round 3, 4.88; Round 4, 4.84; and Round 6, 4.70. As with the previous group, the average took a big dip from Round 3 to Round 4. However, unlike the previous group, the average did not shoot back up in Round 5.
- No fewer than four players chose the number 4 in every round, and it was chosen by 10 players in two different rounds.
- Even after learning the rules by playing a practice round in Round 1, several folks chose surprising numbers in Round 2. Among them: 100; 102; 1,000; 1,900; and 10
^{100}.

I hope you enjoyed this diversion. We now return you to our regularly scheduled programming…

### Is Car Talk Invading My Turf?

On Saturday, my friend Mark Stevens emailed me the following joke:

What is the ratio of an igloo’s circumference to its diameter?

Eskimo pi.

Until now, this joke never appeared on the MJ4MF blog, though a similar joke appeared in a post on Pi Day 2010. This joke does, however, appear in the list of 57 conversions in the “Conversion Chart” on pages 65‑67 of *Math Jokes 4 Mathy Folks*.

The subject line of Mark’s email was “Car Talk Mathy Joke.” I initially thought Click and Clack were pilfering my material, but a quick search for “conversions car talk” revealed that they had posted a list of 37 conversions on the Car Talk web site in May 2000. Not that I could have done anything, anyway. The list that appears in *MJ4MF* is not original. Conversions like this have been floating around the Internet for at least a decade.

However, at least one of the conversions in *MJ4MF* was a Vennebush original:

16 ounces of Alpo = 1 dog pound

In looking through the Car Talk list, I noticed one conversion in their list that was absent from mine:

The first step of a one-mile journey = 1 Milwaukee

(You have to put a certain drawl on the right side so it reads as “one mile walky.”)

My favorite joke of this ilk, which did not appear on the Car Talk list…

2000 mockingbirds = 2 kilomockingbirds

I rather enjoy these corny jokes. In fact, I used the following joke last month at the Virginia Council of Teachers of Mathematics’ (VCTM) conference:

Some people are frustrated by metric conversions, but not me. For instance, if you want to know how many televangelists are equal to one expatriate poet, the conversion is rather simple…

### Mathy Animals

Many animals are proficient at math…

- Rabbits, because they multiply.
- Sea anemone, because they divide.
- Snakes, because they can be adders.
- Beavers, because they work with natural logs.
- Flamingoes, because they balance.

What insect is good at math?

An account-ant.

And a joke funny enough to share, even if it’s only pseudo-mathy…

A dog in a front yard is tethered to a sign that reads, “Talking Dog for Sale.” The man walks up to the dog. “You talk?” he asks.

“Sure do,” the mutt replies.

“So, what’s your story?”

The dog looks up and says, “Well, I was adopted by a mathematician when I was just a puppy. Recognizing my gift, he offered my services to the NSA. I would attend meetings with spies and world leaders. None of them expected me to be listening, so they would discuss all kinds of secret plans in front of me. For years, I was one of the country’s most valued spies. I heard a lot of amazing things. In fact, I was privy to many conversations about secret codes, and I co-authored several papers about cryptography. When I retired, I wrote a memoir about all of my adventures, which will be published later this year.”

Just as the dog finishes his story, the owner steps onto the porch. The man, completely amazed by the dog’s story, asks the owner what he wants for the dog. The owner says, “Ten dollars.”

The man says, “But, that dog is amazing! Why are you selling him so cheap?”

The owner replies, “Because he’s a frickin’ liar. He didn’t do any of that stuff.”

### Blown Out of Proportion

In preparing a workshop about proportional reasoning for the 2011 NCTM Annual Meeting, I came across the following from “Learning and Teaching Ratio and Proportion: Research Implications” by Cramer, Post, and Currier, which appears in *Research Ideas for the Classroom*, edited by Douglas T. Owens. The authors discuss the following problem, which they presented to a class of pre-service elementary teachers:

Sue and Julie were running equally fast around a track. Sue started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?

The authors claimed, “Thirty-two out of 33 pre-service elementary education teachers in a mathematics methods class solved this problem by setting up and solving a proportion: **9/3 = x/15**.”

I wanted to be surprised. Sadly, I was not.

Participants in my workshop (all current middle or high school math teachers) were asked to solve the same problem. Several incorrect answers were suggested, among them 3, 15, 27, and 45. Less than 40% of the attendees obtained the correct answer, 21.

This is frustrating, as proportional reasoning is extremely useful in analyzing real-world phenomena. In fact, it’s even applicable to language arts, as evidenced by the following graphic from my presentation:

The teachers in my workshop aren’t the only ones who have difficulty with proportional reasoning. Students have endless trouble, too…

Teacher: Today, we will discuss inverse proportion. Here’s an example: “If it takes 6 days for 2 men to finish a task, how long will it take 3 men to complete the same task?” The number of men needed is inversely proportional to the number of days required. Consequently, 3 men will be able to complete the task in 6 × 2/3 = 4 days.

Student: Oh, I see! I think I’ve got a real-life application of this. If it takes 6 hours for 2 men to hike to the top of a hill, then it will only take 4 hours for 3 men to hike to the top!