## Posts tagged ‘algebra’

### Questions and Answers

I’ve been teaching my sons how to tell time on an analog clock. The following is a recent conversation:

Me: The little hand is the hour hand, and the big hand is the minute hand.

Eli: What’s the third hand for?

Me: That’s the second hand.

Alex: Why is the third hand called the second hand?

Here are some questions that I’ve been asked recently for which I *did* know to respond…

How good are you at algebra?

Vary able.Why is simplifying a fraction like powdering your nose?

It improves appearance without changing the value.What do you get when you cross geometry with McDonald’s?

A plane cheeseburger.

### Observations

Some things I’ve noticed…

- Algebra is
*x*sighting. - Rational people are partial to fractions.
- Geometricians like angles… to a degree.
- Vectors can be ‘arrowing.
- Calculus teachers can go on and on about sequences.
- Translations are shifty.
- Complex numbers are unreal.
- Most people’s feelings about integers are positive.
- On average, people are mean.

### Math Humor of Mike Vecchione

I’m not necessarily a fan of comedian Mike Vecchione. His recent Comedy Central special merits only 3.5 stars (out of 5). But he’s a Penn State grad, so that’s worth an extra half‑star, and he included a math bit in his act that cracked me up. I’ve provided an abridged version of the transcript below (it’s abridged so as to keep it PG‑13).

I suspect you’ll be no more amused by the transcript than a blind person would be by a Three Stooges routine. Without inflections and body language, it might not translate well. Still, I think it’s worth a read — but you will probably enjoy the live performance more, if you can tolerate a little off‑color humor.

Without further adieu…

My dog broke a mirror. I freaked out. I was like, “Oh, my God. Is he gonna get 7 years bad luck or 49 years, just because he’s a dog?” I’m not a math guy. I’m not a math guy.

You know what? I shouldn’t say that, because I am good in math. I was good in math, good in English. You know what I was bad in? Algebra. I was bad in algebra — ‘cause I like my letters in words, I like my numbers in problems. I don’t like ‘em mixed.

[…]

I can solve problems, just not the way the teacher in algebra wanted me to solve problems. She put the problem on the board:

x+ 7 = 15. Solve the problem. You want me to solve the problem? I went up, I erased the board. I’m like, “Problem solved, bitch! Let’s get some pizza, and stop playing these games, baby. Let’s do some fractions, take the reciprocal, flip the script. Let’s make that fraction so filthy and improper.”She just kept putting the problems up on the board. I just kept following her and erasing the problems. Then she yells at me.

I’m like, “Number 1, I like to attack the problem, not the person. That’s the first rule of problem solving. And B, you kinda seem like you’re a trouble maker, because you got to come up with all these fake problems, and it’s really cutting into our pizza time.”

And she’s like, “You can’t list things 1 and then B. It’s 1 and 2, or A and B.”

And I’m like, “Oh, you don’t like it when I mix numbers and letters together? Like you do in algebra, you hypocrite?”

She was trying to teach me:

Don’t be afraid of the letters. She’s like, “Don’t be afraid ofx. It’s a variable, an unknown. That’s all it is — a variable or an unknown.”I’m like, “Not to me. To me, X is the rating of a movie that I enjoy watching in the comforts of my own home.”

[…]

I got an F in algebra. I took it home. My mom was furious. She’s like, “You failed algebra?” I’m like, “That’s not what that means. That’s a F. It’s a variable, an unknown. We don’t know what F means. It could mean I failed, but it probably just means I’m fantastic.”

### Presentation at Central Connecticut State University

Yesterday morning, I presented a math and technology workshop at Central Connecticut State University. The attendees were teachers who are testing a new Algebra I curriculum. As part of the workshop, the attendees participated in my favorite online activity, and they attempted my favorite problem.

To ensure that I provided appropriate activities, I reviewed a draft of the new curriculum ahead of time. The vocabulary list for the first unit contained the word *hendecagon*. Guessing that most attendees were unfamiliar with the term, I provided a diagram:

Seriously, a *hendecagon* is an 11‑sided polygon. The Susan B. Anthony dollar coin is probably the most common example of a hendecagon in everyday life. (As it turns out, there is no connection between the number 11 and Susan B. Anthony, but the reverse side of the coin contains an image adapted from the Apollo 11 mission insignia.)

(For the record, I’d be interested in creating more silly images like the one of the hendecagon above. Are there other math terms that lend themselves to similarly silly drawings? If you think of one, post a comment.)

At the end of the workshop, I gave away several copies of *Math Jokes 4 Mathy Folks*. To determine who would get the copies, we played my favorite game:

- On a piece of paper, write down a positive integer.
- Share your number with a neighbor (for verification later, if necessary).
- The winner is the person who wrote down
*the smallest integer not written by anyone else*.

The rule for determining a winner often confuses folks, so let me give an example. Let’s say 20 people chose the following numbers:

4, 1, 7, 11, 4, 6, 5, 2, 1, 3, 2, 1, 8, 1, 4, 7, 9, 2, 3, 18

- The smallest integer in the list is 1, but it was written by four people. Hence, none of them would win.
- The next smallest integer is 2, but it was written by three people, so none of them would win, either.
- The number 3 was chosen by two people, and the number 4 was chosen by three people — no winners.
- Which brings us to the number 5. The person who chose 5 is the winner, because she wrote the smallest integer not written by anyone else.
- Although the folks who wrote 6, 8, 9, 11, and 18 wrote numbers that weren’t written by anyone else, they did not choose the smallest numbers not written by anyone else.

As with many things, this game may not sound very fun with a text explanation. But play it a few times. Or better yet — play it the next time you go to dinner with a large group of friends, and the losers have to buy dinner for the winner.

I learned this game from my friend Richard Rusczyk, who referred to it as “Harold’s Game,” because he learned it from our friend Harold Reiter. Despite the name Richard has given the game, I do not believe that Harold is the creator.

### (Lack of) Math Videos

I just saw a math video that showed a technique for multiplying two-digit numbers. I’ll share this trick with you in a minute. But I am dismayed that this video — along with so many other math videos I’ve seen on the web — are willing to *show* these tricks but not *explain* them. So I’m going to provide the trick, which I thought was pretty cool, along with an explanation as to why it works.

Before I do, though, I have to vent. Math videos on the web are a real problem. The ones I’ve seen generally do very little to promote conceptual understanding. Instead, the videos present skills without explanation, and the implicit message is that you don’t need to understand why it works — just do exactly as the videos say, and you’ll get the right answer. The issue is that students then see mathematics as a series of disconnected rules they need to memorize, instead of seeing it as the beautifully interconnected discipline that it is.

Arrgh.

Okay, I’ll now step down from my soapbox and share the trick with you…

Take two numbers, both between 10 and 99, that meet the following criteria:

- Their tens digits are the same.
- The sum of their units digits is 10.

Here are some problems that meet these criteria:

- 22 × 28
- 74 × 76
- 33 × 37
- 55 × 55

I will now state the obvious — there are only 81 multiplication problems to which this trick applies. Still, I think it’s fun to explore the math underlying the trick.

To find the product, then, follow this process:

- Take the tens digit, and multiply it by one more than itself.
- Multiply the units digits.
- Finally, concatenate (put together) these two results to make a three- or four-digit number. (As noted in the comments below, if the result of Step 2 is a one-digit number, you’ll also need to put a 0 in front of it.)

Let’s take the first problem above, 22 × 28, to see how this works in practice.

- The tens digit is 2, so multiply 2 by one more than itself: 2 × 3 = 6.
- The units digits are 2 and 8, so multiply them: 2 × 8 = 16.
- Consequently, 22 × 28 = 616.

More generally, the trick could be explained with symbols. If the tens digit of each number is *n*, and the units digits are *p* and *q*, then we’re trying to find the product of (10*n* + *p*)(10*n* + *q*). Using the FOIL method from algebra gives

(10*n* + *p*)(10*n* + *q*) = 100*n*^{2} + 10*n*(*p*) + 10*n*(*q*) + *pq *= **100 n^{2} + 10n(p + q) + pq**

But since the criteria implies that *p* + *q* = 10, this becomes

100*n*^{2} + 10*n*(*p* + *q*) + *pq* = 100*n*^{2} + **100 n** +

*pq*

This can be further refined to

100(*n*^{2} + *n*) + *pq*

and the important point, then, is realizing that *n*^{2} + *n* = *n*(*n* + 1)*.* Which is just another way of saying that we’re going to take the tens digit, *n*, and multiply it by one more than itself.

And that’s it. The first part of the expression above, 100(*n*^{2} + *n*), gives the hundreds (and perhaps the thousands) digit, and the second part, *pq*, gives the tens and units digits.