Archive for: February 2008

If you’re new here, you may want to subscribe to my RSS feed. Thanks for visiting!

original photo from Richard Masoner Edited by Brian

Not sure what got me thinking about this today, but I was musing about the commutative property, and how it applies to addition and multiplication. (Yeah, it was a pretty boring morning.)

Specifically, I was thinking about the word, “augend”. The augend of an addition problem is the first of the series of addends. It’s not a word that is usually taught, and I was wondering why not.

You should be aware that addition and multiplication have the commutative property, and subtraction and division don’t. That just means 4+6 is the same as 6+4, and 3*7 is the same as 7*3, but 8-2 and 2-8 are not the same, nor are 9/3 and 3/9.

So it doesn’t really matter which of the numbers is placed where in addition, so you don’t need to specify which is the augend, and which is not. You can call them all addends. Same with multiplication - you can call them all multiplicands. (Not so with minuends, subtrahends, dividends and divisors, though - they don’t commute.)

But I think we should teach about augends. There are several good reasons why, and you can read about them at a lesson I just put up at Names of the numbers in basic arithmetic operations. That lesson is all about why we give the different parts of arithmetic problems their names, (like dividend, divisor, and quotient) and why it makes sense to learn them.

Now that I feel like I’ve cleared this all up for you and me, I’ve got something that I’m not so clear on. Maybe some kind reader has some insight about it she or he’d like to share. It’s this:

Since you must differentiate the names of the parts of subtraction or division problems, what happens if a problem has more than two terms, like 8-3-2? Is there a name for the third term? What if there are fourth or fifth terms?

I assume that they are called, “the second (secondary?) subtrahend, the third (tertiary?) subtrahend, and so on, but I’m not sure.

Anybody got any insights?

You may want to check Names of the numbers in basic arithmetic operations first, though.

By the way, if anybody can write me and tell me why I chose the image that I used for this post, I’ll send them a free Math Mojo e-booklet. (Use the contact me near the top right navigation bar on this page.)

Update: You don’t have to write about that anymore - we have a winner! Mark (see below) got the booklet.

To clarify: The big dog in the picture is “Doggy Daddy,” and the little dog at the door of the train is “Auggie Doggy.” (They are Hannah-Barbera cartoon figures.) They are about to enter a commuter train. Get it? Augend/Auggie, commutative property/commuter train? (Groan.)

Original photo by didbygraham

Q: When is a Compromise not a Compromise?
A: When it’s a Red Herring

We talked about “red herrings” in the previous posts about “The Case of the Missing Dollar.”

I occasionally do after-school presentations of MathMagic for the C.R.O.P. program in rural upstate New York. I’ve been participating in the program for years, as a local artist (magician). The program pays a very small honorarium, and a travel budget (Monday I drove 120 miles round-trip for peanuts) Most of the artists do it out of love of bringing their art to children who otherwise may never get exposed to it. It is a labor of love to all concerned, but it is a great mission.

The Creating Rural Opportunities Partnership (CROP) After School and Summer Program is a program which does just what it is named.

In it’s mission statement:

The goal of CROP is to provide intellectual development and opportunities for academic achievement for students grades K-8 with a strong focus on middle school.  In addition, CROP provides enrichment, health, wellness, life skills, recreational and cultural opportunities for 1260 middle school and primary grade students, community members and parents through a 21st Century Community Learning Center Partnership.

Over the last few years, funding for this excellent program has dwindled. Sound familiar? Art and after-school programs are typically among the first to get their throats cut when the government feels it needs to cut costs.
(more…)

Original Photo by Norsehorse Edited by Brian

Ah, I love it when readers beat me to the punch!

The comments to the original post pretty much sum up the paradox and it’s solution very well.

Khaled’s and Mark’s comments illustrate perfectly one of the things I wanted to point out about this puzzle. That point is:

Just because something is phrased a certain way is not reason to assume that that phrasing is the best way to represent the problem. And one way to critically examine the situation is to reframe it in a mathematical equation.

Khaled said, “Interesting how, once you assume that you can implicitly trust a given source, you can be led through any logic, or illogic, and have a lot of trouble pulling yourself back to a critical mindset.”

How true. Then Mark gave a good method to understand how to see where the paradox lies when he said, “I started to write an equation, because properly written equations can solve all counting problems, but then realized that this was pointless, because adding 2 dollars to the 27 dollars the guests paid did not reflect what happened.”

Exactly! The question was phrased to lead you to believe that because the facts were a certain way (which it accurately represented) you had to see it in a certain way (which was anything but accurate).

(more…)

Original Photo by Norsehorse Edited by Brian

There’s a braintwister that’s been going around the internet, well, probably ever since there was an internet. It’s actually probably thousands of years old in one version or another. You may have seen it phrased like this:

Three men go into a motel. The man behind the desk said that the room costs $30. So each man paid $10 and went to the room.

Later, the desk clerk realized that the room was only $25. So he sent the bellboy to the men’s rooms with five one-dollar bills.

The bellboy couldn’t figure out how to split five dollars evenly three ways, so he gave each man one dollar, and kept the other two for himself.

This meant that the three men had each paid $9 for their rooms, which makes a total of $27 dollars. Adding the two dollars that the bellboy kept would make a total of $29 dollars.

So where is the other dollar?

My advice to anyone trying to solve anything like this, or trying to think about anything at all, for that matter, is not to jump to conclusions.

Want to give it a try and add your thoughts in a comment? Go for it! I’m not asking for the solution, just some thoughts about the meaning of the puzzle - how it relates to life, logic, decision-making and understanding your world. I am not putting this up as a trivial puzzle.

My comments will be in the next post.

(Note: When I originally posted this, there were a few typos and other mistakes in it. If you busted your head over it till now, please accept my apologies. It should be correct now.)

I found this on a weird little personal blog:

Why has math been hated by some?

Because it requires them to think and forces them to give the correct and exact value. Math has a clear distinction of right and wrong. Most people love to speak about any issue, but hate to accept that they’re wrong.

That’s the beauty of math… right is right and wrong is wrong.

Concise and true, isn’t it? So many people hate the way their parents, teachers, politicians, and salespeople waffle and prevaricate, yet they don’t like their Ideas to be held up to scrutiny. They are going to grow up just like the people they are complaining about now, unless they learn a system of honest critical-thinking. Math and logic are just the ticket.

Math is the great equalizer. You don’t have to be rich or privileged to excel at it. Some of the greatest mathematicians started out as sons or daughters of poor, rural families. Curiosity and a local library are all you need to get you on your way.

Once again I am going to suggest the greatest introductory math book I know, “The Realm of Numbers,” by Isaac Asimov, even though it is out of print. Google it. Get it.

It is written for non-math people. It will take you from counting to algebra, in plain terms, without “worksheets.”

Don’t take my word for it. It’s a great book. Go now, and google it.

Hotcha!

Brian

Are you a homeschool parent struggling to teach your child math? Or are you just frustrated by the way your kid’s school teaches math? You’re definitely not alone, and you’re in great company.

Here is part of a story from a father who faced the same thing. It’s a comment left by Mark, a reader at the MathNotations blog.

You should read the entire article, then scroll down to the comments where the exchange between Paul Michael Goldenberg and myself (Brian) begins. Read them to understand the background to the great comment by Mark, which is partially reproduced here:

“I had a 4th grader who was being just totally crushed. I tried to help with math homework, but the assignments were chaotic. He just finally refused to even try to do his math homework. To see tears in his eyes and protest, “I’m not good in math, Dad!” broke my heart.

“I took him home, and found out that he could do not complete a 10 x 10 multiplication table or do any long division. I concocted a non-stressful systematic build-from-a-foundation scheme. We used a variety of manipulatives, including an abacus for computation. (I guess you and I think alike, eh?) I taught him basic algebra using a balance and weights: keep the pans level, and that’s an equation analogue.

“Upshot: he started calculus at age 16.”

Amazing! What a motivation for you to try your own ways (with or without Math Mojo methods) to help your child. Of course, not everyone can know as much math as Mark (I sure don’t!) but you can do plenty with the methods you can find on this site, and in Mark’s comments (click the link to MathNotations, above, to get there).

So head out there and read them now.

Wait! The author of that comment just left a terrific comment below. Make sure you read it if you are trying to teach your child math.

(Or “The Seamus on Mitt Romney”)

Sometimes you read a story that just grabs you. I got one forwarded to me today that, at first glance, seems to have nothing to do with math. But since Math Mojo readers know that math is more meaningful than that stuff they shoved down your throat in school, I think you’ll appreciate this one. Bear with me.

In the summer of 1983, Mitt Romney took a vacation with his wife and five sons, to his parents’ cottage on the Canadian side of Lake Huron. The trip from Boston was twelve hours long.

According to an article in the Boston Globe by Neil Swidey and Stephanie Ebbert, Globe Staff, June 27, 2007:

“…Before beginning the drive, Mitt Romney put Seamus, the family’s hulking Irish setter, in a dog carrier and attached it to the station wagon’s roof rack. He’d built a windshield for the carrier, to make the ride more comfortable for the dog…

“… As the oldest son, Tagg Romney commandeered the way-back of the wagon, keeping his eyes fixed out the rear window, where he glimpsed the first sign of trouble. ”Dad!” he yelled. ”Gross!” A brown liquid was dripping down the back window, payback from an Irish setter who’d been riding on the roof in the wind for hours.

“As the rest of the boys joined in the howls of disgust, Romney coolly pulled off the highway and into a service station. There, he borrowed a hose, washed down Seamus and the car, then hopped back onto the highway. It was a tiny preview of a trait he would grow famous for in business: emotion-free crisis management.”

Yeah, otherwise known as “compassionate conservatism.” You know that to “conserve” means “to not use,” or “to use as little as possible.” To conserve your compassion for what? Your cronies? Your sons, who you conserve for your political agenda, but not to fight in a war that you support? Certainly not for your dog…
(more…)