Uh, this is an off-topic piece. One of the benefits of working from home is that I can listen to my wife teaching our children, and every now and then, I drop in and explain some aspect of the topic further. Out of the corner of my ear, I heard my wife Ruth explaining Greatest Common Factors and Least Common Multiples to our fifth child, Jonathan.

When I was a kid, I was kind of a prodigy with math (I am not so now), and when we went through it in school, I remember tutoring my classmates on the two topics. I always thought the two concepts were related, but I never understood how, until it struck me last week.

Consider the numbers 60 and 144. What are their Greatest Common Factors and Least Common Multiples? To start, let’s factor the two numbers:

Then, let’s place the common factors in the intersection set.

The greatest common factor is the product in the intersection set, in this case, 3×2x2 = 12. The product of the union set (just multiply across) is the least common multiple — 5×3x2×2x3×2x2 = 720.

When I realized this, I drew it out for Ruth and Jonathan, and told them “Look at the ravings of a madman.” But later that evening, Ruth came to me and thanked me for it, because it worked with Jon, and clarified it to her.

As a mathematician, I am nothing great, but my intuition has been a great help to me at many points. This was one of them.

Update Saturday Afternoon

As F comments, “Which has the consequence that LCM*GCM = number1*number2.”

I should have written that myself, but didn’t.  Thanks for pointing that out.  It is an application of the rule that:

set A + set B = union of A&B + intersection of A&B

As we were driving down the highway Monday evening, back from our oldest daughter’s symphony concert at U-MD, my wife and I began talking about teaching children about money.  We homeschool, so we have to consider a lot in training our children for the real world.

Some of my children have an interest in the market, some don’t. Personalities differ, but you want to give them some core knowledge that everyone can use. There have been people in our home school get-togethers who when they find out I am an investor, they ask “Do you know of any good books on the stock market for kids?” Lamely, I suggest the out-of-print book by Ken Fisher’s son, Clayton, which is pretty good, but I didn’t think it was definitive.  One has complained to me about the Stock Market Game, which seems to teach speculation, not investment.

That’s true of most stock market contests — the only exception I can think of was the Value Line contest back in 1984 . I managed to place in the top 1%, but not high enough to win. That contest forced you to pick 10 stocks from ten different groups for six months. The stocks were sorted by price volatility deciles, so you had to pick some volatile stocks and tame stocks. The stocks were equal weighted, and there was no trading. Great contest — I would love to run something like that. I have suggested it to The Street.com, but no dice. Hey, maybe Seeking Alpha would like to try it! Nominal prize money, but there would be bragging rights!  (Abnormal Returns, this could work for you as well…)

My wife tells me to think about it. Well, today, as I’m going through my personal e-mail, I run across a note from the Home School Legal Defense Association promoting the National Financial Literacy Challenge. Timely, I think. They are having a competition based off of the national standards published in 2007 by the Jump$tart Coalition for Personal Finance.

So I look at the standards, and I think, “These are pretty detailed… how can you turn this into a usable curriculum?”  I print them out and read a little bit of them to my wife Ruth, who says, “Typical for those that set standards, and aren’t teachers; you can’t work with that stuff.”  My wife was a high school teacher, and despite that hindrance, she still homeschools well.  But she knows the troubles that come to public school teachers as mandates come down from on high.

She asked me, “What would you recommend, then?”  I thought about it and said that the personal finance book that I reviewed recently, Easy Money, would be a good book for high school seniors to read.  It’s not a complex book at all.  Afterward I would discuss it with them.  She asked me why I hadn’t done that for our older two children and I said, “It was published after they went to college.  I’ll ask them to read it this summer.”

For investing, I still think that Buffett’s Annual Reports are understandable to most teens.  Marty Whitman is easy to read as well.  But I always liked Ben Graham, and I think The Intelligent Investor is accessible to the average teenager.  Good investing is not complex… but often we make it so.

Full disclosure: if you enter Amazon from my links and buy anything, I get a small commission.  It is my substitute for the tip jar, and it doesn’t increase your costs at all.

In the center of the picture to the left are my friends Bill and Margie Wright. They have been dear friends of our family for eight years. Their kids have played with ours; they have ten, we have eight.

They’re special people. Margie has homeschooled all of their children (so far), and Bill has built a thriving enterprise, Wright Manufacturing, which makes the best commercial lawn mowers in the world. (Full disclosure: I own a little less than one percent of his private firm.)

Wright Manufacturing is unusual because it is a very innovative firm. They manage the business well with a series of principles borrowed from some of the best Japanese manufacturers, but then marry that to the exceptional innovative engineering talent of Bill Wright and his staff. Many of his competitors license his patented technologies.

Oh, the person on the far right of the picture is, yes, President Bush. He came to visit Bill’s factory today, and gave a brief talk on the economy, and the proposed stimulus package. Here’s what he said:

1:55 P.M. EST

THE PRESIDENT: Let me tell you why I’m here. This man started his own business. He’s a manufacturer, he employs over a hundred people, and he represents the backbone of the American economy. And today I talked about our economy, and the fundamentals are strong, but there’s uncertainty. And there’s an opportunity to work with Congress to pass a pro-growth package that will deal with the uncertainty.

Any package has got to remember that jobs are created by small businesses. A good package will have incentives for investment. The package we passed early in my administration helped him. He bought some equipment and made his firm more productive, kept him in business. And that’s the same spirit that needs to be in this next growth package.

The other thing is, is that we got to make sure that we benefit consumers. We want our consumers out there spending, and the best way to do that is broad-based tax relief. Now, this plan ought to be broad-based, it ought to be simple, and it ought to be temporary.

I had a conversation, Congressman, with the leadership on both sides of the aisle yesterday, and I was encouraged by what I heard. And I believe we can come together on a growth package very quickly. We need to. We need to get this deal done and get it out and get money in the hands of our consumers and our small business owners to help this economy.

I’m optimistic, I truly am. One reason I’m optimistic is because I understand we got all kinds of Americans just like this man here, working hard to provide a living for folks and to make a product people want.

And so, while there’s some uncertainty right now, if we act quickly and in a smart way that helps growth, we’re going to be just fine.

Anyway, thanks for letting me come by. I’m proud to be — I love the entrepreneurial class in — I love people who have a dream and work hard to achieve the dream.

And so — a fine-looking machine you got here.

MR. WRIGHT: Thank you. It’s a team effort. We thank you, thank you for all your work, too.

THE PRESIDENT: Do you wonder where they got the name “Wright?” That’s his name. And his wife is the co-founder of the company. And this is — it’s really great to be with you.

And, Congressman [DM -- Roscoe Bartlett, another good guy with a big family], thank you for being here. I’m proud to be with your workers. You’ve got some fine workers.

MR. WRIGHT: We’ve got a great team here, don’t we?

THE PRESIDENT: Yes, you do. And if they get a little more money in their pocket as a result of the growth package, it will help make sure this economy continues to grow.

Anyway, thank you all very much.

END 1:59 P.M. EST

Bill, for all of his accomplishments, is a humble guy. He and Margie went through many lean times during the early years, and they bore with it well. The business began as a lawn mowing business, and then broadened out to software for managing lawn mowing businesses, together with grass catching attachments. That started the manufacturing. Bill developed a wide number of innovations, from zero-turning radius mowers, to sulkies, and more.

In my opinion, President Bush could not have picked a better place to visit. It is an example of American manufacturing at its best, and Bill Wright is a great example of American entrepreneurship. I am proud to be a part owner of the firm, and to have been able to help Bill at times on financial issues.

Now, for further coverage of the visit of President Bush:

Bush Wants Fast Tax Aid to Boost Economy

Frederick Residents Voice Concerns About Economy

Frederick Officials Hope They Are Recession-Proof (Video)

As far as I am concerned, the stimulus package is hooey; what stimulus occurs now will be funded by debt and a cheaper dollar later. There is nothing wonderful there. There is something wonderful about the Wrights though, and I am happy to be a small part of that.

Years ago, I created a spreadsheet to help young children learn their lowercase letters once they had learned their uppercase letters.  I created it for my two oldest, and now my youngest is using it.  My wife says that it has made that teaching task simple.

The spreadsheet generates seven unique random letters in uppercase on the left, and scrambles them in lowercase on the right.  Print a sheet, and the child draws lines to match the letters.  Hit the recalc [F9] button, and you have a new sheet.  With a little bit of coaching, the child gets the concept in a week or so.  The sheets are easy to use, and the limited matching doesn’t overload their ability to remember, particularly because a few letters on each page will be easy.

If it works well for you, let me know.

A dear friend of mine introduced me to this game, which I play with my kids every now and then. All you need is a set of Scrabble tiles. You place all of the tiles face down, and swirl them around. The dealer (ordinarily the best player, so it distracts him), flips one tile at a time. When enough tiles are flipped to create a word of at least three letters, the first player to name the word claims it, and takes it for himself.

Play continues, with more tiles flipped, but there are two choices now for the tiles that are face up. You can use the tiles to create new words, or combine them with existing words of yours, or words of your opponents. Suppose your opponent has the word “ham,” and there are an “s” and an “e” on the board. The player that calls out “shame” claims the tiles from “ham” and creates the word “shame” in his field. Stealing the words of opponents is often more effective than creating new words, though there is a balance to be maintained. It is also wise to boost the letters in your own words, which makes them harder to steal. Additional note: the letter orders can be rearranged. If “gun” and “one” have been claimed by players, and there is a “d” on the board, a player could take the two words for the word “dungeon.”

Play ends two minutes after the last tile is flipped. Additional rules:

• No proper nouns
• No foreign words, unless they are in common use in English discourse.
• Appeal to an unabridged dictionary is permitted for words in dispute.
• Blanks are wild cards, but the first time a blank is played, it stays that letter for the remainder of the game.

Scoring: each word gets points equal to the number of letters minus two.

Benefits of the game: children learn to think along multiple lines of strategy and structure words in ways that they don’t commonly consider. It is a real mind-stretcher. An aside: this is a game where speed of thought helps but is not determinative. Having a large vocabulary helps, which benefits the grownups.

An off-topic post for the weekend.  My children have benefited from a card-matching game called “SET.”  The main SET website is here, and the rules are here.

What I find interesting about the game is that the optimal strategy forces you to look for linear patterns early and then shift to nonlinear patterns later in each round.  It’s a very mathematical game, and the cards are functionally equivalent to unique 4-digit base three numbers.  (Don’t tell the kids that, or it won’t be fun any more.)

Full disclosure: I just like the game, and so do my kids.  I have received nothing for mentioning here, but it could be of value to parents and grandparents.

2007 is the transition year at the Merkel household.  Our two oldest children go off to college in the fall, and our youngest starts home schooling at the same time.  As such, this is the last time that we can rely on that we will be able to take all of our children on a trip.  In the late summer of 2006, we went to visit my wife’s parents in San Diego; next week, we visit my parents in Milwaukee.

Now, I have no idea what internet access I might have while there.  If I have good access, I will post in the late evenings.  If not, well, you’ll hear from me next on the 29th.  With that, I sign off.  I have a lot of other things to write about, but little time to do so.  Traveling with eight children is quite a feat, and it will take a lot of my energy to accomplish that.

This is a little out of the ordinary for those who frequent my blog, but here is something that my wonderful wife and I use to help our children learn math. If you did not already know, we homeschool our eight children. We are not big on math drill; we think word problems teach reasoning far better. But to do word problems effectively, the ability to have instant recall of the 100 math facts in addition, subtraction, multiplication and division is crucial.

We have our children work at it until they can get it perfect in five minutes in third grade, and three minutes in fifth grade.

To use the spreadsheet (right-click and use “save link as” to download), select the tab that you want to work on. “little add, and “little subt” are the 64 math facts with no number greater than 10. To use the sheet, hit the F9 key to recalculate the sheet, which places all the problems in a new random order. (You’ll never get 2 sheets the same.) Then hit the print icon. F9, print, F9, print, etc… pretty soon you’ll have a lot of unique sheets for drill purposes.

Use and distribute as you see fit. I just want to see children who are good with their math. And, if it works for you, let me know.