One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements: The Get-Out-of-the-Way Problem.

The Distributive Law, or the Get-Out-of-the-Way Problem

"So do you understand why the distributive law of multiplication over addition is important?"

"Yes. It is important because it provides math teachers with yet another way to torment Thirteen-Year-Olds."

"That's the wrong answer."

"It's a free country--or it's supposed to be."

"The right answer is that the distributive law allows you to rewrite equations--to replace one equation with another that means the same thing--and that the main point of algebra is to keep rewriting and rewriting equations, expanding, factoring, and cancelling, until you get to a form where the answer is obvious."

"Sure."

"No. I mean it. Take... will you believe me if I say that if you fire a cannonball straight up with an initial upward velocity of 640 feet per second, then its height above the earth measured in feet t seconds after liftoff is given by the equation:

h = 640t - 16t² ?"

"If you say so."

"Well, now, let's factor this equation. Do the two terms on the right hand side share anything?"

"They share a sixteen."

"So we can rewrite the equation as:

h = (16)(40t) - (16)(t²),

right?"

"I guess so."

"And do they share anything else? do the 40t and the (16)(t²) share anything else?"

"I guess each is a multiple of t..."

"So we can pull out another factor of t, and rewrite the equation as:

h = (16t)(40) - (16t)(t) = (16t)(40 - t) ?"

"Yep. But you need to get to the point."

"Well, just look at:

h = (16t)(40 - t)

and ask yourself, 'When will the left hand side be zero? It will be zero when the right hand side is zero, and that is...?"

"Well, the right hand side is two things multiplied together, and so it will be zero only when one or the other is zero..."

"Exactly. And the first thing on the right hand side is zero when...?"

"When t = 0..."

"And that's the moment that the cannonball is fired upward. So that makes sense. And the second thing on the right hand side is zero when...?"

"When t = 40"

"So that in the form:

h = 640t - 16t²

the equation doesn't tell you much. But in the form:

h = (16t)(40 - t)

it tells you that, after firing the cannon straight up, you have forty seconds to get out of the way--or else!"

"Hmmm..."

"But the two equations are the same--the distributive law guarantees it."

"So you're saying that people who don't know the distributive law regularly get klonked on the head by their own cannonballs?"

"That's not the point..."

---

Interesting Calculations Index Page

[Semi-Daily Journal]

12:57:50 PM    

Oh Crud. The O'Reilly Bioinformatics Conference has been canceled

12:48:46 PM    

Analyst: Microsoft 'hated because it's misunderstood'. "Prejudices and misconceptions about Microsoft make it hard to evalute the company's merits. The biggest myths about Microsoft are that its desktop products are overpriced, it doesn't respect its customers, and reliability and security are poor. And some think the company is downright evil," writes Rob Enderle over on InternetWeek.com.

"In nearly two decades of studying Microsoft, I've been able to dig through the hype that the company generates, as well as the... [MacDailyNews]

Rob Enderle is on the Rolodex of every single tech writer in America. Remember the name. In almost every tech article written for the mainstream media, when they need a 'tech' source, it is good old Rob. I noticed this a few years ago and I am amazed that he still gets quoted. He has a tendency to provide that juicy quote that reporter love. But he is full of hoey here. Where is the discussion that they are a monopoly that was found to illgeally abuse that monopoly position? How about its long history of stealing other company's software and then either waiting them out or paying them off? It is hated because if MS decides it want to do what you are doing, you are dead and might as well take whatever small amount they want. Getting noticed by MS is a sure way for your company to disappear. Usually by immoral and unethical means. Sometimes not even legal.But with $50 Billion in the bank, they can afford to wait anyone out, even the government, as evidenced by their anti-trust battle.  12:42:13 PM    

9th Circuit Commutes 100 Death Sentences. Following a 2002 Supreme Court decision that ruled only juries, not judges, can decide to sentence a defendant to death,... [TalkLeft: The Politics of Crime]

If it is wrong for judges to decide a death sentence from now on, it should be retroactively applied. It seems to me that to leave the ultimate penalty up to an elected official is not prudent. If you can convince 12 members of a jury that someone deserves to die, then you can make the point that the community wants the death penalty. But wanting judges because juries are too hard to convince is just wrong. And I certainly do not think that judges represent the community. Else, why have juries of our peers?  12:15:58 PM    

Are Automated Essay Graders Sucking The Life Out Of Writing?. A slightly disturbing story about how English essays are increasingly being graded by machines. The folks behind the program claim that the computer's results have a very high percentage correlation to scores given by human graders. Of course, the machine is scoring the essays more on technical points concerning sentence structure and the use of certain key words. The same company that makes the software for grading exams is also offering a "training" application that students can use to get feedback on their own papers. As someone in the article points out, while it's nice to have a system to help teach students proper writing techniques, there is also the fear that such systems will beat out any creativity from students by focusing exclusively on the technical (rather than the creative) aspects of writing. [Techdirt]

This is what we are coming to because of all the standardized tests now required. Computers will grade the essays. Of course, the computer can never be wrong. I would be really interested to see how this works out. Will we just raise people to wite articles that computers like to read?

But, the writing I did in Literature and Composition in High School was pretty formulaic. It had to be for the teachers to grade them all. I still learned a lot about how to contruct an argument. The worry is the follow-thru. Are you just given the grade without any human criticism? That would be bad. But as a tool, this could be quite helpful. As the article says, it all comes down to the human critics that validate the computer.  12:01:27 PM    

Baby Boomers Prefer The Printed Page. I doubt this is a surprise to anyone, but the older you are, the more likely it is you prefer to read your news on paper. It's simply a matter of habit. People who grew up reading newspapers every day are comfortable reading newspapers. Those who are used to only accessing news online don't see why they should bother picking up a paper. Of course, plenty of people seem to straddle both worlds, reading lots of news online, but still flipping through their local paper. While I no longer read any paper newspapers, I do agree that it is a different process, and you probably do end up catching more stories serendipitously than when surfing news online. However, online, you have a much wider field of articles to choose from, and can find stories you would never see in your local newspaper. [Techdirt]

I agree with the need for serendipity but I find myself needing to read the paper less and less each day. First, I hate reading a paper in the morning. But if I wait until evening, the news is all late. I hated it when the Seattle Times went to the morning. I am reverting to the original reason for the paper - the comics.  11:53:21 AM    

Music giant slashes album prices. The world's largest record company, Universal, will cut the price of albums in the US by up to 30%. [BBC News | Technology | World Edition]

Great news. Maybe these guys are finally getting some of the message. $17 is way too much, especially since they are cometing with DVDs for the entertatianment dollar.  11:46:35 AM    

Freedom to Rear Your Children. When is your child not yours? " The case of 12-year-old Parker Jensen from Utah raises this issue once again. Three months ago, Parker was treated for Ewing's sarcoma, a soft tissue cancer that was excised from beneath his tongue.... [Dane Carlson's Weblog]

More news of government intimidation. The parents of Parker Jensen are charged with kidnapping because they took their own son out of the state. Because they were not comfortable with having their 12-year-old son take chemotherapy. This ABCNews.com article says that chemo has serious side effects. Yeah, like sterility and stunted growth. This is not therapy that can guarantee survival. There is no evidence that the tumor has spread. Chemo is just to make sure they got everything.

The state got involved and is trying to take custody of a child so that they can make him take a therapy he does not want. Therapy that will alter his life forever. These are the sorts of moral decisions that our government is totally inept at making. But that does not prevent it from signing out a warrant for the arrest of the parents for kidnapping. What, are they going to put the parents away in prison so they can give chemo to the child? That sounds like a good plan. No, but they sure want to intimidate these parents, and any others that might be listening.

The location of his tumor is very unusual for Ewing' sarcoma. So a big question: Is the diagnosis correct? A valid second opinion would have been great but the parent's insurance company would not pay for experts at Harvard to examine the tissue. Great so we have all this going on, with parent's fleeing the state and the state trying to take away custody of a child because an insurance company would not pay for a second opinion. Our health insurance system is sure screwed up.  11:37:04 AM    

I Would Never, Ever, Ever, Ever Advocate the Violent Overthrow of Any Government

Young Webmaster Heads to Prison for Political Website. Young Webmaster Heads to Prison for Political Website: "The 20 year-old Webmaster of a California-based site called raisethefist.com is to begin a yearlong prison term today. After he is released, Austin will be on probation for three years. As part... [Dane Carlson's Weblog]

As I read this article I went back and forth. What this young man was publishing at his website is aborent to me. Simply calling it political is misleading. But I finally arrived at the fact that this young man was charged with posting material he did not write. The person who wrote it is free. This is going after the publisher, not the writer and is wrong. The author of the piece is still free. Sounds like selective prosecution purely for intimidation purposes.to me.

While I know that the young man is going to be self-serving, just as the FBI will be, what I find troubling here is the continuing intimidation. Taking a crime, for which the normal sentencing range is 6-12 months and being able to add 20 years to it is outrageous. Intent leaves a huge gray area that authorities will use to keep people cowed. This is the old 'inciting to riot' mentality.

This is why so many people are simply afraid to ever come to the notice of authorities. It is too easy today to find something and make it sound really horrendous. A script kiddie fiddles around, makes some stupid comments and he becomes an arch-criminal who almost brought the civilized world to its knees.

While publishing this sort of thing may not be the smartest thing, convicting him because someone else he does not know MIGHT use it has a chilling effect. What would happen if someone took a chance remark I wrote and did something. Would I be responsible and dragged into court, threatened with 25 years of prison unles I copped a plea?

A horrific anti-abortion site was allowed to remain up following the appeal of a civil lawsuit by Planned Parenthood and the ACLU. One judge said 'political speech may not be punished just because it makes it more likely that someone will be harmed at some unknown time in the future by an unrelated third party.' This is a group that put up wanted posters of doctors, with a slash through those that had been killed. I guess it is okay to intimidate doctors but you had better not try to do anything that could intimidate the government. But political speech IS about the government. These sorts of laws open up a real slipery slope. It certainly does not make me feel safer.  11:16:41 AM    

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