Spring 2011

Math 101 - Spring 2011
Reasoning with Data

Section 1

Prof: Robert Winters

Calendar of topics and
homework assignments

Section 1:
Mon, Thurs - 1:30pm
Wed - 2:15pm
in room SCI 396

Tips and Tricks

TEXT
Introduction To The
Practice Of Statistics

Moore/McCabe, 5th Ed.

click for price comparison

This semester, we'll use the text Introduction to the Practice of Statistics, Fifth Edition, by Moore & McCabe, published by Freeman, ISBN 0-7167-6400-8. A sixth edition is also available, but HW exercises will be taken from the fifth edition.

You must have access to a version of Excel either on your own computer or on the college computers. Later versions inexplicably lack the data analysis tools (though this may be remedied at some point), so the older the version the better (preferably pre-2008).

We may at times use the ActivStats software, by Paul Velleman, published by Pearson/Addison-Wesley. This software is used in concert with Microsoft Excel, but most of the necessary tools are available within Excel. The text should be available at the bookstore and elsewhere.

Announcements:

Thanks for attending the course!


IN THE NEWS:

Vote-Counting Error In Wisconsin Points to Incompetence, Not Conspiracy (Nate Silver, NY Times, April 8, 2011)


They got a (sugar) pill for that
Can chronic medical conditions, including depression, be treated with placebos? Recent studies suggest they can.
By Deborah Kotz, Boston Globe, March 7, 2011


We meet every Monday, Wednesday, and Thursday in SCI 396 (1:30pm on Monday and Thursday, 2:15pm on Wednesday).

Course Syllabus (HTML)      Printable Course Syllabus (PDF)

Check the Calendar for a day-by-day schedule of classes and homework assignments as well as data to be used in the homework. Homework solutions will be posted after their due date.


Enlightening Video: The Joy of Stats (YouTube video)


The Tips and Tricks page has explanations of how to do some statistical calculations and create graphs (such as making a histogram in Excel 2008 for the Mac). If you have discovered ways to get your versions of Excel to work for particular tasks, submit your description and we'll post them. There is also provided a link for a way to create normal quantile plots in Excel. A new section on making histograms has been added to the Tips and Tricks page. Additions and revisions are welcome on this and other topics in the course.


STATISTICS IN THE NEWS

Aspirin may combat cancer, study suggests
By Stephen Smith, Boston Globe | February 17, 2010

Could the humble aspirin prevent breast cancer survivors from suffering a second bout of the disease or even dying from it? It is a strategy worth further consideration, suggests a provocative Boston study published yesterday.

Scientists from several Harvard-affiliated institutions reported that women who took aspirin after completing breast cancer treatment were half as likely to die from the disease as women who did not regularly use aspirin. Taking the centuries-old remedy also appeared to significantly reduce prospects that breast cancer would return, according to the study published online by the Journal of Clinical Oncology.

The study's authors, along with cancer specialists not involved in the research, cautioned that while the findings are promising, they are not definitive and do not mean breast cancer survivors should start taking aspirin. The study's conclusions were derived from comprehensive health surveys that registered nurses completed every two years, a form of research that can yield important hints about what works and what doesn't in medicine but can't provide absolute answers.

Still, the research adds to a growing mound of evidence that certain low-cost, tried-and-true medicines - aspirin and diuretics, for example - need to be regarded with more respect by doctors and patients who gravitate toward the newest drugs in the medicine cabinet. Studies have suggested that long-term use of aspirin may reduce the likelihood of developing colon cancer.

"There's an increasing interest in aspirin as a cancer preventative," said Dr. Larry Norton, director of breast cancer services at Memorial Sloan-Kettering Cancer Center in New York. "The bottom line is this study is a tenable hypothesis. But I wouldn't recommend based on it that people go out and buy aspirin."

Researchers, led by Dr. Michelle Holmes of Brigham and Women's Hospital and Harvard, used survey results from the Nurses' Health Study to examine the potential impact of aspirin on breast cancer recurrence and survival in 4,164 nurses diagnosed with the disease between 1976 and 2002. A total of 341 women in the study had died from breast cancer by 2006.

After taking into account other factors that might explain why some women lived longer, the researchers, whose work was underwritten by the National Institutes of Health, concluded that aspirin may have figured prominently in reducing breast cancer deaths, as well as the risk of cancer spread, among nurses who took the medication. And there was a suggestion that women who took the drug on more days of the week had the best chance of beating breast cancer.

The study did not ask women what dose of aspirin they were taking, nor why they were taking it. However, previous findings from the Nurses' Health Study have shown that the top reason women in the study took aspirin was to prevent heart disease.

Scientists can't say for sure why aspirin might forestall cancer's return, but research in laboratories and in animals has found that the ancient drug acts like water dousing the embers of a simmering fire, quieting inflammatory processes that contribute to cancer's spread. Other studies in humans have reached mixed conclusions about aspirin and whether it has cancer-fighting propensities.

"We don't know exactly the mechanism, but there are a lot of hints out there that aspirin has anticancer effects," Holmes said. "It is important to reexamine some of these old drugs and see if there can be additive effects on top of conventional treatments."

While the researchers' report concentrated on aspirin, their findings also suggested other nonsteroidal anti-inflammatory drugs - among the top sellers are Advil and Motrin - may reduce breast cancer recurrence, although that effect was evident only in women who took those medicines 6 to 7 days a week. No link could be established between acetaminophen - commonly sold as Tylenol - and reduced breast cancer mortality.

Usually, patients in the midst of chemotherapy, radiation, and other cancer treatments are discouraged from taking aspirin because of concerns about drugs interacting in potentially dangerous ways. And even when women start taking aspirin after treatment, there are still risks, including stomach bleeding and other gastrointestinal complications, specialists stressed.

"Physicians may opt to use aspirin in a breast cancer survivor but that would be based on their own judgment of risks and benefits," said Dr. Powel Brown, chairman of clinical cancer prevention at M.D. Anderson Cancer Center in Houston.

Gold-standard trials that involve randomly assigning some patients to take a drug while others get a dummy pill are designed to assess risks and benefits rigorously, and yesterday, several specialists called for such a review of aspirin and breast cancer.

"Studies like today's generally shouldn't change clinical practice, they shouldn't lead to a change in women's health decisions," said Dr. Eric Winer, chief of women's cancer at Dana-Farber Cancer Institute. "What they do provide is good evidence to move forward with a study asking whether this finding is real and how it should be used in clinical care."

The American Cancer Society took a measured stance toward the new study, with Laura Hilderley, chief medical officer of the New England division, suggesting women discuss the findings with their doctor before taking aspirin.

Hilderley also offered a sober assessment regarding further aspirin and breast cancer research. Pharmaceutical companies have little interest in research on a drug with no patent protection and low profit margins, meaning government agencies are left to pay for such studies.

"Sometimes," she said, "something like this, because it's not a brand-new discovery in chemotherapy, it takes a while before it gets picked up and funded in a clinical trial."

Stephen Smith can be reached at stsmith@globe.com.


February 11, 2010, 1:58 pm
New Poll Shows Support for Repeal of ‘Don’t Ask, Don’t Tell’
By DALIA SUSSMAN

As the Obama administration proposes repealing the policy known as “Don’t Ask, Don’t Tell,” a new New York Times/CBS News poll finds that a majority of the public support allowing openly gay men and women to serve in the military.

There’s less support, however, for allowing homosexuals to serve openly.

Confused?

The results highlight the importance of wording on the issue. In a test, half of the poll’s respondents were asked their opinion on permitting “gay men and lesbians” to serve, and the other half were asked about permitting “homosexuals” to serve.

The wording of the question proved to make a difference. Seven in 10 respondents said they favor allowing “gay men and lesbians” to serve in the military, including nearly 6 in 10 who said they should be allowed to serve openly. But support was somewhat lower among those who were asked about allowing “homosexuals” to serve, with 59 percent in favor, including 44 percent who support allowing them to serve openly.

Democrats in the poll seemed particularly swayed by the wording. Seventy-nine percent of Democrats said they support permitting gay men and lesbians to serve openly. Fewer Democrats however, just 43 percent, said they were in favor of allowing homosexuals to serve openly. Republicans and independents varied less between the two terms.

As the debate of repealing “Don’t Ask, Don’t Tell” moves forward, these findings illustrate the influence that simple language can have.

The national telephone poll was conducted with 1,084 adults Feb. 5-10 and has a margin of sampling error of plus or minus four percentage points for each half sample. Complete poll results and article will be available this evening at www.nytimes.com.

Also this: http://www.cbsnews.com/blogs/2010/02/11/politics/politicalhotsheet/entry6198284.shtml


February 2, 2010
Vaccine-Autism Study Is Retracted
By THE ASSOCIATED PRESS

LONDON (AP) -- A major British medical journal on Tuesday retracted a flawed study linking the measles, mumps and rubella vaccine to autism and bowel disease.

The retraction by The Lancet comes a day after a competing medical journal, BMJ, issued an embargoed commentary calling for The Lancet to formally retract the study. The commentary was to have been published on Wednesday.

The BMJ commentary said once the study by British surgeon and medical researcher Andrew Wakefield and his colleagues appeared in 1998 in The Lancet, ''the arguments were considered by many to be proven and the ghastly social drama of the demon vaccine took on a life of its own.''

Since the controversial paper was published, British parents abandoned the vaccine in droves, leading to a resurgence of measles. Subsequent studies have found no proof that the vaccine is connected to autism, though some parents are still wary of the shot.

In Britain, vaccination rates for measles have never recovered and there are outbreaks of the disease every year.

Ten of Wakefield's 13 co-authors renounced the study's conclusions several years ago and The Lancet has previously said it should never have published the research.

''We fully retract this paper from the published record,'' Lancet editors said in a statement Tuesday.

Last week, Britain's General Medical Council ruled that Wakefield had shown a ''callous disregard'' for the children used in his study and acted unethically. Wakefield and the two colleagues who have not renounced the study face being stripped of their right to practice medicine in Britain.

For the study, Wakefield took blood samples from children at his son's birthday party, paying them 5 pounds each ($8) for their contributions and later joking about the incident.


Broader Measure of U.S. Unemployment Stands at 17.5%
By David Leonhardt - New York Times - November 7, 2009

For all the pain caused by the Great Recession, the job market still was not in as bad shape as it had been during the depths of the early 1980s recession — until now.

With the release of the jobs report on Friday, the broadest measure of unemployment and underemployment tracked by the Labor Department has reached its highest level in decades. If statistics went back so far, the measure would almost certainly be at its highest level since the Great Depression.

In all, more than one out of every six workers — 17.5 percent — were unemployed or underemployed in October. The previous recorded high was 17.1 percent, in December 1982.

This includes the officially unemployed, who have looked for work in the last four weeks. It also includes discouraged workers, who have looked in the past year, as well as millions of part-time workers who want to be working full time.

The official jobless rate — 10.2 percent in October, up from 9.8 percent in September — remains lower than the early 1980s peak of 10.8 percent.

The rate is highest today, sometimes 20 percent, in states that had big housing bubbles, like California and Arizona, or that have large manufacturing sectors, like Michigan, Ohio, Oregon, Rhode Island and South Carolina.

The new benchmark is a sign of just how much damage financial crises tend to inflict. A recent book by Carmen M. Reinhart and Kenneth S. Rogoff, two economists, found that over the last century the typical crisis had caused the jobless rate in the country where it occurred to rise for almost five years. By that standard, the jobless rate here would continue rising for two more years, through the end of 2011.

Most economists predict that the rate will in fact begin to fall next year, largely because of the federal government’s aggressive response — fiscal stimulus, interest-rate cuts and a variety of creative steps by the Federal Reserve and Treasury Department. Friday’s report showed that monthly job losses continued to slow recently, though the improvement has been gradual.

At the White House Friday, President Obama signed a bill to extend unemployment benefits and a tax credit for home buyers, and said that he was looking at ways to enact more stimulus. On Wednesday, the Fed announced that it expected to leave its benchmark interest at zero for “an extended period.”

Nearly 16 million people are now unemployed and more than seven million jobs have been lost since late 2007.

Officially, the Labor Department’s broad measure of unemployment goes back only to 1994. But early this year, with the help of economists at the department, The New York Times created a version that estimates it going back to 1970. If such a measure were available for the Depression, it probably would have exceeded 30 percent.

Compared with the early 1980s, a smaller share of workers today are officially unemployed and a smaller share are considered discouraged workers.

But there are many more people who would like to be working full time and have been able to find only part-time work, according to the government’s monthly survey of workers. The rapid increase in their ranks and in the officially unemployed has caused the rate to rise much faster in this recession than in the early 1980s. Two years ago, it was only 8.2 percent.

One of the more striking aspects of the Great Recession is that most of its impact has fallen on a relatively narrow group of workers. This is evident primarily in two ways.

First, the number of people who have experienced any unemployment is surprisingly low, given the severity of the recession. The pace of layoffs has increased, but the peak layoff rate this year was the same as it was during the 2001 recession, which was a fairly mild downturn. The main reason that the unemployment rate has soared is the hiring rate has plummeted.

So fewer workers than might be expected have lost their jobs. But those without work are paying a steep price, because finding a new job is extremely difficult.

Second, wages have continued to rise for most people who still have jobs. The average hourly wage for rank-and-file workers, who make up about four-fifths of the work force, actually accelerated in October, according to the new report.

Even though some companies have cut the pay of workers, the average hourly wage has still risen 1.5 to 2.5 percent over the last year, depending on which government survey is examined. Average weekly pay has risen less — zero to 1 percent — because hours have been cut. But average prices have fallen. Altogether, the typical worker has received a 1 to 2 percent inflation-adjusted raise over the last year.

In the other two severe recessions in recent decades, workers with jobs fared considerably worse. At the same point in the mid-1970s downturn, real weekly pay had fallen 7 percent; in the early 1980s recession, it had fallen 4 percent.

It is a strange combination: workers who still have a job are doing better than in other deep recessions, but the unemployment and underemployment have risen to their highest level since the Depression.


People are still evolving, heart study numbers say
By Carolyn Y. Johnson, Boston Globe Staff | October 26, 2009

Charles Darwin famously studied evolution in the Galapagos Islands. Now a team of scientists has chosen a decidedly less exotic locale to study the subject - Framingham.

Residents of the Boston suburb have long participated in a landmark study of their cardiovascular health, which has shown that smoking and high cholesterol increase risk of heart disease. Now data compiled for the heart study are providing evidence of human evolution in action - and have led researchers from Yale University, Boston University School of Medicine, and the University of Pennsylvania to predict that the community's next generation of women will be slightly chubbier and shorter and have lower cholesterol.

Evolution occurs because organisms with advantageous traits are more likely to survive and pass on those traits to their offspring - a process called natural selection. It is widely believed that modern medicine and technology have brought human evolution to a screeching halt, since most people - and not only the "fittest" - can now survive and pass on their genes. But the authors of the new research say their work shows natural selection is still occurring.

"As an evolutionary biologist, I've been aware for some time that people in the medical community have the misapprehension that evolution is not occurring in humans," said Stephen C. Stearns, a professor of ecology and evolutionary biology at Yale and the senior author of the paper published last week in the journal Proceedings of the National Academy of Sciences.

The researchers found that a number of traits were associated with having more children - being shorter and chubbier, having a first child at a younger age, experiencing menopause later, and having lower blood pressure or cholesterol - meaning that those traits would likely become more prevalent in the next generation.

"The way we went at it is to use methods in evolutionary genetics that have been applied to plant and animal populations for about 30 years and simply then treat humans as though they were just another population of organisms," Stearns said.

The researchers took data from the Framingham Heart Study, the long-running research study that started in 1948 and has now followed over 14,000 subjects. They looked at two generations of women and chose a handful of traits to analyze: cholesterol level, blood pressure, height, weight, age at first childbearing, and age at menopause.

They controlled for the baby boom and bust and factors such as smoking and used a statistical method to account for the fact that some of the traits they were measuring varied over a person's lifetime, such as cholesterol and weight. Then, they used the Framingham study's extensive family history data to calculate the extent to which any particular trait is inherited. They also looked at those traits' influence on the number of children women had.

They found that a number of partly inherited traits were correlated with women having more children, and that enabled the scientists to make predictions about the future. The next generation of women would have slightly lower cholesterol, be nearly half a pound heavier, and less than a 10th of an inch shorter. They also predicted that a woman's age at first childbearing would decrease by half a month and that age at menopause would be pushed back a month.

The bottom line, they said, was that evolution is occurring in people.

David Haig, a biology professor at Harvard University, said that he frequently hears people say that evolution has ceased in people.

"I think that's clearly wrong - I hear that all the time, and it's clearly the case that we're not having the same number of children, so there's some ongoing selection," Haig said.

But he pointed out it was possible that some external environmental factor was affecting both a trait and a woman's propensity to have more children.

"Weight has changed dramatically over that time span, so there are clearly environmental effects on weight. But the question is what is responsible for the correlation between weight and the number of children [a woman gives birth to], and it could be that being slightly heavier is increasing your fertility or it could be that some unknown factor is both affecting weight and affecting fertility," Haig said.

The study corroborates a 2001 study in the journal Evolution, in which an analysis of 2,710 pairs of twins found that women who bear children at a younger age have more children, and that the trait is passed from one generation to the next. That suggested natural selection was having an effect.

Even though researchers found evidence that humans are evolving, it is occurring at a slow pace - slower than evolution in Galapagos finches and Trinidadian guppies, and more on par with evolution measured in New Zealand chinook salmon and Hawaiian mosquitofish.

What the researchers cannot definitively say is what is causing certain traits to be advantageous, or what set of genetic variations is causing any of the traits - which are also influenced by environmental factors. The traits they studied seem to account for only about 5 percent of the variation in reproductive success.

"The other 95 percent gives all the people who worry about their free will and decisions plenty of room to play in," Stearns said.

Carolyn Y. Johnson can be reached at cjohnson@globe.com.


Math gets political as 2010 Census nears Efforts launched to count shifting minority groups
By Jesse Washington, Associated Press | March 18, 2009

What seems like a simple question - How many Hispanics are living in the United States? - has become surprisingly complex as the 2010 Census approaches.

Hispanics and other minorities have historically been undercounted in the once-a-decade survey. Advocacy groups are now launching their traditional efforts to ensure an accurate count, but a variety of factors have created new problems for the painting of America's official portrait.

Activists and government officials say fears about immigration enforcement and government snooping are making people more reluctant to share their information.

The economic meltdown and Bush administration budget cuts have slowed funding for the census. Millions of laid-off renters and foreclosed homeowners are on the move.

There are more immigrants here, speaking more languages, than ever before. Some of those immigrants may not know what a census is, or may come from countries where such information is used against rather than for the people.

"This country is just much more complex now, on many different levels," said Terry Ao, director of census and voting programs for the Asian American Justice Center.

The 2000 Census counted 35,305,818 Hispanics in the United States. Hispanic groups estimate that several million more were missed. In 2007, the most recent year available, the Hispanic population had grown to an estimated 44,852,816.

The Constitution mandates that every 10 years, each person living in the country - regardless of citizenship or immigration status - must be counted.

The census results are used to draw congressional districts and allocate hundreds of billions of dollars in federal funding for schools, roads and other services. The data also trickle down to state and local governments for determining everything from the size of hospitals to the placement of bus stops.

On a more emotional level, the census is the measure of our nation, a literal definition of what we are. That can touch nerves left raw by the simmering immigration debate.

Anti-immigration groups don't object to an accurate count, which may provide fuel for their arguments. But they are opposed to the past practice of suspending immigration raids while the census is being conducted. And they have major objections to counting non-citizens when drawing congressional districts.

Steven Camarota, director of research at the Center for Immigration Studies, called the practice "an assault on the 'one man, one vote' idea."

"It transfers political power to the citizens who live in districts with high numbers of illegal aliens," he said. "If you live in Southern California, your vote counts a great deal more than if you live in Michigan or somewhere with lower immigration."

Meanwhile, interest groups point out that everyone suffers if undercounting leads to less funding for schools, roads, or hospitals.


Does Bran Make the Man? What Statistics Really Tell Us (Wall Street Journal, January 27, 2009)

Group censures Iraq death toll researcher (Boston Globe, February 5, 2009)

WASHINGTON - A prominent group of polling researchers has accused the lead author of a 2006 study suggesting massive civilian deaths in Iraq of violating the polling profession's codes and ethics.

The Executive Council of the American Association for Public Opinion Research said Dr. Gilbert Burnham, a Johns Hopkins University professor, had repeatedly refused to cooperate with an eight-month investigation into his research on the Iraqi death toll that made headlines in October 2006 when it was published by The Lancet, a British medical journal.

The widely publicized study headed by Burnham contended that nearly 655,000 Iraqis had died because of the US-led invasion and war in Iraq.

"When asked to provide several basic facts about this research, Burnham refused," the council said in a statement. It noted that the group's Code of Professional Ethics and Practices calls for researchers to disclose their methodology when survey findings are made public so that they can be independently evaluated and verified.

The group said his refusal to fully cooperate with the probe violates the standards of science.

ASSOCIATED PRESS


Study: Two drugs may worsen colon cancer, not help it (Boston Globe, February 5, 2009)
By Stephanie Nano, Associated Press

NEW YORK - Doctors thought that combining two newer drugs that more precisely attack cancer would help people with advanced colon cancer. Instead, it made the cancer worse and made the patients more miserable, a study found.

The surprising findings underscore the importance of doing rigorous studies before rushing to mix these pricey, new-generation drugs, the Dutch researchers and other specialists said. The doctors tried combining Erbitux and Avastin because lab tests and an earlier small study had shown promising results.

"This will stand out as a warning," said Dr. Cornelis Punt, the study's leader. "You have to do the randomized studies to see what really happens."

For the study, Eli Lilly & Co.'s Erbitux was added to standard treatment, which includes Genentech Inc.'s Avastin.

Since both are "targeted" drugs and attack tumors in different ways, the thinking was that the combo would do a better job of keeping the cancer from growing.

But the results show "more is not always better," said Dr. Robert Mayer, of Dana-Farber Cancer Institute. He wrote an editorial published with the study in today's New England Journal of Medicine.

What makes the results even more compelling, Mayer said, is that another similar study reached the same conclusion. That study, released in December, tested another targeted drug that works the same way as Erbitux.

"This is the first time we've seen harm by combining targeted therapies and it tells us we need to be cautious," said Dr. Jordan Berlin, a gastrointestinal cancer specialist at Vanderbilt-Ingram Cancer Center in Nashville.

Berlin, who had no role in the research, stressed that the drugs do help patients, just not when given together.

Colorectal cancer is the nation's second leading cancer killer. The disease was expected to kill almost 50,000 Americans last year although death rates are dropping because of screening and better treatment.

The research was done at hospitals throughout the Netherlands and led by Punt at Radboud University Nijmegen Medical Center. The 755 study patients had colon cancer that had spread. They all received Avastin, also known as bevacizumab, and two chemotherapy drugs. Half of them also got Erbitux, also called cetuximab. They were followed for nearly two years.

The group that got Erbitux saw their cancer get worse sooner, the researchers found. On average, their cancer progressed after 9.4 months compared with 10.7 months for those who didn't get Erbitux. The Erbitux group also had lower quality-of-life scores.

The overall survival in both groups was about the same.

Punt said they are now trying to figure out why the combo didn't work; it could be an interaction between these two specific drugs, Erbitux and Avastin.

After the study began in 2006, it was shown that Erbitux didn't help colon cancer patients who had a specific gene mutation.

The Dutch researchers said their study confirmed that - the worst results were in those with the mutation who got Erbitux.




Frozen waterfall in the Middlesex Fells - Feb 1, 2004


Here's something:  http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html


New Math
Tom Lehrer

Some of you who have small children may have perhaps been put in the embarrassing position of being unable to do your child's arithmetic homework because of the current revolution in mathematics teaching known as the New Math. So as a public service here tonight, I thought I would offer a brief lesson in the New Math. Tonight, we're gonna cover subtraction.

This is the first room I've worked for a while that didn't have a blackboard, so we will have to make do with more primitive visual aids, as they say in the ed biz. Consider the following subtraction problem, which I will put up here: 342 minus 173. Now, remember how we used to do that: But in the new approach, as you know, the important thing is to understand what you're doing, rather than to get the right answer. Here's how they do it now:

You can't take three from two,
Two is less than three,
So you look at the four in the tens place.
Now that's really four tens
So you make it three tens,
Regroup, and you change a ten to ten ones,
And you add 'em to the two and get twelve,
And you take away three, that's nine.
Is that clear?

Now instead of four in the tens place
You've got three,
'Cause you added one,
That is to say, ten, to the two,
But you can't take seven from three,
So you look in the hundreds place.

From the three you then use one
To make ten ones...
(And you know why four plus minus one
Plus ten is fourteen minus one?
'Cause addition is commutative, right!)...
And so you've got thirteen tens
And you take away seven,
And that leaves five...

Well, six actually...
But the idea is the important thing!

Now go back to the hundreds place,
You're left with two,
And you take away one from two,
And that leaves...?

Everybody get one?
Not bad for the first day!

Hooray for New Math,
New-hoo-hoo Math,
It won't do you a bit of good to review math.
It's so simple,
So very simple,
That only a child can do it!

Now, that actually is not the answer that I had in mind, because the book that I got this problem out of wants you to do it in base eight. But don't panic! Base eight is just like base ten really - if you're missing two fingers! Shall we have a go at it? Hang on...

You can't take three from two,
Two is less than three,
So you look at the four in the eights place.
Now that's really four eights,
So you make it three eights,
Regroup, and you change an eight to eight ones
And you add 'em to the two,
And you get one-two base eight,
Which is ten base ten,
And you take away three, that's seven.
Ok?

Now instead of four in the eights place
You've got three,
'Cause you added one,
That is to say, eight, to the two,
But you can't take seven from three,
So you look at the sixty-fours...

Sixty-four? "How did sixty-four get into it?" I hear you cry! Well, sixty-four is eight squared, don't you see? (Well, ya ask a silly question, ya get a silly answer!)

From the three, you then use one
To make eight ones,
You add those ones to the three,
And you get one-three base eight,
Or, in other words,
In base ten you have eleven,
And you take away seven,
And seven from eleven is four!
Now go back to the sixty-fours,
You're left with two,
And you take away one from two,
And that leaves...?

Now, let's not always see the same hands!
One, that's right.
Whoever got one can stay after the show and clean the erasers.

Hooray for New Math,
New-hoo-hoo Math!
It won't do you a bit of good to review math.
It's so simple,
So very simple,
That only a child can do it!

Come back tomorrow night...we're gonna do fractions!

Y'know, I've often thought I'd like to write a mathematics textbook someday because I have a title that I know will sell a million copies; I'm gonna call it Tropic of Calculus.


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Last modified Tuesday, July 5, 2011 1:29 AM .