# Math Models and a Birthday Problem resource

Mathematical models is one of those ideas that students should know, but don't. Even after they've studied them. Ask your class the following: "A coin is tossed. What's the probability that it lands as heads?".

Most students have are quick to say 1/2 but that's wrong--the correct answer is we don't know the probability for any particular coin. We could use experimental probability to estimate it but even that's an approximate answer. The probability of heads that the students think is reality is actually a based on a mathematical model with a "fair coin". Mathematical models are approximations of reality. Unfortunately most students who have had some probability don't know coin flipping is based on a model and think the number of outcomes determines the probability (not realizing the equally likely assumption is an assumption which could be false). Some of these problems invariably trace back to the teachers who have taught them incorrectly.

Mathematician William Feller was a well known expert in probability who wrote a classic book An Introduction to Probability Theory and Its Applications in which you can find (by click on "Look Inside") the following quote on page 19: "As a matter of fact, whenever refined statistical measures have been used to check on actual coin tossing, the result has invariably been that head and tail are not equally likely. And yet we stick to our model of an "ideal" coin even no good coins exist. We preserve the model not merely for its logical simplicity, but essentially for its usefulness and its applicability.".

The coin flipping model has two assumptions built into it:

1. There are two outcomes (heads and tails)
2. The two outcomes are equally likely.

Since its possible for coins to balance on their sides (nickels and quarters more easily than a dime), it's possible (though admittedly remote) for a coin to land on its side. And it seems like most people have had experiences of a dropped coin which lands on its side only to roll away. Heck, it's even happened during a football game. This paper estimates the odds of an American nickel landing on its edge as 1/6000. Tested.com says that a study (broken link) indicates "...the "randomness" of a toss is actually weighted ever so slightly towards the side of the coin that's facing upwards when a flip begins....The paper, written by statistics and math professors from Stanford and UC Santa Cruz, also points out that a perfect coin toss can reproduce the same result 100 percent of the time.".

So coin tossing is a simple example of a mathematical model that students should learn. Another model is the famous "Birthday Problem". As I mentioned in an earlier post:

Answering the Birthday Problem involves creating a mathematical model. The model rests on two assumptions that aren't true and should be discussed with the class:

• there are 365 days in a year (Feb 29th is ignored to simplify the model)
• birthdays are equally likely to be on any given day (Also false. This varies from country to country; in the US birthdays are more towards the middle of the year. Count back 9 months and you've got cold weather. Nothing random there.)

I'm revisiting this post because I ran across a chart showing the distribution of birthdays referred to in the second point above. The Gizmodo post "How Common is Your Birthday" says, "The visualization used data from 1973 to 1999 to chart popular birthdays and figured out when the most popular time to pop out babies were."  So now you have a source to back up my claim and a nice chart to use in the classroom.

If you teach the Birthday Problem then you should get a copy of the chart or bookmark the page above.

Here are some stories that caught my attention this week:

• In an earlier post I pointed out the case of a high school student accused of stealing a backpack who was in prison for almost 3 years because he had been accused of stealing a backpack and would not plead guilty. The charges were eventually dropped and, because he was able to specify the specific dates on two occasions when he was abused, video has surfaced of these events. Democracy Now! has  "Explosive video obtained by The New Yorker depicts extreme violence inside New York City’s Rikers Island jail complex." and interviews New Yorker staff writer Jennifer Gonnerman, who has reported on the issue.
• NY Post reports on, "A copy of the state’s English Language Arts test that students took last week was leaked online Wednesday in an apparent act of sabotage by anti-testing activists......“This is a political act and it will be interesting to see whether [test-creation company] Pearson or the state Department of Education understands it as that or goes after them for civil or criminal liability,” said Brooklyn College education Professor David Bloomfield, who called the post an act of “civil disobedience.”"
• The Washington Post covers the widespread resistance of New York to Common Core testing: ,"Newsday has translated raw numbers into percentages, estimating that over 40 percent of all Long Island 3-8 students refused to take last week’s ELA Common Core state tests. Numbers in some districts reached well over 70 percent, with at least one district exceeding 80 percent. It appears that no more thanseven of the 124 districts on the island will meet the testing threshold of 95 percent. And that is before this week’s math tests, when opt-out numbers are expected to climb, as they did last year...It seems clear that the final 2015 tally will well exceed 200,000 students. New York State will likely not make the minimum 95 percent federal requirement for testing.."
• Shamkir 2015 has ended in victory for Magnus Carlsen. Chessbase has the report here. Anand took second with Caruana and So tied for third. Anand's second place performance has put him at number 2 in the world with a 2803.7 Live Chess Rating. I never thought he'd get there again. At 45 with his peak years ago and he's still in the hunt: incredible!
• My sympathies go out to Nigel Short. Not for his 1.5 - 8.5 versus Kasparov (Chessbase only has part 1 out here) but for the savage "beating" the "PC-police" are inflicting on him. The brouhaha, discussed here, starts with Nigel's comment, "“Men and women's brains are hard-wired very differently, so why should they function in the same way? I don't have the slightest problem in acknowledging that my wife possesses a much higher degree of emotional intelligence than I do. Likewise, she doesn't feel embarrassed in asking me to manoeuvre the car out of our narrow garage. One is not better than the other, we just have different skills. It would be wonderful to see more girls playing chess, and at a higher level, but rather than fretting about inequality, perhaps we should just gracefully accept it as a fact.”" which became sensationalized with The Telegraph's article, " Nigel Short: 'Girls just don’t have the brains to play chess".  You can even see Nigel defending his common sense position on Sky News and being given the absurd argument that he's wrong because J Polgar has a plus record against him. Given that there are distinct differences in the hardwiring  of male versus female brains (not to mention the differences between males and between females) and the study Short points out, it's difficult to believe his comments have become so controversial. Sorry Nigel! You deserve better.
• Huffington Post has an article about Ramanujan whose birthday was April 26th. The brilliant mathematician who was most definitely wired differently than others: "The biggest question is how an untrained teenager, and later young adult who repeatedly flunked out of college in his native south India (generally the area of Madras, today's Chennai), was able to obtain--all on his own--mathematical expressions that later would take some of the world's leading mathematicians years and even decades to ascertain and prove.".
• I've deleted the CTAN Mail Archive link and replaced it with the updates gmane.org. They were notifying about many changes in LaTeX that never made it to the other link. The link is CTAN announcements, located on the sidebar.

# Sagetex: Derivative 5/6 without the Chain Rule

I've added two more problems (Problems 5 and 6) to the Sagetex: Derivatives page. Problem 5 requires the student to calculate the derivative of a product of polynomials using the Product Rule. Problem 6 looks like a Quotient Rule but the derivative is more easily calculated by carrying out division before taking the derivative.

Here are some stories which caught my eye:

• L.A. Times reports 9 students at Venice High School were arrested and 14 accused, "...in connection with a series of sex crimes that began more than a year ago and involved at least two female classmates. All but one of the arrests were on campus; authorities were attempting to locate five other students." No doubt this story will continue to evolve.
• Mathematically challenged: it could cost you your marriage. The New York Post reports "Bride Walks out on groom after he botches simple math problem". In a country of arranged marriages, it was a sign to the bride she had been "misled" about his education. "“The groom’s family kept us in the dark about his poor education,” said Mohar Singh, the bride’s father. “Even a first-grader can answer this.”"
• LaTeX Community has a nice piece on "Why I like the TikZ Math library".
• Welcome to school today: a student intentionally runs into a school security officer "..in what has been described as an attempt to ‘chest bump’ the officer...". This knocks the officer to the ground. The officer gets up, grabs the boy and according to Infowars, which has the video posted: "The video shows Hardin lifting the 13-year-old child into the air with a chokehold. The child kicks and flails around before becoming limp and lifeless. Hardin then drops the student to the ground, resulting in what a doctor has described as “an injury to the brain” due to loss of blood flow.". Be aware of the graphic nature before you watch this.
• It's obvious to "older" people that school quality has dropped dramatically over the years (think Calculus students who struggle with fractions) but at least the students of today are tech-savy, right? According to this Fortune article, no. "American Millennials are among the world's least skilled" reports that a new study shows, "...Millennials in the U.S. fall short when it comes to the skills employers want most: literacy (including the ability to follow simple instructions), practical math, and — hold on to your hat — a category called “problem-solving in technology-rich environments.” Not only do Gen Y Americans lag far behind their overseas peers by every measure, but they even score lower than other age groups of Americans."

# Odds and Ends, February 12, 2015

1. I've added another math article template to the LaTeX page. This one uses the Classic Thesis style.
2. Three chess packages were added to the LaTeX page.
3. The Problem page has another problem, due to Leibniz; he demonstrated that $\sqrt{6}=\sqrt{1+\sqrt{-3}}+\sqrt{1-\sqrt{-3}}$.
4. Carlsen tied with Naiditsch at the Grenke Chess classic with Magnus winning on tiebreaks. Chessbase has the report here.
5. Have you ever gotten tired of keeping your students off the portable distraction devices. It might be addiction! If you saw the recent viral photo of someone checking their cell phone while missing the giant humpback whale nearby or saw the report on Japanese teenage girls spending 7 hours on their phones you will have no trouble believing Science Daily's article on the use of mobile technology by children. "The authors question whether heavy device use during young childhood could interfere with development of empathy, social and problem solving skills that are typically obtained by exploring, unstructured play and interacting with peers....."'Until more is known about its impact on child development quality family time is encouraged, either through unplugged family time, or a designated family hour," added Radesky."
6. Voice of America asks, "Is America in a Math Crisis?". Well known mathematician and author Albert Posamentier asserts, “Elementary school teachers in this country, and Europe as well, are part of that general population and consequently they bring that dislike of mathematics subconsciously, sometimes consciously at other times, to the classroom and as a result the teaching mathematics at the elementary school level lacks motivation, lacks enthusiasm.  The enthusiasm of the teacher  is extremely important in turning kids on to the subject matter". Let's also remember that 1/3 of high school math teachers don't have a degree in mathematics. The problems aren't just limited to elementary school.
7. A 10 year old Chinese girl made the Wuhan Evening News for her poem on the tortures of math: “Math is the root of death and it makes life like hell. It wears out children and worries parents. It expels the vitality of school and slowly takes away life.”. Needless to say, I disagree.
8. An 8th grader who through the American flag out of a classroom window is now in serious trouble: From the article, "The principal called the school resource officer with Rio Arriba County, but because it is a federal offense they referred him to the FBI. “I want to report it to them because it is a federal law, so it’s in their hands,” Archuleta said. Last week, he suspended the 14-year-old last week for 10 days, but he is recommending long-term suspension or expulsion."

# Handouts: Stern-Brocot/listing rationals

In earlier posts I mentioned the Stern-Brocot sequence and how it can be used to create another sequence that lists the positive rationals in lowest form exactly one time each. There was a lot of information on the subject so I put together a PDF containing that information plus a little bit more--it's posted on the Handouts page.  In addition to the Numberphile, the link to the Caulkin, Wilf paper, the tree of rationals, and Sage Code I've included an algorithm along with a dry run (shown above) along with some other information from the Caulkin-Wilf paper and some sample questions/tasks that can help flesh out a lesson to your class:

The "Fact" box above as well as the box containing the Sage Code below were created using the tcolorbox package.

So, if you plan on using the Stern-Brocot sequence there is now 1 resource containing all the components of the Stern-Brocot sequence.

Here are a few things that caught my eye recently:

• The Grenke Chess Classic 2015 has one round to go with Carlsen and Naiditsch tied for first and several others in striking distance. In the even of a tie there will be blitz games to decide the winner of the tournament. The games stream here. (Click "Live")
• In the last post there was an item about the shift in the debate over Common Core. Rather than deal with constant criticism the new approach is to say Common Core is about standards, not content. That's misleading from my experience because standards drive content. So schools look to buy Common Core aligned material which help set the level of difficulty and subject matter. The problem of 4 ways to subtract caused a backlash because it was showing up in the textbooks. And as RT notes in this video,  Common Core has legible handwriting taught in kindergarten and first grade only. This skill, no longer being required at higher levels is replaced with other content. As a result, many schools have stopped teaching cursive. The result are kids who can't read or write cursive. It's a piece with a lot of interesting arguments.

# Python/Sage: Stern-Brocot sequence

In the last post I mentioned Numberphile's excellent video on the Stern-Brocot sequence. The explanation of how the sequence is formed is clear enough but extra time will be needed to make sure students know how the sequence is formed. That would mean discussing the algorithm for how the sequence is formed, using the algorithm to create the sequence and then getting them to calculate some terms of the sequence on their own. I put together a quick Python program to create some terms of the sequence and then show the sequence of positive rational numbers. Having a list of both sequences will help to catch any mistakes you might make from, say, getting distracted/interrupted fielding questions along the way. The code is posted on the Python/Sage page.

Here are some other events that caught my eye recently.

1. The 13th Tradewise Gibraltar Masters tournament has ended with Hikaru Nakamura in clear first place followed by David Howell second and a big tie for 3-11. Now all eyes are on Grenke which will continue with round 4. You can follow the game here. (Click Video)

2. I stumbled onto a PhD Dissertation on "Students' Errors in Graphing Calculator Based Precalculus Classes". It's worth looking at some of the misconceptions and examples that caused students to struggle.

3. Politifact weighs in on the Texas governor's claim, "it takes "more than a minute" to teach a student "how to add nine plus six."". Lost in the claims and counter claims (Common Core is standards not content) is the fact that the new materials which are aligned to common core have this bizarre new math. Remember, for example, the 4 methods of subtraction? It's in the books that teachers have to teach out of and wasn't in books before Common Core.

4. An excellent web page on Using Writing in Mathematics.

# Counting the Rationals

In The Most Important Lesson of High School Math I mentioned that I thought the goal of the high school math curriculum should be to get students to the point that the rational numbers have measure zero (while avoiding such technical speak). The video I linked to (How Big is Infinity) took students through Cantor's diagonal argument that the positive rational numbers can be listed. That list, however, has different forms of the same number (4/2=2/1=8/4=...). But I frecently found a post by Numberphile on the Stern-Brocot sequence. This sequence has a nice property: take the ratio of consecutive terms  $\frac{a_n}{a_{n+1}}$ to create another sequence which lists every positive rational in lowest terms exactly once. A brief paper by Caulkin and Wilf has more of the mathematics. I've put together a tree diagram that was used to explain how the rational numbers are created. You can find the PDF and tex file on the Graphics page. The Stern-Brocot sequence and the rational sequence resulting from the ratio of consecutive terms would be a good example to put into your coverage of sequences.

Some other points of interest:

• I found a minor mistake in the code for the last post. If you look at the screen shot I've got y_coords and y1_coords used in the same IF statement. I've fixed the file and uploaded it.
• I've started a list of LaTeX packages that are important/useful to me when I'm using LaTeX; it's on the LaTeX page. There's been a big increase in packages over the years and I'm finding new packages that weren't around years ago. Since I can't keep always remember the package or the commands, I'm listing the packages which are linked to the CTAN documentation.
• The Gibraltar tournament is into round 7 of 9 and Nakamura is in the lead. Meanwhile, the Grenke Chess Classic 2015 has started. It's got Carlsen, Anand and many other top players. You can follow the games by clicking on the link and then pressing the "Video" tab.