SageTeX: Limits (9)

Limits9I've added another limit problem to the SageTex: Limits page. The problem type has the form \lim_{x \to -3}\frac{\frac{1}{3}+\frac{1}{x}}{3+x}.

Some events that caught my eye:

Sagetex: Limits (8)

Limits8I've added another problem to the Sagetex: Limits page. The problem, shown above, is on using the Squeeze Theorem to establish a limit.

Here are some recent stories that stood out:

  • Vincent Knight, from the blog Un pue de math, has posted 21 videos on "Basic LaTeX using Sagemath Cloud". The videos are posted here.
  • Do you know 4 methods for subtracting numbers? The Erick Erickson Show has an example of Common Core math that is causing headaches for parents. I've been fortunate to not have had to teach from stupid material like this. You can understand why Common Core has lost a lot of support.
  • A Baltimore teacher gets into a fight with a student and some of it is recorded on video. WBAL TV11 has the local news report (which includes video). "Officials said the incident started as a verbal confrontation that turned physical. The question is whether the teacher acted unprofessionally or if she was just overly frustrated with the student, who sources said was the instigator.". The article continues, "City schools spokeswoman Anne Fullerton sent 11 News a statement, saying, "The teacher was treated for injuries and has been placed on administrative leave pending the outcome of external and internal investigations. The student, a minor, was charged with assault and released to a parent."".

Sage: Menger Sponge

MengerSCThe Menger Sponge is a 3 dimensional fractal with infinite surface area and zero volume. I put together some code to create the fractal but it's been plagued with problems. The screenshot above shows it running for the second iteration. It worked for 3 iterations, initially but now it doesn't but strangely enough the same code will render a 2 dimensional picture of the 3 dimensional object:

Menger4I've posted the first 4 iterations of the Menger Sponge on the Graphics page. The code that generated the images is posted on the Python/Sage page. The basic process behind the code is mentioned in this Scientific American article. The output was as a .gif file and it was converted to PDF (and jpeg above) using GIMP.

Some stories that caught my eye:

  • The Atlanta school cheating scandal is back in the news as the trial is going on now. Local news at Channel 2 in Atlanta has a report and video here. Two people have admitted to cheating saying they were told to. At the 1:30 mark, one witness says he didn't realize it was criminal. (?!).
  • An article from the NY Times, here, talks about the new Common Core based tests. The writer here, in NY, looks at the version her child will be taking and brings up a potential problem: "In many ways, it is a better test than the fill-in-the-bubble multiple-choice exams of my youth. With a computer-based test, the questions can be more complicated but still easily graded. Both consortiums also offer paper versions for the time being, because not all schools have enough computers and Internet connectivity."................"

    But for other questions, the test provides a more complex equation editor — rows of buttons including numerals, mathematical operations like add and subtract, a tool to enter fractions — for entering the answer.

    A keyboard and mouse is not a natural way of doing math, and I wondered whether these questions would be more a test of computer interface.

    Laura Slover, the chief executive of Parcc, asked me if I had done the tutorial before taking the practice test. I had not. I asked if it was reasonable to expect that all students would have the time and opportunity to do that. She said the organization encouraged taking the tutorial, and that in the field test last spring, students who practiced beforehand did not find the computers an obstacle.

    Hint to parents: If your children are to take one of these tests, make sure they puzzle out the interface first.".  This sounds like a disaster in making. It's difficult to imagine all these families taking the tutorial before taking the computer based test.


LaTeX: Behavior Log

BehaviorI've added a behavior log to the Procedural page. As part of progressive discipline policies, this is a useful form to have. Do you have a student who you've talked to already about staying on task? Get out the behavior log write his name down, the date, and the problem (eg playing a video game on his cellphone). And then get their signature. If the student refuses then their defiance is good reason to send them to the dean's office. Once the behavior log is signed you've got a record of the student' problems along with the student's admission of guilt.

Having a behavior log gives you a way to de-escalate conflicts with students: just walk back to your desk, get the form out, and come back. You should, of course, include information in your syllabus about how you use the behavior log. For example, 3 behavior problems results in a parent teacher meeting. When students know you have the their signed confession and that there is a consequence/annoyance for the third strike, students are more likely to keep their behavior in check. For those students who just don't care, you have the documentation you need for the parents along with the signature of their little angel admitting guilt on multiple occasions.

Here are some stories that caught my eye.

  • tells us a "war" is coming: "By activating its nuclear option and cancelling its teachers' contract, the Philadelphia School Reform Commission took an action Monday that could remake the city's schools and have national implications.The unilateral step at a morning meeting has already set off a battle.". The article goes on to say,"The terms imposed Monday mean most PFT members will have to pay either 10 or 13 percent of the cost of their medical plan beginning Dec. 15, depending on their salaries. They now pay nothing if they opt for a basic plan. Officials said workers would pay between $21 and $200 per month, beginning Dec. 15.The changes will save the cash-strapped district $54 million this school year, officials said, and as much as $70 million in subsequent years.That money, SRC Chairman Bill Green said, will be invested directly in classrooms, with principals empowered to use the cash as they see fit - to hire a full-time counselor and nurse, perhaps, or to pay for more supplies or after-school programs.Green said district principals, blue-collar workers, and families have stepped up, and it was time for the teachers to do the same."
  • The Las Vegas Review Journal reports on changes in sex education that have upset the local community: "Considered changes include education of homosexuality as early as ages 5 through 8 and giving everyone “respect regardless of who they are attracted to.”Children of that age range also would be taught that “touching and rubbing one’s genitals to feel good is called masturbation.”“You want to teach my 5-year-old how to masturbate?” said parent Julie Butler, referencing the item getting the most attention Monday night."........."Clark County Superintendent Pat Skorkowsky tried to calm the approximately 50 parents by asserting that the curriculum is not proposed for the district’s 357 schools.The district is just gathering community input, he said.".  Public outrage resulted in a public apology. The local news reports say, "In an Oct. 3 letter to parents, Clark County School District Superintendent Pat Skorkowsky said, “I believe that our school district has inadvertently broken trust with many of our families and with some members of our Board of Trustees.”He said the problem was created when the district set up a series of invitation-only community forums to discuss possible changes in sex education curriculum. “We made a misjudgment when we put out a document in these community forums from the Sexuality Information and Education Council of the United States, also known as SIECUS,” Skorkowsky said."....."“I regret that we made these decisions, and that we did not open these initial input sessions to the general public,” Skorkowsky said. “We also should have notified the Board of School Trustees as a whole about the documents that were passed out at the forums.“I am sorry that the trustees have taken criticism over the situation because they did not approve this process as group, and some of them, including Board President Erin Cranor, did express objections to the methodology of the meetings.”"

Sage Interact: the Birthday Problem

BirthdayProbThe Birthday Problem asks the question:  In a group of n people, what's the probability that at least 2 will have the same birthday?

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.)

Under those assumptions the sample space is 365^n, since there are 365 choices for each birthday. Counting the number of ways that nobody has the same birthday is just (365)(364)....(365-n+1), giving us the probability of at least 2 people with the same birthday as the complementary probability: 1-[(365)(364)....(365-n+1)]/(365)^n$.

This simple model yields surprising answers: with a group of only 50 people the probability of at least 2 people with the same birthday is 0.97; this illustrates how counter-intuitive probability can be. So is the model any good? It turns out that although the model is built on incorrect assumptions, experimental evidence validates gives results that are close to the model. That's a basic fact about models; they're based on assumptions that in many cases aren't true. Ultimately, experimental evidence is needed to determine whether the model is accurate. Try the Birthday Problem with each of your classes and find out how well it works with 1 class, with any 2 classes, any 3 classes, and so on.

I've created code for computing the Birthday Problem given n people and plots the probabilities on a graph. You can see the output above; the code is on the Python/Sage page.

Some stories that caught my eye recently:

  1. Education Next has articles on "Rethinking the High School Diploma". Three articles consider the idea of having 2 diplomas: 1 for graduating and the second higher diploma for excellence/mastery. It seems symptomatic of today's culture. You can't enforce standards so everybody needs to graduate and the diploma lacks value. So create a second diploma which will show the employer the student has mastered the basics. An interesting idea.
  2. Businessweek has an article suggesting that the US should station soldiers in schools, similar to air marshalls on a plane. The Call of Duty author has, "...anticipated objections. “The public won’t like it, they’ll think it’s a police state,” he said. But, he went on, “All of these are solvable problems.” Anthony’s address, which was punctuated by videos depicting such future threats as a U.S. drone hacked by Iran and a hotel massacre in Las Vegas, included repeated exhortations to policymakers to learn from the examples of corporations and creative artists in selling potentially unpopular ideas. “When we have a new product that has elements that we’re not sure how people will respond to, what do we do as a corporation?” he asked. “We market it, and we market it as much as we can—so that whether people like it or not, we do all the things we can to essentially brainwash people into liking it before it actually comes out.”". Incredible.....
  3. Huffington Post reports on teachers behaving badly: "Louisiana Teachers Planned Illicit Group Sex With Student: Police". The article states, "Police now say that one of the Louisiana high school teachers accused of having group sex with another teacher and a 16-year-old student had previous sexual encounters with the minor.". There's a video from Huffington Post.

Sagetex: Limit of a rational function

Limits7I've added another problem to the Sagetex: Limits page. There are three different variations of the problem depending on whether the polynomial in the numerator has a degree which is greater than, equal to, or less than the degree of the polynomial in the denominator. Choose which type of rational function you want for your test/quiz. Delete the rest.

The Sage code for the problem contains some noteworthy features. The line


is used to specify that the coefficients of the polynomial are rational numbers. A random polynomial is defined by

N = R.random_element(degree = degree)

and the issue to be aware of is that if, for example, degree = 5 (or any integer) then the polynomial created has degree less than or equal to 5. Therefore, I created a WHILE loop to discard any polynomial with degree < degree. Since the polynomial degree is random, Sage has to convert it to a string of LaTeX code. The rational function, in LaTeX, is created here:  output = latex(N/D) and then inserted into the problem using \sagestr.

Several stories at Democracy Now! caught my eye:

Altermundus: creating simple geometric diagrams

I've added an example to the page on creating lines and rays with the Altermundus package. The additional information is posted here. Start with a typical geometric diagram you might need to construct in high school mathematics.

TrigDiag1I'm calling the diagram "simple" because it consists of line segments, angle markings and text. In order to create the lines, you need 2 points. Therefore, when you look at the diagram you should see 9 points are needed.


I take the bottom left point to be the origin but that isn't required. The building on the left is made up of 4 points:


The building on the right has 5 points:


Once you have the 9 points defined you create the line segments. In addition to \tkzLabelSegment there is \tkzDrawPolygon. The \tkzDrawPolygon takes 3 or more points and creates polygon defined by them. The building on the left and right are \tkzDrawPolygon(A1,A2,B2,B1) and \tkzDrawPolygon(D1,D2,E2,E1), respectively. The dashed lines are created:

\tkzDrawSegments[dashed](B2,H B2,D2 B2,D1)

Be careful here! Note that there is a space after every 2 vertices. Putting a comma between them causes an error. After that the \tkzLabelSegment is used to get text.
\tkzLabelSegment[right=3pt](B1,B2){$100$ ft}

and finally the angles are marked and labelled.

\tkzLabelAngle[pos = 0.75](H,B2,E2){$42^{\circ}$}
\tkzLabelAngle[pos = 1.0](E1,B2,H){$20^{\circ}$}

Note the is a special command to create the right angle mark.

By breaking down a simple geometric diagram into the points needed to create the lines, you can easily decompose the picture into pieces that can be plotted with the macros mentioned. Define the points, create the lines, and add the text. See the page on creating lines and rays with the Altermundus package for the complete file to create the diagram.

Sagetex: derivative as a limit

LimDQI've added another problem to the Sagetex: Limits page. Find the derivative of quadratic function using the limit of a difference quotient. The screenshot is above.

Several stories caught my eye recently:

  1. William Stein, the driving force behind Sage, reports on a, "major 3d update. Print worksheets with embedded 3d graphics. Pan with alt/command. Plots persist between refreshes.". More details can be found here.
  2. The Washington Post has a piece about an "award winning" principal who used to support Common Core and is now opposed to it. She writes about the "Four Common Core 'Flim-Flams'".
  3. Common Dreams reports: "Hundreds of students from high schools across Colorado's Jefferson County school district walked out of classes on Monday, Tuesday, and Wednesday this week, protesting attempts by rightwing members of the school board to amp up what they consider the "positive aspects of the United States and its heritage" within the district's history curriculum while minimizing focus on more progressive aspects of history such as people's movements, the history of struggle, and "social strife."". The article says, "that plan would look at Advanced Placement history courses to make sure materials "promote citizenship, patriotism, essentials and benefits of the free-market system, respect for authority and respect for individual rights" and don't "encourage or condone civil disorder, social strife or disregard of the law."" A passage from a piece on the Breitbart website gives a student opining, "...the nation's foundation was built on civil protest, "and everything that we've done is what allowed us to be at this point today. And if you take that from us, you take away everything that America was built off of."".
  4. This hasn't gotten much publicity: reports, "A secret program to monitor students' online activities began quietly in Huntsville schools, following a phone call from the NSA, school officials say.". According to the school officials they were alerted to a student making threats on Facebook. "The NSA, a U.S. agency responsible for foreign intelligence, this week said it has no record of a call to Huntsville and does not make calls to school systems.". So now Huntsville City Schools are looking at social media sites for gangs, guns, and threats of violence. One school official says, ""There was a foreign connection," said Wardynski, explaining why the NSA would contact Huntsville schools. He said the student in Huntsville had made the online threats while chatting online with a group that included an individual in Yemen.". But it's not just Huntsville. The article continues, "A company called Geo Listening watches social media for school districts including Glendale, Calif. Their web site reads: "Geo Listening's unique monitoring service will process, analyze and report the adverse social media from publicly available student posts... We align our reporting criteria with existing school district procedures and board policy as they relate to student conduct & safety."".

Sagetex: Dartboards and tree diagrams

CombDartsI've added another problem to the Sagetex: Combinatorics/Probability page: "Three darts are thrown at the dartboard. A score is given for each region where the dart lands and the total score is just the sum of the $3$ dart scores. Assume all three darts hit the dartboard. How many different total scores are there? Enumerate them with by drawing a tree diagram."

It's a problem that's illustrates how a tree diagram can be used to organize the solution in a way that anyone can easily check the answer. Typesetting the tree diagram made this a a time consuming problem to create. But the work is done now and the code can be downloaded for you convenience.

Here are some stories that caught my eye recently:

  • Ben Swann alerts us to a "zero tolerance" policy to the extreme.  In Teen Suspended After School Claims Notebook is "Drug Possession" we learn "...a teenager was punished with a lengthy suspension after teachers discovered her folder which contained stories with references to marijuana use." . Her suspension of 10 days was for "...“possession of a controlled substance” despite no drug testing and no drugs in Krystal’s possession". Her written account is considered  possession, apparently, "...although the district’s drug policy posted online provides no specific definition of paraphernalia".
  • CBS San Francisco reports on how outrageous high school behaviour is having an impact: "A Taco Bell restaurant in Antioch has started closing its dining room in the afternoons after managers say it has become a magnet for high school students after class, and fights have been breaking out.“At school you get suspended or something for that, and if you’re not at school you go to the plaza and fight and get away with it,” one student said.". That even includes death threats. Check out the video.

Sagetex: Area under a quadratic using limits

LimAreaI've added another problem to the Sagetex: Limits page. The problem is to find the area under a quadratic using limits. The problem is easy to formulate but explaining the solution is tedious programming. You can see a screenshot of the problem above.

The fourth round of the Chess Masters Final is in the books. Anand has sole possession of first place. The scoring in the event is 3 points for a win, 1 point for a draw. With 2 rounds left and a 4 point lead this tournament is over from a practical standpoint if not a mathematical one. You can follow the games from the site, ChessBomb (on the sidebar) as well as Livestream.

Common Dreams has more on the militarization of the school system: it's even more extensive than you think. From the link: "Turns out San Diego isn't the only school district in the country to get a $700,000 Mine-Resistant Ambush-Protected Armored Vehicle and other military baubles from a federal government overloaded with shiny lethal gimcrackery from its many failed wars. At least 120 schools and colleges in 33 states, including Florida, Georgia, Kansas, Michigan, Nevada and Utah, have gotten such geegaws, according to tenacious reporting by Muckrock and others. Texas - including the town of Cut and Shoot, population 1,000 - tops the list, with at least ten districts gleefully acquiring 15 surplus military vehicles, 64 M-16 rifles, 18 M-14 rifles, 25 automatic pistols, extended magazines, 4,500 rounds of ammunition, armored plating and tactical vests.". ZeroHedge has some quotes from the Wall Street Journal talking about how the Los Angeles Unified School District has gotten "... grenade launchers, M16 rifles and even a multi-ton armored vehicle from the program. But the district is getting rid of the grenade launchers, ".One consequence of spread of weapons is some weapons have been lost, stolen, or otherwise accounted for: "Nine California law enforcement agencies are suspended from the 1033 program, according to Cal OES. The suspensions all stem from lost weapons, including one pistol, 10 M16 assault rifles and one M14 rifle.

The San Mateo Sheriff’s Office has received 78 assault rifles and a truck through the program.  More than 7,700 M16s have been distributed  in California since 2006. In just the last two years, California agencies have also received 41 mine-resistant vehicles."

Next, with all the pressure to get technology into the school system, it's interesting to see the New York Times running an article ("Steve Jobs was a Low-Tech Parent") about how Steve Jobs and various tech CEOs having strict rules regarding the usage of technology in their own family: "...I’ve met a number of technology chief executives and venture capitalists who say similar things: they strictly limit their children’s screen time, often banning all gadgets on school nights, and allocating ascetic time limits on weekends."

Finally, the Daily Caller takes Bill Gates to task on pushing the Common Core onto YOUR kids, but not his. Given the increasingly political nature of Common Core, I think you're going to hear about this in the future.