# Sage: Polynomial Interpolation

In an earlier post I mentioned that it's not that uncommon for math teachers and even education professionals making up academic resources to mess up badly on sequences. For example, the IXL site, which is, in general, an excellent resource for teachers and students makes a common error with sequences both here and here. Experiment with the problems that are randomly created at those 2 links. Try entering a "wrong" answer and you'll get an explanation of what the "right" answer is. But the "logic" they're using is that here's a formula that describes the sequence, therefore the next term is... The problem is that if you follow the same logic you can justify the missing terms of the sequence to be whatever you want. Therefore those problems have no correct answer and should not be given. Moreover, they shouldn't mark other answers as wrong.

Whenever teachers and math professionals are wrong on the math, you've got a teachable moment. This lesson would be for someone teaching about matrices who has gotten through reduced row echelon form. I'll suggest that you start a class by working through the material on the IXL website and have them figure out what the terms of the sequence are. Write the various sequences on the board to refer to later. When they've gotten comfortable ask them what will happen if you put in a term that doesn't seem to fit the pattern. They'll predict that you will be marked wrong. After showing them that you are indeed marked wrong by IXL figure out a formula to justify your answer and have students confirm that it works. You now have a formula that justifies a sequence that IXL was marking you wrong for. Get the class to discuss what it must mean for an answer to be correct (you can find a formula) and for what it means to be incorrect (it's impossible for anyone to find a formula).

Now it's time for math! Elicit that a sequence is a function from the positive integers into the real numbers (or integers, depending on how you teach it). Remind them that this means the sequence -5, 3, 11, 19, 27, ... corresponds to the function where f(1)=-5, f(2)=3, f(3)=11, f(4)=19, f(5)=27. Give them the mathematics known as polynomial interpolation or Lagrangian interpolation, and using one of the IXL sequences written on your board, set up a Vandermonde matrix. Work through the mathematics to create the polynomial. Get those calculators out to solve the matrix equation. And at the end you'll have a polynomial which they'll need to confirm works to generate the sequence. By the end of class your students will have learned about Vandermonde matrices, seen the math behind polynomial interpolation, used the calculator to power through some of the calculations and construct a polynomial that shows that even the "experts" get things wrong. That's a lesson the class will remember long after they've forgotten how to do the math.

But it's easy to make mistake with calculations so I've constructed a Sage program to go through the steps to create an interpolating polynomial using Vandermode matrices. It's posted on the Python/Sage page. A little warning is necessary, however. Usually you just go to a SageCellServer (or SageSandbox on this site) and copy/paste the code and press "Evaluate" to get the code to run. For reasons I don't understand, sometimes you get some sort of I/O Error such as the one below.

That error can pop up at various stages in compilation. The code runs, sometime it's just a matter of pressing "Evaluate" several times.

In order to set the code for your sequence, you need to alter xvalues and yvalues in the code. For example, if your sequence is 3,2,_,0,-1,-2, _,... and someone says that the pattern is to subtract 1 from the previous terms so the missing numbers are 1 and -3 then you'll need to pick the values you want in your sequence. If you choose to complete the sequence as 3,2,5,0,-1,-2,11,... then you'll need to set xvalues = [1,2,3,4,5,6,7] because you have 7 terms in your sequence and yvalues = [3,2,5,0,-1,-2,11] because those are the terms of your sequence. Press "Evaluate" and you'll get a polynomial that goes through those points. So there is a formula for your sequence--it's just not obvious to most people.

Here are some stories that caught my eye this past week:

• Decades ago, a high school degree was normal and few people had college degrees. Yet people could get jobs that allowed them to provide for a family. Now the quality of high school education has been watered down, graduation rates are higher and a college degree is "necessary" to get a job. And that means people are some \$30,000 in debt after college (which is more likely to be 5 years now) and too many students have taken courses in topics that lack rigor and meaning. And while I've taught algebra to many young students outside the US, somehow in the US algebra is too complicated for 18-22 year old students to master--it's actually standing in the way of people graduating. How can you get horribly educated students through college with a pesky math course in the way? Simple--water down standards at the college level. Now Wayne State University leads the charge in dumbing down education, "Up until now, students had to take one of three different math classes before they could earn their degree. Now, depending on their major, students may be able to squeak through college without taking math. The university is leaving it up to the individual departments to decide whether math will be a required part of a degree's curriculum." So in the future, students can graduate high school and college and still not have the math skills of someone who only graduated from high school 50 years ago...and pick up a lot of student loan debt along the way. But at least more people are graduating from college! Progress?
• Now compare the American drive to banish math with this clip from NextShark called "Watch Korean Students Take the American SAT Math Section For the First Time". You'll hear some people talk about Americans as "exceptional"--that probably shouldn't be taken as a compliment. Even the weak Korean students are feeling better about their math now....
• The Columbus Dispatch reports "The State Board of Education is expected to lower minimum proficiency standards on two new high school math tests after results came in lower than expected.The move raises questions about whether benchmarks for new assessments will accurately gauge a student’s readiness for college or a career" Somehow I doubt they care about college readiness...
• Wired.com with an article on "deep learning" that underpins AlphaGo and other computer programs.
• He's gone viral! ABC News with a video on "8TH GRADER GRABS BELLY LAUGHS FOR CANDIDATE IMPERSONATIONS"
• Mental Floss with "15 Observational Facts About Isaac Newton"
• The bit-player blog has a fascinating post on the non randomness of the prime numbers. "These remarkably strong correlations in pairs of consecutive primes were discovered by Robert J. Lemke Oliver and Kannan Soundararajan of Stanford University, who discuss them in a preprint posted to the arXiv in March. What I find most surprising about the discovery is that no one noticed these patterns long ago. They are certainly conspicuous enough once you know how to look for them...For the past few weeks I’ve been noodling away at lots of code to analyze primes mod m. What follows is an account of my attempts to understand where the patterns come from. My methods are computational and visual more than mathematical; I can’t prove a thing. Lemke Oliver and Soundararajan take a more rigorous and analytical approach; I’ll say a little more about their results at the end of this article."
• A May 18th, 2016 interview with graph theorist Maria Chudnovsky on anthonybonato.com. Looking for a female mathematician to inspire the girls in your classes.  Look no further; she's even been in 2 commercials.

# SageTex: Complex Numbers

It's been a very busy week as schools wind down for the year. I've added a new page Sagetex: Complex numbers to the sidebar. I will, of course be adding problems to the page in the coming weeks.

Here are some stories that caught my eye this week:

• Mathematician Dr. Ken Ono has been written about in Quanta Magazine.
• Dr. William Stein is leaving academia to start a company around Sagemath. His pdf slideshow can be found on this Reddit thread. He has a talk about this in a video posted here.
• Ars Technica reports, "Google's DeepMind AI division will face off against humanity's number one Go player, Ke Jie, according to Chinese state news agency Xinhua. Ke Jie is currently the top-ranked Go player in the world."
• DurhamLocalNews has a piece on how "Legislation recommended Wednesday by a Senate committee would require North Carolina schools to offer traditional high-school-level mathematics classes alongside newer “integrated” classes that arrived with Common Core...If the bill were to become law, districts would have to offer both math class sequences to students starting in the 2017-18 school year."
• Click2Houston tells us a creepy story about a math teacher who enjoys being with kids a little too much: "The video shows Gregg Gustafson wrestling and tickling minors at a student's home when the parents were not around...One of the students told the police Gustafson gave him several wedgies that broke his underwear. The boy said he was wrestling with Gustafson and that he would not let him tap out.The student posted two videos of his broken underwear. Gustafson allegedly asked the boy to get rid of the evidence."
• Washington Post on the new school report "Many people in education and the workplace don’t think some of the English Language Arts and math standards — which are being used in most states — are what students and workers need to be successful in college and career." One of the findings is that calculators are prevalent in math class. That's exactly what I'm seeing--but you'll see the teacher teaching how to work a problem and then they jump to using a calculator, so that the students quickly forget the skills they were supposed to learn. Whose "best practice" is that? More disturbingly, "There may be disagreement across K-12, college, and workforce about which mathematics topics are important to success in postsecondary STEM coursework and STEM careers. In K-12, there may also be disagreement about when these topics should be introduced in the mathematics curriculum." Shouldn't these issues have been worked out before forcing them upon everyone?
• Ars Technica with an interesting article on "That ’70s myth—did climate science really call for a “coming ice age?”"
• EAGNews on how "For a mere \$6.5 million, New York City’s School Construction Authority renovated a former clothing store on Fifth Avenue into a “state-of-the-art” facility for 18 students to participate in its new, free pre-K program....The renovations cost the city about \$362,222 per pupil, which was about \$160,000 per seat more than the next most expensive renovation in Staten Island, for creating partitions, pouring concrete slabs, adding sprinkler systems and HVAC, and other upgrades." It's easy to spend public money.
• ZeroHedge on the embarrassment that is the new generation. But it's not just America. "In the western world, political correctness is often taken to absolutely ridiculous extremes in attempt to keep people from being exposed to anything that could remotely be considered “offensive”.  For instance, just consider a couple of examples from the United KingdomThis hyper-sensitivity has prompted the University of East Anglia to outlaw sombreros in a Mexican restaurant and caused the National Union of Students to ban clapping as “as it might trigger trauma”, asking youngsters to use “jazz hands” instead. Could you imagine banning clapping? But this is actually happening.  Anything that might make someone feel the least bit “uncomfortable” is now being labeled as a “micro-aggression”, and at schools all over America “safe spaces” are being set up where young people can avoid anyone or anything that may make them “feel uncomfortable, unwelcome or challenged”.And this is not a fringe movement at all.  These “safe spaces” are being established at some of the most prestigious universities in the entire country, and in at least one case a “safe space” included “calming music, cookies, Play-Doh and a video of frolicking puppies”
At Brown University – like Harvard, one of the eight elite Ivy League universities – the New York Times reported students set up a “safe space” that offered calming music, cookies, Play-Doh and a video of frolicking puppies to help students cope with a discussion on how colleges should handle sexual assault. A Harvard student described in the university newspaper attending a “safe space” complete with “massage circles” that was designed to help students have open conversations.Are you kidding me?...Now that we have defined “safe spaces” and “trigger warnings”, I am going to define a term that I used in the title of this article.

“Wussification” is the act of turning someone into a “wussy”.  And urbandictionary.com defines “wussy” in the following manner… A person with no guts. A person who whines all day and sits around and cries like a little baby for years over nothing. Will blow anything out of proportion and create drama to forget about their sad miserable lives.If our young people need cookies, Play-Doh and videos of frolicking puppies to deal with the challenges in their lives right now, what in the world are they going to do when the things really start falling apart in America?The real world can be exceedingly cold and cruel, and our young people need to be equipped to handle whatever life will throw at them."

• RT with the latest on "A breakthrough in the study of a mysterious Hellenistic clockwork device which was found at the bottom of the Aegean Sea more than a century ago has led researchers to conclude it was designed for philosophers to peer into the future.Dubbed the ‘Antikythera mechanism’, after sponge divers hauled the bronze mechanism from a shipwreck off the coast of the Greek island in 1900, the machine has been described as an ancient computer because of its advanced technological design."
• SOTT has an interesting hour long interview "Stefan Molyneux and Dr. Duke Pesta on the dangers of Common Core"

# 1,1,2,3,5,....

Contrary to what you might be thinking, this post is not about the Fibonacci sequence; it's about an all too common mistake that teachers and test makers make. I've found this error to be very common at the high school level but what brought it back to mind was when I was exploring the IXL.com website recently. Let me start by saying I actually think the website is an excellent resource for teachers. Just find a skill that your students are deficient in and send them to the IXL site for practice on that very topic. If you get your class registered you can even get a lot of analytical data on how your students are doing. That said, the website has some problems, too. You might not think that an educational site would teach students incorrectly but you'd be wrong. A quick look around the website reveals some problems in probability, irrational numbers, and sequences. I'm going to focus on sequences. Sequences are covered in many grades so there are are many different links that are "teaching" students incorrectly, such as here or here. If you experiment (the problems get randomly generated) you'll have no trouble coming up with the type of wrong answer that I've already talked about in my post "...is Ambiguous". In that post I showed that given a formula for the first k terms of the sequence you can produce a formula to justify whatever real number you wanted to be next. But there are some variations you will see on the IXL site that have a slightly different format:

• Find the next move of a recursive sequence (e.g. 1,1,2,3,5,...)
• Find more than one missing term from another sequence (e.g. _, 3, 11, _, 27, 35)

Since neither of those is covered by the previous post (I'm ignoring the fact that recursive formulas can be rewritten as an algebraic formula--because we don't need that formula to create our function, unlike last post) I'm revisiting the issue. If these mistakes are so common then the subject should be the foundation of a lesson--to make sure students learn the correct theory and, hopefully, future teachers and educational sites won't teach students incorrectly. That, by the way, is one reason why I'm so adamant that you can't have teachers without a bachelors degree teaching math in high school. It doesn't mean that they all will know this (many still won't); it's just that there are many such pitfalls that a teacher with a poor math background will make. For example, I've tutored students recently that had a certified and tenured teacher "teach" that there are 6 ways to form a committee from 3 people. Turns out she had taught students about permutations for over a week using a word "committee" that should only be used in talking about combinations. Needless to say, I had to fix what she broke. But I digress.

The first type of problem is illustrated by the sequence 1,1,2,3,5,... which you might recognize as the Fibonacci sequence. Yes, that's AN answer, but it's not THE answer. In fact, THE answer doesn't exist; it's possible to produce a polynomial p(x) such that p(1)=1, p(2)=1, p(3)= 2, p(4)=3, p(5)=5 and p(6)= whatever we want. That is, ... is still ambiguous for reasons beyond what I mentioned in the earlier post.

Similarly, in the second case we can pick whatever missing numbers we want and produce a polynomial to go through them. Like a statistician, we can produce any answer we want. Unlike a statistics there is real math underlying the method. Don't believe me? For the sequence 1,1,2,3,5,.... I'm going to claim that the 6th term is -3 and give you a formula to prove it. Try this:

Notice that I've got some help getting the answer. Sage is going through the calculations to make things quick and avoid mistakes. You can see p(1)=1, p(2)=1, p(3)= 2, p(4)=3, p(5)=5 and p(6)=-3. Also notice I said "the polynomial"; there are other polynomials that would work but in the context used above it's about THE polynomial that comes about by following  a particular sequence of steps. Suppose, instead of -3, I want p(6)=11. No problem:

The same method works with the sequence:_, 3, 11, _, 27, 35. It looks like THE formula is 8x-13, so that the first term is -5 and the fourth term is 19. But it's not THE answer, it's just AN answer. Here's one such polynomial.

And there is no reason that the terms have to be integers; it just makes the polynomial nicer. In this one pi is used as a missing term in the sequence. You can click on the image below to make it bigger.

The bottom line is that an nth degree polynomial can be created to go through any sequence of points [where the x values are different]. As a result, ...is ambiguous and it's a shame that many teachers and even educational websites keep making the same mistake. Going forward, I'll look at how Sage can help us compute these polynomials.

Here are some stories that caught my eye over the last week.

• Truth in American Education reports "Even though the Bill & Melinda Gates Foundation acknowledged that the Common Core State Standards and related assessments have not accomplished what they had hoped they are bringing the standards, assessments and accountability movement to higher education.....Of course we have to have standardized tests to go with the common standards

“It may be difficult to list everything students should know and be able to do,” the book said, “but when faculty are asked to focus on essential elements they are quite ready, willing and able to define priorities for student learning in their disciplines.”

One of the project’s goals is for the white papers to be used for the creation of tests, or assessments, that colleges can use in a standardized way. However, those possible assessments must be voluntary, the book said, and based on multiple measures rather than a simple box-checking, multiple choice test.

College faculty, the article noted, may be forced to adopt measures like these." I get the feeling that success isn't measured by academic performance but by the amount of money spent on standardized testing.

• The Valley Patriot explains "The Ugly Truth About Common Core Education" with such observations as "Private testing companies and textbook publishers are pushing for spending millions of dollars pushing Common Core because these corporations see the possibilities of hundreds of millions of dollars in profits generated from the new tests, computers, preparation guides, and Common Core-aligned textbooks.... The new ELA texts contain graphic pornographic depictions, which should cause deep concern for every parent. One especially graphic text in the curriculum is Toni Morrison’s The Bluest Eye, which graphically depicts rape, incest, and pedophilia. Is this what we want our students to read and “analyze” in English class? Dr. Sandra Stotsky, who wrote a large part of the ELA Framework for Massachusetts, which made Massachusetts the highest achieving state in the nation, resigned in disgust from the Common Core Advisory Team and is now actively opposing the implementation of Common Core."
• Graduation is approaching and you might think it's time to reward academically superior students. Not in this Texas school. Kens5 has the story: "National Honor Society (NHS) stoles are frequent sight at high school graduation ceremonies around the country, but one Plano Senior High School student is frustrated that he won't be allowed to wear one when he puts on his cap and gown next month. According to school practices, students are not allowed to wear NHS regalia....KellyAnn Frederick says a National Honor Society sponsor claimed school administrators want everyone to feel included in graduation and not single students out." Since said teenager wants to be singled out for his hard work the issue is apparently that the students who didn't excel will feel badly.
• The Dayton Daily News reports "Ohio students performed dramatically worse than expected on two state exams this spring according to preliminary results, and state officials want to soften the scoring system so more students pass....An Ohio Department of Education memo leaked by Wagner shows that only 24 percent of students scored proficient or better on the Geometry test, and only 21 percent did so on the other sophomore math option — Integrated Math II. Students had to get roughly half of the questions correct to be proficient....So Jim Wright, director of ODE’s office of curriculum and assessment, on Wednesday proposed softening the scoring system so that 52 percent of students would pass the Geometry exam and 35 percent would pass Integrated Math II." The rhetoric of higher standards sounds great until your numbers look bad. The result? "Wagner calls the system “class warfare,” pointing out that for years, test scores have closely tracked income levels, with poorer students scoring worse." Luckily some educators are actually interested in standards, "“We’re increasing the rigor as we should. We should make sure that every child has the ability to perform at something after high school,” Gunlock said. “Hiding behind the fact that they all got a diploma even if they weren’t qualified doesn’t make any sense to me. … The idea that this is holding back poor kids is baloney.”". The point was to raise the academic bar because student achievement was poor. The fact that the DOE wants to pass them anyway shows you how academic performance is not a priority.
• A surprising result in Shakmir 2016. Tournament favorite Caruana lost in the penultimate round and drew the final round to tie for first with Mamedyarov--Mamedyarov, having beaten Giri in the last round then beat Caruana in the playoff to win the title. Chessbase has the report here "Tied for first, both Fabiano Caruana and Shakhriyar Mamedyarov faced each other in a Playoff shortly after the end of Mamedyarov's game against Giri. In the first mini-match of games at 10 minutes plus 3 seconds, Fabiano Caruana achieved a decisive advantage in both games, but each time failed to deliver the killing blow and the Azeri managed to escape. With two draws and a score of 1-1, the momentum had to seem to swing to Mamedyarov's side, if only for psycholgical reasons: escaping 'certain defeat' twice in a row had to be a confidence booster, feeling the opponent is just not firing all cylinders at this point. Furthermore, Caruana is not known to be the best blitz player (his FIDE Blitz rating is not a fluke), an area where Shakh might now feel he had he edge.Whatever the truth of it, Mamedyarov won the first game in a very tense battle that had everyone on the edge of their seats. Vishy Anand, watching online, was no less riveted and commented about their great fighting spirit, "We are not worthy!". Game two saw Caruana in a must win situation with black, but being blitz, anything goes. After reaching a clear advantage in a rook and opposite-colored bishop ending, it was the opinion of grandmaster pundits that Fabiano was going to win it and level the score, but everything went south for the American and after a wild scramble he was suddenly worse with no chance of winning whatsoever, and the players shook hands." Hou Yifan had a disappointing last place finish. Playing with the men is quite different when every game is a battle. Judith Polgar's place as a (former) top 10 player in the world still puts her as the best woman ever.
• Gen Whiny is on the move in Yale University yet again. According to Reason, "Some Yale University students are demanding changes to the English Department curriculum: specifically, they don't think it should feature so many English poets who were straight, white, wealthy, and male. "It is your responsibility as educators to listen to student voices," the students wrote in a petition to the faculty. "We have spoken. We are speaking. Pay attention....Students should feel comfortable challenging the notion that a Shakespeare or a Milton deserves his place in the canon: in fact, that sounds like an excellent subject for a classroom discussion facilitated by a professor. But professors need to actually teach students about these pivotal figures before those discussions can be had....In a brilliant piece for Slate, Katy Waldman eviscerates the idea that non-white students have nothing to learn from dead white poets: ...

But even if you disagree, there’s no getting around the facts. Although you’ve written that the English department “actively contributes to the erasure of history,” what it really does is accurately reflect the tainted history we have—one in which straight white cis-men dominated art-making for centuries—rather than the woke history we want and fantasize about. There are few (arguably no) female poets writing in Chaucer’s time who rival Chaucer in wit, transgressiveness, texture, or psychological insight. The lack of equal opportunity was a tremendous injustice stemming from oppressive social norms, but we can’t reverse it by willing brilliant female wordsmiths into the past.

In tangentially related news, Nicholas and Erika Christakis—who declined to protect students from offensive Halloween costumes—have finally resigned their positions at Yale." The adults in the room are now gone.

• The Federalist explains how the Gen Whiny came to be, "Forty-five years later: By this time, we had instituted a wildly successful pogrom in higher education, eliminating most of the conservative faculty and driving the rest underground. We ran amok, instituting freethinking, progressive reforms at every level of the system....Gradually, new codes of conduct were instituted to ensure everyone was comfortable at all times. ...The enlightened former freaks who now inhabit these campuses have become increasingly hypersensitive and nasty, spitting tacks at people for all manner of imaginary crimes such as “cultural appropriation.” Recently, a white guy got hammered for wearing dreadlocks. This is deeply ironic because, as I recall, we hippies were masters of cultural appropriation. Hookahs, Nehru jackets, bead curtains, reggae, Eastern religions, sitar music, Tibetan prayer flags, chakras, ethnic food, dashikis, Rasta shoulder bags, ironically worn military apparel, mandalas, henna tattoos, muumuus, hand-woven Guatemalan tunics, pyramid power, Maori tattoos, macramé... Meanwhile, at Emory University, some fascist from outside the perimeter jumped the razor wire and wrote “Trump 2016” on the sidewalk in chalk —a situation that could easily be remediated with a bucket of water. Instead, the student government allocated emergency funding for counseling sessions, and the college administration issued the usual limp apologies and assurances. Students were afraid to attend classes because they might sit near someone with a different worldview.....Today’s outraged, privileged, fragile snowflakes conjure up utterly trivial nonsense to consider as an affront: microaggression. This can include using the wrong one of more than 50 gender pronouns, sideways glances, snort-chuckling, eye rolling, resigned sighing, and even merely existing in proximity to a person with raw sensitivity. "

# Sagetex and Graph Theory

I've added some information to the Graph Theory, Sage, LaTeX page. For more complicated graphs, typesetting the placement of each individual vertex is too time consuming. Sage can help us out. Some of the important commands are located here. Many of the named graphs are posted here. SageTeX lets you use the power of Sage in both creating the graph and calculating the various parameters you want. You can find more details posted on that page along with working code to get you started.

Here are some stories that caught my eye over the last week:

• WTVR.com, CBS 6, reports "The mother of a Virginia middle school student said she was angry, frustrated, and mad after her son was handcuffed, suspended from school, and charged with stealing a 65-cent carton of milk from the cafeteria. As it turned out, Shamise Turk's son Ryan was on the school's free lunch program and the milk -- was free.....Police said the larceny charge was because Ryan tried to "conceal" the milk, a claim Ryan's mother denied. The school spokesperson said Ryan was suspended for theft, being disrespectful and using his cell phone in school."
• Money.CNN says, "For years, America's college campuses swelled with more and more students. But enrollment peaked in 2010 at just over 21 million students. Attendance has dropped every year since....The two types of colleges with the biggest declines in enrollment are: community colleges and for-profit universities. Those schools draw heavily from low-income and minority households."
• Science Daily with a piece on titled "40-year math mystery and four generations of figuring" which "So, what is the Kelmans-Seymour Conjecture, anyway? Its name comes from Paul Seymour from Princeton University, who came up with the notion in 1977. Then another mathematician named Alexander Kelmans, arrived at the same conjecture in 1979. And though the Georgia Tech proof fills some 120 pages of math reasoning, the conjecture itself is only one short sentence: If a graph G is 5-connected and non-planar, then G has a TK5."
• Oregonlive reports "Portland Superintendent Carole Smith apologized Friday for allowing students and teachers at two schools to continue to drink tainted water after tests showed unsafe levels of lead...At Rose City Park, that meant students were free to drink lead-laced water for eight school days after district officials knew about the toxic results."
• ZeroHedge reports "Federal investigators revealed another blow to Detroit Public Schools this week. Meet Carolyn StarkeyDarden - the system’s former grant-development director - who has just been charged on suspicion of obtaining nearly \$1.3 million by lying about children’s tutoring services....Carolyn Starkey Darden set up a company and allegedly ran a scheme between 2005 and 2012 in which she submitted fake invoices for tutoring services that were never provided to students, according to charges filed by the U.S. Attorney’s Office in Michigan’s Eastern District." This is so typical of the public school system where huge amounts of money can be easily skimmed by the corrupt. Because it's almost never about the kids.
• RT on the 8 year old who brought a loaded gun to class. "It is unclear to whom the gun belonged and what made the boy bring it to school. The NYPD said there were no arrests made so far. Authorities believe that the boy might have had a dispute with another child, and they suspect his teenage brother had something to do with this incident, WABC reported....This is the sixth time authorities in New York confiscated a firearm inside a city school since March. In one case, a 14-year-old boy brought a loaded 9-millimeter gun to Middle School 61. "
• The Third Shakmir tournament is underway with chess superstars Caruana, Kajarkin, Radjabov, Hou Yifan and more. The games can be followed live here.

# Sage Interact: Generate Discrete Data

In an earlier post I designed a Sage Interact to generate data for continuous distributions. This week I've added a Sage Interact to generate data for discrete distributions. A screenshot of the Interact is shown above. To use the Sage Interact, copy the code posted on the Python/Sage page and paste it into any Sage Cell Server. Press "Evaluate" to start the interact. Pick the distribution you want to generate data for, adjust the parameters and the Interact will create your data as well as calculate standard statistics for the data. This is a quick way to generate examples for quizzes, tests, and lessons.

Here are some stories that caught my eye in the last week:

• the74million.org has a surprising piece on "Connecticut’s Shame: In One of America’s Richest Counties, a High School Has Been Failing for 50 Years". The school is so bad there isn't really any way to get worse "Even to the jaded, Bassick’s achievement statistics are disturbing. Last year, only 15 percent of students tested proficient in language arts on the new Common Core-aligned state tests. The percentage of students who met that benchmark in math? Zero." Good thing they have certified teachers getting the most out of them.
• Segregation--it wasn't that long ago. Actually, it's still going on. RT reports, "A northwest Mississippi school district has been ordered by a federal district court to "consolidate its secondary schools" that have long been separated along racial lines some six decades after the US Supreme Court ordered school desegregation.....This decision serves as a reminder to districts that delaying desegregation obligations is both unacceptable and unconstitutional,” said Principal Deputy Assistant Attorney General Vanita Gupta, head of the DOJ's Civil Rights Division. " In what way was the 50 year delay shown to be unacceptable?
• Sputnik news notes that "A former elementary school paraprofessional in the Atlanta area must report to jail by 6:00 on Friday evening after being charged with reckless endangerment for hanging a five-year-old student by his belt from a classroom blackboard."
• Reason.com closes the chapter on a teacher who used the N-word in class. "Andrea Quenette, the University of Kansas communications professor subjected to protests and a formal investigation after offending her liberal students, was cleared of wrongdoing. But she is still ultimately out of a job."
• EAGnews with a horrific tale of student animal behavior that was impossible to imagine decades ago. "South Fort Myers High School officials believe as many as 25 boys had “inappropriate activity” with a 15-year-old female student inside of a bathroom on campus Tuesday. Students told NBC 2 the incident occurred in a girl’s bathroom after classes ended and involved the school’s football team." The NBC 2 link has a video and there's another link here with a little more depth, "While she would not pinpoint why the different students were on campus after hours, she said South Fort Myers students involved in extracurricular activities — athletics or otherwise — participate in a study hall from 1:47 to 2:30 p.m. This is roughly the time frame for when the bathroom activities took place, Chandler said." So about 45 minutes in which we have dozens of students unaccounted for and no adult apparently around the area to see or hear what was happening.
• The Greeley Tribune on the decline of teachers in Colorado. "Since 2010, UNC has watched enrollment in its teacher preparation programs plummet from a high of nearly 4,000 in 2011 to just 2,900 last year. The state is expected to graduate only 2,000 next year, and it needs twice that amount, education officials say...There are states that are worse off than Colorado (Oklahoma, California). There are states that are doing better (Connecticut). But every state has seen its list of needs increase as the number of people pursuing teaching as a career has decreased." The school system is run by many people who aren't up to the challenge but there is little consequence for mismanagement and poor quality. Take the Connecticut situation above: what will be the consequence for poor performance? Will the school close for having 0% proficiency in math? How did the school get to a 0% proficiency rating with "professionals" running the school? Do teachers still get pay increases (above the rate of industry) for such an awful showing? Money will keep coming in from taxes, and the system will continue on. When there's no consequence for failure, expect more of it.

# Coin Tossing references

The subject of coin tossing keeps coming up, no doubt because it is something the average person can relate to--no abstruse definitions like with limits (in Calculus). And that forces me to go back and find references for some of the basic points in the arguments that should be known but aren't. I've decided it's about time to accumulate the basic points and along with references:

The probability of flipping heads on a coin is not 1/2. The assumption that flipping heads on a coin is 1/2 is a mathematical model and not reality which is akin to using 3.14 for pi. Coin tossing is a deterministic process in physics as demonstrated by a coin tossing machine, "To make his point, Diaconis commissioned a team of Harvard technicians to build a mechanical coin tosser -- a 3-pound, 15-inch-wide contraption that, when bolted to a table, launches a coin into the air such that it lands the same way every single time. Diaconis himself has trained his thumb to flip a coin and make it come up heads 10 out of 10 times. But what he really wanted to know was whether unrehearsed tosses -- by ordinary folk who flip coins with unpredictable speeds and heights and catch them at different angles -- would show that the outcome of the act was, in fact, random." Persi Diaconis, Susan Holmes, and Richard Montgomery are authors of the article "Dyanmical Bias in the Coin Toss" (.pdf). There is a Numberphile video with Diaconis (about 8 minutes) that gives a brief overview and there is a YouTube lecture by Diaconis (about 55 minutes) with more detail. One of the main assumptions is that you start the coiin with the heads side up is

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.

The first assumption isn't always true. The Abstract of paper (by Murray and Teare) mentions the odds of an American nickel landing on its edge is about 1/6000.

The deterministic nature of coin flipping can be found in the Phys.org article "Heads or tails? It all depends on some key variables" which says:

"But first, here's what the researchers concluded: Using a high-speed camera that photographed people flipping coins, the three researchers determined that a coin is more likely to land facing the same side on which it started. If tails is facing up when the coin is perched on your thumb, it is more likely to land tails up. How much more likely? At least 51 percent of the time, the researchers claim, and possibly as much as 55 percent to 60 percent -- depending on the flipping motion of the individual.In other words, more than random luck is at work."

I've put this post on the Other page for future reference. If you teach probability in school this is a good topic to "go beyond" the basic curriculum. Too many students learn that the probability of flipping heads is 1/2 and not that the probability of flipping heads on a fair coin is 1/2. And the coins around us in the real world aren't fair.

Here are some stories that caught my eye this week:

• What do you do with your spare time? According to Sott.net, "A 15-year-old boy believes he has discovered a forgotten Mayan city using satellite photos and Mayan astronomy. William Gadoury, from Quebec, came up with the theory that the Maya civilization chose the location of its towns and cities according to its star constellations. He found Mayan cities lined up exactly with stars in the civilization's major constellations. Studying the star map further, he discovered one city was missing from a constellation of three stars. Using satellite images provided by the Canadian Space Agency and then mapped on to Google Earth, he discovered the city where the third star of the constellation suggested it would be."
• ZeroHege has an article showing just how deep the economic problems bite, "According to Bloomberg, a new survey by Discover Financial Services found that 48% of parents think their child should pay a portion (if not all) of the cost of attending college, up from 39% four years ago. And just how will potential students pay that portion? Why, student loans of course. 32% of respondents said they would ask the bank for help, while 27% plan to rely on family savings, 4% said they would dip into retirement funds, and 3% even indicated that they may refinance their home to pay for their kids college."
• USNews has an excellent article on just how badly our schools are letting us down, and there is an economic price. "One in four who enter college immediately after high school graduation must pay college-level prices for high school-level classes....But before you write off inadequate high school preparation as a function of a student's family background or the type of college they attend, know this: Nearly half of first-year remedial students come from middle-class, upper middle-class and wealthy families. Forty percent are enrolled at public and private four-year colleges...We already knew that high schools typically underserve students from low-income families and communities, but apparently they're doing poorly with wealthier students as well. It turns out that all students are susceptible to the leaky K-12-to-college pipeline – no one is immune. This should be a wake-up call for all."

# tkz-graph: more options

My confusion over the way to implement graphs in LaTeX prompted an earlier post where I started a Graph Theory, Sage, and LaTeX page. The first post suggested the Normal style for creating a basic graph: it's quick, clean and saves on printer ink. But using tkz-graph and tkz-berge you gives you a lot more control, if you need it. The two packages have enough differences in their approach that I thought a page of templates to serve as a starting point would be useful for me. Graph theory and discrete math, unfortunately, don't have much place in the educational curriculum so I've gone a little bit lighter on the details.

As you can see from the screenshot above we can change various aspects of the graph: the vertex color, the text color in the vertex, the color and thickness of the edges and even add labels. You can download the template and experiment with the code yourself.

Here are some stories that caught my eye over the last week:

• Start with the Detroit teacher strike. Detroit is a poster child for what's wrong with public education. EAGnews has the coverage, "Detroit Federation of Teachers members ditched their students to protest about pay today, despite data that shows they’ve been ineffective at improving the district’s worst-in-the-nation student academic performance....The combined results for students of all grades tested last spring shows a mere 2.9 percent met basic proficiency standards for science, 7.9 percent reached that threshold for math, 8.1 were proficient in Social Studies, and 14.6 met standards in English Language Arts." It's difficult to argue that teachers are professionals when their results are this bad.
• KOTAnews reports that Wyoming is dropping out of common core. Red Alert News claims that North Dakota is the 9th state to reject Common Core. The Salt Lake Tribune says "On Wednesday afternoon, Republican gubernatorial candidate Jonathan Johnson called for the end of SAGE testing in Utah schools.Not to be outdone, Gov. Gary Herbert issued his own call hours later, urging the state school board to abandon both SAGE and Utah's statewide education standards for math and English."Today I am asking the State Board of Education to consider implementing uniquely Utah standards," Herbert said in a letter to the board, "moving beyond the Common Core to a system that is tailored specifically to the needs of our state."".
• Be careful about doing math in a public place, especially if you seem like a foreigner. SOTT.net on "A woman sitting next to an Ivy League economist told flight crew she had security concerns about the man, after seeing him write in a foreign script. It turned out to be a differential equation.". How stupid have are we?!?
• The Washington Post has a fascinating article on "Education activists are increasingly becoming concerned about the computer grading of written portions of new Common Core tests....The standard PARCC contract indicates that this year, Pearson would score two-thirds of the students’ writing responses by computers, with only 10 percent of these rechecked by a human being.  In 2017, the contract said, all of PARCC writing samples were to be scored by machine with only 10 percent rechecked by hand...This policy appears to contradict the assurances on the PARCC scoring FAQ page that says,“Writing responses and some mathematics answers that require students to explain their process or their reasoning will be scored by trained people in the first years.”...The Pearson and AIR contracts also promised studies showing the reliability of computer scoring. ...According to Les Perelman, retired director of  a  writing program at MIT and an expert on computer scoring, the PARCC/Pearson study is particularly suspect because its principal authors were the lead developers for the ETS and Pearson scoring programs. Perelman said:  “It is a case of the foxes guarding the hen house.  The people conducting the study have a powerful financial interest in showing that computers can grade papers.”....Indeed, research shows it is easy to game by writing nonsensical long essays with abstruse vocabulary......Unable to analyze meaning, narrative, or argument, computer scoring instead relies on length, grammar, and arcane vocabulary to do assess prose....On April 5, 2016, the same day we sent the letter, Rhode Island Education Commissioner Ken Wagner ...claimed that “the research indicates that the technology can score extended student responses with as much reliability- if not more reliability- than expert trained teacher scores …..”". In an educational system that creates a pathetically weak "standard" of teacher certification that has little to do with quality resulting in most students being unprepared for college even though graduation rates are rising, computer scoring makes perfect sense. Design an algorithm that can be programmed to deliver whatever percentage of good scores you want. It's not really about education, it's who gets the dollars and how to deliver product for as little money as possible.
• It's not really math, so perhaps we shouldn't be too surprised how badly statistics is doing in predicting the elections. ZeroHedge reports, "How poor have the election forecasters been this year?  It is a topic many are discussing given the large number of upsets we've had during the Primaries.  For example, statistician Nate Silver (who started the campaign season proclaiming Trump had <2% chance of being nominated) by March 1 predicted with 94% probability that Trump would win Alaska (he lost). Silver then predicted on March 8 with >99% probability that Clinton would win Michigan (she lost).  Silver again predicted on May 3 with 90% probability that Clinton would win Indiana (she lost).  But there is another issue besides being wrong, which is how much model flip-flopping is occurring just up to these elections. The most proximate example is Silver stating this past Sunday that Cruz had a 65% chance to win Indiana; the next day (Monday, the eve of the election) and with little new data, he "adjusts" that to Trump having a 69% chance to win!  That's horrible! ". Low level math has you churn out answers supported by work for people to check. In statistics you show your data (which may have been massaged or manipulated) and argue your case. The essence of math is proof, and the nontheoretical statistics we encounter most every day has a high BS content. Statisticians like to claim how good it is "...if it's done right", meaning their way. But when the integrity of the data itself isn't open for inspection, statistics is open to widespread abuse in a way that math can't be.
• The Hechinger Report addresses the high math failure rates at universities. "A few years ago, administrators at San Diego State noticed high “D-F-W” (grades D and F, and withdraw) rates — 35 to 50 percent — for math courses, according to Michael O’Sullivan, chair of the math and statistics department. In 2014, the newly elected O’Sullivan, along with frustrated faculty, decided to overhaul the program. ....The changes at San Diego State and in other colleges’ math classes are similar to components of the Common Core Standards for Mathematical Practice...As of this year, San Diego State has implemented all seven pieces, said Michael O’Sullivan. It is too soon to know long-term results, of course, but for now the professors are happy that this semester’s Calculus II midterm grades increased by five to eight percent compared to previous years, according to Ricardo Carretero, professor of applied mathematics."
• Your funny money is no good here. ABC13 reports, "Now 14, Daneisha was hoping to eat that day's lunch of chicken tenders with her classmates using a \$2 bill given to her by her grandmother when she was stopped by the long arm of the law. "I went to the lunch line and they said my \$2 bill was fake," Danesiah told Ted Oberg Investigates. "They gave it to the police. Then they sent me to the police office. A police officer said I could be in big trouble."Not just big trouble. Third-degree felony trouble. And that's just one of eight counterfeiting charges investigated against high- and middle-school students at Fort Bend ISD since the 2013-2014 school year....Then the Fort Bend ISD police investigated the \$2 bill with the vigor of an episode of Dragnet, even though at that school 82-percent of kids are poor enough to get free or reduced price lunch.The alleged theft of \$2 worth of chicken tenders led a campus officer...to the convenience store that gave grandma the \$2 bill...... The \$2 bill wasn't a fake at all. It was real....The bill so old, dating back to 1953, the school's counterfeit pen didn't work on it...."He brought me my two dollar bill back," Joseph said. He didn't apologize. He should have and the school should have because they pulled Danesiah out of lunch and she didn't eat lunch that day because they took her money.""

# SageTeX: First Derivative Test

This week I've added another problem to the SageTeX: Derivatives page. The screenshot is above. Given a polynomial students need to create a table to show when the function is increasing or decreasing. Then they have to determine the local extrema.

Filling out the table makes use of Sage's ability to calculate derivatives. Here's a small snippet of the code. Note the indentation has been lost.

if df(0)>0:
a13 = '+'
a14 = "increasing"
else:
a13 = '-'
a14 = "decreasing"

Depending on whether the derivative is positive or negative we can fill out the chart. And the same logical reasoning that allows you to fill out the chart by hand is the same logical flow that the Python code goes through--but without the errors we humans are prone to make. And of course, recompiling can generate lots and lots of problems with an answer key, faster and more accurately than any human could.

Here are some stories that caught my eye this week:

• ZeroHedge has the latest on the Chicago Pension Scandal, "Take the example of two union lobbyists who substitute taught for one-day in the public schools and then started collecting over \$1 million of lifetime public ‘teacher’ pension payout – despite a state law expressly designed to stop them. And now take all the other 7,499 educators. The retirees in question paid so little into their own retirement (breaking even on their cost basis within the first 20 months of retirement) that taxpayers now face a \$900 million bill just to keep the pension payments flowing!...The fraud appears to be focused on the city of Chicago. Some examples:
• Northern Illinois school districts are driving the majority of \$100,000 pensions. In fact, 6,706 pensions for over \$800 million in annual payouts were conferred by districts in the Chicago metropolitan suburban area. Only 793 six-figure pensions totaling \$95 million in annual payouts were conferred by school districts in the rest of the state. Yet, income-taxpayers across the whole state guarantee the retirement annuities for everyone.
• The Top 100 All-Time pensions: #1 \$302,991 (Lawrence Wyllie at Lincoln-Way CHSD) to #100 \$200,812  (Michael Radakovic at Aurora East USD 131). Read the Top 500 All-Time IL teacher pension list.
• The Top 5 school districts conferring six-figure pensions are Palatine TWP HSD 211, Palatine (449); Township HSD 214, Arlington Heights (419); Consolidated HSD 230, Orland Park (196); Northfield TWP HSD 225, Glenview (188); Maine TWP HSD 207, Park Ridge (180)."
• You thought hoped her 15 minutes of fame were over. But it isn't--she's baaaack! Melissa Click was interviewed, shamefully enough, by the Chronicle of Higher Education. Reason.com dissects it all: "More:

While Ms. Click acknowledges that she was certainly frustrated that day, she says she was simply trying to protect the black student protesters. Everything she has come to stand for since the video came out—intolerance, anger, mouthiness, and dismissiveness—is exactly the opposite of who she says she really is. Focusing on her behavior, she says, is a way to take attention away from the demands of Concerned Student 1950, the group of protesters.

"I’m not a superhero," Ms. Click says. "I wasn’t in charge." But she’s taken the fall. "When it got out of control," she says, "I was the one held accountable."

And not by accident. Click was held accountable because she committed assault. If her behavior is drawing attention away from the student activists, that's entirely her own fault.

The Chronicle story also includes some biographical details that uncritically accept Click's I-am-a-hero narrative....Is this the profile of a woman who has overcome great adversity, or the profile of an intolerant ideologue firmly convinced of her own greatness?.....But she doesn't deserve sainthood, either. She did a very bad thing, and her revisionist attempts to explain away her criminal behavior should be rejected. Assault is wrong, even if the person committing it has a minor in women's studies."

• An article in the Tennessean says, "Police handcuffed multiple students, ages 6 to 11, at a public elementary school in Murfreesboro on Friday, inspiring public outcry and adding fuel to already heightened tensions between law enforcement and communities of color nationwide.The arrests at Hobgood Elementary School occurred after the students were accused of not stopping a fight that happened several days earlier off campus. ....Murfreesboro police didn't say what state law the kids violated, but parents of several of the arrested children say the kids were charged with "criminal responsibility for conduct of another," which according to Tennessee criminal offense code includes incidents when a "person fails to make a reasonable effort to prevent" an offense."
• NPR reports on the latest report card of public high schools, "This tiny downtick was statistically significant in mathematics, but not for the reading test....NAEP scores are comparable across decades — back to 2005 for math and all the way back to 1993 for reading....According to research by Ho and others, just under 40 percent of students score at college and career ready levels on NAEP....One is that in 2015 the nationwide high school graduation rate was 82 percent, not 40 percent. That leaves a potentially large group of kids who got diplomas but who weren't ready to succeed in college. Who is right: their high schools or NAEP?
...On the other hand, he says, "the less-than-charitable view would be that graduation is just a lower standard than college readiness. If you get right down to it, the reading and math required by NAEP, the ACT, the SAT, colleges and careers is much greater than what high schools are saying is sufficient." High schools with dumbed down standards to increase graduation? Shocking!
• Most of us aren't particularly excellent at what we do, but imagine if we could compete against high school students. We'd look really good then. Vice News continues "A few weeks ago, Jonathan Nicola was the star player on his high school basketball team, with a coach who believed he had a shot at going pro. But now that it's come out that the 17-year-old is actually a 29-year-old man, his ambitions — whatever they may have been — have been put on hold, and he's begging the Canadian government to send him back home to South Sudan."
• The Huffington Post looks at standardized testing, "Turns out, academic conformity sells, and business is booming: As of 2011, Gaston Caperton, the president of the College Board,nonprofit owner of SAT, was paid \$1.3 million. Richard Ferguson, formerexecutive officer of ACT Inc., made roughly \$1.1 million. Meanwhile, The National Board on Educational Testing and Public Policy at Boston College reported that the value of the standardized testing market was anywhere between \$400 million and \$700 million........Standardized testing isn’t just about every student meeting the same standards. It is about every student, school, and administrator paying for the same product." So, so true. And public education means there companies have less people to persuade. Give an incentive for those at the top and a lot of money flows to the company. A good article which is worth reading very carefully.
• Chessbase reports on Fabiano Caruana winning the US Chess Championship. The US Women's Chess Championship was won by Nazi Paikizde after the favorite to win, Irina Krush, had an uncharacteristically bad tournament and the tournament leader, Tatev Abrahamyan, lost the final round. Some very ugly chess by the women but fighting chess that kept me riveted as never knew who would make the final mistake to lose. The tournament was followed by the Ultimate Blitz championship which featured Kasparov, Nakamura, So, and Caruana. Nakamura won, Kasparov was only .5 points behind but the highlight was a So-Kasparov brilliancy. The Chessbase report quotes Yasser Seirawan as saying "Wesley's game against Kasparov will go down in history as one of the greatest blitz games ever played. I will remember that game for the rest of my life." and Kasparov said "It reminded me of games Morphy played against amateurs.". It's that brilliant. Make sure you check out the game at the Chessbase link.
• It looked like Carlsen was going to easily win the Altibox Norway Chess tounament. A loss in the penultimate round kept the issue in doubt but winning the final round gave Carlsen first place. Chessbase has the story here.
• Yet another Common Core defector wanna-be: Michigan. Truth in American Education says, "Michigan’s Common Core Repeal Bill just made it over a a major hurdle. The Senate Education Committee voted to pass SB 826, a bill that would repeal Common Core and replace them with Massachusetts pre-Common Core standards."

# Graph Theory, Sage, and LaTeX page

I've added another page to the website. It's still in the beginning stages but it will cover graph theory, \$latex \LaTeX\$ and Sage. You can find it here and it's listed on the sidebar or click here. From this documentation at the Sagemath website you can see the way Sage works with \$latex \LaTeX\$ to produce graphs. This method is quite different than what you'll find in the Altermundus tkz-berge package despite the fact that Sage uses that package. The information I just posted has to do with the creating a graph in the simplest and cleanest way. I opted for the 'Normal' representation of graphs to allow for a choice of labels inside or outside. A vertex is created as follows:

\Vertex[LabelOut,Lpos=270,L=\$v_1\$,x=0,y=0]{R1}

where R1 is what the vertex will be referred to by the program and the factors which determine the label are handled in the options:

LabelOut to put the vertex label outside the vertex. Lpos is the label position. It's the angle of rotation around the center of the vertex (0 is east of the vertex, 90 is north of the vertex, etc). L= will be the label, and x=, y= handle the placement of the vertex. Simple and efficient. You can read more on the new page where I've posted the code for two 'Normal' graphs, with labels out and with labels in.

Here are some stories that caught my eye this past week:

• 11alive covers Tennessee phasing out Common Core. "Common Core standards ignited a political brawl last year when state lawmakers, who saw the standards as federal overreach, pushed to scrap them. In response to cries for state-specific standards, Gov. Bill Haslam authorized a review of the state's English and math standards."
• St Louis Post Dispatch reports "Missouri became the latest state to adopt a new set of education benchmarks to replace the national Common Core standards, ditching the benchmarks Tuesday following conservative backlash.". Common Core is not so common anymore.
• EAGnews has a piece on  Tennessee students "Parents and community leaders in Murfreesboro are fuming after police arrested numerous elementary students at school over a fight that occurred off campus several days prior....Several students at Hobgood Elementary School, ranging in age from about 6 to 13 years old, were then handcuffed and arrested, and hauled to a juvenile detention center before their parents were even notified, WKRN reports.....“Here’s the catch, the girls were just bystanders in the video. They were not fighting, they were not instigating it, they were just standing in the video. And there was actually no technical fight in the video,” he said. “But the claim and the accusation is that they didn’t stop the fight, so they should be arrested.”"
• RT has a piece where the title says it all: "Teacher tackles pupil to the ground in ‘deeply disturbing’ classroom attack".  A violent, profanity laden assault that will hopefully lead to jail time.
• But it's better overseas, right. Not all the time. RT has video footage of  "A teacher in China bore the brunt of a vicious classroom uprising after he tried to forcibly collect a student’s exam paper in a shocking case of school violence caught on camera."
• Don't even think about messin' with Colorado schools. ABC news reports, "A suburban Denver school district is arming its security staff with military-style semiautomatic rifles in case of a school shooting or other violent attack, a move that appears unprecedented even as more schools arm employees in response to mass violence elsewhere."
• Chalkbeat reports  "Facing widespread backlash after years of controversies and testing glitches, one of the world’s largest testing companies is taking an unusual approach to quieting critics: It’s opening its doors." The Washington Post covers some of the problems here.
• The Altibox Norway Chess 2016 tournament has 4 rounds in  the books. Magnus in first but with Kramnik, Giri, and MVL lurking around the tournament is wide open. You can follow it here. Click on the "Live Streaming" button at the top of the page to follow it live.
• There are just 2 rounds left in the US Chess Championship. You can watch it live here. Caruana leads for the championship section, and Abrahamyan for the women. Irina Krush doesn't control her fate anymore as she competes for another title.

# Altermundus: circle-circle intersections

I've added information on getting the intersection of 2 circles using the tkz-euclide package. The tkz-euclide package gives a macro \tkzInterCC to find the intersection of 2 circles. As there are multiple ways to input a circle, the macro can be used in different ways.

\tkzInterCC(D,B)(A,C) \tkzGetPoints{M}{N}

finds the intersection of circle centered at D containing point B along with the circle centered at A containing point C. The two intersection points are recovered with \tkzGetPoints macro. Likewise

\tkzInterCC[R](A,1 cm)(B,1 cm) \tkzGetPoints{M1}{N1}

finds the intersection of circle centered at A with radius of 1 cm along with the circle centered at B with radius 1 cm. Note the R option.