Odds and Ends: July 9,2014

Several odds and ends:

  1. I've updated the Sage Cheat sheet for version 6.2; correcting errors, typos, and any deprecated commands. You can find it on the Python/Sage page (at the top).
  2. The OP/ED page of the NY Times has letters related to the common core article mentioned in a previous post. The letters are here.
  3. Wisconsin is gearing up for a war on common core. "If you think there was a fight over Common Core educational standards in the Capitol earlier this year, think again. Interviews with dozens of candidates show that it was just a warm-up for the war coming next year over those standards."...."WisconsinEye interviews with more than 70 candidates in Aug. 12 primary elections show that scrapping Common Core standards, or reasserting the authority of local school boards to set their own academic goals, is a top priority of Assembly Republicans. Most Democratic candidates defend Common Core."

Sagetex: Random Bipartite Graphs

RandBiSummer's here and I've had the luxury to just play around. I try to put extra math into the curriculum through either warm up problems, extra credit problems, or examples in the  curriculum that I follow. Discrete math figures prominently in the extra math. In my opinion, discrete math deserves a prominent place in the high school curriculum; it certainly deserves a bigger share of the curriculum. But in a top down system you can't add extra material to the curriculum that the average math teacher has never seen and expect it to be implemented properly.

Graph theory is ideal because it requires little mathematical background to understand and solve problems yet it also provides a way to introduce theorems (many of a basic nature that high school students can understand) along with counterexamples thereby providing a basic primer into the sort of considerations that mathematicians are looking at. In the past, geometry had been the topic for introducing proofs but it's dry as dust and now common core has ripped out much of the proof content so that your typical student can go through math and with very limited exposure to theorems/proofs/counterexamples.

Graph theory would be a way to get that back in a way that is much more interesting than the dry "2 parallel lines are cut by a transversal" proofs that have bored generation after generation. You can even connect to matrices through the adjacency matrix of a graph. Topics like recurrence relations and generating functions complement the curriculum of calculus (power series) and probability. Unfortunately, common core has de-emphasized proofs and matrices and upped the exposure to statistics, so I think it's my duty to undue the damage of the curriculum.

Which brings us back to the current post. Bipartite graphs which have the same number of vertices in each partite set provide a way of determining whether a suitable job can be found for each person so that everyone has a job (see the "Matching" section for the link above). One partite set with vertices, say, A,B,C,D have edges to the other partite set of vertices (a,b,c,d) representing jobs based on whether the person is qualified to do the job. The question becomes, Is there an assignment of jobs so that everyone has a job they're qualified to do? If yes, then there is a (perfect) matching for the graph. If no then there needs to be an explanation as to why no such assignment is possible (Hall's Marriage Theorem).

As a teacher the problem I usually face is finding lots of examples that can be used in the classroom or on a test/quiz. Sage/sagetex is the tool that works best for me. The code generates bipartite graphs so that the number of vertices in each partite set is equal and the probability of an edge being selected for the picture is 1/2.  The code is posted on the Graphics page along with sample output (the PDF shown above). The number of vertices is a random number set between 4 and 8 here: N = Integer(randint(4,8)). The probability of the edge being chosen is set here: if Integer(randint(0,1))==1:

It would be relatively simple to choose a larger number than 1, such as 9, and draw the edge if the number chosen is between 0 and 7 but don't draw the edge if the number chosen is 8 or 9. That would make the probability of the edge appearing to be 80%.

Through setting the parameters to your liking and repeatedly compiling the code you can generate many examples that you can use. If you're particularly ambitious you can add extra code spit out example that have (or fail to have) a perfect matching. Of course, Sage can help with that.

Odds and Ends: July 1, 2014

FlowChartA few minor changes:

  1. I've added the flowchart/tree diagram (PDF and tex file) to the Graphics page. When talking about linear systems I've found students slow to pick up the vocabulary of consistent/inconsistent and dependent/independent. The flow chart helps students to remember the difference.
  2. The important graph from the last post, has been posed as a problem: Find the derivative of f(x)=x^2\sin(1/x^2) if x \neq 0 and f(0)=0 at 0. I think it would be well suited as a warm up problem at the beginning of class to motivate the subsequent discussion.
  3. The NY Times has a good article on the difficulties with Common Core.

A very important graph

SinIx2sinEveryone studying mathematics needs to know some typical functions and some important characteristics. At the lowest level you need to know the trig functions and their (as applicable) domain, asymptotes, midlines, periods, etc and for exponentials and logarithms you should know the domains and asymptotes. At a slightly more advanced level, students should be aware of the  function \frac{\sin(x)}{x} and how its limit exists at 0 (and is equal to 1) even though the function isn't defined there. Or that x\sin(1/x) is a function whose limit at 0 exists and can be determined using the Squeeze Theorem. The graph show above is also a function that needs to filed away but, surprisingly, it isn't well known. I say that because I have trouble finding it in a typical calculus text (or the lesson it teaches). That's a shame; the function is f(x)=x^2\sin(1/x^2) if x \neq 0 and f(0)=0.

Take the derivative and you'll get f'(x)=x\sin(1/x^2)-(2/x)\cos(1/x^2). And the derivative at 0? Chances are overwhelming that your students will say the derivative doesn't exist at 0 because f'(0) isn't defined. But that's wrong. That won't be apparent by looking at the graph of the derivative:DSinIx2sinFor the typical students who dive for a calculator so they can avoid thinking for themselves, they'll get to see that the calculator isn't going to help out here. They'll have to use the definition of the derivative (and the Squeeze Theorem). That's good practice. Ultimately this function shows that derivative can exist at a point even if the formula $f'(x)$ doesn't exist there. For all these reasons, this graph should appear in every calculus book.

The PDFs have been posted on the Graphics page.

Odds and Ends: Jan 22, 2014

Several issues to mention today:

  1. I've added 4 PDFs to the Graphics page. These illustrate how a graph can be discontinuous. More on that later.
  2. Magnus Carlsen now holds all world chess titles: World Chess Champion, World Rapid Champion, World Blitz Champion. The domination is complete.
  3. Common Core is becoming a conservative litmus test: The linked article notes, "Common Core standards initiative is now a “hot button issue within the GOP” that has even earned the nickname “ObamaCore,” one that “lays bare the political divide.”" and "Common Core standards will grow even more intense as their full impact will be experienced next fall when they become more widely implemented. For now, however, even U.S. Senate primary races in states such as Mississippi, Tennessee, and Louisiana have Common Core as a central issue."

The 4 PDFs on continuity deserves a comment. In teaching accelerated Precalculus I actually spend about 1/4 of the year covering 3 chapters in Calculus. As it was my first year teaching that in a US public school I "stuck to the script" and followed the book closely. The book mentions that points where a function is discontinuous is because

  • the function is not defined there (removable singularity)
  • the limit doesn't exist at that point
  • the limit exists there but differs from the function value

and goes on to introduce removable/nonremovable discontinuity and jump continuity. Those examples are shown in the PDFs I added (Discont1 through 3). But, strangely enough, the book never mentions essential discontinuity. If your book is the same, now you know why I have 4 pictures [Edit: It's \sin\left(\frac{\pi}{x}\right)]. My book then, would have to classify that as a nonremovable discontinuity which is not a jump discontinuity. Why not mention the term and use it to flesh out the idea of continuity: that essential discontinuities are the worst type of departure from continuity and that knowing information about the discontinuities of a function will be important in representation of functions as infinite series and Riemann integrability. I see no reason why the book should fail to mention the term essential discontinuity.

Sagetex: Combinatorics/Probability (8/9)

Comb8I've added 2 more problems to the Sagetex: Combinatorics/Probability page. The problem above has the form:

A high school committee of 6 is to be made from 32 boys and 41 girls. Within this set of students there are 2 senior boys and 3 senior girls. How many committees of 3 boys and 3 girls are there that contain at least one senior boy and one senior girl?

It's worth noting an interesting aspect of this problem: in order to properly count, the various cases need to be listed and this means the output of the solution will be dynamic in the sense that the number of cases to be listed changes in the solution will change depending on our specific number of senior boys and girls. In the case of 2 senior boys and 3 senior girls we need to throw out committees with

  • no senior boy and no senior girl
  • no senior boy and 1 senior girl
  • no senior boy and 2 senior girls
  • no senior boy and 3 senior girls
  • no senior girl and 1 senior boy
  • no senior girl and 2 senior boy

Change the number of senior girls or senior boys and that list given in the solution changes.

The output of the various cases is handled in the sagesilent block. We need to avoid counting no senior boy and no senior girl twice, so no senior boy and no senior girl is calculated here:

NoSeniors = binomial(girls-Sgirls,3)*binomial(boys-Sboys,3)

and 2 FOR loops create the rest of the cases. Here the committees with 0 senior girls are all created

for i in range(1,Sboys+1):
output += r"$0$ senior girls and $%s$ senior boys: $C(%s, 3)\cdot C(%s,%s)\cdot C(%s,%s)$\\"%(i,girls-Sgirls,Sboys,i,boys-Sboys,3-i)
remove += binomial(girls-Sgirls,3)*binomial(Sboys,i)*binomial(boys-Sboys,3-i)

A similar loop handles 0 senior boys. Note that 3 is hard coded. That's the number of girls (and boys on the committee).  Increasing that number will make too many cases to enumerate for a test. Likewise

Sboys = Integer(randint(2,3))
Sgirls = Integer(randint(2,3))

keeps the problem size more reasonable as well; 6 cases is enough and, depending on the ability of your class, it might be more appropriate to decrease the number of cases; i.e., make the hard coded 3 a two.

Odds and Ends: June 14, 2014

Just a few odds and ends to mention

  1. Norway Chess 2014 is over and it's Car---Karjakin winning it for his 2nd time in a row. Carlsen takes second and Grischuk third. Some interesting, spirited, fighting chess along the way but lots of blunders along the way. Aronian blundering a queen in round 3, Aronian, Caruana, and Topolov in Round 5, Giri's self demolition in Round 7, Carlsen's epic blunder in a completely winning position in Round 8; it makes you wonder how much rating inflation is out there. It doesn't seem like the Fischer, Karpov, Kasparov, Botvinnik, Capablanca, etc. made mistakes as often despite having lower ratings.
  2. China Smack had a Hong Kong elementary school admissions test math question posted a short while back. While it took me about 20 seconds to get it, there are apparently a lot of adults that struggled. From the sample comments below, I liked this one "I tried various numerical/mathematical sequences, and even used calculus,". D'oh! This might destroy your stereotypes of people who are good at math; my experience teaching in Asia meant I wasn't surprised...
  3. Gov Bobby Jindal vetoed Common Core legislation. The tide seems to be turning as "Common Core" evokes a pained response in more and more people. More states are having second thoughts. Leaving aside content issues, the implementation in my state has been very poor. Imagine knowing there's a hard deadline years in advance and (years after the deadline) we still don't have Common Core textbooks. If that happened in a private school someone would lose a job; in the public school system---well, nothing happens. No incentive or reward for doing your job well and no real punishment when you do a terrible job. Think of Michelle Rhee finding all the "lost" textbooks and supplies in that warehouse. Who lost a job for mismanagement?

Sagetex: Combinatorics problems (6/7)

Comb6I've added 2 more problems (along with the solution) to the collection of randomized problems on the Sagetex:  Combinatorics/Probability page. The two new problems have the following form:

Problem Type 6: A committee will be made from 5 English teachers, 3 math teachers, and 4 history teachers. If the committee must have 2 teachers from different disciplines then how many committees are there?

Problem Type 7: How many nonnegative integers less than 100000
contain the digit 5?

Handouts: Notes on Induction

InductionNotesI've added some notes on weak and strong induction; they're posted on the Handouts page. The notes are a stripped down version that you'll need to flesh out. While I've given a motivation, definitions, example problems, and then an assortment of problems, I've left out a lot that you'll probably want to fill in. A high school level induction to induction shouldn't be the clean sanitized version you'll find in the notes; it should include the process of getting to the answer. That is, rather than present the proof as given in the examples, I would explain the scratchwork that gets you to the answer which is then cleaned up into the final version given in the notes.

The notes are a hack of the LeGrand Orange book template to 1. Get an article version 2. change the color 3. add green and red boxes to highlight information.

The large assortment of problems contains problems of varying difficulty so you'll need to work them out to decide what level is best for your students.

TCEC Season 6: Game 18

A look at the brilliant game 18.