Wednesday, June 29, 2016

Algebra II (Common Core) June 2016 Question 13

According to the Chicago Tribune (see here) the Ferris wheel on the Navy pier in Chicago was replaced with a new Ferris wheel that gives rides of 12 minutes and 3 revolutions. Opening on May 27, 2016, it is now named the Centennial Wheel.

I point this out due to question 13 on this year's New York State Regents Exam in Algebra II (Common Core) from June 1, 2016.:

This question shows an attempt to make mathematics "real world". The only two concerns I have are, firstly, why use information that is factually incorrect and, secondly, how "real world" would a Chicago Ferris wheel be to New York State students. At least use Coney Island or Great Escape!

Mathematically speaking, this question should be very simple. It is really just asking for the smallest value for H(t), and for an exam incorporating graphic calculators it is a snap.

Here is a GeoGebra model for one "car" on this wheel:

A more meaty question for Algebra II students would have been to identify the arc length between the position at time 0 and the bottom point on the trip.

Moral of the story: New York State, please stop hiding mathematics behind a veil of weakly designed "applications". 

Tuesday, June 28, 2016

Algebra II Common Core question 20 (my apologies to the "non-math" readers)

Here is a question from the Algebra II (Common Core) regents Exam from New York in June 2016. This question struck me as soon as I saw it. My comments are below.

Firstly, my immediate reaction was to recognize that there were three visible places where the tangent is horizontal, which is the max for a polynomial of degree 4, so, in essence, the graph could only zoom down and down in both directions were it continued. For an even degree polynomial, this is only possible with a negative leading coefficient.

I did a little canoodling with calculus and GeoGebra and obtained a function with a graph pretty close to the question in hand:

How many Algebra II students have a solid grounding in derivative calculus? Probably, like, none of them? So my reasoning process is no good (even though it is correct).

At this point something hit me: a graph of a pH-based function can not be equal to a polynomial. Polynomials have domains covering all the real numbers (the entire x-axis, if you wish), but pH values only range from 0 to 14. Also, since pH values can be less than 6 and more than 10, for the function to even have a degree 4 polynomial as a good approximation , the oxygen consumption of the snails would have to negative, meaning that the snails actually created oxygen instead of consuming it. If that were the case, load up the Mars mission with lots of snails in some very basic (or acidic) compartments!

I still cannot see how an Algebra II student would have any comfort (or understanding) of this question.

But, aha! It is multiple choice. Choice 1 is given (degree 4), choice 3 can be seen (2 humps), and choice 4 can be seen (two pieces where it descends left-to-right), so it ca only be choice 2 that is incorrect.

This question does give more evidence to the idea that one of the latest trends has been to go "real world examples" and risk losing touch with mathematics. Especially aggravating when the mathematics of the solution has absolutely nothing to do with pH values. The content amounts to built-in obfuscation 

Also, the graph takes an immediate jump from x = 0 to x = 6 without using any of the accepted symbolic notations (such as zig-zag or "squiggle").

How is an Algebra II student supposed to reason this question without falling into the "there's only one choice left" scenario?

Monday, June 27, 2016

How curvy would you like it?

Here we have a graph of both the Algebra I and Algebra II Common Core Regents exams in New York State for June 2016.

Almost identical.

Although Algebra II is awarded an average of 1.39 more points than Algebra I gets, with a maximum 4 points more, and a minimum of 0. Algebra II never lags behind Algebra I.

(Please note, I am not commenting on the relative difficulty of the tests.)

What interests me is the "closeness" of the two graphs.  Both give a fast track to a scaled score of 65 (minimum passing grade) and then slow down as they approach the top end.

Easy to pass, hard to ace.   It certainly appears to narrow the achievement gap.

Thursday, June 23, 2016

Algebra II (Common Core) Conversion

The horizontal axis is based on a student's response to 24 multiple choice questions (2 points each), 8 free response questions worth 2 points each, 4 free response questions worth 4 points each, and 1 free response question worth 6 points, for a total of 86 points. The vertical axis will give the grade as reported in the student's records and report cards.

The curve used here adheres very closely to a cubic equation which is included in the graphic.

I cannot help but notice the "fast track" to level 3, which is the "passing" level for current students. The raw score takes a very large jump before getting to the "mastery" level 5.

Please take note that a student who can answer 8 multiple choice questions, and must guess at the other 16 of them, will tend to get 12 correct (the 8 plus one fourth of the remaining 16), for a total of 24 points. That student needs to get 1 point out of all the free response questions in order to pass.  A student who knowledgeably answers 12 of the multiple choice questions, and guesses at the other 12, will tend to get 15 correct (the 12 plus one fourth of the other 12) for a total of 30 points, and at that point has already passed.

I will save my judgments for later, after I have had a chance to see the test itself.

Please take note that level 2 exists as a "safety net" for Special Ed. students, and its is a fairly narrow band. It seems the real goal in the scale is to get to level 3 quickly.

More later.

Wednesday, June 22, 2016

Summer is here!!!!

In honor of the end of the school year for New York State schools, I decided a little cartwheeling would fit the mood. Please take note, the earth is not a sphere, but an ellipsoid, so using a circle would be incorrect!


As is usual, this was created in GeoGebra, and can be found here.

Tuesday, June 14, 2016

United States of America or United People of America?

As we are progressing through this mess of a campaign I cannot but feel a bit disappointed by the National Popular Vote interstate compact. Those of you who are unfamiliar with it ought to find out about it. The electoral college "safety valve" would become a meaningless appendage, and any "hanging chads" would involve the whole country in a close election, and not just one state. Many of you probably think that the chads of 2000 decided the election, but that is overlooking the facts of the other 49 states.  (Do not crazy over the winning jump shot if the team already had 100 points.)

Please be aware that the NPV movement only needs states with electoral votes totaling 270 to "sign on", and it becomes moot for all the other states. Once the votes are counted, the "270" states electoral votes would be committed, and, as they say, that's the show, folks.

Should the NPV become the standard, a recount would involve all states and in each state it would take place under the laws of that state. A contested result could (actually, it might have to) involve court cases in each state, even in a state that voted hugely in favor of one candidate. Political paralysis would be quite possible.

What this National Popular Vote really attacks is the fact that we are the United States of America. This NPV push basically shoves states aside.  There is not even a national agreement on how to vote (see this). How can we say to the states that they can decide whether to use paper ballots, direct recording, punch cards, etc., and then tell them that their results may not matter anyway? Your state voted for candidate B? Ignore it, cause candidate A got more votes in other states.

In addition, what if a candidate has 270 supposed electoral votes before the west coast polls have closed? Should those voters just "skip it", as clearly the east coast votes are worth more?

New York and its governor signed off on the NPV in 2014.  Could this have been an overreaction stemmed by the fact that NY went Gore but the electoral college went Bush in 2000? I hope not. After all, since popular vote count was tallied  (starting 1824), 16 presidents have been elected while getting less than a majority (see here). In 2000 both Bush and Gore received less than 50%.  Analyzing further, Bush received over 50% of the votes in 26 states, Gore exceeded 50% in 14 states (plus D.C.).

In California and New York, Gore received 2,998,097 more votes than Bush. In the rest of the country Bush was ahead of Gore by 2,545,202. (see here).  Some people left the campaign believing that the Supreme Court stole the election for Bush. People could equally think that Gore almost stole it from Bush thanks to the media capitals of New York and California.

A lesson to be learned is that the electoral college helped limit the mess to Florida thanks to its less than stellar voting procedures. But lets keep in mind that the electoral college ultimately went with the winner in 26 states instead of the winner in 14 states. If NPV had been in effect, Gore might have been the winner, but not until recounts and courts had spoken in many many other states, in addition to Florida.

Interesting to note, in The Federalist Papers #68 (Alexander Hamilton) stated
It was equally desirable, that the immediate election should be made by men most capable of analyzing the qualities adapted to the station, and acting under circumstances favorable to deliberation, and to a judicious combination of all the reasons and inducements which were proper to govern their choice. A small number of persons, selected by their fellow-citizens from the general mass, will be most likely to possess the information and discernment requisite to such complicated investigations.
This just gives one sense of the body in why the Constitution included an electoral college, and did not  go for a general popular vote.

Interesting note: if deadlocks force the vote into the House of Representatives, the District of Columbia gets no vote, and each state gets one vote. When the electoral college met in March, a deadlock would go to a newly elected Congress. Now the electoral college meets in December. Would the deadlock go to the existing Congress or wait until the new Congress? Stay tuned...