Who says you must use tangent?

I am beginning an analysis of the questions recently posted in the Fall 2014 New York State Common Core Sample Questions for the Regents Examination in Geometry (Common Core) You can find it here.

Here is the first question posed, along with sample solutions. (My part continued below.)
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The answers to the above include, in addition to the correct choice, some key "distractors".  Replace "tan" with "sin" above and you obtain answer choice (3). Switch the 69 and 102, and you will get choice (4). Replace 69 with 59 and 102 with 86, and your result is choice (2). 

Please take note that should you make one of these errors, you will have a 0% chance for a correct answer, whereas if you randomly make a selection, your chances are 25%. 

Also, please recognize that the correct solution to this question begins with careful and accurate reading. The ability to read the question is paramount. The best math student is blinded if they cannot read the question. 

During my years in the classroom I many many times discovered students who could do such problems perfectly fine if I read the problem to them, but could not handle it on their own. In constructing tests, care has to be taken in controlling how much of the grade should be dependent on the ability to read, and how much actually is dependent on that ability.

Please take note that a student who has absolutely no knowledge of tangent can solve this problem perfectly should they begin by using the Pythagorean Theorem to find the hypotenuse length, then properly apply either sine or cosine. The phrase "tangent must be used" is mathematically incorrect.

My final note is that to obtain feedback about a student's skills and knowledge, I would never make such a question "multiple choice".  This type of question creates easy, but hardly useful, data.

Carefulness must accompany caring

Errors happen. That is inevitable. Mistakes abound.  New media have fact checkers, writers have proofreaders, weight lifters have spotters, and so on. Nobody is perfect.

So why do so many errors that are avoidable remain undetected?  I have compiled a gathering of those I have seen while I have had a camera with me (see here). I have limited my collection to printed media, either hard copy or digital. No audio or video here. The picture below I have not included in my collection because I have not yet convinced myself that it is an error. It might have been the intention to write "french fries" while showing mashed potatoes.

What amazes me is that each of these is (was) made by more than just one person. Perhaps just one made the initial error, but the non-detection of it makes others equally responsible, if not more so.

Note I do not call them careless errors. Carelessness implies that one just doesn't care, and, not being a mind reader, I cannot make that claim in any of these cases. Who knows whether or not the initial mistake was called by indifference. However, the failure to rectify the mistake is a sign of indifference. In each case (except the automobile photo) a business either saw the error and did not care about it, or was careless in not. (The automobile photo I place in a different category since the sign was inside the car over a couple weeks while its owner was home and had actually moved the care once or twice.)

These items become newsworthy as they indicate a sense of complacency, which brings to mind Thomas Edison's quote “We shall have no better conditions in the future if we are satisfied with all those which we have at present." We have to jump start our culture and society into being more careful, not just more caring.

Albert Einstein said “Whoever is careless with the truth in small matters cannot be trusted with important matters.”

Samuel Johnson stated "It is more from carelessness about truth than from intentionally lying that there is so much falsehood in the world."

Where have you gone Joe DiMaggio, er, I mean decent cable TV?

Back on September 12 of this year Dane Wigington had an article online titled The Weather Channel Switches To Reality Shows (see it here). I do not wish to repeat it, but I will add to it the comment that this weekend the channel outdid itself. A whole weekend of "Prospectors", a show whose content I do not know and care not to know. I turn on The Weather Channel to find out about the weather. That wish is met with decreasing frequency, and this weekend almost not at all.

Finding weather on The Weather Channel is about as difficult as watching shows about travel on The Travel Channel. Perhaps I have missed something: maybe the word "travel" has now morphed into a synonym for "food", as most of its shows revolve around some unhealthy or just strange aspect of eating.

I will not even go into looking for history on The History Channel.

Maybe big business is really truly trying to dumb down society. Does intelligence scare them?

Fresh?

A few days ago I wrote a post about taking thought-provoking photos. Here is one I took a couple years ago. Needless to say, it was not taken in the US. I cannot think of a single place in the US where live chickens would be allowed to roam in front of a fast food restaurant, and I cannot imagine a Colonel Sanders franchisee tolerating live fowl outside the front door.  This picture was taken, however, near downtown George Town in Grand Cayman.

I know this photo is not unique (just do a Google image search of"Grand Cayman KFC") but I thought it thought provoking anyway. It at least got me checking on chicken in the wild in US towns and cities. 

What happened?

Whenever I travel I try to take as many photos as I can that are worthwhile but not find-able all over the web. No sense taking pictures that have already been taken. I look for unique pictures. Even more so, I like them to be though-provoking.

This one I took while walking on a dock in St. Thomas, USVI, a bit over a year ago. I can tell you that the hat was floating, although some people to whom I have shown the picture thought I had caught it in midair. That led me to consider how we describe what we see, and how we extrapolate in the process.

Anyone who describes the hat as in midair must presume some person, or some thing, as the thrower, unless perhaps it was windblown.  Even knowing it is floating, the question could still be asked: How did it get there?

So, if I was to ask you to write the history of this hat, as best you can, how would you respond?

FOILed again?

The statement in red here is one I came across while taking another look at Common Core standards. I decided to Google the statement. I did discover that Google will not let you use more than 32 words between quotes, so I Googled just the 1st sentence in quotes (so Google looks for an exact match), and got 14,100 hits. Listed below are just a few of them.

There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y).

2014 Mississippi College- and Career Readiness Standards for Mathematics

https://www.illustrativemathematics.org/standards

http://www.inghamisd.org/downloads/iisd_curriculum_math/feb_2013_cd_update_20130306_154552_1.docx

State of Nevada Dept. of Education

It's a bit misleading. The second sentence says "may have a better chance". Anyone who really and truly understands what binomial multiplication is and why it is, WILL have a better chance. A much much better chance.

It's also interesting that the statements refer to "a mnemonic device" and gives it a bad flavor. I suspect the device the authors had in mind was the old FOIL method for binomial multiplication, and the impetus behind the statements was the knowledge that there are teachers who taught only FOIL, and nothing more. (Could it be in use here only because F-O-I-L actually spells a word in English? Could this be why US math is falling behind?!? Imagine!?!) Do you know the meaning of FACE? Does it help or hinder in music studies?

I find it odd that a statement which embodies such a major thrust in educational philosophy is phrased in the context of binomial multiplication.  Multiplying binomials is something most people never do in their lives. However, if binomial (and polynomial) multiplication where actually dealt with carefully and fully they would be recognized as the foundation behind most algorithms for multiplication. Multiplying a pair of two digit numbers is actually a binomial multiplication. Multiplying 564 and 37 is actually multiplying a trinomial and a binomial. (I find it interesting that my spell-checker just marked "trinomial" as a misspelled word. For the record, it does not acknowledge "pf" as a possible error.)

The 4th grade standards include this: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

The arrays and area models referred to here are exactly the same devices that are, or could be, used to describe the distributive process applied when multiplying polynomials. It would be nice if our educational system would recognize this and capitalize this, and perhaps gather 4th grade and high school teachers together to carefully comb out more connections. I an willing to bet that a substantial number of 4th grade teachers are not tuned in to the distributive rule as it relates to polynomial multiplication. A few visuals are here.

Whatever happens with Common Core, let's at least recognize that it doesn't have all the right answers, but it could be spurring people into asking the right questions. I hope.

GeoGebra no later than middle school...

This is just a continuing saga of options that teachers and students now have that I hope they are using.  with the technology we have now students can be introduced to new things in dynamic ways making them accessible as never before.

I just made this in GeoGebra as an example of check boxes in case I needed one at the AMTNYS conference presentation next week.