From circles to ellipses?

Many people seem to think of circles as simple things and ellipses as "oh, that's math and I was no good at it...".

Here is a bit of a perspective on that created in GeoGebra.

If you know nothing of ellipses beyond "oval", start by just using the control buttons at the top. If you are comfortable with ellipses, but a bit of a skeptic, experiment with the sliders as well. If you really want to know what is happening, download the complete file here.

Regardless of where you stand, GeoGebra does give a great dynamic approach to mathematics.

math is an exploration!

Any study worthy of its name involves decoding and analyzing and learning from the work of others. A high school trig student has all the math skills needed to create a file like this. A lower grade student might be encouraged to learn about sines and cosines due to their "hidden" use here.

At worst: it's a a neat design! (find the file here.)

Let's make mathematics great again...

This is a magical question from the New York Regents Exam in Geometry (Common Core) of June 2016. Based on a NYSED decision, both (1) and (3) are accepted as correct, which means that item III is both always true and not always true. Magic!!!

For the visually-motivated. here is a quick GeoGebra file based on this question.. Unfortunately, due to space limitations, I had to put limits on the dilation scale factor and the translation components (computers do not like to accept infinity as a number!)

This question was one of 3 in this exam in which multiple answers were deemed acceptable (after initial scoring, grading, and graduations were all over.)

I have two concerns over this fiasco:

1. A student seriously confused by question 14 (not be the mathematics) might have, in following the directions to indicate the BEST choice, left the answer blank. That would have been correct, as there cannot be multiple BEST answers. They would get no credit, yet a student who was clueless and put down a random selection would get credit.
2. Many teachers use old regents exams as study/review/practice for exam time. What will NYSED do to ensure that future use of these exams is predicated on information regarding these mess-ups? It is now three questions (14, 22, and 24) on this exam that have been acknowledged as invalid.

Be aware that on question 14 two answer selections are accepted as correct, on question 22 all 4 answer selections are being accepted as correct, and on question 24 all 4 answers plus a blank are being accepted as correct. See these links: question 24, questions 14 and 22

June 2017 NYS Geometry Regents Question 24 has no correct answer!

July 25, 2017 note:  I have just received word that NYSED has deemed all 4 answers to be granted credit for this question.

Here is a question from last months new York State Regents Exam in Geometry (Common Core)

This question, to be answered best, requires you to identify three which are sufficient. The one left over would be the answer. Wouldn't it?

The Regents "powers that be" have decreed that choice "2" is the correct answer. They are wrong. There is no correct answer: all 4 choices are sufficient for the task.

A key item in analyzing this question is recognizing that point C in the diagram is between points A and E and between points B and D. This information is embedded in the diagram.  Should a blind student be taking this test, this information must be verbally communicated (maybe using the sense of touch is possible, but I will not deal with that here.)

If the diagram has been totally omitted from the question, all bets are off. As a matter of fact, a better question would have been to leave out the diagram and leave out the word "not" in the question.

Although this does not constitute a proof, it illustrates what a proof for choice "2" would be dealing with. I did not use the same point names, but the lengths are correct. Points A, B, E can be dragged.

A more detailed proof has been written and the link can be found at https://www.change.org/p/nysed-mark-q-24-on-the-june-2017-geometry-regents-as-correct

Learn something new...

Math students (or anybody, for that matter) should try to learn something new everyday.

This might help meet that need for somebody out there.

Meanderings on a rainy afternoon

Just a bit of doodling in Geogebra, trying to keep from getting mentally rusty.
Students should be working with GeoGebra.
It's free, it's fun, and it's educational. A trifecta.
The file can be found here.
 

Cavalieri's What?

Here is question 27 from the New York State Geometry Regents exam of June 2016.

One thing specifically stands out in this question. Cavalieri's Principle.

Students in New York do not have to remember the quadratic formula, how many feet in a mile, or even the Pythagorean Theorem. Those items are all on the Common Core High School Math Reference Sheet with which each student is supplied. 

But Cavalieri's Principle? That stands out. After all, everyone should know the main success of Felix Cavalieri, lead singer of the Lovin' Spoonful.

Oh, sorry, Wrong Cavalieri.

This Cavalieri is actually the gentleman who proved the formula for the volume of a general prism (which can be found on the reference sheet). 

I will admit that when I first read this question, I said to myself "what the heck is Cavalieri's Principle?" After a little research I was reminded that I had confronted it somewhere in my mathematical meanderings.

It does please me that NY has risen its standards and now expects students to remember and be able to apply Cavalieri's Principle. Now if they could only do that with the other items on the reference sheet...

NYS Regents Algebra 1 Questions 18 and 19

The above question bothers me.  The statement of the problem makes no claim about any domain, and the answer key's choice of (2) actually requires that the problem be interpreted by the student as a question regarding a recursive sequence. This question appears to result from a mashup of two different issues: recursive sequences on the one hand, and definition of function on the other.

Obtaining the answer that is deemed "correct" requires that the student assume that every result from the formula (normally associated with the concept of "range") can then be used as an input into the formula (normally associated with the concept of "domain"). 

Although the topics of recursive sequences and function definitions are both legitimate for first year algebra, this question still does not belong.

Here is question 19:

This question can be answered by simple substitution and calculation. A student need show no skill regarding solving linear equations to answer it correctly. 

Compare that question to this from Ninth Year Mathematics from 1969 (the one I took)

(The [5,1] indicates 5 points for solving and 1 for checking. ) Admittedly, this question was 1 of 7 ten pointers out of which the student had to answer 4. To see what the choices were, check out the exam here. 

For that matter, compare June 1969 with June 2017 and then tell me that our standards have really been raised. (Keep in mind that hand held calculators of any kind did not exist back then.)

The Cubic Regression Rules!

The actual exam for the June 2017 New York Algebra 1 (Common Core) has not been released, and teachers are strictly forbidden to communicate via the web on the exam until after June 30, so as I begin my yearly review (as a NY citizen who happens to be a retired teacher), all I can comment on is the conversion chart. Here I have plotted it and included (dotted line if you look closely) the cubic regression curve based on the data contained in the conversion chart itself.

I have always looked at level 1 as the failing range, and level 2 as the "safety net" where Special Ed passes but non-Special Ed does not. That might have changed by now. Needless to say, a regular ed- student has to get to level 3 to be considered "passing"

I noticed that, in comparison to the conversion chart for last June, every raw score from 4 to 21 received a scaled score 1 or 2 points higher this year.  No raw score from 24 and up got any boost this year, and a few lost a point on the conversion side.

Here are June 2016 and June 2017 plotted together:

Needless to say, the cubic regression is pretty much the same. I do wonder why NYSED prefers it so much.

Here I have added (in green) the conversion chart for Algebra II from 2016:

Ever since NYSED started using the raw-score to converted-score approach to scoring these exams, there has been battle between those who liked the old "straight-line" approach and NYSED itself. In case you are wondering, the straight-line approach is based on the concept that if you answer 50% of the test correct, your score is 50. Answer 75% correct, get a 75, etc. If I added the graph of the straight-line approach, it would be a straight line from bottom left (0,0) to upper right (86, 100). 

What the heck- here it is:

It is easy to see that almost everywhere the scaled score exceeds the 'straight-line" score, with a minimal gain at the top end, and increasingly large gains as one heads from the bottom towards that magical mystical number known as "65".

I am partially from the "old school". When I took the Regents Exam in algebra 1 (I believe it was properly called "9th Year Mathematics" back then) the exam consisted of 30 short-answer questions (2 points each) and 7 Part 2 ten-point questions, of which you had to answer 4.  There was a bit of a top end hammer since a student capable of answering all 7 part 2 questions could get credit for no more than 4 of them. (If a student answered more than 4, only the first 4 answered would count)

Thinking back to those tests, the typical 9th Year Mathematics exam contained a dozen multiple choice questions, each with 4 choices. The random guesser would, on average, get 3 correct out of 12, contributing 6 points towards passing. That student would have to earn 59 points out of the remaining 76. In last year's Algebra I exam there were 24 multiple choice questions, with 4 choices each, so the random guesser would average out with 6 correct, for 12 raw score points. To get to the minimum passing score that student would need 15 out of the 36 remaining possible points. Which is more difficult, 59 out of 76 or 15 out of 36? Hard to tell, recognizing that a student who has to guess on all of them probably doesn't know much of the course.

The other end does intrigue me: the student who gets all the multiple choice correct, be it by guessing or by knowledge or some combination of the two.

In 9th Year Mathematics the multiple choice got you 24 points on your way to a minimum passing grade of 65. In Algebra I last year, the multiple choice gets you 48 points on your way to a minimum passing score of ....27. For that matter, a student who can successfully answer 11 out of the 30 multiple choices gets 22 points. Now that student guesses  on the other 13 multiple choice questions will, on average, get 3 or 4 correct. Even at 3, those student now has 6 more points, and has now earned a passing raw score of 28. the student has "passed" the exam without going beyond the multiple choice questions. If that student had guessed by filling in choice 2 or choice 3 for all guesses, he/she had a solid advantage. (There were 5 answers of "1", 7 answers of "2", 7 answers of "3", and 5 answers of "4". I will have to see if it is a trend to put correct choices in the middle slots.)

What does appear to be a trend to me is a trend to make exams in such a manner that passing scores are in easier reach while top scores are harder to obtain. Some may say it appears to narrow the "achievement gap", but appearances can be deceiving.

Pictures worth 1000 words.

Last week I happened to see a car in Vienna with mathematics all over it. Due to a lot of pedestrians the best picture I could take was of the spare tire case. This picture was taken right near a jewelry store and a Bitcoin store, so in my mind a Tom Clancy novel could start with it. The Casino Wien is just around the corner, the Hotel Sacher is a block away (Sacher of Sacher torte fame). The Vienna Opera is across the street from the Hotel Sacher. The options for starting a novel of intrigue are plentiful. (I erased out the license plate for the privacy of the car owner, although I would question anyone driving a Mercedes covered in math who had a desire for privacy.)

Here is another picture taken, this time in Kunsthistorisches Museum in Vienna. I believe a middle school math lesson could be constructed around these buttons, starting with the task of placing the buttons in numerical order. For some reason the U.S. seems to avoid negative numbers in elevators, let alone decimals. Could there be a greater comfort level with numbers over there "across the pond" than here at home?

Bezier for Young People

Bezier Curves are generally not confronted by k-12 students at all. Their equations can be complex, and in the absence of dynamic geometry, the topic in general can be mind boggling. Historically, algebra has been the driving force for graphing. With dynamic geometry, that can be reversed.

If you can mentally stomach (how's that phrase?) the idea of a point on a line steadily sliding from one location on that line to another location on that line, you've mastered the necessary skills. Visually, a point sliding from one place to another would look like this:

Being able to picture this is all you need to know. Here it is

Jack and Jill

Here is a short Geogebra example I put together as an example of how GeoGebra could be used to introduce young students to mathematics that they might never see unless they got into a precalculus class that included polar graphs.

Among the concepts used (but not named) are midpoint, rotation, rotational speed, and 3-leafed rose.
The only geometric term used is "circle".

Even so, the situation can be used as a springboard lots of questions.
 

Among those questions could be:

• What if they walked the same speed?
• What if Jill walked faster?
• What if they walked in the same direction around the circle?

Denying younger students the opportunity to ask these questions and explore their solutions is to do a disservice to those students.

Creative solutions

Thanks to www.Boston.com

Co

Why? Why? Why?

Here is question 3 from the January 2017 New York Regents Exam in Algebra I (Common Core),

This question shocked me when I first saw it.

To get a taste of what such a correlation coefficient is, I suggest you take a look at Wolfram. New York State's EngageNY has an "introduction" to correlation coefficients (find it here) that says "It is not necessary for students to compute the correlation coefficient by hand, but if they want to know how this is done, you can share the formula" and then shows

It should be noted that this formula the bounds of summation and variable subscripts are omitted, so it is a meaningful formula only for those "in the know", and I suspect that very few Algebra I students fit that description. 

Is this a case of presenting a "magic button" on a calculator as a key to answering a complex question? Is that how math should be taught?

New York's modules include the following:

Take note of the phrase "use technology".  That essentially means the student should plug in the numbers, hit the necessary buttons, and find a result. It is more like following a recipe, and most people recognize that a recipe becomes unnecessary after it has been followed a number of times: not because it is not being followed and not because it is understood, but because it is remembered.

Here is a question from the August 2016 Algebra I Regents;

I would suspect that after enough repetitions of the recipe some students might have caught on and realized that the choices offered make this question a bit easier than it would have been had the data been a bit different and the choices involved some seemingly random 4-digit decimals.  Perhaps that might be why questions such as this have, over the years, been categorized as "cookbook" problems.

The next question here is from the Algebra I Regents exam from August 2015:

There is absolutely no way any student will successfully answer this question without following a recipe on their graphic calculator. I do not know why NYS left all the space on the page as all that is asked of the student is to write down one equation and then a two digit decimal together with one word.

Here is a similar question from the January 2015 Algebra I exam:

The big difference here is the 60% increase in time entering the data (16 values instead of 10) and the explanation of part (b). But again, it's enter two columns of data, hit a couple more keys on the calculator, and read off a result.

A similar process is involved in the next question, the big difference is that it asks the student to pick an item from a different line in the calculator's display:

Do all these questions belong in Algebra I?

Are these items here only because of the lobbying efforts of the companies that sell the graphic calculators?

I would much rather see students get the flavor of least squares analysis by using FREE software such as GeoGebra. For an example, check out the Least Squares Demonstration here.

While correlation coefficient seems to be an easy testable item (when proper calculator is present), it should be recognized that it is merely a measurement of how well a regression line actually "fits" the data.  The true mathematical questions begin with the regression line itself: what it is, why it is, why it is useful, when is it useful,  how we know it is the "best" fit, what do we do when this is not useful, and many others. For a sense of the current state of regression analysis, just visit Wikipedia.

Please take note: correlation coefficients and a slew of its tag-a-longs could fit in high school mathematics, but not in Algebra I.  Dynamic geometric approaches to concepts such as least squares could be well developed in an Algebra I course, using software such as Desmos or Geogebra or other equivalent packages. 

The biggest problems with high school mathematics occur when students are expected to know and do things mindlessly. Recipes need to be avoided. Magic buttons on calculators need to be avoided like poinson needs to be avoided.

The true value and meaning of the quadratic formula comes only to one who tires of completing the square. Completing the square obtains its true value and meaning when one tires of trying to find factors (especially when they do not exist!). The real meaning of factoring comes to one who is fed up with constantly having to guess and check. There is a hierarchy to the knowledge and skills of mathematics. Jumping too quickly to a higher level does a disservice to students.

Imagine what would happen if we limited single variable quadratics to finding intersections with the x-axis on a graphic calculator, and skipped over everything mentioned in the previous paragraph! It makes just as much sense as tossing regression analysis into Algebra I only because it can be done on a graphic calculator. Jumping too quickly to a higher level does a disservice to students, especially when that jump takes place largely do to the mere availability of a handheld calculator.

Is Kindergarten too old?

Here is a very simple dynamic graph, showing a point sliding down a radius in the same amount of time that it takes the circle to do one rotation around its center. Simple to see and explain.

It is the explanation part that makes this valuable. No matter what grade level, it could be used. From simple use as an introduction to the words circle, radius, rotation, and speed (imagine if children started learning these words in kindergarten) to a PreCalculus class being asked to generate parametric equations so that they could continue the graph on their graphic calculators.

The file itself can be found here.

If you wish to read more about this type of structure, check out this at Wikipedia.

Be creative in your use!

Here is a sketch that could basically be used with any grade, from a visual with elementary students, to a "can we make it ourselves" with middle school students, to a model for exploration for upper levels.

With an elementary class, I would leave out all the text, and create a step-by-step show, from first circle to tangent line to second circle to midpoint to trace, but not using sophisticated language. With middle school students I would use the basic geometric language and do a step-by-step as well. Upper students who are familiar with Geogebra could be shown the graphic and asked to recreate it. Those unfamiliar could be guided through it. Precalculus students could be challenged to determine an equation that could be graphed on a graphic calculator.

Adjustments to the file are easily made.

The main point is that this technology should not just be used as crutch with old curricula, but should also be used as an avenue for new approaches to mathematics education.

The complete file can be found here.

The Magic Formula From Heaven

Here is a statement that absolutely scares me. It is from Lesson 19: Interpreting Correlation from  the NYS Common Core Mathematics Curriculum.

It is not necessary for students to compute the correlation coefficient by hand, but if they want to know how this is done, you can share the formula for the correlation coefficient given below.

It demonstrates, in a nutshell, a massive hypocrisy in the implementation of Common Core math here in NY. What starts out, in the lower grades, as a powerful attempt to make students consciously aware of what they are doing, has morphed into the old and disastrous "give them a formula or a calculator and they will be happy." When doing basic arithmetic the emphasis on understanding by expecting multiple methods sometimes seems like overdoing it. Here, understanding is just tossed aside.

Take a look at this Wikipedia page (https://en.wikipedia.org/wiki/Pearson_correlation_coefficient), read it carefully, and then tell me if anything related to this concept actually belongs in Algebra I. If that does not convince you, check out the Wolfram page. For a bit more, check out this one.

What really bugs me is that using a calculator to compute the Pearson correlation coefficient bypasses any attempt to teach understanding. No student will come out of this with any knowledge other than "this is what they told me in school".  That is exactly the situation that "teaching for understanding" was supposed to avoid.

Why not let students actually get the opportunity to explore topics such as least-squares regression using software such as GeoGebra? Here is a small visual example of what GeoGebra can do. The labeled points can be dragged and the sliders control slope and y-intercept of the line. The best fit is the line that gets the sum of the squares as small as possible.

We should keep in mind that there is no need to have a line of best fit if we can see all the data plotted in front of us. A best fit line can help us if it can be understood as a model of prediction. If that is the goal, then it would make sense to have some strong connection between the modelling line and the correlation coefficient. Check out what New York State students experience and see if that connection is solidly made.

Does mathematics need to be more wordy?

The January 2017 New York regents exam in Algebra I (Common Core) contains a question with a model response that I do not get.

Here are the directions:

Here is the question and the model response:

The student has shown in 2 steps how to convert one equation into slope-intercept form, and you can see that the equation ends up identical to the first equation in the question. In answer to "Is he correct?" the student answers "No." In explanation, he states that the two equations are for the same line.

The model response scoring states: Score 1: The student wrote an incomplete explanation.

Mathematically, this model response nailed it. For some reason it only gets half credit. Was it not verbose enough? Is there a minimum number of words required?

You can see the entire group of model responses here.

Is understanding a language too difficult?

Here is a list of words:

1. noun
2. verb
3. adjective
4. adverb
5. subject
6. predicate
7. direct object
8. indirect object
9. clause
10. pronoun

I could continue the list, but these will do.

You might ask, so what about these words? In answer, I can state with conviction that the knowledge of these terms, their meanings, and their usages, was key in the development of my personal understanding of our language.

Even though I spent years teaching and learning mathematics, I still use the concepts of adjective and noun when discussing fractions. Many of you might remember the terms "numerator" and "denominator", which are fancy words for the ideas of "how many?" (adjective) and "of what?" (noun).

The simple fraction \({\textstyle{2 \over 3}}\), properly read "two thirds" gives us a quantity (two) and a denomination (thirds), thus answering completely the question "How many, of what?".

This morning I made a discovery: none of these words is used, at all, in the January 2017 New York state regents exam in English Language Arts (Common Core) . 

I found that atrocious.  How can one even discuss the "art" of language without involving its grammatical structure? 

Basing an entire curriculum on grammar would be a waste of time, just as doing nothing but arithmetic under the name of mathematics would be useless drudgery. But to totally ignore grammar amounts to designing a skyscraper while ignoring the hidden structure that holds it up.

Here is question from the New York regents Comprehensive Exam in English from August 1978:

How would students react in this day and age?

I guess it won't matter too much, as long as the gun is around to take care of the grizzlies.

January 2017 New York regents exam question 24 is not nice

Here is a question from the January 2017 New York regents exam in Algebra II (Common Core)

Please take note that, according to the statement, the height of the box is "x-1".

That fact alone is enough to force any modeling function to only use x values larger than 1.

Similarly, the width being "5-x" forces the modeling function to use values for x that are less than 5.

This graph includes values for x that must be excluded, so it is not a good modeling function.

In addition, words such as "fastest" when used without a reference create an implication of time as a parameter. Combine the lack of a time factor in this question together with the issue of a box with a changing volume (the first sentence in the question refers to "boxes" where as the rest of the question refers to a single box), and confusion in the mind of the test taker is to be expected.

This question could have been written quite differently and perhaps have been restored.

When the day comes when these tests can be truly computer-based, a question such as this might be accompanied by a visual such as that shown here:

In the meantime perhaps New York could stop its insistence on testing questions being "real world".