Many people seem confused by the process of selecting a President.

At no point has this country been other than a republic of states, and the US Constitution specifies the means for those states to elect a President. That process is the Electoral College.

No matter what is on your ballot when you vote, you are not voting for a President. You are voting for an elector, a member of the Electoral College. ALL your voting takes place within your state. There is no vote for a national office.

Tomorrow the Electoral College meets to vote for the next President.

I pray and hope that it exercises this task with utmost care and diligence, and does not merely make its selection because of a misconstrued sense that the "public" voted for one person or another. The public did not vote for anything more than the electors.

Electoral College: this is your opportunity to confirm the wisdom of our Founding Fathers.

## Sunday, December 18, 2016

## Monday, December 5, 2016

### Let Math Test Questions Test MATH!

Here is a portion of the New York State Grade 5 Mathematics Reference Sheet:

Here is a question from New York State's 2016 Grade 5 Mathematics Test Released Questions:

Thank goodness that New York does not allow for calculators on this test, as the only mathematics being tested is the division of 240 by 4. Everything else is reading comprehension. Careful patient reading, as any student familiar with aquariums (aquaria, if you wish) would know that they size is typically communicated in gallons (at least in the US) and would then question the strangeness of this so called "pet store."

If scores on this test are used to measure (determine?) students' mathematics knowledge and skills, the test should be focused on those skills and that knowledge.

I entered the first sentence of the question into Microsoft Word and let it supply the readability statistics, and it measure the sentence at the Flesh-Kincaid Grade level of 7.6. I then took the same sentence to https://readability-score.com and it came back with these:

Does this question belong in a MATH test?

Here is a question from New York State's 2016 Grade 5 Mathematics Test Released Questions:

If scores on this test are used to measure (determine?) students' mathematics knowledge and skills, the test should be focused on those skills and that knowledge.

I entered the first sentence of the question into Microsoft Word and let it supply the readability statistics, and it measure the sentence at the Flesh-Kincaid Grade level of 7.6. I then took the same sentence to https://readability-score.com and it came back with these:

Please, then, at least accept the possibility that a student's reading abilities might, just might, have a bigger effect on their answering of this question than does their mathematical talents.

By the way, the results on readability for the previous sentence is here:

## Saturday, December 3, 2016

### Math without GeoGebra is like a day without sunshine!

It might be the right time of year to push the use of GeoGebra as a classroom presentation tool. I created this interactive file back in March of this year, and pull it out now with a request for teachers to send in ideas for modifications of this file, or suggestions for new files (send to dave(at)davemath.com. This file can be found here.

Any and all GeoGebra files can be downloaded and modified.

GeoGebra is totally dynamic and its use is limited only by your imagination!

## Tuesday, November 15, 2016

### Never be satisfied with guess and check!

(Please read my last blog entry either before or after this one. They go together.)

\[\begin{array}{c}\left( {nx + m} \right)\left( {px + s} \right)\\nx\left( {px + s} \right) + m\left( {px + s} \right)\\np{x^2} + nsx + mpx + ms\\np{x^2} + (ns + mp)x + ms\end{array}\]

The above should be recognized by all secondary teachers of mathematics as a generic example of using the distributive property (of multiplication over addition, as it is frequently phrased) to multiply a pair of binomials.

Simple and to the point. Anyone can do it.

What I wish to point out to those who do not notice is that \[(np)(ms) = (ns)(mp)\]

This might seem like a bit of obvious but but seemingly irrelevant trivia, except for the fact that this little fact is the key that unlocks what I believe to be the way that quadratic factoring should be done (and taught).

Take a close look at an actual example (using numbers).

\[\begin{array}{c}(2x + 3)(5x + 7)\\2x(5x + 7) + 3(5x + 7)\\10{x^2} + 14x + 15x + 21\\10{x^2} + 29x + 21\end{array}\]

This is how the multiplication of binomials should look. (Forget that mnemonic FOIL. Forget it now and forget it forever.)

Now we will see the same steps displayed in reverse order.

\[\begin{array}{c}10{x^2} + 29x + 21\\10{x^2} + 14x + 15x + 21\\2x(5x + 7) + 5(3x + 7)\\(2x + 5)(3x + 7)\end{array}\]

Take note the first step involves separating \(29x\) into \(14x + 15x\). Why choose 14 and 15? Because they add to 29 AND multiply to 210, the product of 10 and 21. The rest is just applying that same old distributive law.

Take a look at one from scratch, say \(4{x^2} + 43x + 63\)

Our first step will be to find two numbers that add to 43 and multiply to 252, which is the product of 4 and 63.

\[\begin{array}{c}\left( {nx + m} \right)\left( {px + s} \right)\\nx\left( {px + s} \right) + m\left( {px + s} \right)\\np{x^2} + nsx + mpx + ms\\np{x^2} + (ns + mp)x + ms\end{array}\]

The above should be recognized by all secondary teachers of mathematics as a generic example of using the distributive property (of multiplication over addition, as it is frequently phrased) to multiply a pair of binomials.

Simple and to the point. Anyone can do it.

What I wish to point out to those who do not notice is that \[(np)(ms) = (ns)(mp)\]

This might seem like a bit of obvious but but seemingly irrelevant trivia, except for the fact that this little fact is the key that unlocks what I believe to be the way that quadratic factoring should be done (and taught).

Take a close look at an actual example (using numbers).

\[\begin{array}{c}(2x + 3)(5x + 7)\\2x(5x + 7) + 3(5x + 7)\\10{x^2} + 14x + 15x + 21\\10{x^2} + 29x + 21\end{array}\]

This is how the multiplication of binomials should look. (Forget that mnemonic FOIL. Forget it now and forget it forever.)

Now we will see the same steps displayed in reverse order.

\[\begin{array}{c}10{x^2} + 29x + 21\\10{x^2} + 14x + 15x + 21\\2x(5x + 7) + 5(3x + 7)\\(2x + 5)(3x + 7)\end{array}\]

Take note the first step involves separating \(29x\) into \(14x + 15x\). Why choose 14 and 15? Because they add to 29 AND multiply to 210, the product of 10 and 21. The rest is just applying that same old distributive law.

Take a look at one from scratch, say \(4{x^2} + 43x + 63\)

Our first step will be to find two numbers that add to 43 and multiply to 252, which is the product of 4 and 63.

\(\begin{array}{c}(1)(252)\,\,\,add\,\,to\,\,253\\(2)(126)\,\,\,add\,\,to\,\,128\\(3)(84)\,\,\,add\,\,to\,\,87\\(4)(63)\,\,\,add\,\,to\,\,67\\(6)(42)\,\,\,add\,\,to\,\,48\\(7)(36)\,\,\,add\,\,to\,\,43\end{array}\)

Take note: the numbers on the left are no more than counting, 5 was skipped because it is not a divisor of 252, and we stop at 7 because we found the numbers we need. With these numbers we can continue:

\(4{x^2} + 7x + 36x + 63\)

\(1x(4x + 7) + 9(4x + 7)\)

\((1x + 9)(4x + 7)\)

Please please recognize that this process is highly programmable. It is actually a system based on action, not a fallback "guess and check" that is pushed on kids far far too often. Also, take note that I do accept "1" as a meaningful numeral, and do not always jump at the chance to avoid writing it.

Here is a sample with negatives: \[6{x^2} - 7x - 5\]

We start by taking pairs of factors of \[ - 30\]. Since the two numbers must add to a negative, we will make the larger number in each pair negative and keep the smaller one positive.

\[\begin{array}{c}(1)( - 30)\,\,\,add\,\,\,to\,\, - 29\\(2)( - 15)\,\,\,add\,\,\,to\,\, - 13\\(3)( - 10)\,\,\,add\,\,\,to\,\, - 7\end{array}\]

We got our two numbers pretty quickly this time, and we can continue.

\[\begin{array}{c}6{x^2} - 7x - 5\\6{x^2} + 3x - 10x - 5\\3x(2x + 1) - 5(2x + 1)\\(3x - 5)(2x + 1)\end{array}\]

Please please remember that the special cases with leading coefficient 1 are special only to someone who knows the whole story. They are not special to a student seeing them for the first time. What may be seen by those in the know as a shortcut cannot be seen by beginners as a shortcut. A shortcut is never meaningful unless and until a longer route is known.

If you would like a seemingly endless list of practice problems (with check-ability), check this out.

## Friday, September 16, 2016

### Just a Simple Protractor

Sometimes the simple things misunderstood create academic mountains where there should not even be molehills. Take proper use of a protractor.

A simple exercise using a good display can most probably stop these problems before they begin.

I suspect that is some teacher out there who could profit by using something as simple as this.

Find the original here.

A simple exercise using a good display can most probably stop these problems before they begin.

I suspect that is some teacher out there who could profit by using something as simple as this.

Find the original here.

## Thursday, September 15, 2016

### Some thoughts on "progress"

So now the Massachusetts Turnpike is joining the movement to "cashless" tolls, which I see as a euphemism for "no more salaries and sick days and health insurance and no more PEOPLE!" Such is progress. This progress will have 3 levels of charges from least expensive to most expensive: Massachusetts EZ-Pass, out-of-state EZ-Pass, and the rest. They justify higher tolls by claiming additional costs, but..Free Ride: New Mass. Pike tolling system will allow you to bypass tolls in Springfield, Worcester points out that in many instances short runs on the Mass Pike will be toll-free. Seems to me that charging "others" more will help subsidize this.

Speaking of the "others", another article, from WVBC5, states "For drivers who lack any type of electronic tolling transponder, the cost of driving from New York to Logan Airport will nearly double from its current $7.10 to $13.40 plus a 60-cent billing fee." Doesn't that sound like a strong push to sell EZPass? Be advised that to get an EZPass, automatic deduction must be started from an account (could be a credit card), and the tolls are actually prepaid with the requirement of an initial payment of at least $20. They actually take your money up front.

FYI: With the advent of "cashless", the Mass Pike toll for the entire eastbound run, end-to-end, for EZPassMA holders will go from $6.60 to $7.65. Talk about the economies of progress. People will pay more even though people will be losing jobs.

I have a NY EZPass, and I got it originally out of a wish to get in shorter lines.

It won't be long before before your driverless Uber takes you via cashless tolls to visit your college student son or daughter who is just months away from huge debt and no job. Such is progress.

It has been a while since I began to see "self checkouts" at grocery stores and Walmart and Target, etc., giving you the opportunity to perform the cashiers job at no pay and with no discount. Take a moment to read Self-checkouts: Who really benefits from the technology? courtesy of CBC.

Amazon's attempts on drone delivery are disguised as "faster delivery" but are actually just an attempt to cut back on the need to pay people such as deliverymen. They are already making their "human" delivery less efficient by frequently bypassing UPS and FedEx and opting for USPS, whereby your items could be ordered Thursday afternoon, arrive at your local post office distribution center Friday, then sit until Monday until your local post office delivery time. Drone technology just might be accepted if the customers see it as a 'speed-up" in delivery. Delivery times to my home have gotten appreciably longer despite using Amazon Prime.

I see more and more instances of people losing jobs not because of jobs becoming obsolete (such as farriers with the advent of the automobile) but rather because technology has found a way to keep the job but make it robotic. The only people who can consider some of these changes as progress are the investors, as they see payroll and benefits decrease, increasing their own bottom line.

To read some opposing viewpoints on this issue, check out these two articles:

How Technology is Destroying Jobs from MIT Technology Review

Automation and Technology Increase Living Standards from www.heritage.org.

## Friday, September 9, 2016

### Get the Cart Behind the Horse

Many times during the flash of time when I was in the classroom (even though sometimes it felt like an eternity), I questioned what I was doing as I was doing it, and in the process it seemed like I was living the distinction between school and education.

One thing that always bugged me was that I had to repeatedly refer to the commutative property of addition before students had adequately internalized the facts that in addition the order of the two numbers is irrelevant, but with subtraction order matters. Similarly with multiplication and division. In my school, students would become versed in arithmetic well before getting bogged down in the legalese vocabulary of the laws of arithmetic. (Just as we expect people to be able to drive legally and safely without needing to quote government statutes on transportation)

One thing that always bugged me was that I had to repeatedly refer to the commutative property of addition before students had adequately internalized the facts that in addition the order of the two numbers is irrelevant, but with subtraction order matters. Similarly with multiplication and division. In my school, students would become versed in arithmetic well before getting bogged down in the legalese vocabulary of the laws of arithmetic. (Just as we expect people to be able to drive legally and safely without needing to quote government statutes on transportation)

As another example, I refer to the distance between two points on the coordinate plane. For those non-mathematically inclined, that amounts to placing two dots on a grid such as shown here, and determining the distance between them using the grid's scale.

Quite frequently you would see a textbook begin by giving the formula, then include a short dissertation on where it came from (normally making reference to the Pythagorean theorem), and then jump into an application or two.

Just like the real world! First the formula is found, then it is verified to be true, then it is used! That is exactly how life goes, isn't it? Of course not!

In my personal school I would never make mention of the distance formula until the time came to compare the distance method to that for slope. As a matter of fact, I would introduce the formulas together, AFTER students already had a solid grasp of calculating distances and slopes on the coordinate grid. Prior to that I would make copious use of the phrases "change in y", and "change in x" interlaced with references to "rise" and "run" until the students picked out their seeming interchangeability, at which point we would shorten them to Î”x and Î”y, at which point there would be discussion of the Greek letter delta (any connection to a river's delta?).

Somewhere buried withing the classroom conversation would be the discovery that in the question of distance the signs of Î”x and Î”y were irrelevant, but in the case of slope they were very relevant. To help us with this issue, we would establish a tradition of referring to one of the points as (x

_{1,}y_{1}) and (x_{2, }y_{2}). Only then, when appropriate, would we actually convert our methods for distance and slope into actual formulas. The formulas would appear as short hand for what they were already doing, not as a prescriptive rule to be mindlessly followed. (Could you imagine if a student driver's first lesson was all about the cruise control buttons?)## Tuesday, September 6, 2016

### Welcome to a new school year

I have a creation here that I made today with my usual goal in mind: hoping to help teachers teach and students learn. My hope is that somewhere sometime this school year, some teacher or some student might find this helpful. That is the "some" of my intentions.

The original file can be found here: https://ggbm.at/p48kEWCp.

The original file can be found here: https://ggbm.at/p48kEWCp.

## Thursday, September 1, 2016

### Part of Math is Just Being Amazed.

This I put together this morning just as a thought provoker. It shows the essence of a relationship between circles and ellipses that I have not seen in any book, be it textbook or not.

The basic relationship is accessible to anyone who knows what midpoints of segments and rotation around a point mean. The file shown here can be downloaded at https://ggbm.at/nmNGpVs6

## Friday, August 26, 2016

### Mathematics is much more than you see in school!

This is a revisit of a GeoGebra file I made a couple years ago. I put it here as another step in my push to get students to experience higher level math early on. With the exception of the sentence in red, the entire process uses only basic geometric concepts: points, circles, rotations, segments, midpoints.

The file itself can be found here. Spread the news!

The file itself can be found here. Spread the news!

## Tuesday, August 23, 2016

### Mathtype into Word into Google Docs into Blogger: What a Country!

I have been unable to embed animated gifs into Microsoft Word, but I found Google Docs would handle it. So I created a document in word (with MathType), copied it into Google Docs, embedded the animated gifs (which had been created with GeoGebra and then edited for size in Fireworks), and here it is. The formatting of the equations needs adjustment, but the process seems to work.

This document is for a presentation at the Association of Teachers of Mathematics of New York State scheduled for Rye, NY, in November.

This document is for a presentation at the Association of Teachers of Mathematics of New York State scheduled for Rye, NY, in November.

## Friday, August 12, 2016

### Mathphobics can learn this too!

Here is another file that just might help some teacher or student somewhere as the new school year begins. Find my original here.

## Wednesday, August 10, 2016

### 3...2...1..GO!

Schools will be starting around the country soon, and there is never a chance to "start over". Even Olympians often get to restart. Not teachers or students!

With the goal of helping somebody somewhere get a smooth start, I made this little file.

If one person finds it helpful, it will have been worthwhile. The file can be found here.

## Friday, August 5, 2016

### NBC fails

NBC cannot broadcast opening ceremonies of the Olympics live.

Probably so that they can be sure to get massive ads to fit in.

## Thursday, August 4, 2016

### Math is Fun

Although I could fill pages with comments on the 2016 US presidential campaigns, I choose this time to stick with my GeoGebra efforts. Here is a file I wish I could have used when I was in the classroom. Using basic geometric concepts out of any high school geometry class (circles, rotations, segments, midpoints) it targets a graphing question which would be extremely hard, if not impossible, to introduce without dynamic geometry. Anyone can experience it, even "math-haters".

This file (find it here) took around an hour to create, but has the benefit of reuse. It only needs to be created once. By posting it as a public file, anyone anywhere with access to the web can use it.

I hope people do use it, and come to appreciate how, properly used, programs such as Geogebra can transform and revitalize the classroom.

Remember: math is a game. Go out and have some fun!

## Tuesday, July 26, 2016

### Do you know the state capitals?

It appears to me that when many people hear the name "GeoGebra" they respond with a "what?" and when you indicate it is a combination of "geometry" and "algebra" they immediately categorize it as something from mathematics.

Wrong!!!

GeoGebra can be used in many non-mathematical ways, but I must admit that behind any computer application is the world of symbolic logic, which is hard-core mathematics. I guess if you are reading this, you are using mathematics, whether you want to or not.

Here is a quickie that I put together this morning that would most likely not be used in a mathematics classroom. The possibilities are endless. Restricting GeoGebra explorations to just math teachers and math students will do nothing but strengthen "the Wall" that exists between mathematics and the rest of the world.

Here is a quickie that I put together this morning that would most likely not be used in a mathematics classroom. The possibilities are endless. Restricting GeoGebra explorations to just math teachers and math students will do nothing but strengthen "the Wall" that exists between mathematics and the rest of the world.

A couple of years ago I put together a file dealing with the states and a map, which you can find here). As of today that file needs some tweaking to make it work with the latest html coding standards, but it gives the idea anyway. Enjoy!

## Tuesday, July 19, 2016

### Citizenship at conception?

I have just been reading the GOP platform as posted here.

One part I find completely befuddling. The GOP has tied its entire anti-abortion stance to the 5th Amendment's reference to "life, liberty, and property". The section of the platform is entitled "

**The****Fifth Amendment: Protecting Human Life",**yet is entirely about abortion.
If they wish to take that stance, they had better be prepared to support and/or accept and/or 'deal with"the following (list incomplete):

- tax dependency status for all unborn children, starting at conception.
- "conception certificates" validating location of conception in the event that the child(ren) are born in a different country
- the need for other countries to adopt similar "conception certificates", as the "anchor baby" issue will probably move from country of birth to country of conception.
- citizenship at conception
- the counting of unborn children in the census and the impact on apportionment, etc.
- legal issues when pregnant mother is imprisoned (as innocent fetus ought not to be imprisoned)
- airlines, restaurants, and other such industries charging more for pregnant women, as they are not individuals, but rather are legally 2 (or more) people.

Again I sense a party that has run to a position in order to gather votes without thinking of the overall impact of such a position. All political parties adopt positions in order to get votes, which makes it very difficult to determine what the politicians really think, because it is their real day-to-day thinking 9or lack of it) that controls their daily actions.

## Tuesday, July 12, 2016

### Just a couple of circles....

I hope to create a group of stand-alone basic files that could be an aid to any geometry teacher/student. This one shows how an equilateral triangle is constructed using just circles (compass, if you wish) and a straightedge to draw the sides.

It is my hope that through posts such as this I can help demonstrated the utility of GeoGebra in the high school classroom. This file can be found here.

## Friday, July 8, 2016

### If only....

Years ago I was wishing that my materials (textbooks, workbooks, etc.) allowed me the option of introducing the trig graphs BEFORE doing any right triangle trig. Almost all teaching addresses the special case of right triangle trig before dealing with the whole enchilada. Why?

This sketch would have helped me greatly. The dynamic nature of this sketch enables the teacher and learner many more opportunities than static textbooks.

The file can be found here.

This sketch would have helped me greatly. The dynamic nature of this sketch enables the teacher and learner many more opportunities than static textbooks.

The file can be found here.

## Thursday, July 7, 2016

### Algebra I (CC) June 2016 question 24: Could (Should?) it be better? (update July 20, 2016)

The question above seems sort of innocuous. It appeared on the June 2016 New York State Regents Exam in Algebra I (Common Core).

It seems to overlook the option of a student not to set it up this way:

A student who begins with this approach would then be slightly befuddled as to what the question is asking. This student could easily fall into the "guess trap", with a 75% chance of picking the expected answer.

The question could be improved by changing "should" to "could". That would urge this student to tweak their thoughts by allowing for the possibility of different approaches.

Exam writers should go out of their way to create questions that do not penalize students for applying good mathematics just because it does not meet a certain "prescription".

Update: Take note that in order for two integers to multiply to 156, all a student needs to recognize is

At no point is an equation necessary. Equations should not be viewed as a first resort, but as a last resort. Number sense should rule.

Suppose the question had used the number 6 instead of 156. Absolutely no change in question style. Would you still tell students that they

It seems to overlook the option of a student not to set it up this way:

A student who begins with this approach would then be slightly befuddled as to what the question is asking. This student could easily fall into the "guess trap", with a 75% chance of picking the expected answer.

The question could be improved by changing "should" to "could". That would urge this student to tweak their thoughts by allowing for the possibility of different approaches.

Exam writers should go out of their way to create questions that do not penalize students for applying good mathematics just because it does not meet a certain "prescription".

Update: Take note that in order for two integers to multiply to 156, all a student needs to recognize is

156 = 2 * 78 andThen recognize that 12 and 13 are consecutive integers. The student needs to remember to include negatives as well.

156 = 3 * 52 and

156 = 4 * 39 and

156 = 6 * 26 and

156 = 12* 13 and

At no point is an equation necessary. Equations should not be viewed as a first resort, but as a last resort. Number sense should rule.

Suppose the question had used the number 6 instead of 156. Absolutely no change in question style. Would you still tell students that they

**should**be using an equation?## Tuesday, July 5, 2016

### Algebra 1 (CC) question 29: Mathematics is a language!

Here is a model response to question 28 from the June 2016 New York State Regents Exam in Algebra I (Common Core):

Side note: Reading Comprehension Passage B on the Regents Exam in English Language Arts (Common Core) is a translation of a poem originally in Chinese. I have nothing against translations, and I am currently reading

But: translations on an English Language exam? Aren't there enough sources originally written in English? (Douglas Hofstadter wrote an excellent book on the issue of translations [

The student only got half credit because he/she "did not write an explanation".

**But there is an explanation. It is written in the language of mathematics!**

The student clearly answers "yes" gives the reason saying, in essence. because that is what the solutions are, and then gives sound mathematical support for that claim.

The only flaw I might see is that "x" has drifted above the fraction line in one step, but in the pressure of an exam with limited time to carefully proofread and edit, I would excuse that.

In truth, this is a very very good response.

Side note: Reading Comprehension Passage B on the Regents Exam in English Language Arts (Common Core) is a translation of a poem originally in Chinese. I have nothing against translations, and I am currently reading

*Crime and Punishment*for the third time with different translations.But: translations on an English Language exam? Aren't there enough sources originally written in English? (Douglas Hofstadter wrote an excellent book on the issue of translations [

*Le Ton beau de Marot*]. I highly recommend it.)## 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

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:

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:

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!

## 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.

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

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...

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...

## Wednesday, May 25, 2016

### Be careful out there!

The question given here is from the January 2016 Algebra II/Trig Regents in New York State

I have one major issue with this question.

A student needs to know nothing about circles and their equations. The point (3,-5) makes only one of the four choices true. All a student had to know was which number was x, which was y, and then substitute them in and do the arithmetic.

Yes, the student had to know something. But they had to know nothing about circles.

Please take note that the pair (0,-2) does not satisfy any of the equations.

I can guess at the design behind the question: the student has to know the standard form of the equation of a circle, what to substitute in for its parameters, and how to "wrap up" to identify the radius and complete the solution. As an "fill-in-the-blank" questions this would be fine.

As multiple choice, with the choices it has, it stinks.

## Tuesday, May 17, 2016

### Just a Power Breakfast...

The other morning, while at my local diner for pre-golf breakfast, I sat at a rocky table. I was talking with a man I knew at the next table, and he grabbed an unused paper placemat and started folding it. After the customary 5 or 6 folds (I don't think he got to seven folds), he handed it to me and I stuck it under the table leg and the rockiness was solved.

But, aha!, that was not the end! I, being the dastardly nasty math teacher type, asked him a simple question: If you folded just like you have been folding, but do it 100 times, how thick would it be?

The immediate response was "100 times the thickness of the paper". At that point I said "not even close, you've got to go higher". He asked how high, so rather than telling him I took out my cell phone calculator, punched in a guesstimate for the placemat thickness (I took it for about 300 to the inch) and multiplied it by 2 to the hundredth power. He agreed that exponents could be used for repeated doubling, so we calculated it out. The number is not relevant, but its magnitude (about 11 billion

*light-years*).
He made the usual "no way!" response. Then his tablemates got into the conversation. One of them said "you must be a math teacher", and I said I had been. That's okay.

What befuddled me, however, is that they took the answer and the method for obtaining it as indicative of an esoteric nature of math, mentioning how bad they were at math. All I was doing was multiplying by two!

Then it hit me. My question was out of their comfort zone, yet they wanted to stay in their comfort zone. It was easier for them to reply based on their gut feelings, rather than thinking the problem through.

Many people get to the point where a problem like this is not really a problem. The others have to think about it, and serious thinking while eating breakfast in a diner surrounded by friends and relatives is not an easy task.

Does Whitehead's famous quote apply to individuals as well? "

*Civilization advances by extending the number of important operations which we can perform without thinking of them*."
I wonder; is a school classroom the proper environment for serious thinking? It is a good environment for factual learning and algorithmic learning (how to solve quadratics, write a complete sentence, learn arithmetic, learn how to spell, for example.) But deep, serious, thinking? Especially on a timed test? My best mathematical thinking is done alone, with smooth jazz or light classical playing, with the opportunity to take my dog out when the thinking is in the mind and not yet ready for putting pencil-to-paper. In college I used to go into an empty classroom in the evening and make major use of the chalkboards when working on assignments. Class time was for acquiring the specifics which, presumably, would give a helpful knowledge base for the assignments. Deep thinking in a class full while at a small lecture desk? No way.

Thinking is what we have to do when we don't know what to do. If we know what to do, there's no problem. Yes, we would like future generations to be good problem solvers, but, more than that, we need to prepare them with a factual and knowledge base so that many potential problems will not be problems.

Stanford has a course called Introduction to Mathematical Thinking. The course description states

Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today’s world. This course helps to develop that crucial way of thinking.

Why do they use the box analogy this way? Why don't they just say that they will be addressing mathematics as it sits in a larger box? They put down high school math while ignoring that high school students are not professional mathematicians, and they fail to mention that "professional mathematicians" is a gross generalization. Many mathematicians I know work in very small boxes anyway. One quick web click and I find a representative of this group listed with

Specialties: Abstract algebra, Galois theory, geometryInterests: Commutative rings and algebra

Most professional mathematicians are equally specialized. (Just as you would not go to a dermatologist when you need a pacemaker) Stanford also overlooks the fact that all those "professional mathematicians" were once high school students.

I would like to believe that Stanford's description is referring to mathematicians hired by business and industry to apply their special skills and talents in very specific arenas. If that is what they meant, it is not what they said.

Back to breakfast. My diner neighbors, at least one of them if not more, added that they "never saw any use in the math they were seeing in school". They missed the fact that they will never use what they do not know. Yes, I do not know the Greek language, and naturally I have never been able to use that knowledge. Would my life be different if I had learned Greek? Maybe yes, maybe no, but doors of opportunity are and were definitely closed to me. I cannot say what was on the other side of those doors since I could never go through them.

## Monday, May 16, 2016

### Time is of the essence

On January 22, 2015 I posted a blog about the FSA Mathematics Reference Sheets.

After thinking about the issue for over a year, it dawned on me that another part of the FSA reference sheets is just as insulting to students. That part contains the following:

Florida appears to have decided that expecting students to remember these bits of information is tantamount to cruel and unusual punishment. New York must have thought differently, as none of the time conversions appear on its reference sheets.

It is fascinating when you compare the reference sheets from different states. Texas, for example, (see here) has no conversion facts on its reference sheets. None. At all.

What intrigues me is that each state is claiming to be aspiring to "higher standards." Are we raising standards when we no longer expect students to know that there are 60 minutes in an hour? It's even more strange when we tell them, as fact, that 52 weeks is 1 year. That is flat out wrong.

For clarification, 52 weeks is 364 days, and no year is 364 days in length. To obtain that fact, I had to know that 1 week is 7 days, which must be something Florida expects students to remember, as it is left out of their reference sheet. When I was young I remember being told that there were "52 weeks IN a year" and also being told that it was not an exact match: 52 was as many as you get get without exceeding 1 year. We were also told to use that piece of information only when estimating.

The current wave seems to look at remembering as unimportant and unnecessary. The "experts" tell us to "teach children how to think".

Speaking of thinking...

The clip above is from the New York State reference sheets for high school math (see it here). The rectangles are mine. I used them to indicate the conversions that are expressed incorrectly.

Each of those conversions that are boxed are approximations expressed as if exact. There is no side comment clarifying anything (such as "rounded to the nearest hundredth"). In the absence of a side comment, there are symbols commonly used that mean "approximately equal", such as "≈". Florida could have escaped detection by stating "52 weeks ≈ 1 year".

I do commend Ohio for proper use of "≈" on its reference sheets when stating a value for Ï€.

before i sign off on today's missive, I wish to encourage you to read Ben Johnson's blog post from early 2010. Read it and ask yourself if you would rather have a generation that remembered that an hour was sixty minutes, or a generation that has to look it up.

## Friday, May 13, 2016

### Baseball is Math!!

As I continue my experiments with GeoGebra 3D, I thought about that lace pattern. Rather than reinventing them, I borrowed from http://paulbourke.net/geometry/baseball/, and used them in this graphic. Nothing fancy. Just getting a handle on GeoGebra 3D features.

One of the features I am having issues with is getting a 3D version (the kind you have to view with special glasses) to save and embed successfully. Wish me luck!

## Thursday, May 12, 2016

### See what one midpoint can do for you!

When you first see the image below, you will see a red ellipse. That ellipse is the result of letting two points (the blue and red X's) move at the same speed in opposite directions around their respective circles and tracing the midpoin between them. Start Animation to watch!

Change the speeds and you will change the graph. Experiment!

To modify, expand, adjust, alter, or just plane change the file, get it here.

## Tuesday, May 10, 2016

### Using GeoGebra to pose questions

I truly wish I could have had dynamic geometry such as GeoGebra to pose problems for my classes.

I suspect this problem could have fit in as a challenge to a precalculus class. It is probably also a good question for math teachers.

Well, is it?

I suspect this problem could have fit in as a challenge to a precalculus class. It is probably also a good question for math teachers.

Well, is it?

## Sunday, May 8, 2016

### What would they do?

Every once in a while i just try to imagine what young elementary school students might come up with if given half a chance to acclimate themselves with a program such as GeoGebra. What if...

## Thursday, May 5, 2016

### An elliptical rose?

Every once in a while I try to create something that just acts as a"tug" to get the user moving in the direction of dynamic math. Here is a little file that just demonstrates something that a lot of teachers might never think of because without dynamic geometry it is massively ugly.

It just involves controlling the rates at which two points travel on an ellipse, and tracks the midpoint of the segment connecting them. Many PreCalculus books include sections on polar graphs, based heavily on trig. This takes those shapes and modifies them. A sketch based on circles can be found here.

If you would like to play around with this file, get it now!

## Tuesday, May 3, 2016

### WAVING to the crowd!!!

I was experimenting today with GeoGebra 3D, trying to see how I could simulate a "ripple tank". The Geogebra part is not too bad (so far), but getting the 3D file embedded in a web page is a nightmare. Getting browsers, devices, operating systems,. and GeoGebra to get all in sync is not nice.

At any rate, I took a rather large automated gif copy of part of a file, doctored it up to shrink it down in size a bit, and put it here.

For its parent file, you will have to wait. It is still in "production."

Enjoy!

## Monday, May 2, 2016

### Monkey in the middle!

Here is just another demonstration of a relationship between an ellipse and two special concentric circles. Two points will rotate around the circles at equal speeds in opposite directions. The midpoint between them will be traced. Those rotating points will start at the top of the circles if the vertical axis is longer than the horizontal axis. Otherwise the rotating points start "due east" from the center.

Play around.

The original file is here. Download it and experiment. GeoGebra is cool!

## Thursday, April 28, 2016

This little sketch shows an example of how individual points can help one gain control of smooth curves. The concept has numerous applications, is based on high school mathematics, and has a logic to it that can be understood by anybody with a bit of number sense and some familiarity with coordinates.

This file can be found here.

## Tuesday, April 26, 2016

### Time for a change?

I have gradually come to the opinion that the introduction to mathematics visually should come from circles. For years we have striven in schools to begin with segments and triangles. Sure, those things are important, but let's face it: fun time as a kid involves circles (merry-go-rounds, hula hoops, etc) and other curved shapes. What kid starts laughing after quickly walking a straight line? Let them spin in a circle, though...

Here is just some doodling from this morning as I was experimenting in Geogebra. My complete file is here.

Here is just some doodling from this morning as I was experimenting in Geogebra. My complete file is here.

## Tuesday, April 19, 2016

### Some thoughts on primary day in New York (there is some math in here anyway)

The following is from the April 11, 1997 edition of The New York Times, (Find article here). The event was a speech Donald Trump gave as Principal for a Day at a public school in New York City.

Note that he did not even say that he would randomly draw 15 names: no, he would "pick" 15 people. Also, he would take them to his store, so that his money would be paid to his store. At least Andres Rodriquez made everyone aware that Trump was nothing more than a selfish rich guy.

Make sure you read the article. There is more to the story.

While reading a Maureen Dowd article from Nov. 17, 1999, it hit me: "Trumpster" does indeed rhyme with "dumpster." It does. It's a fact. And I am sure that many Trump fans would hate me for saying it. I guess it is possible to hate the truth.

According to 2014 data (see here) the 21st Congressional district had 127,262 registered Democrats and 181,832 registered Republicans. The 5th Congressional district had 301,082 registered Democrats and 35,339 registered Republicans.

In the 2016 primary (voting today) each district will have 3 delegates up for grabs in each district. That gives the 35,339 people in district 5 the effective voting power of the 181,832 from district 21.

Under primary rules in effect, a candidate needs 50% of the vote to get all 3 delegates. The specific numbers are probably different this year, but not by much.

So 18000 people in district 5 (partly in Nassau County, partly in Queens) gets a candidate 3 delegates. Compare that with 91,000 required in district 21 (parts if not all of 12 northern NY counties).

Any wonder why upstate gets largely ignored?

Also: Donald Trump cries out about a "rigged" system. I haven't heard him say anything about this.

Mr. Trump glided to the microphone.

''First of all, who likes Nike sneakers?'' he asked. All 300 fifth graders raised their hands. Mr. Trump leaned in to drop the bombshell. ''If everybody puts their name on a piece of paper right now, I will pick 15 people and I'll take you to the new Nike store that I just opened at Trump Tower.''

The fifth graders erupted in frenzied excitement at the promise of a trip to what Mr. Trump described as the ''inner city called 57th and Fifth.'' But a little while later, 11-year-old Andres Rodriguez had a question.

''Why,'' asked Andres, whose father is dead and whose mother cannot work because of a bad leg, ''did you offer us sneakers if you could give us scholarships?''

Note that he did not even say that he would randomly draw 15 names: no, he would "pick" 15 people. Also, he would take them to his store, so that his money would be paid to his store. At least Andres Rodriquez made everyone aware that Trump was nothing more than a selfish rich guy.

Make sure you read the article. There is more to the story.

While reading a Maureen Dowd article from Nov. 17, 1999, it hit me: "Trumpster" does indeed rhyme with "dumpster." It does. It's a fact. And I am sure that many Trump fans would hate me for saying it. I guess it is possible to hate the truth.

According to 2014 data (see here) the 21st Congressional district had 127,262 registered Democrats and 181,832 registered Republicans. The 5th Congressional district had 301,082 registered Democrats and 35,339 registered Republicans.

In the 2016 primary (voting today) each district will have 3 delegates up for grabs in each district. That gives the 35,339 people in district 5 the effective voting power of the 181,832 from district 21.

Under primary rules in effect, a candidate needs 50% of the vote to get all 3 delegates. The specific numbers are probably different this year, but not by much.

So 18000 people in district 5 (partly in Nassau County, partly in Queens) gets a candidate 3 delegates. Compare that with 91,000 required in district 21 (parts if not all of 12 northern NY counties).

Any wonder why upstate gets largely ignored?

Also: Donald Trump cries out about a "rigged" system. I haven't heard him say anything about this.

## Friday, April 15, 2016

### What's your sine?

I rarely post twice in one day, but today I am.

My last post was all about using GeoGebra as a tool where the "teaching moments" arise in the creation of the file.

This example is more about the idea of teacher use of GeoGebra to create presentation files. Think of it as an interactive dynamic PowerPoint show,

Enjoy.

PS: This one can be found here.

### Can YOU make This?

There is not much to this graphic, other than the fact that it was made in GeoGebra (using a clip-art graphic from here).

There is, however, a lot of mathematics involved in its creation. This diagram is not intended to teach, show, or explain anything. The creation of this diagram (similar to yesterday's) could make for a great lesson involving geometry, algebra, systematic thinking in general, and target-based thinking. For this diagram, I will withhold my source so that, if you just want to copy it, you really have to search it out.

Carpe diem.

## Thursday, April 14, 2016

## Tuesday, April 12, 2016

### From 2 Circles: A Little Magic

Things that would be next to impossible with the old "chalk talk" become quite teachable using GeoGebra. That is not to say that files like this can be made by young students: rather, their teachers can use files like to help students discover math!!

f

With precalculus students I would aim to give them specific circles, and ask them how they can use those circles to pinpoint an equation for the ellipse.

With 8th grade students I would use a file such as this to embed in their brains some basic geometry and basic geometric vocabulary. I might even give them a graph paper with two circles drawn, a compass, protractor, and straightedge, and ask them to recreate this process by hand.

Same file, same subject, totally different strategies. GeoGebra, creativity, and time: the key ingredients..

f

With precalculus students I would aim to give them specific circles, and ask them how they can use those circles to pinpoint an equation for the ellipse.

With 8th grade students I would use a file such as this to embed in their brains some basic geometry and basic geometric vocabulary. I might even give them a graph paper with two circles drawn, a compass, protractor, and straightedge, and ask them to recreate this process by hand.

Same file, same subject, totally different strategies. GeoGebra, creativity, and time: the key ingredients..

## Monday, April 11, 2016

## Thursday, April 7, 2016

### Did you know?

There are many many people who know a lot about ellipses who do not know this.

Be the first on your block to be aware of it!

Be the first on your block to be aware of it!

## Wednesday, April 6, 2016

### Can you have a 3.55-leafed rose?

Back when I was in the classroom, as a student and a teacher, I was never satisfied with the explanations why y=cos(nÎ¸) has "n" leaves when n is odd, but "2n" leaves when n is even. I wish I had GeoGebra back then.

This even helped me internalize how a polar graph relates to a Cartesian graph, helping me picture the x-axis collapsed down to a single point.

At any rate, a school where nobody is working with GeoGebra is a school missing out on a huge resource.

This file can be found here.

;;

This even helped me internalize how a polar graph relates to a Cartesian graph, helping me picture the x-axis collapsed down to a single point.

At any rate, a school where nobody is working with GeoGebra is a school missing out on a huge resource.

This file can be found here.

;;

## Thursday, March 31, 2016

### Doesn't memory matter anymore? (Revisited)

Here are two images from PARCC's math reference sheets. (F you do not know what PARCC is, find out. Everyone should know). Look at them closely, and tell me what you notice.

Did you get it? The thing that stands out?

Look for "1 mile = 5280 feet".

It is in both. It also happens to be in all the PARCC reference sheets for grades 6, 7, and 8 as well.

For some reason PARCC expects students every year to use this information. Heck, they think it is important enough to put it on a reference sheet.

Evidently what they do not expect is students to remember that 1 mile is 5280 feet.

Ditto for for "1 quart = 2 pints", "1 pound = 16 ounces", "1 ton = 200 pounds", and others as well.

Take note. All is not lost!

They have left out "1 yard = 3 feet".

My guess is that with its emphasis on problem solving and the availability of calculators, PARCC presumes that a student can figure that out by dividing 5280 by 1760.

I think back to my years in school, before calculators, when answering the question "What is 5280 divided by 1760?" correctly showed an ability to do something.

Those who belittle that ability, please take out a piece of paper and a writing utensil, and without using any electronic devices, divide 34785 by 4638.

For more thoughts on this topic, please visit this.

## Wednesday, March 30, 2016

### If only I had this in class...

GeoGebra has a steep learning curve at first, but the benefits are many.

Here is a little example demonstrating how control buttons and check boxes can help you control what is displayed. while maintaining interactivity of the display.

I might have used this in class by asking students to first give what they think an angle bisector (or median) might be. I could have projected this on the board, had the class give its best, and then show what the results actually are.

When discussing compass and straightedge constructions, project this on the board, demonstrate the construction, then let compare work with the Geogebra results.

The only limitations are your imagination together with the skills you have learned.

The file can be downloaded here.

## Tuesday, March 29, 2016

### Let me know what you think of this

GeoGebra has massive functionality and flexibility.

Here is something that I could never have done in class without GeoGebra. It compares a couple methods for generating ellipses.

Play with it and let me know how I could improve it. The file can be found here.

## Monday, March 28, 2016

### Sine, Sine, Everywhere a Sine

I just made this up as an aid in teaching/learning the relationship between the sine of an angle and the unit circle. I have left it to the teacher to take the final step and actually point out the angle.

Do not deny your teachers and students the opportunity to learn and use GeoGebra!

Do not deny your teachers and students the opportunity to learn and use GeoGebra!

## Thursday, March 24, 2016

### Give GeoGebra a Chance

Dynamic Geometry can be used to gain efficiency and minimize careless errors in class. Here is another that I wish I had had in class. Imagine how much more I could have focused on the students in class by having presentations dynamic and almost error-free. Imagine students having access to the same dynamic files from home.

Here is an example that shows how the measures of the angles of a triangle can be found if the lengths of the three sides are known. Besides the basic algebra, the student would need to know the Pythagorean Theorem and basic right triangle trig definitions. Taking the last step, to the angle measures themselves, could use a trig table, a calculator, or even Google. (But Google will give angle measure in radians, not degrees, so its use should be guided. If you wish, you can google "calculator", flip it into degree mode, and do the inverse calculation).

This file can be found here.

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