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.

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!

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.

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.

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.

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

156 = 2 * 78 and 
156 = 3 * 52 and 
156 = 4 * 39 and 
156 = 6 * 26 and
156 = 12* 13 and 

Then recognize that 12 and 13 are consecutive integers. The student needs to remember to include negatives as well. 

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?

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):

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.)