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...