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.

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.

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.

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.

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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.

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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.

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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.

 

Message in a Bottle

Just an attempt at making a "catcher". Put together in GeoGebra.

The original is here.

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!!
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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..

A Post Without Many Words

Find original here.

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!

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(nθ) 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.

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