My graphic from yesterday has been extended to include foci and the related hyperbola with their asymptotes. I will leave it to your imagination and knowledge to identify how the foci, asymptotes, and hyperbola can be connected to the animation.

Thoughts, opinions, and ideas about Mathematics and other things from a retired teacher.

## Wednesday, December 10, 2014

## Sunday, December 7, 2014

### More on Ellipses!

An expansion on an earlier post (December 5)

Every ellipse can be constructed by rotating two points on concentric circles at the same rotational speed in opposite directions, and tracing the midpoint of the segment between them. This sketch shows the process for horizontal and vertical ellipses.

Every ellipse can be constructed by rotating two points on concentric circles at the same rotational speed in opposite directions, and tracing the midpoint of the segment between them. This sketch shows the process for horizontal and vertical ellipses.

## Saturday, December 6, 2014

### Know the Quadratic Formula!!!

The Common Core does correctly propose that each student should

The entire process is here.

*Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q**that has the same solutions. Derive the quadratic formula from this form.*(see here)The entire process is here.

## Friday, December 5, 2014

### Ellipses a different way

Every ellipse can be constructed by rotating two points on concentric circles at the same rotational speed in opposite directions, and tracing the midpoint of the segment between them. Here, the two points Drag1 and Drag2 are movable on the x-axis. This particular file is restricted to horizontal ellipses.

## Wednesday, November 19, 2014

### Are you a math hater?

Here is a creation by Markus Hohenwarter, the CEO of GeoGebra. (see his page here.) Below I have shown a "doctored" version, where I have shown all the hidden parts of his creation.

Quite often mathematics is what goes on behind the scenes. The math-phobics actually hate that which makes their lifestyle possible!

Quite often mathematics is what goes on behind the scenes. The math-phobics actually hate that which makes their lifestyle possible!

## Monday, November 17, 2014

### A quick GeoGebra-based illusion

This sketch was created to show how easily our eyes and brain can mislead us.

A simple rotation is perceived differently if key anchor segments are removed.

The complete file is here.

A simple rotation is perceived differently if key anchor segments are removed.

The complete file is here.

## Friday, November 14, 2014

### Polar Roses

This is a continuation of my investigations which I have called "floating midpoints". Enjoy

## Saturday, November 8, 2014

### Limacons and cardiods without trig!!!

You have control over the location of point E along the x-axis from 1 TO 4. Point N rotates around point E on a circle of radius 1. The blue circle always contains the origin and is centered at point N. The blue dotted line is tangent to the green circle (that N rotates on). The red dotted line is the perpendicular from the origin to the blue dotted tangent line. The point of intersection of those two lines is traced.

Although more complicated than some of my other sketches, it still shows what can be created WITHOUT USING TRIGONOMETRY that, in our textbooks, is usually not even mentioned until well after trig is covered in detail.

For the full file, visit http://tube.geogebra.org/material/show/id/266203

## Friday, November 7, 2014

### So what is a focus and directrix?

It is quite possible to use GeoGebra to give students the "flavor" of a topic before getting bogged down in textbook details. This particular sketch can give the flavor of the concepts of focus and directrix, their meaning and visual placement, well before any need for calculation arises. In short, GeoGebra can be a motivator.

## Thursday, November 6, 2014

### Sierpinski Triangle Generated Randomly

This sketch will gradually create the Sierpinski triangle by randomly moving, at each step, halfway towards one of the 3 vertices, chosen randomly.

## Friday, October 10, 2014

### Join as at AMTNYS 2014 in Syracuse

Attend my session at the AMTNYS (Association of Math Teachers of New York State) in Syracuse (Nov. 9-11) and you can find out how to make sketches like this, as well as see what I can show you about animations in GeoGebra. For conference details, visit the AMTNYS conference site here.

## Monday, October 6, 2014

I have titled this one "A Heart As Good As Any" and it can be found (and downloaded and edited) here.

Circles with radii 1, 3, and 6, centered at the origin at 3 units below. Two points rotating counterclockwise at the same rotational speed, one clockwise at twice that speed. Those points are connected to form a triangle, and two angle bisectors shown. The trace shows the path of the point of intersection of those medians, the incenter.

## Wednesday, August 20, 2014

## Tuesday, August 19, 2014

### Math is Fun!!!!

There is no need to hold off complicated plotting for higher grades. Young students can learn a lot while playing and exploring with GeoGebra. Here is just a little sample. Have fun!

## Tuesday, August 12, 2014

### Warm Up Your Brain for September

Can you recreate this?

Can you determine parametric equations for the boundary of the painted region?

It is time to start exercising brains for the start of a new school year!

This file involves two points rotating around a triangle, in opposite directions, one of them moving three times as fast as the other. The segment between those two points is one side of an equilateral triangle. The triangle becomes a "paintbrush" and its trace shades in a region.

Can you determine parametric equations for the boundary of the painted region?

It is time to start exercising brains for the start of a new school year!

This file involves two points rotating around a triangle, in opposite directions, one of them moving three times as fast as the other. The segment between those two points is one side of an equilateral triangle. The triangle becomes a "paintbrush" and its trace shades in a region.

## Wednesday, July 9, 2014

### Centroids can be fun!!

Any school that does actively use GeoGebra is missing out on an amazing tool!!!

the full file can be found here.

the full file can be found here.

## Monday, June 9, 2014

## Friday, May 23, 2014

### A sewing machine got me doodling...

Imagine what you could do if you only starting playing around with Geogebra!!

## Wednesday, May 21, 2014

## Friday, May 16, 2014

## Thursday, May 8, 2014

### What is mathematical about a boat?

Everything that was used to create the Geogebra file involves knowledge that should be gained by a high school trig student.

(I am assuming that in the age of technology the concept of "phase shift" has become an integral part of a trig course.)

I did use the built-in tangent line function, rather than go through the task of using calculus operations to find the same result.

For those who want to see how the tangent line is actually identified, make sure to pay attention in your Calculus class.

As long as the notion of tangent lines is on the table, I suggest you take a look at this (the original is here)

## Wednesday, April 9, 2014

### Traveling Salesman for Beginners

The Traveling Salesman problem is, simply stated, the problem of finding the best route between locations.
Here the only consideration is straight line distance, in the attempt to visit each point and end up back where you started.
Drag the points to see what happens.

## Friday, March 28, 2014

### Using a Protractor

Everybody need to know how to use a protractor!!!

The GeoGebra page itself is here.

The GeoGebra page itself is here.

## Wednesday, March 19, 2014

## Friday, February 28, 2014

## Tuesday, February 25, 2014

## Monday, February 24, 2014

### 4 Leafed Rose via Midpoints

Point D does 3 counterclockwise laps for every lap that point C does. Point E traces the path of
the midpoint of segment CD.
The red graph is the polar equation r=cos(2Î˜)

the file is at http://ggbtu.be/m89785

the file is at http://ggbtu.be/m89785

## Thursday, February 20, 2014

### Something special about the law of sines?

The Law of Sines says that, in any triangle, the sine of an angle, divided by the length of the opposite side, is constant.

or If you give up, click here!!!!

A question not asked in high school texts is this: is there something special about the value of that constant. Does it measure something? Is it a clue about anything?

or If you give up, click here!!!!

## Tuesday, February 18, 2014

## Thursday, February 13, 2014

### 12? or 13?

On this snowy day in the great northeast, it is time to practice our counting skills. Just count the number of people in this picture. After it moves, count again. Then go figure...

I got this a while back and have had it on my web page for years. I am still searching for a "neat" solution. Good luck!!
## Monday, February 10, 2014

## Thursday, February 6, 2014

### Some practice, some play

Two parts today. the first part is for some practice for first year Algebra students. For the original GeoGebra file, click here. The second (see below) is just some doodling.

This may be a cartoon hat, a sketch of the space shuttle from underneath, or a side view of a bell. Whatever you may see or think of, it comes from just moving circles.

If you want to see the GeoGebra file itself, click here.

This may be a cartoon hat, a sketch of the space shuttle from underneath, or a side view of a bell. Whatever you may see or think of, it comes from just moving circles.

If you want to see the GeoGebra file itself, click here.

## Wednesday, February 5, 2014

### Curiosity

Everyone is a student and the only thing controlling that is their curiosity. Doodling itself can be a stimulus to curiosity. Here is my latest GeoGebra doodle from this morning. For a copy of the GeoGebra file, click here.

This second one is not a doodle. It is using a specific equation which I had seen somewhere on the web. When I can find it again I will give credit where credit is due. This is just a doctored version of the original. My GeoGebra file is here

## Sunday, February 2, 2014

## Saturday, February 1, 2014

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