If only... (updated)

What I could have done in the classroom if I had had GeoGebra, a smartboard, and a tablet/laptop of every student. I would toss the textbook, and go.

Here is a sketch easy to create, but full of mathematical questions.
I have included the option of showing some parametric info just as a "spur".

The blue point, the center of the blue circle, rotates on the green circle.
The red point rotates on the blue circle. The red and blue points complete one rotation at the same time.
Does this generate an ellipse?

Start with parametric equations as above.
\(\begin{array}{l}x = n\cos ( - \theta ) + m\cos (\theta )\\y = n\sin ( - \theta ) + m\sin (\theta )\end{array}\)

Use negative angle identities.
\(\begin{array}{l}x = n\cos (\theta ) + m\cos (\theta )\\y =  - n\sin (\theta ) + m\sin (\theta )\end{array}\)

Factor
\(\begin{array}{l}x = \left( {n + m} \right)\cos (\theta )\\y = \left( { - m + n} \right)\sin (\theta )\end{array}\)

Rewrite:
\(\begin{array}{l}\frac{x}{{n + m}} = \cos (\theta )\\\frac{y}{{m - n}} = \sin (\theta )\end{array}\)

Square and add
\({\left( {\frac{x}{{n + m}}} \right)^2} + {\left( {\frac{y}{{m - n}}} \right)^2} = {\cos ^2}(\theta ) + {\sin ^2}(\theta )\)

Use Pythagorean identity
\({\left( {\frac{x}{{n + m}}} \right)^2} + {\left( {\frac{y}{{m - n}}} \right)^2} = 1\)

Rewrite
\(\frac{{{x^2}}}{{{{\left( {n + m} \right)}^2}}} + \frac{{{y^2}}}{{{{\left( {m - n} \right)}^2}}} = 1\)

We now have a standard equation for an ellipse.

Birth of an Ellipse (Revisited)

As we begin the new year, I am revisiting one of my past GeoGebra creations. This shows how an ellipse results from tracking the midpoint as the endpoints of a segment rotate around two circles with the same center. The points rotate in opposite directions, but complete an orbit in the same amount of time. 

This is just one of my examples intending to show how topics generally left for later in high school can be introduced much earlier. This example reinforces concepts such as circle, rotation, segment, and midpoint while generating an ellipse. The equations shown in the process can be eliminated or hidden. They were included for those who wish to connect this with higher concepts in Algebra II or later.

This is also my first post to be shared with Facebook, which I have joined, at least for now.  Consider this a test!

normal probability cumulative density function

If the title of this post seems awkward, imagine my feelings when I found it on page 36 of New York State's Algebra II Fall Sampler from Fall 2015.

I encourage you to look at that link, and keep in mind that just a few minutes ago I Googled that phrase (in quotes) and got 3 results, with the Fall Sampler being the second in the list. Here it is. ( I clicked the "If you like" at the bottom, and picked up one more link.)

The Common Core states that students should be able to "Use calculators, spreadsheets, and tables to estimate areas under the normal curve."

The area under a curve is a whole subject in and of itself (part of Calculus), and its discovery (or invention?) generally begins with rectangle approximations, trapezoidal approximations, limits, continuity, etc. Are these all part of Algebra II? I know that the process of approximation of the area of a portion of the Cartesian plane bounded above and below by continuous functions of x is very highly programmable. A lot of mathematics is involved in creating such a program. Expecting high school students to use such a program while remaining ignorant of the mathematics involved in its creation seems to be a disservice to those students.

The whole business about Common Core seemed predicated on understanding mathematics. Does someone who has mastered the art of pushing buttons on a calculator understand this concept? And whose idea was it to base a high school sample question on "normal probability cumulative density function".

Limacon with inner loop

Continuing my push to use Geogebra to introduce younger students to "fancy math", here is a plot similar to my last two blog entries. Although the shape is determined by the relative speeds of Jack and Jill, just as in the previous two postings, this one was shifted a bit to the right just so that the origin is more centered.

The original file is here.

I will not be posting here for a couple of weeks as a different project will intervene. Merry Christmas!

5-Leafed Rose the simple way

Here is an extension of yesterday's post, where I have adjusted the graph to get 5 leafs instead of 3.
This file can be found here.
Below is a bit of mathematics that I would have discussed in a class. with the goal of graphing in parametric mode on a graphic calculator. For students not yet up to that, I would adapt this so that they can experiment with different rotational speeds, reverse directions, etc., and would not include the pre-plotted graph.

 \[\begin{array}{c} A = ( - .75,0)\\ Jack = ( - .75 + \cos ( - 2t),0 + \sin ( - 2t))\\ C = (.75,0)\\ Jill = (.75 + \cos (3t),0 + \sin (3t))\\ {\rm{red}}\,{\rm{point = }}\frac{{Jack + Jill}}{2}\\ {\rm{red}}\,{\rm{point = }}\frac{{\left( {\cos ( - 2t) + \cos (3t),\sin ( - 2t) + \sin (3t)} \right)}}{2} \end{array}\]

Going in Circles Can be Fun!!

A major push of mine is using dynamic geometry such as GeoGebra to introduce younger students to more interesting mathematical concepts.
Here, based on knowledge of circles, rotations, and midpoints, a student can learn something about 3-leafed roses. The file can be found here.

I would love to see how to do this in Desmos, or how to make it web-friendly in Nspire.

Matrix Transformations in GeoGebra

GeoGebra provides a great avenue for traveling through the world of matrix transformations of the plane. Here is just a little example. Imagine if our teachers could compile a folder full of such sample, ready to pull out and share at a moments notice.

This one can be obtained  here. I hope it helps somebody somewhere.