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.

Bezier for Young People

Bezier Curves are generally not confronted by k-12 students at all. Their equations can be complex, and in the absence of dynamic geometry, the topic in general can be mind boggling. Historically, algebra has been the driving force for graphing. With dynamic geometry, that can be reversed.

If you can mentally stomach (how's that phrase?) the idea of a point on a line steadily sliding from one location on that line to another location on that line, you've mastered the necessary skills. Visually, a point sliding from one place to another would look like this:

Being able to picture this is all you need to know. Here it is

Be creative in your use!

Here is a sketch that could basically be used with any grade, from a visual with elementary students, to a "can we make it ourselves" with middle school students, to a model for exploration for upper levels.

With an elementary class, I would leave out all the text, and create a step-by-step show, from first circle to tangent line to second circle to midpoint to trace, but not using sophisticated language. With middle school students I would use the basic geometric language and do a step-by-step as well. Upper students who are familiar with Geogebra could be shown the graphic and asked to recreate it. Those unfamiliar could be guided through it. Precalculus students could be challenged to determine an equation that could be graphed on a graphic calculator.

Adjustments to the file are easily made.

The main point is that this technology should not just be used as crutch with old curricula, but should also be used as an avenue for new approaches to mathematics education.

The complete file can be found here.

Math without GeoGebra is like a day without sunshine!

It might be the right time of year to push the use of GeoGebra as a classroom presentation tool. I created this interactive file back in March of this year, and pull it out now with a request for teachers to send in ideas for modifications of this file, or suggestions for new files (send to dave(at)davemath.com.  This file can be found here.

Any and all GeoGebra files can be downloaded and modified. 

GeoGebra is totally dynamic and its use is limited only by your imagination!

  

Never be satisfied with guess and check!

(Please read my last blog entry either before or after this one. They go together.)
\[\begin{array}{c}\left( {nx + m} \right)\left( {px + s} \right)\\nx\left( {px + s} \right) + m\left( {px + s} \right)\\np{x^2} + nsx + mpx + ms\\np{x^2} + (ns + mp)x + ms\end{array}\]

The above should be recognized by all secondary teachers of mathematics as a generic example of using the distributive property (of multiplication over addition, as it is frequently phrased) to multiply a pair of binomials.

Simple and to the point. Anyone can do it.

What I wish to point out to those who do not notice is that \[(np)(ms) = (ns)(mp)\]

This might seem like a bit of obvious but but seemingly irrelevant trivia, except for the fact that this little fact is the key that unlocks what I believe to be the way that quadratic factoring should be done (and taught).

Take a close look at an actual example (using numbers).
\[\begin{array}{c}(2x + 3)(5x + 7)\\2x(5x + 7) + 3(5x + 7)\\10{x^2} + 14x + 15x + 21\\10{x^2} + 29x + 21\end{array}\]
This is how the multiplication of binomials should look. (Forget that mnemonic FOIL. Forget it now and forget it forever.)

Now we will see the same steps displayed in reverse order.
\[\begin{array}{c}10{x^2} + 29x + 21\\10{x^2} + 14x + 15x + 21\\2x(5x + 7) + 5(3x + 7)\\(2x + 5)(3x + 7)\end{array}\]

Take note the first step involves separating \(29x\) into \(14x + 15x\). Why choose 14 and 15? Because they add to 29 AND multiply to 210, the product of 10 and 21. The rest is just applying that same old distributive law.

Take a look at one from scratch, say \(4{x^2} + 43x + 63\)

Our first step will be to find two numbers that add to 43 and multiply to 252, which is the product of 4 and 63.

\(\begin{array}{c}(1)(252)\,\,\,add\,\,to\,\,253\\(2)(126)\,\,\,add\,\,to\,\,128\\(3)(84)\,\,\,add\,\,to\,\,87\\(4)(63)\,\,\,add\,\,to\,\,67\\(6)(42)\,\,\,add\,\,to\,\,48\\(7)(36)\,\,\,add\,\,to\,\,43\end{array}\)

Take note: the numbers on the left are no more than counting, 5 was skipped because it is not a divisor of 252, and we stop at 7 because we found the numbers we need. With these numbers we can continue: 

\(4{x^2} + 7x + 36x + 63\)

\(1x(4x + 7) + 9(4x + 7)\)

\((1x + 9)(4x + 7)\)

Please please recognize that this process is highly programmable. It is actually a system based on action, not a fallback "guess and check" that is pushed on kids far far too often. Also, take note that I do accept "1" as a meaningful numeral, and do not always jump at the chance to avoid writing it.

Here is a sample with negatives: \[6{x^2} - 7x - 5\]

We start by taking pairs of factors of \[ - 30\]. Since the two numbers must add to a negative, we will make the larger number in each pair negative and keep the smaller one positive.

\[\begin{array}{c}(1)( - 30)\,\,\,add\,\,\,to\,\, - 29\\(2)( - 15)\,\,\,add\,\,\,to\,\, - 13\\(3)( - 10)\,\,\,add\,\,\,to\,\, - 7\end{array}\]

We got our two numbers pretty quickly this time, and we can continue.

\[\begin{array}{c}6{x^2} - 7x - 5\\6{x^2} + 3x - 10x - 5\\3x(2x + 1) - 5(2x + 1)\\(3x - 5)(2x + 1)\end{array}\]

Please please remember that the special cases with leading coefficient 1 are special only to someone who knows the whole story. They are not special to a student seeing them for the first time. What may be seen by those in the know as a shortcut cannot be seen by beginners as a shortcut. A shortcut is never meaningful unless and until a longer route is known.

If you would like a seemingly endless list of practice problems (with check-ability), check this out.

Math is Fun

Although I could fill pages with comments on the 2016 US presidential campaigns, I choose this time to stick with my GeoGebra efforts.  Here is a file I wish I could have used when I was in the classroom. Using basic geometric concepts out of any high school geometry class (circles, rotations, segments, midpoints) it targets a graphing question which would be extremely hard, if not impossible, to introduce without dynamic geometry. Anyone can experience it, even "math-haters".

This file (find it here) took around an hour to create, but has the benefit of reuse.  It only needs to be created once. By posting it as a public file, anyone anywhere with access to the web can use it.

I hope people do use it, and come to appreciate how, properly used,  programs such as Geogebra can transform and revitalize the classroom. 

Remember: math is a game. Go out and have some fun!

 

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!

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.

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.

Give GeoGebra a Chance

Dynamic Geometry can be used to gain efficiency and minimize careless errors in class. Here is another that I wish I had had in class. Imagine how much more I could have focused on the students in class by having presentations dynamic and almost error-free. Imagine students having access to the same dynamic files from home.

Here is an example that shows how the measures of the angles of a triangle can be found if the lengths of the three sides are known. Besides the basic algebra, the student would need to know the Pythagorean Theorem and basic right triangle trig definitions. Taking the last step, to the angle measures themselves, could use a trig table, a calculator, or even Google. (But Google will give angle measure in radians, not degrees, so its use should be guided.  If you wish, you can google "calculator", flip it into degree mode, and do the inverse calculation).

This file can be found here.

Go Figure Again!

This is just an adaptation of yesterday's entry.

I continue my quest for the recognition of dynamic geometry, such as GeoGebra, as a major tool in the learning of mathematics. There is no need for geometry to be taught the same way it has for decades!

GeoGebra as a Presentation Tool

I made this this morning just to be an example of how GeoGebra can be used to create dynamic presentations for teachers.
It is pretty simple to follow and requires no knowledge of GeoGebra. It can be downloaded from GeoGebratube here.
Enjoy!

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.

Competition is good

I am making a slow inroad into a comparison between GeoGebra and the TI Nspire . I have previously blogged about the capability to plot polar graphs using just basic geometry (see here for an example.)

It will be a while before I get up to speed, but in the meantime I will proceed just as I did at first using Geogebra: generating an animated gif. Any interactivity in my blog may come much later (that is, if TI has allowed for actively embedding documents.)

In the meantime, it's a start!  Here two points are rotating in opposite directions, one twice as fast as the other, and the midpoint between them is traced.

Happy Labor Day!

Just a little message made in GeoGebra!

A Rose is a Rose is a Rose is a .....

Dynamic Geometry can be a great avenue for introducing seemingly complicated math topics to the younger students in lower grade levels.  It also can be used to deal with topics from different perspectives.
Here I am using GeoGebra to show how rose graphs, normally not seen in school until polar graphs are dealt with, could be introduced much much earlier.

In this graphic points M and n will be rotating around the circle in opposite directions. My next post will show what happens in some cases when the rotation of one of the points reverses direction.

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Motivate students and hold them accountable..

John Metallo has a short article in the Albany Times Union of Saturday. August 15, 2015. The entire article can be found here. In under 20 sentences, he pretty much nails the issue regarding student success in school.

The victory of Jason Day in the PGA Championship has reminded me that the road to success has three major parts: decent opportunities, good guidance (instruction), and the will to succeed. Far too often the latter is ignored when it comes to education "reform". I suspect that is because the powers-that-be feel they have no power to change the students' "will", but see an easy route via changing the "guidance" and "opportunities". You know the old theory of "change what you can."

Our New York governor seems to be all over "accountability" for teachers and schools, yet somehow absolves students of any responsibility in the matter.  I fully believe that students have the ultimate responsibility, but I also feel that the adults in their world can do much much more to help them get motivated.

The famous quote (prayer, call it what you wish) goes like this:

God, grant me the serenity to accept the things I cannot change,

The courage to change the things I can,

And the wisdom to know the difference.

This, as a code for life, might seem self-evident, but it is anything but. Taken literally, it can lead to a feeling of complacency. It presumes wisdom, but how do we measure our own wisdom? What if we sense, incorrectly, that there is something that we cannot change? Unless we acknowledge our lack of wisdom, we would be falsely serene.

I believe that, as a culture, we have underestimated our powers to improve student motivation. We have become complacent in that regard, passing the buck to causes we perceive as beyond our control.

Perhaps it is time to address the issue of focusing on student failure over and above school failure. Successfully eliminating student failure would, in essence, eliminate failing schools. Yet, doing such is impossible without addressing student motivation (the "will to succeed").

I hope that I am doing a small part with my work in GeoGebra and my inclusion of some of them in my blog postings. What I do know is that I would make sure that students spent whatever time they could working with GeoGebra, not because it is a panacea for mathematics instruction, but because it could be one of the best motivating factors we have.

Make sure you know all sides of it and see it from all angles!

This was created with the specific goal of supplying self-practice for students in which they can check their own results. It might also be adapted for classroom use.

It continues my quest to see dynamic geometry (such as GeoGebra used here) become an integral part of school mathematics.