GeoGebra with a Purpose

(Lost the source of this!)

I got to give a whiz-bang 60 minute (with an option for 30 extra minutes) intro to GeoGebra at the New Tech network conference this week. 50 plus tech-savvy teachers... so it was good. I am always worried that people expect me to tell them about GeoGebra for an hour, when purpose is to get them started using it on the spot, in ways that make sense of their potential use. (Note that if you are in driving distance, I am more than happy to come do this at your school. No GeoGebra lectures, however.)

Purposes. So what are the ways that people make use of it? Oh, let me count them:

1. World's best graphing calculator. (A little weak on statistics and CAS, but that's improving quickly.) For you and your students. For algebra, calculus, or geometry.
2. Mathematical image editor. For uses in reports, papers, handouts or assessments.
3. Demonstration tool. Project a great visualization on your screen to show to or discuss with students.
4. Focused mathematical activity for students.
5. Open-ended inquiry tool. Pose a question and let students investigate.

Requirements. The AMAZING thing about this tool is that with version 4.0, all of these are accessible to teachers in that 60-90 minute start up.

1. Open the program, start typing equations on the input bar.
2. Needs some quick familiarity with the tool bar to make your image, then File > Export > Graphics View As A Picture.
3. GeoGebraTube. If you have not looked at this, you are missing out. 14,000 sketches and counting; free accounts, search, likes, tagging and you can collect them in teacher mode or show collections in student friendly mode. This is why you need minimal expertise to start using the program deeply. If you can run YouTube and you are a teacher, you can do this.
4. See #3. 
5. Students today are geared for this kind of tool. You give them access, they'll figure things out about it that I don't know.

Really, any training beyond that first 90 min. is about if you want to become proficient in number 5, or if you want to be designing your own activities. Some teachers are doing that anyway by the end of an hour, most by the end of a half day.  Once you start using it, there's a big danger of being sucked in by the possibilities of what you can make. The power of dynamic examples is as much greater than static electronic images as static electronic images were than hand drawn. (My opinion. No research. Actually yes research, but they would never quantify so crazily.)

After my session, I got to go to Geoff Krall's (@emergentmath) session on formative assessment. He was using the MARS  MAP (Mathematics Assessment Resource Service - Mathematics Assessment Project) materials.  In particular, he used the Ferris Wheel lesson to get us collaborating and specific in discussion.

As we discussed, it really got me thinking about how I would use the task early on. It would make a good project or assessment, I think, but what about as an inquiry? The basic problem was to make a symbolic model [find a, b and c for a+b*cos(ct)] for the height of a car on a specific Ferris wheel. Then there was a card sort which got students comparing context, equation and graphs.

I've given many explorations before that got students experimenting with parameters to see the effect on graphs, but I love the idea of tying it to a context. That doubles up on the intuition they can apply - physical and visual. If the students had access to that, they might be able to do enough trials to start to generalize. Even without much trigonometry understanding, it's a nice context for graph transformations. For me, these kind of thoughts now lead to GeoGebra. I made a quick sketch, with the Ferris wheel in a 2nd graphics window, and was delighted to find that even the 2nd window worked on GeoGebraTube.

But since then I thought it would be worthwhile to develop a bit more. Both to familiarize myself with using the 2nd graphics window and to make the single model into a reusable activity. I knew I wanted to have either a customizable or random Ferris wheel, some animation of the situation and a way for the students to enter the equation.

That bore some thought: sliders, input boxes for parameters or an input box for the function?  Sliders are best for seeing continuously linked examples, but can make a problem like this too easy! The input boxes for the parameters helped support the idea of structure, require some thinking before making a new guess, and don't require as much typing as entering the whole function. Plus you can isolate one parameter and just adjust that. That might be a positive or negative.  It feels like a support for learners early in this, by encouraging them to focus on one parameter at a time.

The trick to working on two graphics views is the advanced tab of object properties. You can use any tools from the main window. Just select the tool and then use it in the 2nd graphics window. The objects you make there show up in the algebra view. But when you edit things, or create them in the input bar, they migrate or appear in the first graphics window. The solution is in the object properties, advanced tab; just check the box you need. Note that you can have something appear in both ... there just has to be a cool use of that.

I don't think there's anything else too tricky about the sketch. I used the Function[ , , ] to get the modeled equation to move with the tracing point, the ZoomIn[1] command on the button to clear traces, and the UpdateConstruction[] command to reset the Ferris wheel dimensions. (I had slick graphics window dimensions based on the Ferris wheel, but then the ZoomIn[1] command doesn't work. Ultimately I thought it was better to see the Ferris wheel changing sizes anyway.)




 The sketch is on GeoGebraTube: Teacher page for download or Student worksheet for in browser use. You have to click in the main window to get the animation button to show.

Two Final Problems

Trig Problem 2

For my preservice high school teachers' "final" (really a last Standards Based Grading opportunity), there were two problems that while similar in many respects were quite different in results. All of the problems were listed by one standard, but typically could be used for other standards. It's the student's responsibility to describe what standards they are demonstrating, though I will help if it demonstrates something well that they need.

Trig Problem 2. (Standard: Law of Sines, Law of Cosines and applications)

Figure out some of the missing information in the diagram.



The pictures were made in GeoGebra, which I highly recommend for mathematical image creation, as well as more active uses.




Geometry Problem 1. (Standard Lines: parallel, perpendicular, properties of angles)

Find more angles.

Geometry Problem 1

Similarities: visual, finding connections, geometry, students have previously done and been assessed on similar problems.

Have to love easy-to-draw memes.

Differences:  throughout the semester students saw trigonometry as something difficult, and had much less confidence on them.  Students were very successful with the angles problem, able to find all the angles, and be able to justify their results. Why vertical angles are congruent, why there are 180º in a triangle, etc. On the "trig" they quickly resorted to visual inference (like the angles at A were all 60º), supposition, and ignored contradictions (such as finding that the length of CD was less than 6 units), and did almost no extension to other standards from circle geometry.

It was fascinating to read their work, and I wish we had more class time to look at the results. It felt like direct confirmation of the Van Hiele levels, and convicted me that as much time as we devoted to trigonometry, I need to find more ways to increase their experience.  While I thought the circle diagram was more subtle, I didn't realize the great difference in how students would see it. Only one student realized CD must be 6 units, which is the entry to me for many of the possible values that can be determined.

Tau-ism: amputating Pi to Tau

Some of the best and beautiful mathematics has been created centuries ago. Open any 18th century mathematics book or journal ( or older if you can read Latin ), do the same with a book or journal of any science of your choice, or compare it to a newspaper of the same period if that's what you prefer. My point is that the vast body of mathematics produced by generations before us is still accessible and thus usable. This accessibility is a unique characteristic of the science of mathematics.

I am not at all for borrowing words from common language and redefining them in the context of mathematics. Words like ring, group or space got an entirely different meaning, words like programming, number or weight may even confuse mathematics students, the full initiation in the language takes years. Personally, I would rather have used a new word for the concept of group, i.e. symmetry transformations, ( at least something that includes symmetry ). Suppose we would implement that word -now-. Before we know it we would have changed the language of math completely. Making the past of mathematics inaccessible for future generations. Something nobody really wants.

Now, what if, we would redefine Pi?! Michael Hart ( wisely ) calls it an immodest proposal in his 'The Tau Manifesto'. Naturally, I thought it was a joke, some sort of parody, but he seems to be serious. Naturally, I am flabbergasted, appalled by such an idea. It is like amputating pi to tau ( See the video for a detailed explanation. )

Mathematics is beautiful, but not perfect. If our ancestors made choices we wouldn't have made today we have to live with them. The cost of change outweighs the benefits manifold. Science includes safekeeping the discoveries of the past.

The video below has been viewed almost 400,000 times in one month.

Relation between Phi and Pi

$$\phi = \frac{1+\sqrt{5}}{2}= 2\cos {\frac{ \pi}{5} }$$
$$\pi = 5 \arccos \frac{\phi}{2}$$
( Who can improve on Euler's identity by adding $\phi$ to it in an elegant fashion? )

I watched the BBC Horizon documentary "What is Reality?" The constants in physics seem nothing more than carpets to stash away the dust, i.e. stuff we don't understand  yet. It looks as though there are no beautiful equations in physics: physicists make them look beautiful by creating all sorts of constants. - Forgive my ignorance, my knowledge of physics is limited. But when I heard the lead scientist of Fermilab explaining that they don't know -what mass is- I was flabbergasted. They "need to find the Higgs-boson particle" first. Then he talked about the pure ecstasy and euphoria he experienced when they found the last quark. They are completely obsessed by a particle that may not exist, they look and live like heroin-addicts, caring about one thing only: Higgs-boson. - ( Forgive me, I am jealous! )

Back to mathematics. What are the fundamental constants in mathematics? I am not sure. I suppose Euler's Identity is an excellent start with 1, 0, i, $e$ and $\pi$. Given a URM, then $e$ and $\pi$ become 'computable' to any decimal precision. So in that sense one might argue that $e$ and $\pi$ are not fundamental constants. 0 and 1 are, of course. Because they are part of the definition of a URM, think of the zero and successor instructions. But what about geometry? In geometry $\pi$ is a constant: the ratio of a circle's circumference to its diameter.

Trig Visualizing

Rebecca Walker and I modeled a lesson for our secondary student teachers on trigonometric equations, based on the first chapter of the Precalculus book from the very interesting CME Project curriculum.  While it has some interesting applications, this curriculum really does a good job of letting the mathematics be the context and addressing mathematical habits of mind.  The lead developer is Al Cuoco, who has a great history of interesting math and math ed work.

The lesson is a bit of a stretch, because we're just touching on one section, using a bit of information from three or four.  We did unit planning one week, lesson planning the next week, and finally the lesson.  The TAs read The Teaching Gap, so then we connected it to the idea of lesson study, and a discussion both about how to revise this lesson, and why lesson study might work as professional development.

We have two Geogebra sketches to help with visualization.

 As a sketch or a webpage.  This sketch supports visualizing sine and cosine with unit circle connections.

As a sketch or a webpage.  This sketch lets you invert trig functions using the Unit Circle representation.













This is my first attempt at a WCYDWT.  When I was making these sketches (don't worry, I disinfected them before posting) I had a bad cold, so was constantly reheating my tea.  Watching it go round and round.  Thinking, "so when do we know a position and want to know the angle, with possible multiplicities...hey, wait a second."  If I was using this, I think I would start with the video, and use that to motivate the idea of solving for information based on the circle position, as well as how periodicity relates to multiple solutions.

This has to be the world's most boring video.  Enjoy!

Here's a slightly more polished version of the handout we used with the sketches.  There was some discusssion with the student teachers as to whether the inverse trig or the algebraic solutions part should come first.  I think they could be switched, depending on what you wanted to emphasize with the students and how strong their trig background is.  Also, the handout is written as if the teacher is demonstrating with the computer, which is what we wanted to model for them, (no lab is no reason to no have technology) but the ideal would be to have the students have access to the sketches.


Solving Trig Equations

Trig Visualizing

Rebecca Walker and I modeled a lesson for our secondary student teachers on trigonometric equations, based on the first chapter of the Precalculus book from the very interesting CME Project curriculum.  While it has some interesting applications, this curriculum really does a good job of letting the mathematics be the context and addressing mathematical habits of mind.  The lead developer is Al Cuoco, who has a great history of interesting math and math ed work.

The lesson is a bit of a stretch, because we're just touching on one section, using a bit of information from three or four.  We did unit planning one week, lesson planning the next week, and finally the lesson.  The TAs read The Teaching Gap, so then we connected it to the idea of lesson study, and a discussion both about how to revise this lesson, and why lesson study might work as professional development.

We have two Geogebra sketches to help with visualization.

 As a sketch or a webpage.  This sketch supports visualizing sine and cosine with unit circle connections.

As a sketch or a webpage.  This sketch lets you invert trig functions using the Unit Circle representation.













This is my first attempt at a WCYDWT.  When I was making these sketches (don't worry, I disinfected them before posting) I had a bad cold, so was constantly reheating my tea.  Watching it go round and round.  Thinking, "so when do we know a position and want to know the angle, with possible multiplicities...hey, wait a second."  If I was using this, I think I would start with the video, and use that to motivate the idea of solving for information based on the circle position, as well as how periodicity relates to multiple solutions.

This has to be the world's most boring video.  Enjoy!

Here's a slightly more polished version of the handout we used with the sketches.  There was some discusssion with the student teachers as to whether the inverse trig or the algebraic solutions part should come first.  I think they could be switched, depending on what you wanted to emphasize with the students and how strong their trig background is.  Also, the handout is written as if the teacher is demonstrating with the computer, which is what we wanted to model for them, (no lab is no reason to no have technology) but the ideal would be to have the students have access to the sketches.


Solving Trig Equations