Geogebra CUE Conference Resources

This is an online version of a presentation that I gave at the 2011 CUE Conference in Palm Springs. This presentation is designed to give an overview and some of the functions of Geogebra. For more resources please check out the BrokenAirplane Geogebra page.

Click on the link to start using Geogebra. Then click on download. You will be presented with two options.

• Choose Webstart if you would like to install Geogebra on your computer. This is the best option as it will not require the Internet but it requires administrative access on your computer. Talk to your school's IT person about getting this installed.
• Select Applet Start to run Geogebra from within your browser without installing anything.

The first example is in reference to helping students understand Slope. This is a critical concept for students and also crucial to completing my Slope Art project. I have my students create "sliders" which allow them to change the values in the slope-intercept equation y=mx+b. Students can move these values back and forth to see the effects it has on the equation. To learn how to use sliders, check out my Interactive Math post about Geogebra.

In the following example, I use sliders once again but to demonstrate how to make any quadratic equation and see the effect of changing the coefficients of a,b, and c.

Geogebra is just as powerful with Geometry as it is with Algebra and you will find it excellent for creating proofs and constructions.

To create the perpendicular bisector (see the pictures on the right):

1) Use the new point tool and the line tool to create three points, A and B and make a line through them. 

2) Create a new point C and place it more than halfway on line AB.

3) Using the compass tool, create a circle with a radius from A to C and the center at A. Then create another circle with the same radius and with the center at B.

4) Then create new points at the intersections of the two circles and connect them with a line. You have created a perpendicular bisector!

Often, students need or want to see a concept again. Click on the "View" menu and then "Navigation Bar for Construction Steps" this will put a playback control at the bottom of Geogebra that you can replay for your students again and again. There is a way to animate movements as well in Geogebra.

Geogebra can explain math in a way that paper cannot convey. Here is a link to a visualization of the Unit Circle and its relation to Trigonometry.

While there are many more features, one that has been beneficial to me as a math/science teacher has been the ability to easily and clearly work with data. While Excel and other software provides this functionality, it is often too complex for younger students. With Geogebra you can quickly make graphs and equations that can be manipulated and modified.

There is so much that Geogebra can do I hope that you are interested in learning more. Be sure to check out the resources on BrokenAirplane as well as links to other great sites about Geogebra. Thanks for stopping by.

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Sine, Cosine, and the Unit Circle in Geogebra

I have received some requests for this applet so I decided to post it for anyone who wants it. Seeing the connection between the trigonometric functions and the unit circle unifies a lot of mathematics. It also makes sense of these functions that must typically be memorized or looked up.

If you have not yet used Geogebra, be sure to use my getting started with Geogebra post. It has helped quite a few teachers start using Geogebra. It really does not take that long to learn.

Customize this applet for you:

• If you would like to see it full screen, double click on the animation. 
• To turn off the animation, right click the slider and uncheck Animation On. 
• See how I made this applet by clicking on View and then Construction Protocol. You can use the forward and backward buttons on the bottom to step through the construction as well.
• If you would like to see the values of Cosine and Sine change as x changes, go to View and Algebra View.
• To change any aesthetics or animations you can right click on any of the Dependent Objects in the Algebra View.
• Finally, if you turn on both Algebra View and Spreadsheet View from the View menu and then right click Cosine or Sine in Algebra View, you can turn on Trace to Spreadsheet to collect the values cycled through from pi to 2*pi.

Enjoy, and if you have any suggestions for other applets you would like to see, let me know. A special thanks to Guillermo of Mathematics and Multimedia for his post on Embedding into a Blogger post.

Be sure to check out all of the BrokenAirplane Geogebra resources.

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Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Share Math with Students and the World

If you have been playing with Geogebra learning the basics, sliders and animation, and hopefully some of your own, it won't be long before you want to share this with your students.

Of course you can show it on the projector if that is an option for you, but what happens when your students go home and need to remind themselves? Or perhaps you would like them to play with a concept before they come to class the next day. Whatever the reason, there is going to arise a need to share Geogebra files with your students outside the classroom.

Exporting a Worksheet
Luckily, Geogebra was built for easy sharing. Once you have created your Geogebra Worksheet that you want to share, click on "File" and then move your mouse over "Export". This brings up the sharing options.

There is the option to take a screenshot of your worksheet which can come in handy, but you probably want to share the whole thing, interactivity and all. Click on Export as "Dynamic Worksheet as Webpage (HTML).

From there add in your information to describe what the worksheet is about. I would recommend that you don't leave this section blank as I have seen many unlabeled worksheets that have made no sense at all. It is a great idea to add instructions and descriptions to make the worksheet as helpful as possible.

Don't be afraid to check out the advanced tab as well. These options allow you to restrict certain options like right-clicking, reseting the construction, hiding the tools, etc. This is not for the purposes of restriction but to ensure that the student only does what you want them to do. If you give too many options for a construction where you only want them to drag the sliders, they could get confused.

Finally, click "Export" and decide where to save the file and what it should be called and like magic, your Geogebra Worksheet is there all packaged into one HTML File. This makes it easy to email the file to your students. If you have a website, you can add this as a page by placing it in your website's folder and linking to it.

It used to be much more difficult to share Geogebra files on the web, but now it is all inside one self contained file. Keep in mind that if your computers have updated JAVA, the files can be used without access to the Internet. This came in handy one day when the network was down.

If you are going to embed the file into your blog or anywhere else, you can follow the directions on Kate Nowak's Blog but keep in mind that if you resize the file you are going to not shrink down the file but cut off some of it. That is why I usually will provide the actual file or link to the file by using the Geogebra Uploader.

Here is a link to the Quadratic Coefficients Exercise mentioned in the previous post on Sliders and Animation in Geogebra.

If you are interested in embedding the sketch into your blog, there are some excellent instructions on the Mathematics and Multimedia blog.

Community
Geogebra has been around for long enough that there are quite a few people using it. That is the power of Open Source, people who might not previously have been able to learn and contribute to this project can freely use it on any computer in the world. In fact the problem is almost too many resources.

Let the community support you right away in using Geogebra. What are you planning on teaching today? Trigonometric functions, quadratics, Derivatives, Addition? Type in _________ (your topic) geogebra worksheet and I would be shocked if you couldn't find something that you could immediately use in your classroom.

Here are some great resources that you can always rely on for great Geogebra Content:
Geogebra Wiki - The main site keeps a great collection of Worksheets that you can add to covering a wide variety of topics. There may be a Geogebra Institute in your area. Click the link to check it out. There is also a YouTube Channel dedicated to news and features in Geogebra.

Geogebra Math - Created by Linda (of Math247) and Maria (of Natural Math), legendary in the math community for their innovations and contributions. You will love these Worksheets.

Mathematics for Middle School - This is another blog where it is clear that the author (Andreas) is committed to creating great resources to share with educators and her students. The focus is on arithmetic and algebra which is so important as these abstract concepts demand visual explanation.

Mathematics and Multimedia - This blog will really take you deep into mathematics. The tutorials not only demonstrate topics but also proofs or concepts that cover a large span of topics. Guillermo has done a masterful job with this blog.

I will continue to do Geogebra tutorials and topics and if you have any ideas, suggestions, or questions about what you would like to see, let me know and I will either directly message you in the comments or Twitter or create a post to help you out. Remember to sign up for the CUE Conference in March and I will see you there!

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Make Math Move with Sliders and Animation in Geogebra

It is incredible that within the first few hours of posting the Geogebra Basics post, hundreds had already checked it out via Twitter, Facebook, and the BrokenAirplane RSS feed. There are so many people who are interested in learning more about using Geogebra in their classrooms and I hope you are getting the hang of the basics and perhaps tried a few things of your own.

The next two features, are one of the most requested and often used features for Geogebra because they promote interactivity and understanding. Once you get the hang of it, your students will never look at an equation the same.

Sliders

This is one of my favorite features in Geogebra. Graphs/Functions show changes over a range of numbers, but what happens when you change the equation? These relationships are the core of the equations but students often miss it as they try to plot the graph.

Step 1: Open Geogebra (steps 1-3 in the basics post).

Step 2: Look at the bottom of the Geogebra window. On your left you will see the word "Input" and next to it is a text box to type in equations, variables, and more.

Step 3: Lets create some coefficients that we can play with. In the Input Bar, type in m=0 and hit "Enter" on your keyboard. It is important that you follow these instructions literally, so do not use an uppercase M (although you could use any letter in the future but for the example follow carefully)

After hitting "enter" you will notice under "Free Objects" a new object m which is set equal to zero.

Step 4: Create one more free object by typing "b=0" into the input bar and hit "enter".

Step 5: Next to the free objects "m" and "b" you will see a clear bubble. It is actually an ON/OFF switch for the slider. Click on the bubble for both of them and you should see a slider appear in the graphing window.

Step 6: Back in the input bar, type in "y=m*x+b" and hit enter. Note: The asterisk between the m and x is the multiplication symbol. You must type it this way as you will get an error if you try to just use "mx".

Step 7: It may have seemed like nothing happened, but in fact you just graphed y=mx+b, the equation of a line. The reason you can't see it is because you have your slope (m) set to zero and it is covered up by the x-axis.

So lets see that line and the power of sliders. If you look at the "m" slider, it says m=0. Click and drag the ball on the slider back and forth. Watch as the line changes its slope and imagine your students going "Ahhhhhhh" as you explain and show slope changing. The visual demonstration gives the student the mental connection they need to understand.

You can adjust the "b" slider as well and see the y-intercept change. Although it is called the "y-intercept" students still seem to get confused until you show them that its value determines where the line hits the axis and adjust it to show them how it can change.

As students begin to see these relationships, they can begin to apply their knowledge to new situations and visualize the graphs in their heads before they have to plot them by hand. Imagine a classroom where students and not calculators see the relationships between variables and coefficients.

If you really want a fun demonstration, create three new sliders "a", "b", and "c" (step 3). Then enter in the equation y=(a*x^2)+b*x+c (the "^" symbol comes from holding shift and pushing the number "6") and you can show your students the relationship of the variables and coefficients in a quadratic. Seeing these numbers change,  become positive and negative, and zero really makes it all the clearer that math can represent real things like data and physics.

Changing the range of these sliders can be really useful and you can do this by right clicking on the slider ball and choosing object properties. Changing the interval will choose the range of numbers over which the slider will show. There are many other settings you can change in this menu like color and thickness, and much more. Be sure to try changing the size or color of your lines as well to make them easier to see from far away if you are showing this to a large group.

Animation

It can be very useful to have your students see the sliders move through all of the numbers over and over. This frees you up to walk around the room but also shows an animation of math moving which can be very beautiful and help promote understanding with trigonometric and polar functions.

To activate animation of a particular slider, right click the ball of the slider you wish to animate, and select "Animation On". By default it will cycle through all of the positive numbers and then go through the negative numbers over and over. You can change this in the slider's "object properties" menu and under the "slider" menu and "Animation" setting change the speed of the animation. The three options for an animation are: Oscillating, which is the default back and forth cycling through all of the numbers; and Increasing or Decreasing which will only once go through the positive or negative numbers. It might be necessary to slow down the speed so narration and explanation are possible.


Play around with those, and I do mean play. This is software that really encourages fun in math and I hope that these features will help your students to see that. In the next post, we take a more global look and see how to share Geogebra files with your students and the World! Also, there are an enormous amount of resources out there that I want to share with you from educators already making great stuff with Geogebra. See you next time.

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7 Steps to Interactive Math with Geogebra: The Basics

It's the beginning of the year, and I am so glad I took the break from posting. It gave me a chance to read some great books, work on a few side projects which you will hear about soon enough, get the house finally in order, and spend some wonderful time with my family.

The FIRST Robotics Competition will begin this Saturday and I can't wait to tell you all about this years competition challenge.

Since I wasn't posting over the holidays I was not able to give you a present. So although it is belated, I wanted to share with you one of my favorite pieces of free and open source software, Geogebra (Geometry + Algebra). I remember seeing this software for the first time years ago and seeing the potential for Educational Technology. In fact, I will be speaking at the 2011 CUE conference in Palm Springs this year entitled "Geogebra for Interactive Mathematics".

Should you start using Geogebra? Well tell me if you have experienced this:

• Difficulty graphing on the Whiteboard because of time or quality of drawing skills.
• Confusion with your students about the relationships of variables and their coefficients (e.g. what happens when "m" or "b" changes in y=mx+b).
• Students forgetting a long proof or steps to solve an equation.
• And all of the other frustrations that come with the limits of reality to explain abstract concepts.

If you nodded your head to any of the above, then you need Geogebra. Don't worry though, it is free and easy to get started.

Getting Started With Geogebra

Step 1: Go to www.geogebra.org and click download.

• UPDATE: Since this post was written, there is now a Geogebra for Elementary students. It is the same functionality but with bigger fonts and an easier user interface. 

Step 2: You have 3 options for using Geogebra to accommodate as many possible situations as possible.

• Webstart - This is the option to choose if you want to be able to use Geogebra offline. It requires you to install the software and so make sure you talk with your school's IT department about getting this setup. Good option if you can get it just in case the web is down.
• Applet Start - Allows you to use Geogebra fully in your web browser as long as you are connected to the web.
• Offline Install - Use this option to download a Geogebra installer and share with a student if they do not have Internet access.

Most of the time you will use the Applet Start. This requires up to date Java which most classrooms have if the IT department works on them at least once a year.

Step 3: Open Geogebra. It will look the same no matter how you open it and on which computer, which is a beautiful thing when you are working with students.

Let's look at the construction tools that you will use most often. These tools are used together with others to make objects and manipulate them.

These buttons also store additional tools. I will describe each button and the additional tools in that menu. To see the additional tools, just click on the little arrow in the lower right of each button.

From left to right, you will find:

1. Move, Rotate, Record to Spreadsheet
2. New Point, Intersect two objects, Midpoint or Center
3. Lines, Segments, Vectors, and Ray Tools
4. Perpendicular, Parallel, Tangent, Best Fit
5. Draw Polygon, or Regular Polygon
6. Circle Tools
7. Ellipse, Hyperbola, Parabola
8. Angle, Distance, Area, and Slope
9. Reflections and Translations
10. Sliders, Text, Images
11. Move, Zoom, Show/Hide

A lot of features, and it could seem overwhelming at first. But like any powerful software like Audacity or Inkscape, it is important to play around and get the basics and over time you will grow deeper in your understanding. Most important is that you believe in yourself and your ability to do this! I promise.

Step 4:Lets play around a bit to get the hang of the interface. Geogebra opens up with a nice graph all ready for you so lets add some dots. Click on the New Point Button and then click anywhere on the graph.

You'll notice that a new point has been created and labeled "A". This is also reflected on the left under "Free Objects". This means that you can change or move the point if you want.

Now create another point anywhere you want. A new point labeled "B" is created and the coordinates for both A and B are shown on the left-hand side.

Step 5: Lets turn those points into a line. Click the appropriately named "Line through Two Points" button on the right of the new point button. Then click on Point A and then Point B on the graph. A new line is formed labeled "a" (lowercase to differentiate it from points and you can change all of the names to whatever you would like later).

If you did the previous steps correctly, you should also see the equation for your line under dependent objects (because it depends on Point A and B). It defaults to standard form (Ax + By = C) but you can change that to slope intercept form by right clicking on the line and choosing y=mx+b. You can see this in the picture below but remember that our points and lines are going to differ so my numbers are not the same as yours.

You are probably ecstatic over how much clearer and precise your math demonstrations can be. Let me show you a couple more things today and next time we will go a bit more advanced.

Step 6: Lets make a perpendicular line. Click the next button to the right called "Perpendicular Line". It asks you to click on a point and a line, so click on either Point A, Point B, or create a Point C with the add a new point button from step 4. Then click on Line A.

Your perpendicular line has been created. If you would like to "undo" a step, click Ctrl+Z or choose it from the Edit Menu, but followers of this blog know the power of using Keyboard Shortcuts.

You may need to shift or zoom out, and you can easily do so by clicking on the "Move Graphics View" button. Just make sure you click the "Move" button (the furthest left) in order to click on objects again.

Step 7: As amazing as Geogebra is, you might ask yourself why my CUE Conference demonstration refers to using Geogebra to make "Interactive Mathematics". Well, usually once you have printed out a worksheet or drawn something on the board, it is done and it is difficult if not impossible to play around with it. There are expensive SMART boards that can create Interactive Math but you can do it for free with a computer and Geogebra.

Making sure the "Move" button on the far left is selected, select, hold, and drag Point A. You should see the entire line and it's perpendicular line shift around as well as the equations and points on the left. Check out the video below to see Geogebra in action.

Pretty amazing huh? Math does not have to be dead, now students can play with it, and play they will. Let them loose on Geogebra and they will create amazing art, drawings, and constructions in no time.

In the next couple of posts, I will show you even more amazing things in Geogebra. We should be able to cover:

• Sliders and Animation
• Embedding and Sharing Geogebra files with your students and the World
• And some great resources from other Educators that you can immediately start using in your classroom.

If I could enlist in your help, I want to ensure that the CUE conference and these posts are helpful and relevant to you. If you have a suggestion for a Geogebra topic (or any topic in general) please leave it in the comments, send me an Email, comment on Facebook, or tweet me @BrokenAirplane. Thanks!

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Math-o-mir allows you to create math faster than ever before

As a teacher and a math geek I love to create math, play with it, share it with others. This is why I tell everyone I can find about Geogebra and Python for their classrooms. The difficulty is if I want to create a worksheet, project, or lesson involving math equations, I had three options for formatting:

1. Paper/Pen (messy and not easily modified)
2. LaTex (difficult and time consuming)
3. Microsoft Word/Open Office Math (useful but felt like I was using a program, not "doing" math)

When I am doing math with my students I want them to have that same rush I feel every time I pick up a napkin and start doodling. I believe I have found that software with Math-o-mir. I am grateful to Sue and the MathFuture Google Group for directing my attention to this amazing piece of software. 

It takes a few minutes of playing with the software before you get the hang of it. But once you do you will find yourself making math as fast as you can type it. With so many built in math functions and symbols this will serve Elementary through University. The formatting is very flexible, you can position it as you need to, change the font, and change just about anything you want with just a couple of clicks.

I recommend that after you install the software you push "F1" and read the very well written tutorials/ documentation. My favorite is that it will allow you to draw a diagram or plot the equation in seconds!

Last and most important in my opinion, you can select which parts to copy and paste into your document or presentation as an image (I would like to see in the future it copied as a .png file so it is not blurry if you make it larger). However, the software was designed to be a "Math Notebook" so you and your students can use the program to do all of the work you need.

Yet another example of technology making our lives easier. My gratitude for Danijel Gorupec for creating such powerful software and releasing it to the world for free.

Download Math-o-mir