Wednesday, July 2, 2014

Geogebra Series I: Angles in Standard Position

This past year my math department started using Dropbox to share files by course team, and the way it improved collaboration was very exciting.  One of the course team leads set up a folder for each unit, with subfolders for the normal items like assessments and lesson plans, but she also added a subfolder for common core materials.  The teachers for this course team (Algebra II) started posting Geogebra files in the common core folder, and we would discuss how to use the presentations at course team meetings.  Feedback after these lessons was mixed.  Some teachers thought the presentation was extremely helpful, and others thought the students still needed more practice.  Our team knew that it would be ideal to observe each other using the presentation, but that doesn't always fit into the schedule.  We started using screencasts and teacher guides to help out with this situation.  In this series I will share some of the teacher guides and screencasts for Geogebra tools I have used over the years.  I think the questions that we choose to ask can make all the difference in how useful the tool is, so if you have a question to add to this list, please share.

I chose angles in standard position first because it was the first tool I used to introduce a concept to students, and it made such a big difference in level of understanding from the previous year.  For this lesson it is helpful if each student has a calculator so that they can all participate (any type is fine).

A student version of this tool is available at the link below.

Angles in Standard Position Tool

Our team went through the process below when introducing angles in standard position.

1.  Enter an angle in degrees into the positive or negative angle input box, then hit enter.  Do not put the negative sign for negative angles.  You also don't need to include the degree symbol.  We'll start with a 200 degree angle.

2.  From here the teacher can talk about the definition of an angle in standard position, and identify the initial ray and terminal ray by color.  A quick review/reminder of the quadrants and the 90, 180, 270 and 360 degree angles fits well at this point in the lesson.

3.  Type 300 degrees (or similar) into input box, but don't hit enter yet.  Have students predict what the angle will look like, and discuss how they know.  Then hit enter and call on a few students  to give the formal name for the blue and yellow rays, as well as to describe the procedure for graphing an angle in standard position.

4.  Now we are ready for the fun stuff.  Type 2000 degrees into the positive angle input box, and let students predict what will happen.  Where will the angle terminate? How do you know?  How do you graph an angle bigger than 360 degrees?  Discuss with your partner.  I'll give a couple of minutes for students to discuss in pairs, and then we will share out before we reveal the answer.

5.  Spend some time talking about why Geogebra graphs a 2000 degree angle as a 200 degree angle.  Have students predict some other angles that will be graphed the same as a 200 or 2000 degree angle.  Write some predictions on the board, have students justify answers.  Introduce positive coterminal angle vocabulary.

6.  Show students how negative angles are graphed.  I start with -160 degrees on the same diagram from step 5 so that they can see that we can have negative coterminal angles as well.

7.  At this point in time I hand out this problem set, and I model the first problem with them so that they know how I want the work to be shown.  This includes showing work for how to find a positive and negative coterminal angle, as well as a diagram of the angle.  We get fancy with the diagram and color code, because I am not convinced that they really understand the concept until I see initial and terminal rays along with the number of rotations and the direction of rotation.

8.  I also model the second problem with the class, which asks students to graph a -1160 degree angle in standard position.  The worksheet asks for a positive and negative coterminal angle, and so I encourage students to find the one that GeoGebra will give (between 0 and 360 degrees for all cases).  Students then complete the rest of the problem set, using Geogebra to verify answers.

This tool shows a positive angle larger than 360 degrees being sketched dynamically.  I am guessing there is a better tool out there to demonstrate this visual, and if you know of one please share!

In addition to being a powerful visual aid, Geogebra is a great tool to check for understanding.  I try to keep the emphasis on the formative assessment aspect of a presentation when possible, and I love that the tools can be accessed quickly for review at the end of a class or on another day.

Useful Links:

How to graph a 2000 degree angle with Geogebra
How to graph a -1160 degree angle with Geogebra
Geogebra Tools from Making Math Visual blog.
Geogebra Wiki.  Includes tutorials on how to get started with Geogebra.
Some Things I Wish I Knew When I Started Using Geogebra.


  1. How to they show the number of rotations, positive, and coterminal angle all on the same graph? Or do they draw them separately?

    1. For the applet on this page (Angles in Standard Position Tool), you can enter one positive angle and one negative angle to display at a time. I couldn't figure out how to show the number of rotations, so we just had students write the number of rotations down on the paper. We had the students create their own diagrams for some of the classes, and we showed them screencasts to walk through the steps. The screencasts are on this page under useful links (graph 2000 degree and -1160 degree angle with Geogebra). Hope this helps! If not, let me know if you have other questions.