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Volume 14. Adventures in Dynamic Geometry

This volume collects most of the collaborative dynamic-geometry curriculum developed for the VMT Project from 2012-2015.

Specifically, it includes:

* The construction crew game -- the game version available as a GeoGebraBook in GeoGebraTube.

* Explore dynamic geometry together -- a version that focuses on dependencies, based on analysis of student work.

* Topics in dynamic geometry for virtual math teams -- The version with the most advanced topics.

* Dynamic-geometry activities with GeoGebra for virtual math teams -- The first full curriculum. It includes tutorials on VMT and GeoGebra.


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table of contents

Here are the historic versions:

1. The original version was developed for use with teachers in Fall 2012. Their students worked on the first 5 activities in Spring 2013. It is a workbook with 21 topics for collaborative dynamic-geometry sessions.

Stahl, G. (2012). Dynamic-geometry activities with GeoGebra for virtual math teams. Web: http://GerryStahl.net/elibrary/topics/activities.pdf.

2. The next version was developed for use with teachers in Fall 2013. It includes 18 topics, including a number of advanced, open-ended investigations, as well as 10 tutorials or appendixes. The 18 topics involve 79 GeoGebra tabs, which are shared figures or tasks for groups to work on using GeoGebra. The teachers worked on many of these.

Stahl, G. (2013). Topics in dynamic geometry for virtual math teams. Web: http://GerryStahl.net/elibrary/topics/geometry.pdf.

3. The third version was developed for use by teachers with their students in Spring 2014. It is more tightly focused on the notion of designing dependencies into geometric constructions. It is envisioned for a series of 10-12 hour-long online sessions of groups of 3-5 students who may not have yet studied geometry. It is especially appropriate for an after-school math club. It includes 12 topics with a total of 34 GeoGebra tabs, plus 6 tutorials or appendixes. The introduction from this version is included below.

Stahl, G. (2014). Explore dynamic geometry together. Web: http://GerryStahl.net/elibrary/topics/explore.pdf.

4. The fourth version was developed for use by teachers in Fall 2014. It is focused on identifying and constructing dependencies in geometric figures. It includes a total of 50 GeoGebra activities, divided into 13 topics. The first and last topic are for individual work. Six topics form a core introductory sequence and five topics are optional for selection by the teacher.

Stahl, G. (2014). Construct dynamic geometry together. Web: http://GerryStahl.net/elibrary/topics/construct.pdf.

This pdf version corresponds to the GeoGebraBook at http://ggbtu.be/b140867

5. The fifth version was developed for use by teams of students in Spring 2015. It takes a game approach. It includes a similar sequence of 50 GeoGebra challenges, divided into 13 topics. It is still focused on identifying and constructing dependencies in geometric figures.

Stahl, G. (2015). The construction crew game. Web: http://GerryStahl.net/elibrary/topics/game.pdf.

This pdf version corresponds to the GeoGebraBook at http://ggbtu.be/b154045.

The Virtual Math Teams (VMT) Project has developed a collaboration environment and integrated a powerful dynamic mathematics application into it, namely the open-source GeoGebra, which integrates geometry, algebra and other forms of math in a dynamic computational environment. The project made the incorporated GeoGebra multi-user, so that small groups of students can share their mathematical explorations and co-construct geometric figures online. In support of teacher and student use of this collaboration environment, we have developed several versions of a set of activities to systematically introduce people to dynamic geometry, including core concepts from Euclid, standard geometry textbooks and the Common Core Standards for Geometry.

The topics for VMT with GeoGebra are available for free download in several versions. They all include GeoGebra tasks to work on collaboratively and tutorials on the use of VMT and GeoGebra software. The best version is the active GeoGebraBook version at http://ggbtu.be/b140867. Here, you can try out all the activities yourself. (The future version of VMT-mobile will allow you to do those same activities collaboratively with chat in persistent rooms.)


introduction

Topics in Dynamic-Geometry for Virtual Math Teams is a set of topic statements for use with the Virtual Math Teams with GeoGebra (VMTwG) collaboration software.

Dynamic geometry is a new form of mathematics—and you can be a pioneer in it, exploring, discovering, and creating new insights and tools. Dynamic geometry realizes some of the potential that has been hidden in geometry for thousands of years: by constructing dynamic-geometry figures that incorporate carefully designed mathematical relationships and dependencies, you can drag geometric objects around to investigate their general properties.

In addition, the approach of these topics allows you to take advantage of the power of collaboration, so that your team can accomplish more that any one of you could on your own—by chatting about what you are doing, and why, as well as discussing what you notice and wonder about the dynamic figures. Working in a group will prevent you from getting stuck. If you do not understand a geometry term or a task description, someone else in the group may have a suggestion. If you cannot figure out the next step in a problem or a construction, discuss it with your group. Decide how to proceed as a group.

The topics have been designed for everyone interested in geometry. Students who have not yet studied any geometry can use the topics to prepare them for thinking geometrically. Students who are in the middle of a geometry course or have already completed one can use the topics to gain a deeper appreciation of geometry. Even experienced geometry teachers can use the topics to gain a new perspective on an ancient subject.

To harness the truly awesome power of collaborative dynamic geometry requires patience, playfulness, and persistence. It will pay off by providing skills, tools, and understanding that will be useful for a lifetime. You will need to learn how to construct complicated figures; this will be tricky and require practice. You will need to think about the hidden dependencies between dynamic points, which make geometry work; this may keep you up at night. You will even create your own custom construction tools to extend GeoGebra; this will put you in control of mathematics.

Collaborative problem solving is central to this set of topics. The topics have been selected to offer you the knowledge, skills, and hands-on experience to solve typical geometry problems, to explain your solution to others, and to think more like a mathematician.

These topics present a special approach to dynamic geometry, which may be quite different from approaches to learning geometry that you are accustomed to.2 The approach here stresses the importance of understanding how dependencies are constructed into geometric figures. Dragging points around can help you to discover and visualize the dependencies that maintain relationships within a figure. However, you should also know how to construct those dependencies yourself, and perhaps even be able to define your own custom tools to make those constructions easier. This will give you a deeper understanding of geometry as a mathematical system. This approach may take more time and thought, but it will also be more fun and more rewarding.

The approach in this document is built around 12 core topics that provide a coherent experience of collaborative dynamic geometry. In addition, there is an introductory topic for individuals to do on their own to get started and then a transition topic for individuals to do at the end to start to explore the rest of GeoGebra. There are also extensions to most of the core topics for individuals or groups to explore further what they are interested in. Finally, several open-ended advanced topics present new areas of mathematics beyond the core topics.

Each topic is designed for an online team to work on together for about one hour. The sequence of topics introduces student teams to GeoGebra and guides them in the exploration of dynamic geometry, including triangles, quadrilaterals, and transformations. Ideally, everyone should spend 12 hour-long, online, synchronous, collaborative sessions working on the core topics. An instructor might want to select which of these topics to use and which to skip or make optional if it is not possible to do all 12 or if there is time available to do more than 12. A virtual math team might choose which topics they want to explore.

Several “tours” are included at the end of this document. They provide tutorials in important features of the VMT and GeoGebra software. Teachers and students can take the tours when they want more information on using the software. To work on topics outside of a group, individual teachers or students can download the single-user desktop version of GeoGebra and then download selected .ggb files.3

The pages of this document can be distributed to students as worksheets to keep their notes on and in case the instructions in the chat room tabs become erased. The whole document can be distributed to team members to serve as a journal for their sequence of topics and to provide access to the tutorials. It is helpful if everyone has a printed or electronic version of this booklet next to their computer when they are working on the topics.

To get started, it is important that everyone do the Warm-Up Topic on the computer they will be using well before the first collaborative session. This makes sure that everyone’s computer is properly set up and that they are able to enter VMT chat rooms. It also provides a valuable introduction.

The topics and tours have been designed for a wide range of users. Students who have not previously studied much geometry should focus on the main points of each topic and make sure that all team members understand the constructions. Teams of experienced math teachers may engage in more in-depth discussion on implications, conceptualizations, and pedagogy of the dynamic-geometry topics. If something is unclear in the topic instructions, discuss it in your team and decide as a group how to proceed. Pace your team to try to complete all the core parts of a topic in the time you have together.

Here is a general procedure you might want to follow: Before the time assigned for a group session, read the topic description in this document. It might suggest watching a brief video or taking a tour at the end of this document as important preparation. Think about the topic on your own. Then, in the group session, discuss the topic and work together on the various tasks. Discuss what you are doing in the chat and respond to the questions posed in the topic. Mark down in your copy of this document what you noticed that surprised you and what you wondered about that you want to think more about later. Do not just rush though the topic; discuss what is important in it. The point is to learn about collaborative dynamic geometry, not just to get through the topic steps. When the session is over, try to work on your own on the extra parts that your team did not get to. Report to your team what you discovered. Maybe the team can get together for an extra session on the rest of the topic.

This set of topics offers a unique opportunity to experience deeply topics that have fascinated people since the beginning of civilization. Take advantage. Be creative. Collaborate. Explore. Chat. Reflect. Enjoy!

Footnotes:

2. The approach is discussed in detail from various perspectives in "Translating Euclid: Designing a Human-Centered Mathematics". See www.GerryStahl.net/elibrary/euclid

3. The standard single-user version of GeoGebra is available for download at: www.GeoGebra.org The GeoGebra files to be loaded into GeoGebra tabs in VMT chat rooms can be downloaded. For instance, download the .ggb file for Topic 7.3 at: www.GerryStahl.net/vmt/topics/7c.ggb as well as from the VMT channel of GeoGebraTube at: www.geogebratube.org/collection/show/id/4531