General Tech Tool: Geometer's SketchPad & SketchPad Explorer for iPad

Geometer's SketchPad is a tool I was first introduced to a few years ago, but shied away from because I had no prior knowledge and limited technology available to me at the time. While I still have yet to implement this in the classroom, it is one of the tools I am looking at using for next year as we build a more blended-PBL classroom style of teaching at our school. Since I am not as familiar with this tool, I will leave it to a classmate of mine, Terry C. to explain and evaluate. I'll cover the iPad version after!

Original CEP Post

Specific Math Activities using Geometer's Sketchpad (GSP)

Overview

Curator: Terry C

Name of Tech Tool: The Geometer's Sketchpad (GSP)

Brief Description of Tech Tool: The Geometer's Sketchpad is a dynamic, interactive software that supports the teaching of mathematics through the creation, visualization, and manipulation of geometric shapes, mathematical models and graphical representations relevant to geometry, algebra, and calculus. The software is developed and sold by McGraw-Hill Education.

Additional Information

A multitude of resources, ranging from lesson plans and activity guides to tutorials and professional development/training sessions are available for Geometer's Sketchpad.

McGraw-Hill Education maintains several sites that offer tutorials on how to use the software, free classroom activities, and an online community to share new ideas on how to use Geometer's Sketchpad in the classroom:

• Geometer's Sketchpad Resources
• Geometer's Sketchpad Resources Center
• Sketch Exchange - Online Community

A community of Geometer's Sketchpad users have also posted their own lessons and activities for the software on the web:

• The Math Forum @Drexel: Sketchpad Resources at the Math Forum
• Kutztown Unversity: Geometer's Sketchpad Lessons
• Math Bits: Working with Geometer's Sketchpad
• The University of Georgia: GSP Lessons
• Technological Education Consulting: Sketchpad Sketches
• Nate Burchell's Dynamic Geometry Site
• University of Virginia Center for Technology and Teacher Education

Numerous videos and video tutorials for Geometer's Sketchpad are also available on YouTube:

• Geometer's Sketchpad YouTube Videos

System Requirements

Geometer's Sketchpad Installation Guide and Licensing Agreement

For Version 5:

Windows system requirements:

Pentium-based system or equivalent

Windows XP or later

300 MB free disk space

Optional: Internet connection*, Adobe® Reader or Adobe Acrobat®

Macintosh system requirements:

Intel-based system (Note: PowerPC users may use Sketchpad Version 5.04 or earlier, available on the download page.)

Mac OS 10.4 or later

300 MB free disk space

Optional: Internet connection*, Adobe® Reader or Adobe Acrobat®

• An internet connection is required for downloading Sketchpad 5, deregistering Sketchpad 5, and watching Sketchpad Movies.

Geometer's Sketchpad currently is not compatible with iPads and Android devices. Sketchpad Explorer is available as a free iPad app in the iTunes store.

SketchPad Explorer

Curator: Kristen (Me)

Overview: SketchPad explorer is the iPad version of Geometer's Sketchpad. While it does not have all of the same capabilities, it does provide a good amount of activities to explore using this dynamic modeling software. 

Technical Costs & Considerations: Aside from owning or having access to an iPad, there is no cost to download this app. 

Additional Information: I have downloaded the SketchPad Express app and find that I am very much enjoying it and the flexibility it gives me to model mathematics and to play with the modelings they provide. This is a great resource to take advantage of the Representing and the Automating affordances of technology. The app is easy to use and a free resource for you to use in your classroom. After following a brief tutorial and exploring some of the cooler features of mathematics that SketchPad can demonstrate, you get the opportunity to browse through the library and find various activities already created just for this app. Topics include:

• Elementary Mathematics
• Geometry: Constructions
• Geometry; Ellipse Constructions
• Algebra: Dynagraphs
• Algebra: Slope
• Sketch Exchange Community Site
• The Sketch Exchange Community site allows you to browse lessons created by other SketchPad users that are specifically geared toward use with the iPad app. Simply browse with the appropriate search, and you'll find a vast treasury of activities to use.

How this Supports & Supplements PBL/PrBL:

Geometer's SketchPad and it's iPad counterpart SketchPad Express both provide a dynamic workspace that allows students to get visual and study geometry synthetically (without coordinates), while manipulating their ideas to round out their reasoning. With Geometer's SketchPad, you have more freedom to create your own models than with the Express version, but the Express version has some very interesting lessons that require critical thinking and analysis to truly gain an understanding. In my opinion, I like Geometer's Sketchpad best for PBL because it allows for the user to create more independently and to model and play with their own thinking to draw conclusions. However, I like SketchPad Express a lot for PrBL as it provides solid lessons that are truly inquiry based placed into a dynamic environment where students can manipulate the tools and build their thinking.

Geometry Tech Tool: Illuminations-Geometric Solids

Overview

Name & Link to Tech Tool or Tool homepage: Illuminations - Geometric Solids

Brief Description of Tech Tool: From their website:

This tool allows you to learn about various geometric solids and their properties. You can manipulate and color each shape to explore the number of faces, edges, and vertices, and you can also use this tool to investigate the following question:

• For any polyhedron, what is the relationship between the number of faces, vertices, and edges?

This tool could also be used to explore other relationships among 2D and 3D figures while focusing on nets of solids and can be adapted to fit a variety of needs from pre-k through high school. This tool provides a virtual manipulative to help students develop spatial reasoning regarding these figures.

Technical & Cost considerations: As with other Illuminations tools, lessons and activities, the resources are free. This applet is iPad and tablet compatible. It also runs in a variety of web browsers on computers and Macs.

Evaluation

Description of Learning Activity

For lower elementary students, this applet could be used to explore and identify the various 3D figures and the 2D figures that make them up. They could also practice counting the number of 2D shapes and relating that to the name for the 3D figure.

Upper elementary students could practice exploring the nets that are created as they "unfold" each of the 3D shape through the virtual manipulative. Students could then practice designing their own nets to be printed out to see if their net creates a 3D figure. Students could compare and contrast their findings with one another to see if they can come up with any generalizations for nets.

Students in the middle grades could expand on the activities for the upper elementary students and could complete the Geometric Solids Exploration worksheet to try to come up with Euler's formula on their own. They can then try to develop an informal proof for it.

Students in high school could expand upon the middle grades activities to develop Euler's formula and a formalized proof for it. They could also explore and establish their own criteria for creating "working" nets and "non-working" nets and generate their own examples for this using the "My Own Net" feature and printing it out.

1. Learning Activity Types

• LA-Present - (read or attend to) presentation of new content/ideas
• LA-Present-Demo - This tool could be used by a teacher to demonstrate what a net is in relation to a 3D figure.
• LA-Explore - This tool is particularly useful for helping students to explore the various relationships between 2D and 3D figures at any level.

2. What mathematics is being learned?

NCTM Standards

NCTM-Geo-analyze - analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;

NCTM-Geo-visualization - use visualization, spatial reasoning, and geometric modeling to solve problems.

Proficiency Strands

• 
  PS-conceptual understanding Students develop a conceptual understanding of the 2D relationships to 3D models through experimenting and visualization with this tool
• PS-strategic competence Students build strategic competence through their continual trial and error with the creation of nets that create 3D figures.
• PS-adaptive reasoning Students' reasoning is likely to adapt as they continue to explore various combinations of polygons that create nets of 3D figures.
• PS-productive disposition Students will likely struggle with net creation at first, but as they print out and examine more nets from the program, they will become better at creating nets for 3D figures.

Additional comments on what is being learned

This tool is accessible to students at all levels of learning. Younger students made need assistance in using the tool, but once the students have learned basic mouse control, they should be able to use the manipulative to analyze the characteristics of 3D figures in terms of 2D figures to give them a solid foundation for their conceptual understanding of the relationship between the two.

Older students will have to work strategically and diligently on their ideas as they develop Euler's formula as well as when creating their nets to ensure that they create a solid figure. Both of these ideas may take several attempts for the student, but each time, they should be able to adapt their reasoning and work through until they find success. This may require more teacher encouragement and feedback depending on the students' mindset about mathematics.

These features help students to visualize the relationship between the 2D and 3D figures that is outlined in both the CCSS and the NCTM standards, with the upper grades being able to reach the level of developing argumentation about the relationship and proving their conjectures.

3. How is the mathematics represented?

This tool is a virtual manipulative that can take the place of traditional cut and tape nets in the math classroom. This tool could be particularly beneficial for visual learners in helping them to deepen their conceptual understanding of the relationship between 2D and 3D figures and to improve their visual-spatial reasoning. However, for more tactile and kinesthetic learners, this tool may be more of a starting point for a student to see the process of unfolding a solid to create a net, and then the student can use the "My Own Net" feature to create their own nets to print out and try working with.

4. What role does technology play?

This tool provides a great advantage in initially learning and exploring the relationship between 2D and 3D figures as students can continually fold and unfold their 3D figure, as well as color the various aspects of the figure to see just how the net is a mapping of the 3D figure.The ability to look at the figure from multiple perspectives and as a filled solid or a transparent solid is really unique and provides perspectives you don't get with a hands-on manipulative.

Affordances of Technology for Supporting Learning

• Representing Ideas & Thinking - This tool enables the user to manipulate the way the 3D figure is being represented as they explore the relationships between 2D & 3D figures.
• Capturing & Creating - This tool allows the student to create their own nets in an attempt to create a net of a solid figure.

5. How does the technology fit or interact with the social context of learning?

This tool can be used by individuals or partners to support the students' exploration as they work with the applet. Students can play around with the tool individually and form their own conjectures about the various relationships between the 2D and 3D figures that they are able to create and can then share out these ideas with partners and continue their individual exploration from there. Alternatively, students can begin their explorations in a partnership and continually discuss the math that is unfolding before their eyes as they work with this applet. Partner work may prove beneficial for promoting good discussion when using the "My Own Net" explorations.

6. What do teachers and learners need to know?

Users should be comfortable with basic mouse control and should be familiar with the terms: face, edge and vertices when using this applet. Users should also be provided access to a printer to get the most out of their experience and explorations with the applet.

How it Supports & Supplements PBL/PrBL

The various representations of 2D/3D figures comes up quite often in my projects in the geometry classroom, as students are often sketching buildings, bridges, sculptures and other designs and trying to determine how to best construct them. This applet is useful in the classroom because it allows students to get a good visual representation of the way in which the two models are related to one another. I've given students what I consider simple nets before, and they've had no idea what it would become. This applet helps those students to see the changes that happen in a net to see how it maps onto a 3D figure. After getting a conceptual understanding here, students can practice and implement the relationship in their projects.

Geometry Tech Tool: Illuminations-Congruence Theorems

Original Posting for CEP 805

Overview

Name & Link to Tech Tool or Tool homepage: Illuminations - Congruence Theorems

Brief Description of Tech Tool: This tool allows for students to explore the idea of triangle congruence through combining any three attributes of a triangle. Students create their first triangle and then try to create exact copies of the triangle to determine congruence.

Technical & Cost considerations: This is a free tool provided from Illuminations. The tool works in multiple web browsers on computers.




Evaluation

Description of Learning Activity

This tool can be used for an in-class or at-home exploration of the triangle congruence criteria. Students can work through any combination of three triangle attributes to determine if triangle congruence is possible. This could be used as a precursor for students studying triangle congruence to help them explore the idea and see for themselves why certain combinations work and others don't. This would help students to gain a basic conceptual understanding of congruent figures, and triangle congruence in particular.


1. Learning Activity Types
LA-Explore -Students should use this tool primarily as an exploration tool where they are seeking out combinations to create triangle congruence. Students can work on any combination of elements that they choose and will have the ability to reason through each situation to help see why triangle congruence is or is not possible with their chosen combination.


2. What mathematics is being learned?

NCTM Standards

NCTM-Geo-analyze - analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;


Proficiency Strands

PS-conceptual understanding Students work on developing a conceptual understanding of triangle congruence by developing and testing their own hypothesis'.

PS-strategic competence Students work on strategic competency through experimentation with each of the various combinations of triangle components for congruence.

PS-adaptive reasoning Students work on developing adaptive reasoning through continual trial and error with various combinations. Students should improve in time.

Additional comments on what is being learned

Students will be exploring triangle congruence through their own means. As such, students will likely have their own initial thoughts on which scenarios will work for triangle congruence and which won't work. As students work through their explorations, they will likely get better at analyzing the attributes needed and may become better "guess"-ers in the exploration. This tool alone may not be enough for the students to work on all of the proficiency standards, but through strategic lesson planning and guidance it could be. As a standalone, however, students should develop a basic conceptual understanding of congruence.

3. How is the mathematics represented?

This tool is a virtual manipulative that allows users to change triangle attributes and orientation to attempt to come up with pairs of triangles that are(n't) congruent to one another. While students can choose their attributes, they cannot change the features about those attributes (segment length or angle degree) this could cause some confusion among students about whether or not certain triangles, like obtuse and right, can be congruent. Also, the CCSS calls for the understanding of congruence through rigid motions which is not inherently a part of this tool. This tool could be used as a jumping off point to understand what congruence means in terms of triangle attributes, but then another tool would be better suited for students to practice the rigid motion transformations of congruent figures.


4. What role does technology play?

Technology is incredibly helpful in exploring this difficult topic for the students because it allows for them to create and play with their own combination of triangle attributes and do this many times over. Students can work on manipulating their triangles in any manner they choose, something that would be difficult with a hands-on manipulative. However, as noted above, the tool does not allow the user to change the degree or the length of the initial segments/angles chosen, which leaves some types of triangles under-represented in the exploration.


Affordances of Technology for Supporting Learning

Computing & Automating - This tool helps to automate the process of creating multiple triangles with given criteria.

Representing Ideas & Thinking - This tool is particularly useful for creating multiple representations of triangles with given criteria, allowing for students to change the orientation of triangles in an attempt to create triangles that are(n't) congruent to one another.



5. How does the technology fit or interact with the social context of learning?


I could see this activity and exploration being used in the classroom as an individual or partner activity. Students may benefit from having a partner to discuss their explorations with to come up with better "guesses" as to creating congruent criteria, however students could also write down their thinking if working independently. If working in partners, students could make this into a mildly competitive game where they try to come up with combinations that they think will or won't work to try to trick their partners. If working alone, students could try to find all the possible triangle congruence combinations in the fewest attempts possible and then compare their answers with others in the class.



6. What do teachers and learners need to know?


Users need only basic technology knowledge, such as mouse or clicker control, in order to successfully use this tool. Teachers preparing for this lesson should know the criteria that create congruent triangles and which criteria don't create congruent triangles and why so that they are prepared to guide students through this exploration.


How it Supports & Supplements PBL/PrBL

Congruence can be a difficult subject for students to understand initially, especially as they are posing and testing conjectures. This tool helps students to gain a conceptual understanding of congruence in polygons that they can carry forward with them when they are exploring the properties of polygons in problems and projects.