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__Overview__

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

Students develop a conceptual understanding of the 2D relationships to 3D models through experimenting and visualization with this tool

PS-conceptual understandingStudents build strategic competence through their continual trial and error with the creation of nets that create 3D figures.*PS-strategic competence*Students' reasoning is likely to adapt as they continue to explore various combinations of polygons that create nets of 3D figures.*PS-adaptive reasoning*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.*PS-productive disposition*

### 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.