In this activity, students work in all dimensions by sketching 2D pictures of trees and designing a series of 3D treehouses that can â€śfitâ€ť in them. Using four different solids (rectangular prisms, cones, spheres and cylinders), students will mine surface area and volume data for each design to see which shaped house works best. Using the mathematical nets in Volumize, students figure out how 2D skins can wrap the 3D shapes that make up their treehouses.

*Learning about scale, dimension, mathematical nets, volume and surface area through designing 3D treehouses for a 2D tree.*

**Treehouse Design** (20-40 minutes)

*Treehouse Design*student sheets.- iPad with Volumize app.
- Wifi access to send work to other iPads or to the online project space.

- Treehouse (rectangular prism)
- Tree Teepee (cone)
- Tree Pod (sphere)
- Tree Silo (cylinder)

For each model you will focus on surface area and volume data to help you decide how much material you will need, to determine standing and/or sleeping space inside, and to decipher amounts needed to skin each model so that the covering appears the way you would like on its surfaces. Based on your designs, you will pick the optimal treehouse for you!

**Treehouse Design** (20-40 minutes)

Have students open the app and select â€śGet Building.â€ť

Each student or group needs to start with a sketch of a tree and then take a picture of it. Allow as much or as little time for this as necessary. (Students can sketch their trees at home and bring them in the day of the activity.) Students will then need to set the scale in fairly realistic amounts that allows them to judge if they can stand and sleep in their treehouse.

- Each treehouse should remain somewhat simple. The first treehouse students design will be a rectangular prism. They may also add a rectangular prism as a roof if they wish.
- Once the rectangular prism treehouse is designed, students will then draw the skin for it. They can take pictures of drawings and press and hold the shape to Skin.
- Have students follow their student sheet, adding data to tables as prompted.
- Repeat the process for the next three treehouses: Tree teepee (cone), Tree Pod (sphere), Tree Silo (cylinder), answering questions and filling in data tables for each.
- Engage the class in discussion about how their treehouses compare and contrast.
- Share some of your studentsâ€™ work.

Building with 3D shapes on top of a 2D sketch that students create is a great way to frame a discussion around the concepts of moving from 2D to 3D. Ask students:

- What did you notice about the different designs you came up with?
- What shape(s) seemed to work the best for your needs?

What did you notice as you tried to skin your treehouses with your sketches?

Extensions and Inquiring Further

Mathematical nets are useful for the skinning done in this activity. One possible extension is for students to create their own nets for their treehouses with paper, cut them out and fold them into shapes. Design an activity called Shape City and have students create a city populated with all the 3D shapes that your class creates. Define the scale so that there is a real world approximation.

or:

Building 3D treehouses on top of a 2D image is a great way to introduce and to talk about the differences between these dimensions. The book *Flatland: A Romance of Many Dimensions* by Edwin A. Abbott is a story about a world on a two dimensional plane and all the characters that live there. One character gets a new view of reality when it elevates off the plane and gets a view from above. This story is a classic for middle and high school math students. Assigning this novel as a reading would be a great addition to this activity.

For this activity you will be starting with a hand-drawn two dimensional picture of a tree and then, using the Volumize app, you will build a three dimensional treehouse that fits on to the branches. To get started, make a drawing of a tree in the space provided below. Then youâ€™re ready to bring the tree image into Volumize to get building.

To Do:

1. Use your drawing as the background image and set the scale in a realistic way so that you will be able to stand up and sleep inside your treehouse.

2. Create the first model, a treehouse using a rectangular prism. You can also add a rectangular prism as the roof if you wish. Fit the treehouse onto the drawing so that it looks like an actual treehouse. Rotate the model and examine your design from many angles.

3. In the space provided, draw the skin for your treehouse. What would you like to appear on each of the surfaces? Use colored crayons or markers if desired and available.

4. Once you are finished creating the rectangular prism treehouse, yourâ€™re ready to move to the next model. Go back to the homepage and create a new project and remember to save each design.

5. Repeat steps 2 and 3, creating each of the following: a tree teepee using a cone, a tree pod using a sphere, and a tree silo using a cylinder.

6. Fill in the data tables and answer the questions about your designs.

7. Be prepared to share your favorite models and engage in a class discussion about the activity.

- In the space below, draw a tree leaving a space for the treehouses that you will build in this activity. A small example is shown, but be creative and make a unique tree of your own.

Example tree withspace for treehouse: | Your sketch of a tree, with space for treehouse: |

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- The table below is meant to help you keep track of the surface areas and volumes of each treehouse.

Model | Inside Height | Inside Width | Surface Area | Volume |

Treehouse(Rectangular Prism) |
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Tree Teepee(Cone) |
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Tree Pod(Sphere) |
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Tree Silo(Cylinder) |

- In the space below, make sketches for the skin of each tree model:

- When you added the skins to your tree teepee, tree pod and tee silo, what did you notice about the rounded faces? Did you have to change anything to make sure the skin looked the way you wanted it to?

- Did looking at the nets help you in skinning your models? If so, how?

- If you were going to fill each treehouse from floor to ceiling with donuts that are 3 inches in diameter and 1 inch thick, approximately how many donuts could you fit in each model?

- Assuming you donâ€™t need extra materials for overlapping parts, how much plywood would you need to build each design? Plywood is expensive, what could you change in your design to cut costs for each model?