## The Polyhedron

I. Section Objectives

- Identify polyhedral.
- Understand the properties of polyhedral.
- Use Euler’s formula to solve problems.
- Identify regular (Platonic) polyhedral.

II. Cross- curricular-Cubic Houses

- Use the following image from the Wikipedia website.
- This is Figure 11.01.01
- www.en.wikipedia.org/wiki/File:Rotterdam_Cube_House.jpg
- Use the image of the cubic houses to conduct a discussion with the students about the different parts of the polyhedron.
- Students should be able to identify some of the edges, the vertices and the faces of the cubes.
- Have the students identify what is unique about these houses.
- Brainstorm a list and write them on the board.
- When finished, have the students move on to drawing their own design of a cubic house.
- Have them label the faces, edges and vertices of their house design.

III. Technology Integration

- Have students select one specific polyhedron to research.
- Then have them research this polyhedron as it is connected to a theme such as photography, architecture or nature.
- Ask the students to keep track of the websites that they visit.
- Have the students take notes on where and how they discover their specific solid.
- Then have them share their findings in small groups.

IV. Notes on Assessment

- Assess student work through their house design.
- Did the students identify the faces, edges and vertices?
- Were the students creative in their design?
- Also look at the technology integration section.
- Did the students find examples of their solid according to theme?
- Were any of the results surprising?
- Assess student understanding through their sharing.

## Representing Solids

I. Section Objectives

- Identify isometric, orthographic, cross- sectional views of solids.
- Draw isometric, orthographic, cross- sectional views of solids.
- Identify, draw and construct nets for solids.

II. Cross- curricular-Designs of Polyhedrons

- This is also the Technology Integration section because this activity depends on the technology.
- Use the following website for nets of polyhedrons. This website also contains different patterns for many different polyhedrons.
- www.korthalsaltes.com/
- The patterns can be downloaded in pdf form and printed.
- Students will require access to a computer and printer.
- Have each student select two different polyhedrons to work with.
- After printing out the pattern, have the student create/build a model of each solid.
- Then each student is to draw orthographic, cross- sectional views of each solid.
- Students need to be sure that their work is complete and that the solid is correctly represented.
- Then have the students create their own model of a third solid.
- For this one, they can’t use the already created patterns.
- They must create their own model using what they have already learned.
- When finished, allow time for students to share their work.

IV. Notes on Assessment

- Begin by observing students as they work.
- Do the students understand the difference between the different views of the solid?
- Did the students successfully create each model?
- Is the orthographic, cross- sectional view of each solid accurate?
- Provide students with feedback on their work.

## Prisms

I. Section Objectives

- Use nets to represent prisms.
- Find the surface area of a prism.
- Find the volume of a prism.

II. Cross- curricular-Prism Collage

- Have students use magazines to find pictures of different prisms.
- For example, a triangular prism could be piece of pie, a rectangular prism could be a box, etc.
- Students will need scissors, magazines of all kinds, glue and large posterboard.
- Have students identify the prisms in their collage.
- When finished, allow time for the students to share their work.

III. Technology Integration

- Use the following image of a deck prism
- This is Figure 10.03.01
- www.defender.com/expanded.jsp?path=-1|74081|316411&id=86235
- Have the students go to Wikipedia and research what a deck prism was and what it was used for.
- Then have the students write a short paragraph explaining the purpose of a deck prism.
- After the students have finished this, have them draw a diagram of a ship with a deck prism to illustrate where the deck prism would have been placed and the function that it would serve.
- When finished, allow students time to share their work.

IV. Notes on Assessment

- Assess the student collages.
- Are the pictures in the collage all prisms?
- Did the students label the different prisms?
- Assess student work on the deck prism and the ship.
- Do the students understand the purpose of the prism?
- Is it drawn correctly on the ship?
- Provide students with comments/feedback on their work.

## Cylinders

I. Section Objectives

- Find the surface area of cylinders.
- Find the volume of cylinders.
- Find the volume of composite three- dimensional figures.

II. Cross- curricular-Cheese Press

- Use an image from the technology section to have a picture to work with for the following problem.
- Here is the problem.
- Given the following dimensions, figure out the surface area and the volume of the cylinder of the cheese press.
- Diameter of
- Base Area of
- Draw a diagram to explain your work on both parts of the problem.
- Be sure to show your work.
- Allow time for students to share their work when finished.

III. Technology Integration

- Use the following website for information on a cheese press that makes cheese.
- www.thegrape.net/browse.cfm/4,10188.html
- This will give you an image to work with for the first activity.
- Then research the cheese press.
- How does it work?
- When was it first used?
- What are the necessary ingredients for making cheese?
- How long does it take?
- Write a short essay on the cheese press to accompany your mathematical work.
- Students can also go to this website and watch a video about how volume impacts space flight.
- www.thefutureschannel.com/dockets/hands-on_math/orion_space_capsule/

IV. Notes on Assessment

- Assess student work and diagrams.
- Is it accurate?
- Were the students able to figure out the surface area of the cylinder?
- Were the students able to figure out the volume of the cylinder?
- Provide students with feedback/comments on their work.

## Pyramids

I. Section Objectives

- Identify pyramids.
- Find the surface area of a pyramid using a net or formula.
- Find the volume of a pyramid.

II. Cross- curricular-Pyramids

- Begin by using this image to show students an aerial view of three pyramids.
- This is Figure 11.05.01.
- www.alienworld.files.wordpress.com/2008/09/pyramids.jpg
- Looking at this image will also give students a great idea of what a net of a pyramid can look like.
- Students are going to be working on drawing nets of different sized pyramids.
- Assign them the task of drawing three nets for three different sized pyramids.
- They can choose which type of pyramids they wish to draw too.
- Have the students label each net with the type of pyramid and be sure that the pyramids are proportional.
- Allow time for students to share their work when finished.

III. Technology Integration

- This is a great website to explore the volume of a pyramid.
- www.mathsisfun.com/geometry/pyramids.html
- It is very simple and basic in its approach.
- It would be excellent for a student who is having difficulty with the concepts or who just needs more practice.

IV. Notes on Assessment

- Assess the nets of the pyramids.
- Are the pyramids labeled to show the type of pyramid that they are?
- Are the pyramids accurately drawn?
- Are they proportional?
- Provide students with comments/feedback on their work.

## Cones

I. Section Objectives

- Find the surface area of a cone using a net or formula.
- Find the volume of a cone.

II. Cross- curricular-Sculpture

- Students are going to design their own sculpture using different cones.
- These can be cones that they create out of paper, or cones from nature such as a pine cone.
- Begin by conducting a discussion about cones.
- Be sure to review the parts of a cone.
- Tell students that they need to present work to show the surface area and volume of one of the cones that they create.
- Then let them work on their sculpture.
- Students will need paper, markers, colored paper, a surface to build on and glue.
- When finished, allow time for students to share their work.

III. Technology Integration

- Students can go to the following website and look at many different images of cones in architecture.
- www.fiveprime.org/hivemind/Tags/architecture,cone
- They need to select one cone to work with.
- Cones of specific buildings are named. Students can use this information to research about the specific cone.
- Students should write a short essay on their cone and draw a design to represent the cone that they have chosen.
- Allow time for students to share their work when finished.

IV. Notes on Assessment

- Assess student work.
- Is the work creative?
- Are the cones accurate?
- Is there anything that the student needs to improve upon?
- Provide students with feedback/comments.

## Spheres

I. Section Objectives

- Find the surface area of a sphere.
- Find the volume of a sphere.

II. Cross- curricular-Spheres for Space

- Begin with the article on MIT and space spheres.
- Print the article and either have the students read it silently or read it as a whole class.
- Tell the students that they are going to be designing spheres for this space project.
- The spheres need to be the same size as a volleyball.
- You will need some volleyballs and tape measures for this class.
- Have students measure each model and then build a model of their space sphere.
- Students should create a 3D model and draw a design of their space sphere as well.
- Have students work in groups of three.
- Allow time for students to share their work when finished.

III. Technology Integration

- Begin by having students look at this website which looks at MIT students who are designing spheres to go into space.
- www.spacedaily.com/news/microsat-00e.html
- This is a great video which show architecture and space together.
- The engineers in the video work with spheres and also explain how area and volume impact the design of anything that is sent into space.
- www.thefutureschannel.com/dockets/hands-on_math/space_architecture/
- This is a great place to begin a discussion with students about careers that use mathematics.
- Here is another fun website that looks at spheres.
- www.cotf.edu/ete/modules/msese/earthsysflr/spheres.html

IV. Notes on Assessment

- Assess each work product.
- Were the directions followed?
- Did students accomplish the objective?
- Provide students with feedback on their work.

## Similar Solids

I. Section Objectives

- Find the volumes of solids with bases of equal areas.

II. Cross- curricular-Kaleidocycles

- For this activity, you will need to use the information at the following website.
- www.mathematische-basteleien.de/kaleidocycles.htm
- This website provides pictures and directions of how to make different kaleidocycles.
- It is a great way for students to see how similar solids can be combines together.
- Some of the solids are congruent and some are similar.
- Become familiar with some of the patterns and designs before assigning this to the students.
- Then give them the instructions, by printing or providing technology and let them go to work.
- Students need to select at least two different kaleidocycles to create.
- They can also design their different ones.
- Allow time for students to present their work when finished.

III. Technology Integration

- The activity above integrates technology into the making of kaleidocycles.
- www.maa.org/mathland/mathtrek_11_13_06.html
- Have students explore the website on the math trek.
- Discuss the different aspects of the trek.
- You could even take your students on a math trek around the school or town.
- Have the students make notes of all of the different places where mathematics/geometry can be found.

IV. Notes on Assessment

- For this activity, assessment can be completed through observation.
- Be sure to interact with students as they work.
- Offer assistance where needed.

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