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# 3.11: Surface Area and Volume

Created by: CK-12

## 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.
• $8.5\;\mathrm{in\ high}$
• Diameter of $6\;\mathrm{inches}$
• Base Area of $72\;\mathrm{sq.\ inches}$
• 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.

## Date Created:

Feb 22, 2012

Feb 23, 2012
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