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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.

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