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# K12MATH010: Geometry

Unit 3: Extending to Three Dimensions   You’ve made it to the third dimension! In the first two units, you reviewed some of the important properties of two-dimensional figures and proved many of them using formal proofs. Also, prior to this course, you worked with the concepts of area, perimeter, surface area, and volume, and used formulas to calculate them. This unit expands on these concepts, furthering investigating properties of three-dimensional figures, investigating cross-sections (two-dimensional figures created when you “slice” through a three-dimensional figure), and applying this knowledge in the development of volume formulas and the solution of problems.[](#_ftn1)

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Unit 3 Time Advisory
Completing this unit should take you approximately 9 hours, 30 minutes.

☐    Subunit 2.1: 2 hours

☐    Subunit 3.2: 4 hours

☐    Subunit 3.3: 3 hours, 30 minutes

Unit3 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- Explain volume formulas and use them to solve problems. - Visualize the relationship between two-dimensional and three-dimensional objects. - Apply geometric concepts in modeling situations.

3.1 Exploring Solids   In this subunit, we examine the properties of solid figures. You will see the relationship that exists among faces, edges, and vertices. Additionally, you will examine the cross-sections of solid figures, the two-dimensional figures that are created when you slice a solid figure from different angles. This unit is about visualizing solids and understanding the variety of components. Skills learned here will assist you when you take a calculus course, as calculus works with curved figures. Being able to visualize these figures now and calculate the volume and surface area of the basic solid figures will allow for a more enjoyable learning experience when you reach more advanced math courses. The resources in this section provide explanatory content, practice problems, and some investigatory activities.

3.1.1 Properties of Three-Dimensional Figures   - Explanation: CK-12: Geometry: “Exploring Solids” Link: CK-12: Geometry: “Exploring Solids” (HTML)

Instructions: Please click on the link above, and read the content in the sections entitled “Polyhedrons,” “Euler’s Theorem,” and “Regular Polyhedra.” Take notes and complete the example problems as you go. The solutions are explained following each problem. As you read, consider these questions: What is the definition of a polyhedron? What is an example of three-dimensional figure that is not a polyhedron? How would you summarize Euler’s theorem? What makes a polyhedron regular? Can you make a polyhedron with a face of any regular polygon?

Reading this section, taking notes, and completing the example problems should take approximately 15 minutes.

Standards Addressed (Common Core):

`````` ●
[CCSS.Math.Content.HSG-MG.A.1](http://www.corestandards.org/Math/Content/HSG/MG/A/1)
●
[CCSS.ELA-Literacy.RST.9-10.4](http://www.corestandards.org/ELA-Literacy/RST/9-10/4)
●
[CCSS.ELA-Literacy.RST.9-10.5](http://www.corestandards.org/ELA-Literacy/RST/9-10/5)

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
``````

3.1.2 Cross-Sections   - Explanation: CK-12: Geometry: “Exploring Solids” Link: CK-12: Geometry: “Exploring Solids” (HTML)

Instructions: Please click on the link above, and read the content in the section entitled “Cross-Sections” Complete the example problems as you go. The solutions are explained following each problem.

Reading this section, taking notes, and completing the example problems should take approximately 15 minutes.

Standards Addressed (Common Core):

`````` ●
[CCSS.Math.Content.HSG-GMD.B.4](http://www.corestandards.org/Math/Content/HSG/GMD/B/4)

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
``````
• Did I Get This? Activity: Annenberg Learner: “Session 9, Part C: Cross Sections” Link: Annenberg Learner: “Session 9, Part C: Cross Sections” (Flash)

Instructions: Please click on the link above. This page reviews the concept of cross sections and then provides four problems, accompanied by answers, that allow you to check your knowledge.

Completing this activity should take approximately 15 minutes.

Standards Addressed (Common Core):

●      CCSS.Math.Content.HSG-MG.A.1

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

• Interactive Lab: Shodor: “Cross Section Flyer” Link: Shodor: “Cross Section Flyer” (Java)

Instructions: Please click on the link above. You will find a page with four tabs. Click on the tab labeled “Learner.” Here you will need to download the worksheet entitled “Cross Section Flyer Exploration Questions.” Next, click on the tab labeled “Activity” and use the instructions from the worksheet to guide you.

Completing this activity should take approximately 30 minutes.

Standards Addressed (Common Core):

●      CCSS.Math.Content.HSG-MG.A.1
●      CCSS.ELA-Literacy.WHST.9-10.1e

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

• Checkpoint: Mathematics Assessment Project: “2D Representations of 3D Objects” Link: Mathematics Assessment Project: “2D Representations of 3D Objects” (HTML)

Instructions: Please click on the link above, and scroll down to the section of the lesson entitled “Resources.” Download the PDF file underneath “Lesson (complete).” The assessment is on pages S-1 and S-5, and the pages are titled “Vessels of Water” and “Vessels of Water (revisited).” Complete the questions to the best of your ability. When you are finished, use the solutions on pages T-10 and T-12 to check your understanding.

Completing this activity should take approximately 45 minutes.

Standards Addressed (Common Core):

●      CCSS.Math.Content.HSG-MG.A.1
●      CCSS.Math.Content.HSG-GMD.B.4

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

3.2 Volume   In this subunit, you will look at how to calculate the volume of a variety of solid figures and you will examine the formulas of volume for these figures. The resources in this section provide explanatory content and practice problems.

3.2.1 Volume of Prisms and Cylinders   - Activity: Southern Nevada Regional Professional Development Program: “Volume and Surface Area of Cylinders” Link: Southern Nevada Regional Professional Development Program: “Volume and Surface Area of Cylinders” (PDF)

Instructions: Please click on the link above, and complete the practice activities. You should complete all the problems related to volume (problems 2, 9, 10, 11, and 12), but you may want to complete the surface area problems as well. What are the key words that differentiate a surface area problem from a volume problem?

Completing this activity should take you approximately 30 minutes.

Standards Addressed (Common Core):

`````` ●
[CCSS.Math.Content.HSG-GMD.A.1](http://www.corestandards.org/Math/Content/HSG/GMD/A/1)
●
[CCSS.Math.Content.HSG-GMD.A.3](http://www.corestandards.org/Math/Content/HSG/GMD/A/3)

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
``````
• Activity: Learn NC: “Juicy Juice Box” Link: Learn NC: “Juicy Juice Box” (HTML)

Instructions: Please click on the link above, scroll down to the section entitled “Activities,” and follow the steps. This activity is hands on. You will be investigating juice boxes in different forms. This process will get you thinking about how the volume of a figure changes as the shape changes. Additionally, you will see how very different figures can have the same volume. Get creative and have fun!

Completing this activity should take approximately 45 minutes.

Standards Addressed (Common Core):

●      CCSS.Math.Content.HSG-GMD.A.1
●      CCSS.Math.Content.HSG-GMD.A.3
●      CCSS.ELA-Literacy.RST.9-10.3
●      CCSS.ELA-Literacy.RST.9-10.4
●      CCSS.ELA-Literacy.RST.9-10.5

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

• Explanation: CK-12: Geometry: “Volume of Prisms and Cylinders” Link: CK-12: Geometry: “Volume of Prisms and Cylinders” (HTML)

Instructions: Please click on the link above, and read the content. Take notes and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: How does the process for calculating the volume of a rectangular prism differ from the process for calculating the volume of a triangular prism? How does it differ from the process for calculating the volume of a cylinder? How are these processes related? What does Cavalieri’s principle say about calculating the volume of oblique prisms? How did you approach example 9? What is the process for calculating the volume of a compound figure?

Completing the investigation, taking notes, and completing the example problems should take approximately 45 minutes.

Standards Addressed (Common Core):

●      CCSS.Math.Content.HSG-GMD.A.1
●      CCSS.Math.Content.HSG-GMD.A.3
●      CCSS.ELA-Literacy.RST.9-10.4
●      CCSS.ELA-Literacy.RST.9-10.5

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

3.2.2 Volume of Pyramids and Cones   - Activity: Southern Nevada Regional Professional Development Program: “Volume and Surface Area of Pyramids and Cones” Link: Southern Nevada Regional Professional Development Program: “Volume and Surface Area of Pyramids and Cones” (PDF)

Instructions: Please click on the link above, and complete the practice activities. You should complete all the problems related to volume (problems 2, 3, 5, 8, 10, 13 and 15), but you may want to complete the surface area problems as well. What are the key words that differentiate a surface area problem from a volume problem?

Completing this activity should take you approximately 30 minutes.

Standards Addressed (Common Core):

`````` ●
[CCSS.Math.Content.HSG-GMD.A.1](http://www.corestandards.org/Math/Content/HSG/GMD/A/1)
●
[CCSS.Math.Content.HSG-GMD.A.3](http://www.corestandards.org/Math/Content/HSG/GMD/A/3)

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
``````
• Explanation: CK-12: Geometry: “Volume of Pyramids and Cones” Link: CK-12: Geometry: “Volume of Pyramids and Cones” (HTML)

Instructions: Please click on the link above, and read the content of this section, which begins with an investigation that will allow you to explore volume and the relationship between the volume of a prism and a pyramid. Complete the investigation. Next, read the rest of the section. Take notes and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: What is the relationship between a rectangular prism and a pyramid with congruent bases and equal heights? What is the process for calculating the volume of a pyramid? Of a cone? How do we know that the height of the triangular face of a pyramid is not equal to the height of the pyramid? How can we use the height of the face of the triangle to find the height of the pyramid? Finally, look at problems 26 and 27 in the section entitled “Review Questions.” What is the process for solving these problems? How does the process differ between question 26 and question 27?

Reading this section, taking notes, and completing the example problems should take approximately 30 minutes.

Standards Addressed (Common Core):

●      CCSS.Math.Content.HSG-GMD.A.1
●      CCSS.Math.Content.HSG-GMD.A.3

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

3.2.3 Volume of Spheres   - Explanation: CK-12: Geometry: “Surface Area and Volume of Spheres” Link: CK-12: Geometry: “Surface Area and Volume of Spheres” (HTML)

Instructions: Please click on the link above, and read the sections entitled “Defining a Sphere” and “Volume of a Sphere.” Take notes and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: What are the important parts of a sphere? How do you calculate the circumference of a sphere? What is the process for calculating the volume of a sphere? How is the process for calculating the surface area of a hemisphere different from the process for calculating the volume of a hemisphere? Which process do you find to be more difficult? Why?

Reading this section, taking notes, and completing the example problems should take approximately 30 minutes.

Standards Addressed (Common Core):

`````` ●
[CCSS.Math.Content.HSG-GMD.A.1](http://www.corestandards.org/Math/Content/HSG/GMD/A/1)
●
[CCSS.Math.Content.HSG-GMD.A.3](http://www.corestandards.org/Math/Content/HSG/GMD/A/3)

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
``````
• Activity: Southern Nevada Regional Professional Development Program: “Volume and Surface Area of a Sphere” Link: Southern Nevada Regional Professional Development Program: “Volume and Surface Area of a Sphere” (PDF)

Instructions: Please click on the link above, and complete the practice activities. You should complete all the problems related to volume (problems 1, 2, 4, 6, 8, 9, 12, and 13), but you may want to complete the surface area problems as well. What are the key words that differentiate a surface area problem from a volume problem?

Completing this activity should take approximately 30 minutes.

Standards Addressed (Common Core):

●      CCSS.Math.Content.HSG-GMD.A.1
●      CCSS.Math.Content.HSG-GMD.A.3

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

• Checkpoint: Mathematics Assessment Project: “Calculating Volumes of Compound Objects” Link: Mathematics Assessment Project: “Calculating Volumes of Compound Objects” (HTML)

Instructions: Please click on the link above, and scroll down to the section of the lesson titled “Resources.” Download the PDF file underneath “Lesson (complete).” The assessment is on pages S-1 through S-5 and is titled “Glasses.” Complete the questions to the best of your ability. Pages S-4 and S-5 provide hints to help you answer the questions. When you are finished, use the solutions on page T-6 and T-7 to check your understanding.

Completing this activity should take approximately 45 minutes.

Standards Addressed (Common Core):

●      CCSS.Math.Content.HSG-GMD.A.1
●      CCSS.Math.Content.HSG-GMD.A.3
●      CCSS.Math.Content.HSG-SRT.C.8

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

3.3 Geometric Modeling with Three-Dimensional Figures   In this subunit, you will examine more complex relationships of solid figures, including the ratios created by surface areas and volumes of solid figures, the solid figures created by revolutions on the coordinate plane, and additional connections between surface area and volume. Applications of the relationship between surface area and volume are most often seen in science in understanding a substance’s ability to melt or evaporate or in studying animals and their ability to stay cool. Revolutions around the axis are seen again in calculus but with more complex figures that include curved lines. The resources in this section provide explanatory content, practice problems, and some investigatory activities.

3.3.1 Exploring Similar Solids   - Explanation: CK-12: Geometry: “Exploring Similar Solids” Link: CK-12: Geometry: “Exploring Similar Solids” (HTML)

Instructions: Please click on the link above. Read the section, take notes, and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: How do you determine if two solids are similar? What is the relationship among the scale factor of the dimensions of two similar solids, the scale factor of the surface areas of two similar solids, and the scale factor of the volumes of two similar solids?

Reading this section, taking notes, and completing the example problems should take approximately 30 minutes.

Standards Addressed (Common Core):

`````` ●
[CCSS.Math.Content.HSG-GMD.A.1](http://www.corestandards.org/Math/Content/HSG/GMD/A/1)
●
[CCSS.Math.Content.HSG-GMD.A.3](http://www.corestandards.org/Math/Content/HSG/GMD/A/3)
●
[CCSS.ELA-Literacy.RST.9-10.4](http://www.corestandards.org/ELA-Literacy/RST/9-10/4)
●
[CCSS.ELA-Literacy.RST.9-10.5](http://www.corestandards.org/ELA-Literacy/RST/9-10/5)

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
``````
• Did I Get This? Activity: MoodleShare: “Similar Solids” Link: MoodleShare: “Similar Solids” (HTML)

Instructions: Please click on the link above, and read the content that reviews what was taught in the CK-12 textbook, Geometry. Following the reading is a series of practice problems titled “Review Questions.” Complete the problems, and then check your answers using the key provided.

Completing this activity should take approximately 30 minutes.

Standards Addressed (Common Core):

●      CCSS.Math.Content.HSG-GMD.A.1
●      CCSS.Math.Content.HSG-GMD.A.3

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

3.3.2 Revolutions Around an Axis   - Interactive Lab: k12 Action Math: “Polygon Spin” Link: k12 Action Math: “Polygon Spin” (HTML)

Instructions: Please click on the link above. You will find a page that invites you to create solid figures by rotating or revolving a polygon around the x- or y-axis. Sketch a triangle or rectangle and rotate it around an axis. Describe the figure you create. Is it a cylinder? Is it a cone? Does it have a hole? What is the shape of the whole? Calculate the volume and surface area of the figure that you create. Repeat this exercise three times. If you need additional assistance using the Google SketchUp tool, click here (HTML) for a tutorial.

Completing this activity should take approximately 45 minutes.

Standards Addressed (Common Core):

`````` ●
[CCSS.Math.Content.HSG-MG.A.1](http://www.corestandards.org/Math/Content/HSG/MG/A/1)
●
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
``````

3.3.3 Connecting Surface Area and Volume   - Interactive Lab: Shodor: “Surface Area and Volume” Link: Shodor: “Surface Area and Volume” (Java)

Instructions: Please click on the link above. You will find a page with four tabs. Click on the tab labeled “Learner.” Here you will need to download the worksheet entitled “Surface Area and Volume Exploration Questions.” Next, click on the tab labeled “Activity” and use the instructions from the worksheet to guide you.

Completing this activity should take approximately 30 minutes.

Standards Addressed (Common Core):

`````` ●
[CCSS.Math.Content.HSG-MG.A.1](http://www.corestandards.org/Math/Content/HSG/MG/A/1)
●
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
●
[CCSS.ELA-Literacy.RST.9-10.5](http://www.corestandards.org/ELA-Literacy/RST/9-10/5)
●
[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)
●
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
``````
• Activity: Wikibooks: “Worksheets—Area/Volume Unit” Link: Wikibooks: “Worksheets—Area/Volume Unit” (DOC)

Instructions: Please click on the link above. You will find a page with a list of resources for practicing the calculations of surface area and volume of solid figures. Scroll down to the bottom of the page and select the last file entitled “4-8b Review of Surface Area and Volume.” Here you will find many problems for practice. Complete question 5. Obviously, if you’d like the additional practice, you are welcome to try the others as well; however, question 5 allows you to examine both surface area and volume in the same scenario.

Completing this activity should take approximately 30 minutes.

Standards Addressed (Common Core):

●      CCSS.Math.Content.HSG-MG.A.1

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

• Checkpoint: Mathematics Assessment Project: “Modeling: Rolling Cups” Link: Mathematics Assessment Project: “Modeling: Rolling Cups” (HTML)

Instructions: Please begin by clicking on this link (Flash). Examine the four objects. The task is to determine which would roll in the largest circle. Watch the four short videos to see a demonstration of each. The videos are found on the tabs at the top of the page. You can then use the tab titled “Cup Rolling Calculator” to further investigate what happens. Once you have completed these steps, you are ready for the evaluation. Please click on the link provided above, and scroll down to the section of the lesson titled “Resources.” Download the PDF file underneath “Lesson (complete).” The assessment is on page S-1 and is titled “Modeling Rolling Cups.” Complete the questions to the best of your ability. Pages S-4 and S-5 provide hints to help you answer the questions. When you are finished, use the solutions on pages T-8 and T-9 to check your understanding.

Completing this activity should take approximately 45 minutes.

Standards Addressed (Common Core):

●      CCSS.Math.Content.HSG-GMD.A.1
●      CCSS.Math.Content.HSG-GMD.A.3
●      CCSS.Math.Content.HSG-GMD.B.4
●      CCSS.ELA-Literacy.RST.9-10.3
●      CCSS.ELA-Literacy.RST.9-10.4
●      CCSS.ELA-Literacy.WHST.9-10.1d
●      CCSS.ELA-Literacy.WHST.9-10.1e

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.