 Unit 4: Geometry   Imagine you work for a packaging company. You need to design a brand new package for a new product, but it has to meet certain requirements regarding how much space it takes up and how much materials it takes to make it. This is one of the many applications you will be able to solve at the end of this unit in geometry.

In this unit, you will start your study by working with shapes, the angles that form them, their perimeters, and their areas. The two-dimensional shapes will help you build and describe pieces of three-dimensional prisms and pyramids. Finally, you will take all these skills to solve real-world problems involving surface area and volume.

Completing this unit should take approximately 17 hours and 30 minutes.

☐    Subunit 4.1: 1 hour

☐    Subunit 4.2: 5 hours

☐    Subunit 4.3: 4 hours and 15 minutes

☐    Subunit 4.4: 45 minutes

☐    Subunit 4.5: 2 hours

☐    Subunit 4.6: 1 hour and 30 minutes

☐    Subunit 4.7: 3 hours

Unit4 Learning Outcomes
Upon successful completion of this course, you will be able to: - Identify angle relationships and calculate missing angles.
- Describe two- and three-dimensional shapes.
- Explain the relationship between the area and circumference of a circle.
- Solve real-world problems involving area, surface area, and volume of prisms.
- Construct geometric shapes with given conditions using tools and technology.

Standards Addressed (Common Core): - CCSS.Math.Content.7.G.A.2 - CCSS.Math.Content.7.G.A.3 - CCSS.Math.Content.7.G.B.4 - CCSS.Math.Content.7.G.B.5 - CCSS.Math.Content.7.G.B.6 - CCSS.ELA-Literacy.WHST.6-8.2f

4.1 Angles   Angles are part of the building blocks for polygons and polyhedrons. They are important in constructing buildings and houses to ensure that walls, floors, and ceilings will not break. In addition, angles are used in art, architecture, and design to create visually pleasing designs. In this subunit, you will learn more about angles and angle relationships.

4.1.1 Angle Relationships   - Explanation: CK-12: “Angle Pairs” Link: CK-12: “Angle Pairs” (PDF)

Instructions: Read through the descriptions of the angle pairs. Try to use the vocabulary for the different types of angles to name the angles in the pairs. When you are finished, watch the video review and then solve the 15 practice problems at the bottom of the page.

Completing this activity should take approximately 40 minutes.

-   [CCSS.Math.Content.7.G.B.5](http://www.corestandards.org/Math/Content/7/G/B/5)

attributed to CK-12 Foundation. The original version can be found

4.1.2 Finding Missing Angles   - Activity: GeoGebra: “Missing Angle Problems Using Three Basic Angle Facts” Link: GeoGebra: “Missing Angle Problems Using Three Basic Angle Facts” (Flash)

Instructions: The dots on each of the diagrams are movable with your mouse. Try changing the position of the angles to see how they are related to each other. When you get a new position, you can calculate the value of the other missing angle. Click on the “Show ___” box to see if your answer was right. Try this with all three diagrams. Spend 5 minutes working with each diagram.

Completing this activity should take approximately 15 minutes.

-   [CCSS.Math.Content.7.G.B.5](http://www.corestandards.org/Math/Content/7/G/B/5)

• Activity: XP Math: “Complementary and Supplementary Angle Game” Link: XP Math: “Complementary and Supplementary Angle Game” (Flash)

Instructions: This is a timed game. The first level gives you one angle and asks for the complement. In level two, you are given an angle and asked for the supplement. When you go to type your answer in the gray box, be sure to delete the question mark before you type. Play the game twice.

Completing this activity should take approximately 5 minutes.

4.2 Area of Polygons and Irregular Shapes   Using angles to form shapes creates area within the sides and vertices. Area is a critical measure for determining how much tile or carpet to purchase for a room or even how much paint to buy to cover the walls. In this subunit, you will discover formulas for determining the area of several different polygons. Then you will use all of your skills to work with shapes that were created by combining several polygons together.

4.2.1 Area of Parallelograms   - Explanation: CK-12: “Area of a Parallelogram” Link: CK-12: “Area of a Parallelogram” (PDF)

Instructions: Read about Jillian and the parallelograms in her grandmother’s quilt. Pay special attention to how finding the area of a parallelogram is similar to finding the area of a rectangle. You do not need to watch the video. Then solve the 15 practice problems at the bottom of the page.

Completing this activity should take approximately 40 minutes.

-   [CCSS.Math.Content.7.G.B.6](http://www.corestandards.org/Math/Content/7/G/B/6)

attributed to CK-12 Foundation. The original version can be found
• Activity: Curriki: “Interactive Practice Problems: Area of Parallelograms” Link: Curriki: “Interactive Practice Problems: Area of Parallelograms” (Flash)

Instructions: This is a game. All you need to do is place the numerical answer to the area question in the box and hit the next button. When you hit the “next” button, you will be shown whether your answer is correct and the correct work. Then click the “next” button again to get to the next question. Play the game one time.

Completing this activity should take approximately 10 minutes.

• Activity: XP Math: “Finding Areas of Parallelograms” Link: XP Math: “Finding Areas of Parallelograms” (Flash)

Instructions: In this game, you will be given different parallelograms and you will need to type in the area. However, you will want to be careful how you enter your answer. There are three boxes. The first is for the numerical answer. The next two are for the label. Remember that units need to be squared with areas. Play the game one time.

Completing this activity should take approximately 15 minutes.

4.2.2 Area of Trapezoids   - Activity: Curriki: “Interactive Practice Problems: Area of Trapezoids” Link: Curriki: “Interactive Practice Problems: Area of Trapezoids” (Flash)

Instructions: To play this game, you only need to calculate the area of a trapezoid and enter in the answer where you see the empty box. When you click the “next” button, you will be shown the right answer and all the steps needed to solve the problem. Play the game one time.

Completing this activity should take approximately 15 minutes.

-   [CCSS.Math.Content.7.G.B.6](http://www.corestandards.org/Math/Content/7/G/B/6)

is attributed to Curriki. The original version can be found
[here](http://www.curriki.org/xwiki/bin/view/Coll_MathMastery/InteractivePracticeProblemsAreaofTrapezoids?bc=;Coll_MathMastery.InformalGeometry;Coll_MathMastery.Area;Coll_MathMastery.AreaofTrapezoids).
• Explanation: CK-12: “Area and Perimeter of Trapezoids” Link: CK-12: “Area and Perimeter of Trapezoids” (PDF)

Instructions: Be sure to watch the informational videos to see how to tell which sides are the bases and which side is the height. Solve the guided practice problems and check your answers before you solve the 16 practice problems.

Completing this activity should take approximately 45 minutes.

4.2.3 Area of Triangles   - Activity: Curriki: “Interactive Practice Problems: Area of Triangles” Link: Curriki: “Interactive Practice Problems: Area of Triangles” (Flash)

Instructions: To play this game, you only need to calculate the area of a triangle and enter in the answer where you see the empty box. When you click the “next” button, you will be shown the right answer and all the steps needed to solve the problem. Then click the “next” button again to get to the next question. Play the game one time.

Completing this activity should take approximately 15 minutes.

-   [CCSS.Math.Content.7.G.B.6](http://www.corestandards.org/Math/Content/7/G/B/6)

is attributed to Curriki. The original version can be found
[here](http://www.curriki.org/xwiki/bin/view/Coll_MathMastery/InteractivePracticeProblemsAreaofTriangles?bc=;Coll_MathMastery.InformalGeometry;Coll_MathMastery.Area;Coll_MathMastery.AreaofTriangles).
• Explanation: CK-12: “Area and Perimeter of Triangles” Link: CK-12: “Area and Perimeter of Triangles” (PDF)

Instructions: Watch the instructional videos. The first video is at the top of the page. The second video is listed under Example C. Pay special attention to how the base and the height are related to each other in a triangle. Then solve the 11 practice problems at the bottom of the page.

Completing this activity should take approximately 40 minutes.

4.2.4 Area of Irregular Polygon Shapes   - Explanation: CK-12: “Area of Composite Shapes Involving Triangles” Link: CK-12: “Area of Composite Shapes Involving Triangles” (PDF)

Instructions: Read the example of finding the area of a home plate.
Work through the guided examples, and then solve the 15 practice
problems at the bottom of the page. Because the practice problems
are word problems, you should draw a diagram of the object being
described. You do not need to watch the video.

Completing this activity should take approximately 1 hour.

-   [CCSS.Math.Content.7.G.B.6](http://www.corestandards.org/Math/Content/7/G/B/6)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/geometry/Area-of-Composite-Shapes-Involving-Triangles/lesson/Area-of-Composite-Shapes-Involving-Triangles/).
• Explanation: CK-12: “Area of Composite Shapes” Link: CK-12: “Area of Composite Shapes” (PDF)

Instructions: Watch the guided videos on finding the area of shapes made up of several polygons and circles. While watching, be sure to take note of ways to divide the shape into several smaller shapes that are easier to work with. Go through the guided examples, and then solve the 15 practice problems at the bottom of the page.

Completing this activity should take approximately 1 hour.

4.3 Circles   Circles are not made up of sides, so we cannot call them polygons, but circles are a popular shape used in creating cylinders, cones, and many real-life objects. This subunit takes a look at the relationship among the radius, diameter, circumference, and area.

4.3.1 Circumference of a Circle   - Explanation: CK-12: “Circle Circumference” Link: CK-12: “Circle Circumference” (PDF)

Instructions: Read the example problems in the “Guidance” section of the reading. Take note of how the steps change when you have a diameter or a radius. Then solve the 20 problems at the bottom of the page.

Completing this activity should take approximately 45 minutes.

-   [CCSS.Math.Content.7.G.B.4](http://www.corestandards.org/Math/Content/7/G/B/4)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/geometry/Circle-Circumference/lesson/Circle-Circumference/).
• Activity: Curriki: “Interactive Practice Problems: Equation for Circumference and Diameter” Link: Curriki: “Interactive Practice Problems: Equation for Circumference and Diameter” (Flash)

Instructions: To play this game, you only need to calculate the circumference of a circle and enter in the answer where you see the empty box. When you click the “next” button, you will be shown the right answer and all the steps needed to solve the problem. Then click the “next” button again to get to the next question. Play the game one time.

Completing this activity should take approximately 15 minutes.

• Activity: Curriki: “Interactive Practice Problems: Finding Diameter” Link: Curriki: “Interactive Practice Problems: Finding Diameter” (Flash)

Instructions: To play this game, you only need to calculate the diameter of a circle when given the circumference and enter in the answer where you see the empty box. When you click the “next” button, you will be shown the right answer and all the steps needed to solve the problem. Then click the “next” button again to get to the next question. Play the game one time.

Completing this activity should take approximately 15 minutes.

• Explanation: CK-12: “Diameter or Radius of a Circle Given Circumference” Link: CK-12: “Diameter or Radius of a Circle Given Circumference” (PDF)

Instructions: Continue the story of Jillian to find out how you can work backward to get the radius or diameter when given the circumference. Then solve the 16 practice problems at the bottom of the page. You will want to use a calculator with the pi button.

Completing this activity should take approximately 45 minutes.

4.3.2 Area of a Circle   - Explanation: CK-12: Jack Hatert’s “Area of Circles” Link: CK-12: Jack Hatert’s “Area of Circles” (PDF)

Instructions: Read through the examples with Miguel’s story. You will want to note the changes in the steps when you have the radius or the diameter. In addition, you will also be practicing solving for the radius or diameter when given the area. Complete the 26 practice problems at the end of the reading.

Completing this activity should take approximately 1 hour, 15 minutes.

-   [CCSS.Math.Content.7.G.B.4](http://www.corestandards.org/Math/Content/7/G/B/4)

attributed to Jack Hatert and CK-12 Foundation. The original version
can be found
[here](http://www.ck12.org/user:amhhdGVydEB5c3NjaG9vbHMub3Jn/section/Area-of-Circles/).
• Activity: Curriki: “Interactive Practice Problems: Area of Circles” Link: Curriki: “Interactive Practice Problems: Area of Circles” (Flash)

Instructions: To play this game, you only need to calculate the area of a circle and enter in the answer where you see the empty box. When you click the “next” button, you will be shown the right answer and all the steps needed to solve the problem. Then click the “next” button again to get to the next question. Play the game once; there are five questions total.

Completing this activity should take approximately 15 minutes.

4.3.3 Area of Irregular Shapes (Circles with Polygons)   - Explanation: CK-12: “Areas of Combined Figures Involving Circles” Link: CK-12: “Areas of Combined Figures Involving Circles” (PDF)

Instructions: Read through the examples to see how to work with the area of half a circle. Then solve the 15 practice problems at the bottom of the page. You will want to use a calculator with a pi button instead of 3.14, then round your answers to the nearest tenth. You do not need to watch the videos.

Completing this activity should take approximately 45 minutes.

-   [CCSS.Math.Content.7.G.B.4](http://www.corestandards.org/Math/Content/7/G/B/4)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/geometry/Areas-of-Combined-Figures-Involving-Circles/lesson/Areas-of-Combined-Figures-Involving-Circles/).

4.4 Three-Dimensional Shapes   The two-dimensional shapes that we have studied so far can be joined together to form three-dimensional shapes. We live in a three-dimensional world, so the three-dimensional shapes used have many real-life applications. Now we have the boxes, cans, globes, and even wedges of cheese shapes forming. To further study three-dimensional shapes, we discuss covering the shapes through surface area and filling the shapes through volume.

4.4.1 Review of Three-Dimensional Shapes   - Assessment: Quizlet: “Three Dimensional Shapes” Link: Quizlet: “Three Dimensional Shapes” (Flash)

Instructions: Before you start practicing with the flash cards, unclick the “Both Sides” button in the top right corner. You should now see a three dimensional shape. Name the shape, and then click on the flash card (“Click to flip”) to see the answer. Then click on the arrow to the right to get the next flash card. Rotate through the flash cards three times.

Completing this activity should take approximately 5 minutes.

-   [CCSS.Math.Content.7.G.A.3](http://www.corestandards.org/Math/Content/7/G/A/3)

displayed on the webpage above.

4.4.2 Cross-Sections of Three-Dimensional Shapes   - Assessment: GeoGebra: “Cross Sections of a Rectangular Pyramid” Link: GeoGebra: “Cross Sections of a Rectangular Pyramid” (Java)

Instructions: Move the plane by sliding the Move Plane slider to see
what shape is formed when a cross section is parallel to the base.
Then try adjusting the Move Vertex, Change Size, and Rotate Pyramid
sliders, adjusting the Move Plane slider after each change.  Note
what happens to the plane, which is also referred to as a cross
section.  Write a sentence that gives a summary of your findings.

Completing this activity should take approximately 10 minutes.

-   [CCSS.Math.Content.7.G.A.3](http://www.corestandards.org/Math/Content/7/G/A/3)
-   [CCSS.ELA-Literacy.WHST.6-8.2f](http://www.corestandards.org/ELA-Literacy/WHST/6-8/2/f)

• Explanation: CK-12: “Cross-Sections and Nets” Link: CK-12: “Cross-Sections and Nets” (PDF)

Instructions: Read through the passage to find out what a cross section and a net of a three-dimensional shape look like. Then name the shapes in the example problems at the end of the reading. You do not need to watch the video or complete the practice problems at the end of the page.

Completing this activity should take approximately 15 minutes.

• Activity: GeoGebra: “Cross Sections of a Rectangular Prism” Link: GeoGebra: “Cross Sections of a Rectangular Prism” (HTML)

Instructions: Use the “move the sliders” section to change where the cross section of the prism is located. Make a list of all the different shapes that can be formed and where the cross section is located in regard to the base.

Completing this activity should take approximately 15 minutes.

4.5 Surface Area   Have you ever wrapped a gift? Covering or wrapping something is directly related to surface area. Manufacturers may want to know how much material it will take to create the sides of an object, or how much it will take to cover an object. In this subunit, you will start your discovery of surface area with prisms.

4.5.1 Surface Area of a Rectangular Prism   - Explanation: CK-12: “Surface Area of Rectangular Prisms” Link: CK-12: “Surface Area of Rectangular Prisms” (PDF)

Instructions: Read through the examples of how to calculate the surface area of a rectangular prism. While reading, determine how the formula was formed from the area of the faces. Then solve the first 10 of the 15 practice problems at the bottom of the page.

Completing this activity should take approximately 1 hour.

-   [CCSS.Math.Content.7.G.B.6](http://www.corestandards.org/Math/Content/7/G/B/6)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/geometry/Surface-Area-of-Rectangular-Prisms/lesson/Surface-Area-of-Rectangular-Prisms/).

4.5.2 Surface Area of a Triangular Prism   - Explanation: CK-12: “Surface Area of Triangular Prisms” Link: CK-12: “Surface Area of Triangular Prisms” (PDF)

Instructions: Read through the passage to see how a formula can be used to solve for the surface area of a triangular prism. While reading, try to come up with a method for solving that does not require the formula. Watch the review video to see if your method is similar to the method shown. Then solve the 15 practice problems at the bottom of the page.

Completing this activity should take approximately 1 hour.

-   [CCSS.Math.Content.7.G.B.6](http://www.corestandards.org/Math/Content/7/G/B/6)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/geometry/Surface-Area-of-Triangular-Prisms/lesson/Surface-Area-of-Triangular-Prisms/).

4.6 Volume   To find how much water will fill a pool or how much air it takes to fill a balloon, you need to calculate the volume. Volume is all about the space a shape takes up or the space that can be filled up inside. In this subunit, you will start your discovery of volume through prisms.

4.6.1 Volume of a Rectangular Prism   - Explanation: CK-12: “Volume of a Cube or Cuboid using V = lwh - Overview” Link: CK-12: “Volume of a Cube or Cuboid Using V = lwh - Overview” (YouTube)

Instructions: Watch the video to see how the formula V = l x w x h can be used. While viewing, determine if the location of the length, width, or height changes the value of the volume.

Completing this activity should take approximately 5 minutes.

-   [CCSS.Math.Content.7.G.B.6](http://www.corestandards.org/Math/Content/7/G/B/6)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/resource/video/Volume-of-a-Cube-or-Cuboid-using-V-%3D-lwh---Overview?eid=MAT.GEO.862&rtitle=Volume+of+Prisms&ref=%2Fconcept%2FVolume-of-Prisms%2F).
• Explanation: CK-12: “Volume of a Cube or Cuboid using V = Bh - Overview” Link: CK-12: “Volume of a Cube or Cuboid Using V = Bh - Overview” (YouTube)

Instructions: Watch the video to see how the two formulas used for the volume of cuboids are related.

Completing this activity should take approximately 5 minutes.

• Activity: Curriki: “Interactive Practice Problems: Volume of a Box” Link: Curriki: “Interactive Practice Problems: Volume of a Box” (Flash)

Instructions: Use either V = lwh or V = Bh to solve the problems. Place the answer in the box and press the “next” button to see if you are correct. Then press “next” again to get another problem.

Completing this activity should take approximately 20 minutes.

4.6.2 Volume of a Triangular Prism   - Explanation: CK-12: “Volume of Triangular Prisms” Link: CK-12: “Volume of Triangular Prisms” (PDF)

Instructions: Read the description of how to calculate the volume of a triangular prism. Take note of how the steps are similar to the ones used to calculate the volume of a rectangular prism. Watch the video to see the steps in action, and then solve the 15 practice problems at the bottom of the page.

Completing this activity should take approximately 1 hour.

-   [CCSS.Math.Content.7.G.B.6](http://www.corestandards.org/Math/Content/7/G/B/6)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/geometry/Volume-of-Triangular-Prisms/lesson/Volume-of-Triangular-Prisms/).

4.7 Constructions   Constructions may have been started before we had computers, but they still hold an important place in geometry. By being able to create different constructions, you can gain a better understanding of the geometric properties that form different polygons, angles, and lines.

4.7.1 Creating Congruent Lines and Angles   - Explanation: CK-12: “Constructions and Copying a Line Segment” Link: CK-12: “Constructions and Copying a Line Segment” (PDF)

Instructions: You will need a compass to complete this section. Read through the example of how to use a compass to make a line that is the same length as the original line. Then draw five lines of your own length and copy the steps.

Completing this activity should take approximately 20 minutes.

-   [CCSS.Math.Content.7.G.A.2](http://www.corestandards.org/Math/Content/7/G/A/2)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/section/Constructions-and-Copying-a-Line-Segment/).
• Activity: GeoGebra: “Angle Bisector” Link: GeoGebra: “Angle Bisector” (HTML)

Instructions: While watching the second video, use paper, pencil, and compass to follow the steps shown. Then draw four more angles and bisect them using the same steps. You can check your work by measuring the angles with a protractor.

Completing this activity should take approximately 20 minutes.

• Activity: GeoGebra: “Constructing Perpendicular Bisectors” Link: GeoGebra: “Constructing Perpendicular Bisectors” (HTML)

Instructions: Scroll down to the bottom of the page, and click on the triangle button to play the video. While watching the video, use paper, pencil, and compass to follow the steps shown. Then draw four more lines and bisect them using right angles and the same steps you viewed. You can check your work by measuring the angles with a protractor. The angles created by perpendicular lines should measure 90 degrees.

Completing this activity should take approximately 20 minutes.

4.7.2 Creating Triangles   - Explanation: Utah Electronic High School: “Constructing a Right Isosceles Triangle” Link: Utah Electronic High School: “Constructing a Right Isosceles Triangle” (PDF)

Instructions: While reading through the instructions, use a compass and piece of paper to create your own isosceles triangles. Then try to create larger or smaller copies on your own. Also, think about how the steps could be changed to construct other types of triangles.

Completing this activity should take approximately 30 minutes.

-   [CCSS.Math.Content.7.G.A.2](http://www.corestandards.org/Math/Content/7/G/A/2)

attributed to Utah Electronic High School. The original version can
be found [here](http://share.ehs.uen.org/node/12180).
• Explanation: National Institute of Education Technology and Teacher Education: “Constructing Triangles” Link: National Institute of Education Technology and Teacher Education: “Constructing Triangles” (HTML)

Instructions: Follow along with the steps using your own compass, pencil, and paper. Then try to create your own triangles of different sizes.

Completing this activity should take approximately 30 minutes.

4.7.3 Creating Rectangles   - Explanation: Utah Electronic High School: “Constructing Squares and Rectangles” Link: Utah Electronic High School: “Constructing Squares and Rectangles” (HTML)

Instructions: While reading through the instructions, please use a compass and piece of paper to create your own squares. Then try to create three rectangles using similar steps.

Completing this activity should take approximately 30 minutes.

-   [CCSS.Math.Content.7.G.A.2](http://www.corestandards.org/Math/Content/7/G/A/2)

attributed to Utah Electronic High School. The original version can
be found [here](http://share.ehs.uen.org/node/12910).
• Checkpoint: Illustrative Mathematics: “Eight Circles” Link: Illustrative Mathematics: “Eight Circles” (PDF)

Instructions: Determine the area of the entire object and the area of the shaded region. Then scroll to page 2 to check your work.

Note to Students and Educators: “Popcorn Picker” is a great activity that you can try on your own using popcorn or by just watching the video to see volume and other geometric skills in action.

Note to Educators: “Designing a Community” is a wonderful resource for creating geometry units; it breaks the Common Core State Standards down into daily lessons and activities.

Completing this checkpoint and checking your work should take approximately 30 minutes.