Unit 4: Geometry   You are at the ice cream parlor and need to decide if you want a cup or a cone. The store promises to fill each option with your favorite soft-serve flavor. In this unit, you will learn how to calculate the volume of cones and cylinders, which will help you make the most informed decision about your container choice. This unit also covers information about congruency, similarity, and patterns within shapes and lines. Additionally, you will learn how to find a missing side length of a triangle by using the Pythagorean Theorem.

Completing this unit should take approximately 23 hours.

☐    Subunit 4.1: 8 hours and 45 minutes

☐    Subunit 4.2: 4 hours and 15 minutes

☐    Subunit 4.3: 5 hours and 30 minutes

☐    Subunit 4.4: 4 hours and 30 minutes

Unit4 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Understand congruence and similarity. - Given congruent figures, describe a sequence that will exhibit the congruence. - Verify properties of rotations, reflections, and translations. - Describe the effect of dilations, translations, rotations, and reflections using coordinates. - Given similar figures, describe a sequence that will exhibit the similarity. - Explain and describe angles when parallel lines are cut by a transversal. - Show and explain why the Pythagorean Theorem works. - Apply the Pythagorean Theorem to determine unknown side lengths. - Apply the Pythagorean Theorem to solve real-life problems. - Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - Apply formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems.

4.1 Congruence and Similarity   Think about your best friend. Do you enjoy doing the same activities? Maybe you like to listen to the same music, read the same type books, or play the same games. Possibly you even wear the same style of clothing. Most people hang out with others who are similar to them. The word similar in math has a different meaning than the way people use it in an everyday context. In this subunit, you will work with similar shapes and congruent shapes. These will be important vocabulary words to know and understand as you work your way through this subunit.

• Web Media: M. C. Escher Foundation’s “16 Facsimile Prints” Link: M. C. Escher Foundation’s “16 Facsimile Prints” (HTML)

Instructions: Look at the artwork of M. C. Escher (web search other images as well) through the lens of a mathematician. List some characteristics of the artwork that you think resemble mathematics. Look for commonalities among the different prints made by Escher. Spend 10 minutes browsing Escher’s work. This will lead you into the work you are about to begin with transformations.

Looking at the artwork should take approximately 15 minutes.

• Explanation: CK-12: “Translations, Rotations, and Reflections” Link: CK-12: “Translations, Rotations, and Reflections” (HTML)

Instructions: Read about Tanya in the art museum. Think about the Escher artwork that you just looked at. How are they alike? Read the “Guidance” section. Write down any bold words/sentences in your notebook for notes. Sketch the examples of translations, reflections, and rotations. Although this resource is showing how they are transformed on a coordinate grid, these transformations can take place without the grid as well. Continue to read and take notes about how you can use coordinates to name and identify transformations. Complete examples A, B, and C and be sure you understand the solutions. Add the vocabulary words to your notes and complete the “Guided Practice” section. You will have a lot more opportunities to continue practicing these skills.

Reading this lesson and taking notes should take approximately 30 minutes.

4.1.1 Translations   This subunit will focus on the transformation type called translations. While working with shapes, you will recognize how they can be translated by “sliding.”

• Explanation: CK-12: “Translation and Vectors” Link: CK-12: “Translation and Vectors” (Vimeo and HTML)

Instructions: Start by watching the video under the “Watch This” heading. The instructor explains translations and gives a general overview of transformations. Listen closely at the 2:05 mark as the instructor uses the term “A prime,” which is denoted like this: A’. You will see this often with transformations. Take notes on other important information the instructor discusses in the video. Next, watch the video under “Example A.” Take notes and complete the examples as you watch. Solve the problems in the “Guidance” section, including examples A, B, and C. Be sure you understand the solutions. Solve the “Concept Problem Revisited,” add the vocabulary words to your notebook, and complete the “Guided Practice” section.

Watching the video, reading the lesson, solving the problems, and taking notes should take approximately 30 minutes.

Instructions: As you watch the video, you should understand how the notation works and how the notation affects the placement of the shape on the coordinate plane.

Watching the video and taking notes should take approximately 15 minutes.

Instructions: As you watch the video, you should be able to recognize the translation and denote the movement on the coordinate plane.

Taking notes and watching the video should take approximately 15 minutes.

• Did I Get This? Activity: Khan Academy’s “Translations of Polygons” Link: Khan Academy’s “Translations of Polygons” (HTML)

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice applying translations to polygons until you feel confident that you understand how to translate polygons (practice for at least 10 minutes).

Completing these practice problems should take approximately 15 minutes.

4.1.2 Reflections   This subunit will focus on the transformation type called reflections. While working with shapes, you will recognize how they can be translated by “flipping,” or using a mirror image.

• Explanation: CK-12: “Reflections” Link: CK-12: “Reflections” (HTML and Vimeo)

Instructions: Start by watching the video under the “Watch This” heading. The instructor explains reflections and gives a general overview of transformations. Listen closely as the instructor explains how the corresponding points are equal distance to the line of reflection. Take notes on other important information the instructor discusses in the video. Watch the second video. Take notes and complete the examples as you watch. Solve the problems in the “Guidance” section, including examples A, B, and C. Be sure you understand the solutions. Solve the “Concept Problem Revisited,” add the vocabulary words to your notebook, and complete the “Guided Practice” section.

Watching the videos, reading the lesson, solving the problems, and taking notes should take approximately 30 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Point Reflection” Link: Illustrative Mathematics: “Point Reflection” (PDF)

Instructions: Read and solve the problem on page 1. Sketch the situation, if necessary. Check your solution on pages 2–3.

Completing this activity should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Congruent Segments” Link: Illustrative Mathematics: “Congruent Segments” (PDF)

Instructions: Using the two line segments, answer the question on page 1. Check your solution on page 2. If you see that you did something different from the given solution, sketch the solution on page 2 to help you understand the explanation.

Completing this activity and checking solutions should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Reflecting a Rectangle over a Diagonal” Link: Illustrative Mathematics: “Reflecting a Rectangle over a Diagonal” (PDF)

Instructions: Complete the questions on pages 1–2. It might be helpful to either print the resource or sketch the rectangles and their diagonals on your own grid paper. Question b asks you to make a general statement about lines of symmetry. Use your rectangles from the activity to study how the reflections look and picture how it would look if you started with a square. Check your solutions on pages 3–6.

Completing this activity and checking your solutions should take approximately 30 minutes.

• Checkpoint: Illustrative Mathematics: “Reflecting Reflections” Link: Illustrative Mathematics: “Reflecting Reflections” (PDF)

Instructions: Answer questions a–c on page 1. You will want to sketch the situation on your own graph paper or print this resource. Make sure to pay close attention to what each question is asking you to do. Check your solutions on page 2–3.

Completing this activity and checking your solutions should take approximately 30 minutes.

4.1.3 Rotations   This subunit will focus on the transformation type called rotations. While working with shapes, you will recognize how they can be translated by “rotating/turning.”

• Explanation: CK-12: “Rotations” Link: CK-12: “Rotations” (HTML and Vimeo)

Instructions: Start by watching the video under the “Watch This” heading. The instructor explains rotations and gives a general overview of transformations. Listen closely when the instructor talks about the center of rotation, the direction or rotation, and the degree of rotation. She discusses each of these in all of the examples. Take notes on other important information the instructor discusses in the video. Watch the second video. Take notes and complete the examples as you watch. Solve the problems in the “Guidance” section, including example A, B, and C. Be sure you understand the solutions. Solve the “Concept Problem Revisited,” add the vocabulary words to your notebook, and complete the “Guided Practice” problems.

Watching the videos, reading the lesson, solving the problems, and taking notes should take approximately 30 minutes.

Instructions: As you watch the video you should understand how to denote the rotation of the polygon, take note as to what direction the rotation should occur, and understand how to rotate a polygon on the coordinate plane.

Watching the video and taking notes should take approximately 15 minutes.

• Web Media: Math Warehouse “Rotations of Points, Shapes” Link: Math Warehouse “Rotations of Points, Shapes” (HTML)

Instructions: Scroll down to the graph paper and use the small “Next” button on the top left of the graph paper to walk yourself through how to rotate around a point. Use the “Change Rotation” drop down menu to watch a rotation of 90°, 180°, 270°, and 360°. Read and take notes on the example graphs below the interactive demonstration.

Practicing with the interactive graphs and taking notes should take approximately 15 minutes.

• Did I Get This? Activity: Khan Academy’s “Rotation of Polygons” Link: Khan Academy’s “Rotation of Polygons” (HTML)

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice rotating polygons until you feel confident that you understand how to rotate polygons on the coordinate plane (practice for at least 10 minutes).

Completing these practice problems should take approximately 15 minutes.

4.1.4 Dilations   This subunit will focus on the transformation type called dilations. While working with shapes, you will recognize how they can be translated by “enlarging or shrinking.”

• Explanation: CK-12: “Dilations” Link: CK-12: “Dilations” (HTML and Vimeo)

Instructions: Start by watching the video under the “Watch This” heading. The instructor explains dilations and gives a general overview of transformations. Listen closely as the instructor describes the two key pieces information you need to know when dilating a figure. You likely learned about scale factor in seventh grade. Take notes on other important information the instructor discusses in the video. Next, watch the video under “Example A.” Take notes and complete the examples as you watch. Solve the problems in the “Guidance” section, including examples A, B, and C. Be sure you understand the solutions. Solve the “Concept Problem Revisited,” add the vocabulary words to your notebook, and complete the “Guided Practice” problems.

Watching the videos, reading the lesson, solving the problems, and taking notes should take approximately 30 minutes.

Instructions: Answer the questions on pages 94–98. Check your answers here on pages 102–106, and be sure you understand the solutions.

Solving the problems and checking your solutions should take approximately 45 minutes.

4.1.5 Effects of Transformations   This subunit will focus on how using multiple transformations on one shape can change its location and image. You will work with all four transformation types in this subunit.

• Did I Get This? Activity: Howard County Public School System’s Grade 8 Common Core Mathematics: “Transformation Trio Cards” Link: Howard County Public School System’s Grade 8 Common Core Mathematics: “Transformation Trio Cards” (PDF)

Instructions: Each page has four coordinate planes with a transformation on each plane. You have three tasks for each page: (1) determine which transformation does not fit with the other three, (2) identify what transformation (reflection, rotation, translation, or dilation) is on display for that page, and (3) give a detailed description of the sequence exhibited for each figure (example: the triangle reflected over the y-axis). Check your answers here.

Completing this activity and checking your solution should take approximately 30 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Congruent Triangles” Link: Illustrative Mathematics: “Congruent Triangles” (PDF)

Instructions: Using the two rectangles on the graph paper, answer questions a–c. Check your solutions on pages 2–5. The solution section has a lot of detailed information, so be sure you read and understand the explanations.

Completing this activity and checking your solutions should take approximately 15 minutes.

• Checkpoint: Illustrative Mathematics: “Cutting a Rectangle into Two Congruent Triangles” Link: Illustrative Mathematics: “Cutting a Rectangle into Two Congruent Triangles” (PDF)

Instructions: Using the rectangle on the graph paper, answer questions a–d. Check your solutions on pages 2–3. The solution section has a lot of detailed information, so be sure you read and understand the explanations.

Completing this activity and checking your solutions should take approximately 30 minutes.

• Checkpoint: Engage NY’s Grade 8 Mathematics Curriculum: “Module 2 – Mid-Module Assessment Task” Link: Engage NY’s Grade 8 Mathematics Curriculum: “Module 2 – Mid-Module Assessment Task” (PDF)

Instructions: Answer the questions on pages 115–119. Check your answers here on pages 122–127 and be sure you understand the solutions.

Answering the questions and checking your solutions should take approximately 30 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Triangle Congruence with Coordinates” Link: Illustrative Mathematics: “Triangle Congruence with Coordinates” (PDF)

Instructions: Answer questions a–c on page 1. You will want to sketch the situation on your own graph paper or print this resource. Make sure to pay close attention to what each question is asking you to do. Check your solutions on pages 2–8.

Answering these questions and checking your solutions should take approximately 30 minutes.

4.2 Angle Relationships and Parallel Lines    In seventh grade you learned relationships among angles. You probably remember learning about complementary and supplementary angles. Angles are essential in construction, specifically building strong bridges. However, people use angles in their daily life as they maneuver into a parking spot. Football kickers and soccer players also use angles when they line up an important kick. This subunit will build on that knowledge as you learn that angle relationships and parallel lines can help mathematicians solve a variety of math problems.

Instructions: Sketch the parallel lines and the transversal with the instructor. Label the angles as the instructor discusses them. Take notes on corresponding angles, opposite angles, and supplementary angles.

Watching the video and taking notes should take approximately 15 minutes.

Instructions: Sketch the parallel lines and the transversal with the instructor. Label the 60° angle. Pause the video at the 0:40 mark. Figure out the missing angle measure. Go ahead and fill in all the missing angle measures. Continue watching the video as the instructor discusses how to find the angle measures.

Take note that there is a general mistake in the video. Notice that the 60° angles made by the parallel lines and transversal are actually obtuse angles (presumably 120°). This should be a trigger for you as a mathematician that the original label at the 0:40 is not accurate. Do not try to rely on estimation or measuring with a protractor if you are able to solve using supplementary angles.

As the second example starts, pause the video at the 3:45 mark and consider the question posed by the instructor. Look back to your notes about corresponding angles and opposite angles. Once you’ve made a determination on whether the lines are parallel or not, watch the remainder of the video.

Watching the video and taking notes should take approximately 15 minutes.

Instructions: The purpose of this resource is for you to practice labeling and figuring out missing angles. Start the video and watch as the instructor displays some lines and labels some angles. Pause the video at the 1:45 mark. Sketch the lines in your notebook. You don’t need to measure angles with a protractor, as the purpose is to figure out the missing angles just using the information given.

It you get stuck, watch the video for 45-second segments while the instructor gives you some answers about other missing angles. At the 4:15 mark, the instructor begins talking about the sum of the angles in a triangle. This is important information that you will learn more about in the next subunit.

While watching the star example, pause the video at the 6:12 mark and see how many of the angles you can figure out, including the “?” angle. It you get stuck, watch the video for 45-second segments while the instructor gives you some answers about other missing angles. At the 7:30 mark, the instructor again talks about the sum of the angles in the triangle. Finally, finish the remainder of the video to understand how to find the missing angle.

Watching the video and taking notes should take approximately 30 minutes.

4.2.1 Interior Angles of Triangles   - Explanation: Khan Academy’s “Proof: Sum of Measures of Angles in a Triangle Are 180” Link: Khan Academy’s “Proof: Sum of Measures of Angles in a Triangle Are 180” (YouTube)

Instructions: This video uses the rules of parallel lines cut by a transversal to prove that the sum of the angles in a triangle is equal to 180°. Watch the video and take notes with the instructor.

Watching the video and taking notes should take approximately 15 minutes.

``````-   [CCSS.Math.Content.8.G.A.5](http://www.corestandards.org/Math/Content/8/G/A/5)

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• Activity: James Sousa’s Mathispower4u: “Animation: The Sum of the Interior Angles of a Triangle” Link: James Sousa’s Mathispower4u: “Animation: The Sum of the Interior Angles of a Triangle” (YouTube)

Instructions: This video is an animation that shows you another way to see that the sum of the angles in a triangle is equal to 180°. Watch the video and then cut out your own triangle. Fold in each corner and see if you can prove to yourself that the sum of the angles forms a straight line, which is 180°.

Another method of proving this would involve cutting out a triangle, tearing the three corners off the triangle, and then lining them up to form a straight angle. Try to do this on your own.

Watching the video, completing the activity, and taking notes should take approximately 30 minutes.

• Explanation: James Sousa’s Mathispower4u: “Introduction to the Interior and Exterior Angles of a Triangle” Link: James Sousa’s Mathispower4u: “Introduction to the Interior and Exterior Angles of a Triangle” (YouTube)

Instructions: Take notes on how to find both the exterior and interior angles of a triangle. When the instructor presents an example, pause the video and try to solve it on your own before watching the solution. Pay close attention between the 4:10 and 4:55 marks as the instructor talks about the sum of the exterior angles of a triangle. Take notes on this again at the 6:00 mark.

Watching the video and taking notes should take approximately 15 minutes.

• Explanation: James Sousa’s Mathispower4u: “Interior and Exterior Angles of a Polygon” Link: James Sousa’s Mathispower4u: “Interior and Exterior Angles of a Polygon” (YouTube)

Instructions: Watch the video and take notes on general rules about finding the interior and exterior angles of polygons. At the 2:45 mark, the instructor breaks down how to figure out the sum for interior and exterior angles for a triangle, quadrilateral, and pentagon. Take notes on how to do this and make sure you are able to recall this information for future activities. The two theorems listed on the slide at the 5:45 mark will be important for you to write down and understand.

Watching the video and taking notes should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Tile Patterns I: Octagons and Squares” Link: Illustrative Mathematics: “Tile Patterns I: Octagons and Squares” (PDF)

Instructions: Answer questions a–b on page 1. You can use multiple strategies to answer each question. Consider different ways that you might attempt the problem. Check your solution and read the different explanations on pages 2–3.

Completing this activity should take approximately 15 minutes.

Instructions: Sketch the example and then pause the video at the 0:45 mark. Find the angle in question, and then watch how the instructor solves for the missing angle. Even if you solved the example differently, pay attention to the explanation that the instructor gives for how to solve this example problem.

Watching the video and taking notes should take approximately 15 minutes.

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice solving for missing angles until you feel confident that you understand how to find the angles within parallel lines cut by a transversal (practice for a minimum of 10 minutes).

Completing these practice problems should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Find the Angle” Link: Illustrative Mathematics: “Find the Angle” (PDF)

Instructions: Using the information given, solve the question on page 1. Check your solution on page 2.

Completing this task should take approximately 15 minutes.

• Checkpoint: Illustrative Mathematics: “A Triangle’s Interior Angles” Link: Illustrative Mathematics: “A Triangle’s Interior Angles” (PDF)

Instructions: Answer the question on page 1. Make sure your explanation is general enough to work for all triangles. Read the solution on page 2. Compare the solution to your ideas. Add any information to your answer that would benefit another student trying to understand this concept.

Completing these practice problems should take approximately 15 minutes.

4.2.2 Similar Triangles   - Explanation: James Sousa’s Mathispower4u: “Congruent and Similar Triangles” Link: James Sousa’s Mathispower4u: “Congruent and Similar Triangles” (YouTube)

Instructions: As you watch the video, focus your notes and attention to the angle-angle relationship within similar triangles. Pause the video at the 4:04 mark and take notes on the slide that was just explained. Continue watching the video as it leads into the examples. For each example, pause the video, read and solve the problem, and then continue watching as the instructor explains the answer.

Watching the video should and taking notes should take approximately 15 minutes.

``````-   [CCSS.Math.Content.8.G.A.5](http://www.corestandards.org/Math/Content/8/G/A/5)

attributed to James Sousa.
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• Checkpoint: Howard County Public School System’s Grade 8 Common Core Mathematics: “Angles and Triangles Culmination” Link: Howard County Public School System’s Grade 8 Common Core Mathematics: Schools “Angles and Triangles Culmination” (PDF)

Completing these practice problems should take approximately 30 minutes.

4.3 Pythagorean Theorem Proof   - Activity: Howard County Public School System’s Grade 8 Common Core Mathematics: “Triangle Area Maps” Link: Howard County Public School System’s Grade 8 Common Core Mathematics: “Triangle Area Maps” (PDF)

Instructions: This activity is to help you understand the relationship between side lengths of a triangle. Take this resource as an opportunity to try to discover patterns before you are directly taught the information in future resources.

If you are able, it might be helpful to print the page so you can manipulate the triangles while you do this. A conjecture in math is a statement that is believed to be true based on initial understandings, but is not yet proved. Suppose you overheard another eighth-grade student make the conjecture about the second image that when you have a right triangle, the area of the two smaller squares is equal to the area of the larger square. Look at the triangle area maps. In the second image, focus on the size of the squares drawn off of each side length as you think about the conjecture. Answer the following questions as you use the triangle area maps to verify or disprove the conjecture.

1.    How might you find out if this conjecture is true?
2.    What information would you need to test your idea?
3.    How would you test it?
4.    If you are able to test the conjecture, is it true? If you can’t test it, explain why you’re unable to do so.

Completing this activity should take approximately 30 minutes.

``````-   [CCSS.Math.Content.8.G.B.6](http://www.corestandards.org/Math/Content/8/G/B/6)
-   [CCSS.Math.Content.8.G.B.7](http://www.corestandards.org/Math/Content/8/G/B/7)

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Instructions: As you begin the video, the instructor introduces the Pythagorean Theorem. He makes it very clear which type of triangle the theorem applies to. As you watch and listen, take notes on the triangle type and the location of the hypotenuse. Follow along with the first example given to you. Notice how the instructor writes out each step to help organize his information. This will be helpful to you as you move into more challenging problems and you are solving for different side lengths. The answer to the first example could be shown as ≈ 11.40 if you wanted to solve for the. Be aware that both answers are acceptable.

Follow along on the next two examples. The instructor uses the Pythagorean Theorem to solve for different side lengths then just the hypotenuse. Make sure you understand how the formula works for each example. You will have a lot more opportunity to practice this in future resources.

Watching the video and taking notes should take approximately 30 minutes.

• Explanation: James Sousa’s Mathispower4u: “The Pythagorean Theorem” Link: James Sousa’s Mathispower4u: “The Pythagorean Theorem” (YouTube)

Instructions: In the first 2:45, the instructor shows a proof of why the Pythagorean Theorem works using similar triangles. There are quite a few other proofs available if you do a simple web search. It is important that you understand why the Pythagorean Theorem works as it will be beneficial to you to as you use it throughout math. Just after the 2:45 mark, the instructor works some example problems. Pause the video after he gives each example, and solve for the missing side length using the Pythagorean Theorem. Watch as the instructor shows/explains the answer.

Watching the video and taking notes should take approximately 30 minutes.

• Explanation: James Sousa’s Mathispower4u: “The Pythagorean Theorem and the Converse of the Pythagorean Theorem” Link: James Sousa’s Mathispower4u: “The Pythagorean Theorem and the Converse of the Pythagorean Theorem” (YouTube)

Instructions: Take notes on how the Pythagorean Theorem applies to a right triangle, acute triangle, and obtuse triangle. The converse of the Pythagorean Theorem is stated between 0:35–0:45. The instructor does not make it explicitly clear this is what he is stating. You will have a chance to work with the converse of the Pythagorean Theorem in the next resource. Make sure to take note about the term triangle inequality. When the instructor presents an example, pause the video to solve the problem and then continue watching the video while checking your solution.

Watching the video and taking notes should take approximately 15 minutes.

• Activity: Illustrative Mathematics: “Converse of the Pythagorean Theorem” Link: Illustrative Mathematics: “Converse of the Pythagorean Theorem” (PDF)

Instructions: Remember, the converse of the Pythagorean Theorem states that if a triangle’s sides fit the equation a2 + b2 = c2, then the angle opposite the side of length c is a right angle. Use this as you learn about Pythagorean triples and ancient cultures in this resource. Answer questions a–c on page 1. Read the commentary on the top of page 2. Check your solutions on pages 2–3.

Completing this task should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Applying the Pythagorean Theorem in a mathematical context” Link: Illustrative Mathematics: “Applying the Pythagorean Theorem in a Mathematical Context” (PDF)

Instructions: Read the problem and solve the question below the triangle. A simple “yes” or “no” is not an appropriate answer. You will want to explain your answer with mathematical proof. Check your solution page 2. Read both solutions and be sure you understand them.

Completing this activity and checking your solutions should take approximately 15 minutes.

4.3.1 Pythagorean Theorem Application   - Activity: YouTube’s NFL Network: “Benjamin Watson Tackle Saving Touchdown” Link: YouTube’s NFL Network: “Benjamin Watson Tackle Saving Touchdown” (YouTube)

Instructions: Watch the video completely without pausing. Once the video is over, consider who ran farther: New England Patriot player (white jersey) Benjamin Watson or Denver Bronco player (navy blue jersey) Champ Bailey? Watch the video again, this time pausing at the 1:59 mark to look at where each player starts the play. Consider: How could you use triangles and the Pythagorean Theorem to figure out each of their distances? Don’t solve; just think about it. Continue watching, and listen as Tedy Brushi says, “He ran like 120 yards to get there.” Is Tedy correct? In the next resource you will explore this real-world football play in more detail.

Note: Sometimes YouTube links stop working. If this happens, try following the link given on page 2 in the next resource.

Watching this video should take approximately 15 minutes.

``````-   [CCSS.Math.Content.8.G.B.6](http://www.corestandards.org/Math/Content/8/G/B/6)
-   [CCSS.Math.Content.8.G.B.7](http://www.corestandards.org/Math/Content/8/G/B/7)

displayed on the webpage above.
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• Activity: Illustrative Mathematics: “Running on the Football Field” Link: Illustrative Mathematics: “Running on the Football Field” (PDF)

Instructions: Read about the situation involving the football players from the video in the previous resource. Answer questions a–c. Note that there are hash marks on both the length and the width of the football field diagram to help with your estimation. If you are struggling with solving any part of the problems, it might be helpful to read the commentary on the top of page 2. Once you are ready, check your solutions with the answer on pages 2–3. Think back to the video and the Tedy Brushi comment that “he ran like 120 yards to get there.” Is Tedy accurate? How could you explain to Tedy a more exact number of yards that Ben Watson ran?

Solving the problems and checking your solution should take approximately 30 minutes.

• Did I Get This? Activity: Howard County Public School System’s Grade 8 Common Core Mathematics: “Application Stations” Link: Howard County Public School System’s Grade 8 Common Core Mathematics: “Application Stations” (PDF)

Instructions: Solve each of the application situations and check your answers here.

Solving the problems and checking solutions should take approximately 30 minutes.

• Checkpoint: HippoCampus “Geometry: Building a Slide” Link: HippoCampus “Geometry: Building a Slide” (Flash)

Instructions: On the left side of the screen, scroll down to the “Simulations” heading, and select “Developmental Math – Geometry.” On the pop-out menu select “Geometry: Building a Slide.”  The purpose of this simulation is for you to use the information from the past few subunits to solve this application problem. As the instructor walks you through the situation, sketch the slide and triangle underneath. You should not need to measure or guess any of the answers; rather, use your knowledge about triangles, angles, and the Pythagorean Theorem. At the end of the simulation, the instructor will give you some feedback on areas you might need to review. Take the time to revisit these concepts.

Completing this task should take approximately 15 minutes.

4.3.2 Pythagorean Theorem in a Coordinate System   This subunit will give you the opportunity to use the Pythagorean Theorem on a coordinate plane. The activities in this subunit will help you recognize how the coordinate plane is often helpful when determining side lengths and for organizing information.

• Did I Get This? Activity: Howard County Public School System’s Grade 8 Common Core Mathematics: “Task Amazing Amusement Map” Link: Howard County Public School System’s Grade 8 Common Core Mathematics: “Task Amazing Amusement Park” (PDF)

Instructions: Read The Task at the top of the page. Do not scroll down to any other pages, yet, as that is where you will go to check your solutions. Find the map here and the key for the map here. Use both of these to solve the task given to you. It will probably be helpful if you can print the map. When you have completed the task, check your solution with the possible answers on pages 2–4.

Completing the task and checking your solution should take approximately 45 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Bird and Dog Race” Link: Illustrative Mathematics: “Bird and Dog Race” (PDF)

Completing the task and checking your solution should take approximately 30 minutes.

• Checkpoint: Illustrative Mathematics: “A Rectangle in the Coordinate Plane” Link: Illustrative Mathematics: “A Rectangle in the Coordinate Plane” (PDF)

Instructions: On page 1 is a rectangle on a coordinate plane and questions a–c. Your goal is to answer all the questions using the resources from the entire unit thus far. If you are struggling to get started, the top of page 3 can show you a good first step. Adding a rectangle to the coordinate plane will be beneficial to forming triangles, which you have worked with a great deal in this unit.

Completing the task and checking your solutions should take approximately 30 minutes.

4.4 Calculating Volume   Volume is a three-dimensional measurement used to determine how much space something occupies. Volume can also be thought of as how much liquid can fit in an object. You have experience with volume from your work with rectangular prisms in sixth grade and pyramids in seventh grade.

4.4.1 Volume of Cylinders   - Explanation: James Sousa’s Mathispower4u: “Examples: Determine the Circumference of a Circle” Link: James Sousa’s Mathispower4u: “Examples: Determine the Circumference of a Circle” (YouTube)

Instructions: This is an important review from seventh grade that will help you understand how to work with cones and cylinders. Make sure you understand how to find the circumference of a circle. Take notes on the formula and work the examples with the instructor.

Watching the video and completing the examples should take approximately 15 minutes.

``````-   [CCSS.Math.Content.8.G.C.9](http://www.corestandards.org/Math/Content/8/G/C/9)

attributed to James Sousa.
``````
• Explanation: CK-12: “Volume of Cylinders” Link: CK-12: “Volume of Cylinders” (HTML)

Instructions: Read through the resource and take notes on the bold words/sentences. Make sure you know the formula for calculating the volume of a cylinder and how to use it. Understanding why it makes sense is also important. Do the example problems and the problems in the “Guided Practice” section. Watch the video, focusing your attention mainly on the volume portion of the instruction. Complete the problems in the “Practice” section and check your solutions here.

Reading this lesson, watching the video, working the problems, and taking notes should take approximately 45 minutes.

• Explanation: CK-12: “Heights of Cylinders Given Volume” Link: CK-12: “Heights of Cylinders Given Volume” (HTML)

Instructions: This resource will work with the formula for finding the volume of a cylinder; however, you will be trying to find the height of the cylinder given the volume. Read through the resource and take notes on the bold words/sentences. These types of examples are why understanding the formula for calculating the volume of a cylinder is important. Do the example problems and “Guided Practice.” Do not watch the video as it is the same one from the previous resource. Complete the “Practice” problems and check your solutions here.

Reading this lesson, working the problems, and taking notes should take approximately 30 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Shipping Rolled Oats” Link: Illustrative Mathematics: “Shipping Rolled Oats” (PDF)

Instructions: In sixth and seventh grade you worked with surface area of rectangular prisms. This activity will require you to call on this prior knowledge as you work to answer the questions. Read the questions on page 1. You may want to organize your information in a table (take a quick peek at page 2 for what this might look like). Solve questions a–b on page 1 and then check your solutions on pages 2–3.

Completing this task should take approximately 30 minutes.

4.4.2 Volume of Cones   - Explanation: James Sousa’s Mathispower4u: “Ex: Determine Volume of a Cone” Link: James Sousa’s Mathispower4u: “Ex: Determine Volume of a Cone” (YouTube)

Instructions: The first 0:45 of the video explain the relationship between a cone and a cylinder with equal radius and height. Understanding the relationship is helpful to understanding and remembering the formulas. Copy down in your notebook the formulas for the volume of a cylinder and the volume of a cone. Work the example with the instructor. The instructor goes through two ways to solve for the volume of the cone. Make sure you understand both strategies and the appropriate units used for volume.

Watching the video and completing the examples should take approximately 15 minutes.

``````-   [CCSS.Math.Content.8.G.C.9](http://www.corestandards.org/Math/Content/8/G/C/9)

attributed to James Sousa.
``````
• Explanation: CK-12: “Volume of Cones” Link: CK-12: “Volume of Cones” (HTML)

Instructions: Read through the resource and take notes on the bold words/sentences. Make sure you know the formula for calculating the volume of a cone, make sure you know how to use it, and also it is important that you understand why it makes sense in relationship to the volume of a cylinder. Do the example problems and the problems in the “Guided Practice” section. Watch the video. Complete the “Practice problems,” and check your solutions here.

Reading this lesson, watching the video, working the problems, and taking notes should take approximately 45 minutes.

4.4.3 Volume of Spheres   - Explanation: CK-12: “Volume of Sphere” Link: CK-12: “Volume of Sphere” (HTML)

Instructions: Read through the resource and take notes on the bold words/sentences. Make sure you know the formula for calculating the volume of a sphere and how to use it. It is also important that you understand why it makes sense. Do the “Example” and “Guided Practice” problems. Watch the video. Complete the “Practice” problems and check your solutions here.

Reading this lesson, working the problems, watching the video, and taking notes should take approximately 30 minutes.

``````-   [CCSS.Math.Content.8.G.C.9](http://www.corestandards.org/Math/Content/8/G/C/9)

``````

4.4.4 Real-World Application of Volume   - Did I Get This? Activity: Illustrative Mathematics: “Glasses” Link: Illustrative Mathematics: “Glasses” (PDF)

Instructions: Look at each of the glasses on page 1. Notice how glass 1 is a cylinder and glass 3 is a cone. However, glass 2 is composed of both a hemisphere and a cylinder. In sixth grade you worked with two-dimensional composite shapes. In this activity you will need to work with a shape composed of different three-dimensional shapes. Complete questions a–c on page 1. It will probably be helpful to sketch the shape in your notebook, label the given side lengths, and solve each step with precision. Read the commentary on the top of page 2. Consider the bullet points that are included “to foster a number of discussions.” If you were in a classroom setting, how could you contribute to each question? Check your answers on pages 2–4. Read each of the explanations carefully and compare the student drawings to the drawings in your notebook.

Completing this activity should take approximately 30 minutes.

``````-   [CCSS.Math.Content.8.G.C.9](http://www.corestandards.org/Math/Content/8/G/C/9)