Pythagorean Theorem and Volume

Lesson 16

Math

Unit 7

8th Grade

Lesson 16 of 16

Objective


Solve real-world problems involving multiple three-dimensional shapes, in particular, cylinders, cones, and spheres.

Common Core Standards


Core Standards

  • 8.G.C.9 — Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Foundational Standards

  • 7.G.B.4
  • 7.G.B.6

Criteria for Success


  1. Identify individual three-dimensional shapes in composite shapes made of prisms, cylinders, cones, and spheres (MP.7).
  2. Map out a solution pathway and use relevant formulas and math concepts to solve complex, real-world problems (MP.1 and MP.4).
  3. Identify and determine the measurements needed to solve a composite real-world problem.
  4. Analyze the work of others (MP.3).

Tips for Teachers


  • This lesson is designed to take two days. A potential breakdown of the two days is as follows:
    • Day 1: Complete Anchor Problem #1 as a whole class. Students then choose or are assigned one problem from Anchor Problems #2–#6 to complete in small groups. Students should collaboratively develop a strategy to solve the problem and begin solving. The Target Task can be used on either Day 1 or Day 2.
    • Day 2: Students in small groups complete and finalize their solution. Students prepare a poster or visual representation of their problem and response to be posted around the room. Ensure posters include work that is clear to follow, with precise calculations and units (MP.6). Once all students are done, students do a gallery walk and complete Anchor Problem #7. If the Target Task was not given on Day 1, students complete it on Day 2.
  • Some of the tasks are more challenging than the others. There is opportunity for differentiation if the tasks are assigned to students rather than chosen, or if one or two of the tasks are not offered as a selection to the whole group. 

Lesson Materials

  • Poster paper (1 sheet per small group)
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Anchor Problems


Problem 1

Look at the four figures shown below. 

a.   What do you notice about each figure? What do you wonder about each figure?

b.   What measurements would you need in each figure to determine its volume?

Guiding Questions

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Problem 2

Option #1: Glasses

The diagram shows three glasses (not drawn to scale). The measurements are all in centimeters. 

  • The bowl of glass 1 is cylindrical. The inside is 5 cm and the inside height is 6 cm. 
  • The bowl of glass 2 is composed of a hemisphere attached to a cylinder. The inside diameter of both the hemisphere and the cylinder is 6 cm. The height of the cylinder is 3 cm. 
  • The bowl of glass 3 is an inverted cone. The inside diameter is 6 cm and the inside slant height is 6 cm. 
  1. Calculate the volume of the bowl of each of these glasses. 
  2. Glass 2 is filled with water and then half the water is poured out. Find the height of the water. 

Guiding Questions

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References

MARS Summative Assessment Tasks for High School Glasses

Glasses from the Summative Tasks is made available through the Mathematics Assessment Project under the CC BY-NC-ND 3.0 license. Copyright © 2007-2015 Mathematics Assessment Resource Service, University of Nottingham. Accessed March 20, 2017, 3:12 p.m..

Modified by Fishtank Learning, Inc.

Problem 3

Option #2Shipping Rolled Oats

Rolled oats (dry oatmeal) come in cylindrical containers with a diameter of $$5$$ inches and a height of $${9 {1\over2}}$$ inches. These containers are shipped to grocery stores in boxes. Each shipping box contains six rolled oats containers. The shipping company is trying to figure out the dimensions of the box for shipping the rolled oats containers that will use the least amount of cardboard. They are only considering boxes that are rectangular prisms so that they are easy to stack.

a.   What is the surface area of the box needed to ship these containers to the grocery store that uses the least amount of cardboard?

b.   What is the volume of this box?

Guiding Questions

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References

Illustrative Mathematics Shipping Rolled Oats

Shipping Rolled Oats, accessed on June 8, 2017, 2:01 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 4

Option #3: Comparing Snow Cones

Pablo's Icy Treat Stand sells homemade frozen juice treats as well as snow cones. Originally, Pablo used paper cone cups with a diameter of 3.5 inches and a height of 4 inches.  

His supply store stopped carrying these paper cones, so he had to start using more standard paper cups. These are truncated cones (cones with the "pointy end" sliced off) with a top diameter of 3.5 inches, a bottom diameter of 2.5 inches, and a height of 4 inches.

Because some customers said they missed the old cones, Pablo put a sign up saying, "The new cups hold 50% more!" His daughter Letitia wonders if her father's sign is correct. Help her find out.

  1. How much juice can cup A hold? (While cups for juice are not usually filled to the top, we can assume frozen juice treats would be filled to the top of the cup.)
  2. How much juice can cup B hold?
  3. By what percentage is cup B larger in volume than cup A?
  4. Snow cones have ice filling the cup as well as a hemisphere of ice sticking out of the top of each cup. How much ice is in a snow cone for each cup?
  5. By what percentage is the snow cone in cup B larger than the snow cone in conical cup A?
  6. Is Pablo’s sign accurate? 

Guiding Questions

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References

Illustrative Mathematics Comparing Snow Cones

Comparing Snow Cones, accessed on June 2, 2017, 9:38 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 5

Option #4: Flower Vases

My sister’s birthday is in a few weeks and I would like to buy her a new vase to keep fresh flowers in her house. She often forgets to water her flowers and needs a vase that holds a lot of water. In a catalog, there are three vases available and I want to purchase the one that holds the most water. The first vase is a cylinder with diameter 10cm and height 40cm. The second vase is a cone with base diameter 16cm and height 45cm. The third vase is a sphere with diameter 18cm.

  1. Which vase should I purchase? 
  2. How much more water does the largest vase hold than the smallest vase?
  3. Suppose the diameter of each vase decreases by 2cm. Which vase would hold the most water?
  4. The vase company designs a new vase that is shaped like a cylinder on the bottom and a cone on top. The catalog states that the width is 12cm and the total height is 42cm. What would the height of the cylinder part have to be in order for the total volume to be 1,224 π cm3?
  5. Design your own vase with composite shapes, determine the volume, and write an ad for the catalog.

Guiding Questions

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References

Illustrative Mathematics Flower Vases

Flower Vases, accessed on April 7, 2018, 9:16 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 6

Option #5Tennis & Golf Balls

Tennis balls are packaged into cylindrical cans to ship and sell. Three tennis balls fit into one cylinder so that the height of the three balls is equal to the height of the cylinder, and the diameter of one ball is equal to the diameter of the cylinder. 

  1. The radius of one tennis ball is $${1.3}$$ inches. How much air space is in the cylinder around the tennis balls? 

Golf balls are sometimes packaged into square prisms to ship and sell. Three golf balls fit into one square prism, similar to how the tennis balls fit into the cylinder. 

  1. What is the formula for the volume of the square prism? Give your answer in terms of $$d$$, the diameter of the golf ball. 
  2. The volume of the square prism is $${192 \space \mathrm{cm}^3}$$. What is the diameter of one golf ball?
  3. How much air space is in the square prism around the golf balls?    

                     

Guiding Questions

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Problem 7

Choose a problem that you did not solve already and review the work of your peers on their poster. 

a.   Briefly describe their approach to the problem.

b.   Would you have taken a similar approach? Why or why not?

c.   Do you agree with their solution?

Guiding Questions

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Problem Set

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Target Task


At the beginning of the summer, Alessandra bought a new piece of chalk, shown below. She used it for several weeks until it was almost gone. What was left of the chalk is also shown below. All measurements are given in inches, and the figures are not drawn to scale.

How much chalk did Alessandra use? Give your answer in cubic inches.

Student Response

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Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

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Lesson 15

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Irrational Numbers and Square Roots

Topic B: Understanding and Applying the Pythagorean Theorem

Topic C: Volume and Cube Roots

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