Three-Dimensional Measurement and Application

Lesson 13

Math

Unit 6

10th Grade

Lesson 13 of 18

Objective


Calculate the volume of compound objects and those with subtracted solids. Determine how the volume will be affected by scaling one or more dimensions.

Common Core Standards


Core Standards

  • G.GMD.A.3 — Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
  • G.GMD.B.4 — Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
  • N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Foundational Standards

  • 8.G.C.9

Criteria for Success


  1. Identify the three-dimensional shapes that make up a composite figure—represented in a diagram of three-dimensional figure, as a description, or as a two-dimensional shape revolved around an axis.
  2. Choose formulas appropriately to develop a formula for finding the volume of a composite shape, regardless of dimensions.
  3. Use the modified formula to find the volume of a figure. 
  4. Change one or more dimensions of a figure and describe how the volume is affected by this change.
  5. Identify when an estimate of volume is necessary and apply this to the problem.

Tips for Teachers


In terms of pacing, this lesson could be spread out over the course of two days due to the amount of content it contains and in order to accomplish all necessary applications. 

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Anchor Problems


Problem 1

Figure A shows the water level before the rock is put in the water. Figure B shows the water level after the rock is put in the water. What is the volume of the rock? 

Guiding Questions

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Problem 2

Below is a two-dimensional pattern that will be spun on an axis to create a three-dimensional shape.

  1. Find the volume if this two-dimensional pattern is spun on the $$x$$-axis.
  2. Find the volume if this two-dimensional pattern is spun on the $$y$$-axis.
  3. What is the difference in volume between spinning the figure on the $$x$$-axis and spinning the figure on the $$y$$-axis? 

Guiding Questions

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References

Mathematics Vision Project: Secondary Mathematics Three Module 5: Modeling with Geometry"Take Another Spin" (p. 3)

Module 5: Modeling with Geometry in Secondary Mathematics Three: Integrated Pathway CCSS made available by Mathematics Vision Project et al. in partnership with the Utah State Office of Education under the CC BY-NC-SA 3.0 license. © 2014 Utah State Office of Education. Accessed June 8, 2017, 1:18 p.m..

Problem 3

Given the right square prism below:

  1. How is the volume affected when the height is doubles?
  2. How is the volume affected when the side of the square is doubled?

Guiding Questions

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Target Task


Problem 1

A right circular cylinder with a radius of 1 centimeter has been cut from the center of the right rectangular prism below. Find the volume. 

Problem 2

Two cylindrical containers have the same capacity. A designer wants to increase the volume so that each container could still hold the same amount. He doubles the height of one container and doubles the radius of the other. Will scaling each container as described result in the same volume? Why or why not? 

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.

  • Include problems where students need to find the area of a three-dimensional object, such as a ring, through subtraction.
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Lesson 12

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Lesson 14

Lesson Map

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

Topic A: Area and Circumference of Circles

Topic B: Three-Dimensional Concepts and General Volume

Topic C: Cavalieri's Principle, Spheres, and Composite Volume

Topic D: Surface Area, Scaling, and Modeling with Geometry

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