Students extend their understanding of circles, volume, and surface area into modeling situations, formula analysis, and deeper conceptual understandings.
In Unit 6, Three-Dimensional Measurement & Applications, students derive, describe, and use formulas for area and circumference of a circle, and volume and surface area of three-dimensional figures. In addition, students identify two-dimensional shapes that when spun around an axis will form a particular three-dimensional figure, identify cross-sections of three-dimensional figures, and analyze modeling situations.
In this unit, students build on their previous understanding of circles, volume, and surface area they developed throughout elementary and middle school to extend their reasoning into modeling situations, formula analysis, and deeper conceptual understandings. The primary foundational content students will need to have prior to beginning this unit are knowing the formulas for area and circumference of a circle from seventh grade; knowing the formulas for volume of a cone, cylinder, and sphere from eighth grade; and using Pythagorean Theorem from eighth grade and Unit 4.
The Unit begins with Topic A, Area and Circumference of Circles, where students refresh their understanding of area and circumference to solve problems. If students are proficient at these skills, the first three lessons may be skipped or combined. In Topic B, Three-Dimensional Concepts and General Volume, students build on their understanding of two-dimensional figures to develop an understanding of three-dimensional measurement and dimension through revolving two-dimensional figures around an axis, slices, and volume. Topic C, Cavalieri’s Principle, Spheres, and Composite Volume, explores Cavalieri’s principle with the purpose of comparing volumes of oblique and right figures and developing the underpinnings for the formula for the volume of a sphere, used in eighth grade Geometry. Topic C also challenges students to find the volume of nonstandard three-dimensional figures by adding or subtracting volumes of known figures. The unit concludes with Topic D, Surface Area, Scaling, and Modeling with Geometry, which focuses on modeling situations requiring the use of volume, surface area, density, rates, and unit conversions. Students are required to make general plans for the solution of problems, determine the measurements and formulas necessary to carry out these plans, evaluate the plans, and determine a final solution. The use of appropriate measurement is necessary for the estimates in this topic of the unit.
The material from this unit is foundational to applications in Algebra II and solids of revolutions through integration in Calculus.
This assessment accompanies Unit 6 and should be given on the suggested assessment day or after completing the unit.
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Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
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area of a circle | circumference of a circle |
radius | diameter |
level of precision | point (vertex) |
line (edge) | plane (face) |
polyhedron | right and oblique prism |
right and oblique pyramid | apex |
right and oblique cylinder | right and oblique cone |
revolution around an axis | volume formulas |
Pythagorean Theorem | cross-section |
slice | sphere |
Cavalieri's principle | truncated cone |
truncated pyramid | lateral surface area |
surface area | displacement |
dimenional analysis | density |
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A.SSE.A.1
N.Q.A.3
G.GMD.A.1
Describe and use the formulas for area and circumference of circles to solve problems.
N.Q.A.1
N.Q.A.2
N.Q.A.3
Calculate and justify composite area and circumference of circles.
N.Q.A.1
N.Q.A.2
N.Q.A.3
Solve multistep area and circumference of circles problems involving cost and other rates.
G.CO.A.1
Describe the terms point, line, and plane. Define and classify polyhedrons, specifically prisms and pyramids.
G.CO.A.1
G.GMD.B.4
Define a general cylinder and general cone. Identify two-dimensional shapes that when revolved will form a cylinder.
G.GMD.A.1
G.GMD.A.3
Use volume concepts and formulas to analyze and solve multistep problems with cylinders and prisms.
G.GMD.A.1
G.GMD.A.3
Define and calculate the volume of pyramids and cones. Describe the relationship between general cylinders and general cones with the same base area.
G.SRT.C.8
G.GMD.A.3
G.GMD.B.4
Use the Pythagorean Theorem to find missing measurements and calculate volume of pyramids, prisms, and compound shapes comprised of pyramids and prisms.
G.GMD.A.3
G.GMD.B.4
Describe the cross-sections of prisms and cylinders and make conjectures about volume from the cross-sections.
G.GMD.A.1
G.GMD.A.2
Describe Cavalieri’s principle relating equal area cross-sections and volume, and how this relates to the formulas for volume. Derive the volume of a sphere using Cavalieri’s principle.
G.GMD.A.2
G.GMD.A.3
Identify cross-sections of pyramids and use the relationships between the cross-sections to determine the volume of truncated cones and pyramids.
N.Q.A.3
G.GMD.A.1
G.GMD.A.2
G.GMD.A.3
Calculate the volume of a sphere and use this in the solution of problems.
N.Q.A.3
G.GMD.A.3
G.GMD.B.4
Calculate the volume of compound objects and those with subtracted solids. Determine how the volume will be affected by scaling one or more dimensions.
N.Q.A.2
Use lateral surface area formulas to solve problems.
G.GMD.A.3
G.GMD.B.4
G.MG.A.1
G.MG.A.3
Use the surface area and volume to solve application problems.
N.Q.A.2
N.Q.A.3
G.GMD.A.3
Solve multistep volume and surface area problems with rates and unit conversions.
N.Q.A.2
N.Q.A.3
G.GMD.A.3
G.MG.A.2
Apply density concepts to surface area and volume problems.
N.Q.A.2
N.Q.A.3
G.GMD.A.3
G.MG.A.2
G.MG.A.3
Apply constraints on volume, surface area, or cost to solve design problems with three-dimensional figures.
Key: Major Cluster Supporting Cluster Additional Cluster
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