Curriculum / Math / 10th Grade / Unit 6: Three-Dimensional Measurement and Application / Lesson 8
Math
Unit 6
10th Grade
Lesson 8 of 18
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Lesson Notes
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Use the Pythagorean Theorem to find missing measurements and calculate volume of pyramids, prisms, and compound shapes comprised of pyramids and prisms.
The core standards covered in this lesson
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.
G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
The foundational standards covered in this lesson
8.G.B.7 — Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
Some of the Problem Set Guidance is from Grade 8 (8.G.7). However, these questions will help the students build procedural fluency with using the Pythagorean Theorem to calculate the volume of pyramids, prisms, and compound shapes through their conceptual understanding.Â
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
Below is a right rectangular prism. The length of $$\overline{AB}$$ is 4 inches and the length of $$\overline{BC}$$ is 3 inches. The length of $$\overline{MC}$$ is 8 inches. What is the volume of the right rectangular prism?
Robin made a model of a building. The model is composed of a right square pyramid glued exactly onto a right rectangular prism. The model and some of its dimensions are shown in the diagram below.
Release of November 2013 MCAS Retest Items is made available by the Massachusetts Department of Elementary and Secondary Education. © 2017 Commonwealth of Massachusetts. Accessed Sept. 20, 2018, 2:38 p.m..
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
Below is a right circular cylinder with points $$B$$ and $$A$$ that lie on the edge of the cylinder. The length of $$\overline{AB}$$ is 9 centimeters. What is the volume of the cylinder?
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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Describe the cross-sections of prisms and cylinders and make conjectures about volume from the cross-sections.
Topic A: Area and Circumference of Circles
Describe and use the formulas for area and circumference of circles to solve problems.
Standards
A.SSE.A.1G.GMD.A.1N.Q.A.3
Calculate and justify composite area and circumference of circles.
N.Q.A.1N.Q.A.2N.Q.A.3
Solve multistep area and circumference of circles problems involving cost and other rates.
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Topic B: Three-Dimensional Concepts and General Volume
Describe the terms point, line, and plane. Define and classify polyhedrons, specifically prisms and pyramids.
G.CO.A.1
Define a general cylinder and general cone. Identify two-dimensional shapes that when revolved will form a cylinder.
G.CO.A.1G.GMD.B.4
Use volume concepts and formulas to analyze and solve multistep problems with cylinders and prisms.
G.GMD.A.1G.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.GMD.A.3G.GMD.B.4G.SRT.C.8
Topic C: Cavalieri's Principle, Spheres, and Composite Volume
G.GMD.A.3G.GMD.B.4
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.1G.GMD.A.2
Identify cross-sections of pyramids and use the relationships between the cross-sections to determine the volume of truncated cones and pyramids.
G.GMD.A.2G.GMD.A.3
Calculate the volume of a sphere and use this in the solution of problems.
G.GMD.A.1G.GMD.A.2G.GMD.A.3N.Q.A.3
Calculate the volume of compound objects and those with subtracted solids. Determine how the volume will be affected by scaling one or more dimensions.
G.GMD.A.3G.GMD.B.4N.Q.A.3
Topic D: Surface Area, Scaling, and Modeling with Geometry
Use lateral surface area formulas to solve problems.
N.Q.A.2
Use the surface area and volume to solve application problems.
G.GMD.A.3G.GMD.B.4G.MG.A.1G.MG.A.3
Solve multistep volume and surface area problems with rates and unit conversions.
G.GMD.A.3N.Q.A.2N.Q.A.3
Apply density concepts to surface area and volume problems.
G.GMD.A.3G.MG.A.2N.Q.A.2N.Q.A.3
Apply constraints on volume, surface area, or cost to solve design problems with three-dimensional figures.
G.GMD.A.3G.MG.A.2G.MG.A.3N.Q.A.2N.Q.A.3
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