Explain volume formulas and use them to solve problems
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Arguments
Circle Formulas
Methods of Argument
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5 - Argue (informally for the formulas for circumfer-ence and area of a circle)
5 - Argue (informally for the volume formulas for cylinders, pyramids and cones.)
5 - Argue (dissection arguments, Cavalieri’s principle and informal limit arguments)
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5 - I can give an informal argument for the circum-ference formula.
5 - I can give an informal argument for the formula for area of a circle.
5 - I can give an informal argument for the formula for volume of a cylinder.
5 - I can give an informal argument for the formula for volume of a pyramid.
5 - I can give an informal argument for the formula for volume of a cone.
5 - I can use dissection arguments, Cavalieri’s principle, and informal limit arguments with regard to geometric formulas.
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A. Geometric formulas can be justified through informal arguments by a variety of methods.
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A.1 What are dissection arguments?
A.2 What is Cavalieri’s principle?
A.3 What are informal limit arguments?
A.4 How can geometric formulas be justified through informal arguments?
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(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Argument
Volume Formulas
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5 - Argue (informally for the formula for volume of a sphere using Cavalieri’s principle)
5 - Argue (informally for the formulas of solid figures using Cavalieri’s principle)
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5 - I can give an informal argument for the formula for volume of a sphere using Cavalieri’s principle.
5 - I can give an informal argument for the formulas of solid figures using Cavalieri’s principle.
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A. Formulas for volumes of spheres and other solids can be justified informally with the use of Cavalieri’s principle.
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A.1 How can formulas for volumes of spheres and other solids be justified informally with the use of Cavalieri’s principle?
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Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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3 - Use (volume formulas for cylinders, pyramids, cones, and spheres)
3 - Solve (problems)
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3 - I can use volume formula for a cylinder to solve problems.
3 - I can use volume formula for a pyramid to solve problems.
3 - I can use volume formula for a sphere to solve problems.
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A. Volume formulas for cylinders, pyramids, cones, and spheres can be used to solve a variety of real-world problems.
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A.1 How can volume formulas for cylinders, pyramids, cones, and spheres be used in real-world problem solving?
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Visualize relationships between two-dimensional and three-dimensional objects
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Geometry
Cross sections
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5 - Identify (Shapes of two-dimensional cross sections of three-dimensional objects)
5 - Identify (Three dimensional objects generated by rotations of two-dimensional objects)
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1 - - I can identify shapes of two-dimensional cross sections of three-dimensional objects.
1 - - I can identify three dimensional objects generated by rotations of two-dimensional objects.
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A. The shape of a two-dimensional cross section can be identified from the source of a three-dimensional object.
B. A two-dimensional shape can be rotated to generate a three-dimensional object.
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A.1 What is a cross-section?
A.2 How can the shape of a two-dimensional cross section obtained from a three-dimensional object be identified?
B.1 How can a two-dimensional shape be rotated to generate a three-dimensional object?
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