Students study the basic properties of two-dimensional and three-dimensional space, noting how ideas shift between the dimensions. They learn that general cylinders are the parent category for prisms, circular cylinders, right cylinders, and oblique cylinders, and study why the cross-section of a cylinder is congruent to its base. Next students study the explicit definition of a cone and learn what distinguishes pyramids from general cones, and see how dilations explain why a cross-section taken parallel to the base of a cone is similar to the base. Students revisit the scaling principle as it applies to volume and then learn Cavalieri’s principle, which describes the relationship between cross-sections of two solids and their respective volumes. This knowledge is all applied to derive the volume formula for cones, and then extended to derive the volume formula for spheres. Module 3 is a natural place to see geometric concepts in modeling situations. Modeling-based problems are found throughout Topic B and include the modeling of real-world objects, the application of density, the occurrence of physical constraints, and issues regarding cost and profit.

Content Standard(s):

Mathematics MA2015 (2016) Grade: 9-12 Geometry

35 ) Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. [G-GMD1]

Alabama Alternate Achievement Standards

AAS Standard: M.G.AAS.HS.35- Make a prediction about the volume of a container, the area of a figure, or the perimeter of a figure. Ex: how many cubes will go in one figure vs. another. Limit to cylinder, circle.

Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis

16. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Unpacked Content

Evidence Of Student Attainment:

Students:

Given a circle,

Use repeated reasoning from multiple examples of the ratio of circle circumference to the diameter, to informally conjecture that the circumference of any circle is a little more than three times the diameter.

Divide the circle into an equal number of sectors, and rearrange the sectors to form a shape that is approaching a parallelogram.

Make conjectures about the area and perimeter of the new shape as the number of sectors becomes larger, and relate those conjectures to the original circle.

Given a cylinder, explain how a cylinder could be divided into an infinite number of circles, and the area of those circles multiplied by the height is the volume of the cylinder, and use Cavalieri's Principle to demonstrate that if two solids have the same height and the same cross-sectional area at every level, then they have the same volume.

Given a pyramid or cone, explain that the shapes could be divided into cross-sections, and the area of the cross-sections is decreasing as the cross-sections become further away from the base, and the area of an infinite number of cross-sections is the volume of a pyramid or cone.

Teacher Vocabulary:

Dissection arguments

Cavalieri's Principle

Cylinder

Pyramid

Cone

Ratio

Circumference

Parallelogram

Limits

Conjecture

Cross-section

Knowledge:

Students know:

Techniques to find the area and perimeter of parallelograms.

Techniques to find the area of circles or polygons.

Skills:

Students are able to:

Accurately decompose circles, cylinders, pyramids, and cones into other geometric shapes.

Explain and justify how the formulas for circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone may be created from the use of other geometric shapes.

Understanding:

Students understand that:

Geometric shapes may be decomposed into other shapes which may be useful in creating formulas.

Geometric shapes may be divided into an infinite number of smaller geometric shapes, and the combination of those shapes maintain the area and volume of the original shape.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: GEO.16.1: Define two-dimensional objects and three-dimensional objects.
GEO.16.2: Identify the two-dimensional figures that result from slicing three-dimensional figures as in plane section of right rectangular prisms and right rectangular pyramids.
GEO.16.3: Identify three-dimensional objects generated by rotations of two-dimensional objects (as in rotating a circle to create a sphere).
GEO.16.4: Distinguish between two-dimensional and three-dimensional objects.

Prior Knowledge Skills:

Define three-dimensional figures and nets.

Identify three-dimensional figures.

Select and create a three-dimensional figure using manipulatives.

Define two-dimensional figure, three-dimensional figure, and plane section.

List attributes of three-dimensional figures.

List attributes of two-dimensional figures.

Describe the relationship between two- and three-dimensional figures.

Recognize symmetry.

Alabama Alternate Achievement Standards

AAS Standard: M.G.AAS.10.16 Given a cross section of a three-dimensional object, identify the shapes of two-dimensional cross sections (limited to sphere, rectangular prism, or triangular prism).

Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis

17. Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed.

a. Give an informal argument for the formulas for the surface area and volume of a sphere, cylinder, pyramid, and cone using dissection arguments, Cavalieri's Principle, and informal limit arguments.

b. Apply geometric concepts to find missing dimensions to solve surface area or volume problems.

Unpacked Content

Evidence Of Student Attainment:

Students:

(17a) Given a sphere,

Explain how surface area is the total area for the surface of a sphere, and that if we could "unroll" the sphere and show it as a rectangle, the rectangle would have a width that is equivalent to the diameter of the sphere. Its length would be the same as the circumference of the sphere.

Explain how we could find the volume of spheres by using pyramids., understanding the radius of the sphere would be the height of the pyramid.

Given a cylinder, explain how a cylinder could be divided into an infinite number of circles, and the area of those circles multiplied by the height is the volume of the cylinder, and use Cavalieri's Principle to demonstrate that if two solids have the same height and the same cross-sectional area at every level, then they have the same volume.

Given a pyramid or cone, explain that the shapes could be divided into cross-sections, and the area of the cross-sections is decreasing as the cross-sections become further away from the base, and the area of an infinite number of cross-sections is the volume of a pyramid or cone.

(17b) Given a formula, explain how to solve for the missing linear dimension using opposite operations.

Teacher Vocabulary:

Dissection arguments

Principle

Cylinder

Pyramid

Cone

Ratio

Circumference

Parallelogram

Limits

Conjecture

Cross-section

Surface Area

Knowledge:

Students know:

Techniques to find the area and perimeter of parallelograms,
Techniques to find the area of circles or polygons

Skills:

Students are able to:

Accurately decompose circles, spheres, cylinders, pyramids, and cones into other geometric shapes.

Explain and justify how the formulas for surface area, and volume of a sphere, cylinder, pyramid, and cone may be created from the use of other geometric shapes.

Understanding:

Students understand that:

Geometric shapes may be decomposed into other shapes which may be useful in creating formulas.

Geometric shapes may be divided into an infinite number of smaller geometric shapes, and the combination of those shapes maintain the area and volume of the original shape.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: GEO.17.1: Define Cavalieri's principle, circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone; oblique, radius, diameter, height, and base.
GEO.17.2: Compare surface areas of similar figures and volumes of similar figures to determine a relationship using dissection arguments, Cavalieri's principle, and informal limit arguments.
GEO.17.3: Compare the characteristics and volume of oblique and right solids.
GEO.17.4: Describe the properties of a given object (cylinder, pyramid, prism, and cone).
GEO.17.5: Identify the necessary characteristics of a given solid to solve for its volume and surface area(cylinder, pyramid, prism, and cone).
GEO.17.6: Calculate the surface area of three-dimensional figures (cylinder, pyramid, prism, and cone).
GEO.17.7: Calculate the volume of a cylinder, pyramid, prism, and cone.
GEO.17.8: Calculate the area of a circle.
GEO.17.9: Calculate the circumference of a circle.
GEO.17.10: Calculate the area of the base shape.
GEO.17.11: Identify the relationship of geometric representations to real-life objects.
GEO.17.12: Identify the base shape.

Prior Knowledge Skills:

Define three-dimensional figures, surface area, and nets.

Identify three-dimensional figures.

Evaluate how to apply using surface area of a three-dimensional figure to solving real-world and mathematical problems.

Draw nets to find the surface area of a given three-dimensional figure.

Recall how to calculate the area of a rectangle and triangle.

Select and create a three-dimensional figure using manipulatives.

Define diameter, radius, circumference, area of a circle, and formula.

Identify and label parts of a circle.

Recognize the attributes of a circle.

Apply the formula of area and circumference to real-world mathematical situations.

Define formula, volume, cone, cylinders, spheres, and height.

Discuss the measure of volume and give examples.

Solve problems with exponents.

Recall how to find circumference of a circle.

Identify parts of a circle.

Calculate the volume of three-dimensional figures.

Solve real-world problems using the volume formulas for three-dimensional figures.

Alabama Alternate Achievement Standards

AAS Standard: M.G.AAS.10.17 Compare and contrast the volume of real-world geometric figures.