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**Standards**:

- GMD-A.1. 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*. - GMD-A.2. (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
- GMD-A.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

**Technology-enhanced investigations:**

Mathwords.com: Math Dictionary and *Study.com: Key vocabulary for this domain. Use with GMD-A and GMD-B:

circle | volume | pyramid | limit |

circumference of a circle | cylinder | sphere | solid |

area of a circle | cone | Cavalieri's principle | solid of revolution |

2- & 3- dimensions |
cross-section* | formula | axis of rotation |

Thinking Mathematics: videos on YouTube.

- Formula for the Lateral Area of a Cone
- Formula for the Volume of a Cone
- Formula for the Voume of a Sphere
- Formula for the Surface Area of a Sphere

GeoGebra:

- Perimeter, Area, Volume: interactive applets
- Cavalieri's Principle: interactive applet. Aligns with GMD-A.1, GMD-A.2.
- Cavalieri's Principle and the Volume of a Sphere: interactive applet. Aligns with GMD-A.1, GMD-A.2.

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

- Lesson set: Give arguments for formulas of circumference and area of a circle and the volume of a cylinder, pyramid, and cone: Four video lessons: Relate diameter and circumference, informally prove area of circle, calculate the volume of prisms & cylinders by applying the Cavalieri Principle, relate the volume of prisms/cylinders to pyramids/cones by applying the Cavalieri Principle. Aligns with GMD-A.1.
- Lesson set: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems: Four video lessons: Solve real-world problems involving cones, pyramids, cylinders, and spheres. Aligns with GMD-A.3.

Math Open Reference: Solid Geometry unit on cubes, pyramids, cylinders, cones, spheres, conic sections, prisms, and other polyhedra. Manipulatives are included.

National Library of Virtual Manipulatives: Platonic Solids, Verify Vertices, Faces, Edges of five Solids (Java required)

Wolfram Math World: Circle

Zachary Abel's Blog: Three-Cornered Things: Cavalieri's Principle: Proof for computing the surface area of a sphere with graphical illustrations. Use with GMD-A.2.

National Math and Science Initiative: Lesson: Using Linear Equations to Define Geometric Solids: Use this lesson for geometry within a unit on volume applications. Students write the equations of lines and graph linear equations that define bounded regions, calculate the areas and perimeters of the regions, revolve the planar regions about horizontal and vertical lines to create solids, and calculate the volumes of the resulting solids; they use technology (virtual manipulatives) to create and revolve planar figures about horizontal and vertical lines and model the conceptual understanding using a real world situation. Targeted standards include GMD-A.3, GMD-B.4, GPE-B.7 and MG-A.1. Common Core Math Practice standards 1-7 are also addressed.

Shodor Interactivate: Function Revolution: "This 3-dimensional model allows users to explore functions and their revolution around an axis. Once one or two functions have been rotated around an axis, the applet calculates the surface area and volume for the resulting solid." Aligns with GMD-A.

VideoMathTutor.com video (Free complete lesson): Algebra: Formulas From Geometry. Multiple formulas and concepts are covered. Among those are circles (radius, diameter, circumference, area), prisms (volume), pyramids (volume), cones (slant height, lateral surface area, total surface area, volume), cylinders (lateral surface area, total surface area, volume), and spheres (surface area and volume). Note: Additional formulas address angles (complementary & supplementary), triangles (perimeter and area), isosceles triangles, equilateral triangles, right triangles, Pythagorean Theorem, similar triangles perimeter and area formulas for quadrilaterals (square, rectangle, parallelogram, trapezoid). Some diagonal formulas, too, where applicable are included.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 15 manipulatives addressing GMD-A.1, 5 manipulatives addressing GMD-A.2, and 6 manipulatives addressing GMD-A.3. Among those:

- Archimedes Approximation of Pi: Uses inscribed and circumscribed polygons for a unit circle. Aligns with GMD-A.1.
- Approximate Area and Circumference of a Circle Using Isosceles Triangles: Aligns with GMD-A.1.
- Area of a disk: Shows how area of a circle can be related to area of a triangle using a dissection of the circle. Aligns with GMD-A.1.
- Circumference of a circle: Rolling a circle. Aligns with GMD-A.1.
- Cylinder, hemisphere, cone: Aligns with GMD-A.1, GMD-A.3.
- Cavalieri's Principle: Aligns with GMD-A.1, GMD-A.2.
- Approximating the volume of sphere using cylindrical slices: Exact is the limit of approximations. Aligns with GMD-A.2, GMD-B.4
- Approximating volumes by summation: Exact is the limit of approximations. Aligns with GMD-A.2, GMD-B.4.
- Volume of a parallelopiped: Aligns with GMD-A.2, GMD-B.4.
- Two Proofs of the Equality of the Volumes of a Box and a Parallelepiped: Aligns with GMD-A.2.
- Square pyramid and its right triangles: Aligns with GMD-A.3.
- Ratio of surface area to volume for a cube and a sphere: Aligns with GMD-A.3.

**Multiple Choice:**

Geometry 3-D Shapes
is an **interactive** from Annenberg Learner intended for grades
6-8, although other grades can benefit. Students
can interactively learn about three-dimensional shapes, calculate surface area
and volume, and discover some of the mathematical properties of shapes. It comes
with a glossary, tutorials, and an interactive test.

Khan Academy: Volume and Surface Area unit. Practice questions with videos:

- Solid geometry word problems: Aligns with MG-A.1 and GMD-A.3.
- Density word problems: Aligns with MG-A.2 and GMD-A.3.

Math.com, Prisms, Pyramids, Cylinders, Cones, Spheres: Lessons and multiple choice matching problems:

MathsIsFun.com: Solid Geometry: Content explains properties of solids (platonic, prisms, pyramids, spheres, cones, cyliners) and formulas for their volumes and surface areas. Virtual manipulatives are included to view 3D via a spin or drag (e.g., platonic solids, cylinder, cone) and deconstruct some of those. Following each section are multiple choice problems for using formulas. Aligns with GMD-A.

**Constructed-response:**

AlgebraLab.org: Fill in the blank practice problems, including word problems. Be sure to show the related AlgebraLab documents for lessons and additional practice problems on each topic.

- Circumference and Areas of Circles
- Surface Area and Volume for prisms, cones, pyramids, cylinders, and spheres
- Areas and Perimeters of Triangles and Special Qualrilaterals (parallelograms, rectangles, and trapezoids).
- Areas and Perimeters of Regular Polygons

Khan Academy: HSG-G-GMD includes HSG-GMD.A.1 through HSG-GMD.A.3 Common Core aligned problems.

Also see the Volume and Surface Area unit with multiple choice and constructed response practice questions with videos.

MATHguide.com: Volume of Common Solids: Students learn how to calculate the volume of common solids to include definition of volume, prisms, cylinders, pyramids, cones, and spheres. Following each lesson, there is a quizmaster. Fill-in and check answers. After all lessons, there is a Volume Tour quiz on volumes for all figures. Note: Lesson ideas are provided for teachers who are interested in having students discover the volume formulas. Aligns with GMD-A.3. MATHguide also includes Surface Area of Common Solids.

Shodor Interactivate: Surface Area and Volume: A virtual manipulative allowing learners to rotate prisms accompanies this quiz on finding volumes and surface areas of rectangular prisms and triangular prisms.

**Performance tasks:**

Illustrative Mathematics: Geometry:

- Centerpiece: This task uses the cylinder volume formula and aligns with standard GMDA.3.
- Doctor's appointment: This task uses the cone volume formula and aligns with standard GMDA.3.

Mathematics Assessment Project: High School: Geometry:

- Task 425: Equations of Circles 2
- Task 213: Evaluating Statements About Enlargements (2D and 3D)
- Task 216: Calculating Volumes of Compound Objects

Mathematics Vision Project, Secondary 2 Student Edition:

- Module 7: Circles a Geometric Perspective: This module contains 10 classroom tasks. Standards GMD-A.1, GMD-A.2, and GMD-A.3 can be found within tasks 4-5 and task 10.

**Standards**:

- 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.

**Technology-enhanced investigations:**

Cabri: This is software to understand 3D geometry in the classroom. Use with GMD-B.4.

GeoGebra: 2D to 3D and Cross Section applets, including supporting videos. Aligns with GMD-B.4.

Imagine Learning Classroom (formerly LearnZillion) subscription needed: Lesson set: Identify the shapes of 2D cross-sections of 3D objects and identify 3D objects generated by rotations of 2D objects: Five video lessons: Visualize cross-sections of prisms, pyramids, cylinders, cones; predict 3D results of rotating simple figures. Aligns with GMD-B.4.

National Math and Science Initiative: Lesson: Using Linear Equations to Define Geometric Solids: Use this lesson for geometry within a unit on volume applications. Students write the equations of lines and graph linear equations that define bounded regions, calculate the areas and perimeters of the regions, revolve the planar regions about horizontal and vertical lines to create solids, and calculate the volumes of the resulting solids; they use technology (virtual manipulatives) to create and revolve planar figures about horizontal and vertical lines and model the conceptual understanding using a real world situation. Targeted standards include GMD-A.3, GMD-B.4, GPE-B.7 and MG-A.1. Common Core Math Practice standards 1-7 are also addressed.

WisWeb: Lathe: Learners can create 3D solids of revolution. They first make a line and then rotate it with this applet to see all views. Use with GMD-B.4.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 32 manipulatives addressing GMD-B.4. Among those:

- Slicing a solid of revolution made from part of a sine curve. Aligns with GMD-B.4.
- Plane sections of a cone: Aligns with GMD-B.4.
- Solid of revolution: See a solid formed by rotating region between two curves about the x-axis. Aligns with GMD-B.4.
- Solids of known cross section: Choose a base region and the shape of the cross section, which is perpendicular to the x axis. Aligns with GMD-B.4.
- Approximating the volume of sphere using cylindrical slices: Exact is the limit of approximations. Aligns with GMD-A.2, GMD-B.4
- Approximating volumes by summation: Exact is the limit of approximations. Aligns with GMD-A.2, GMD-B.4.
- Volume of a parallelopiped: Aligns with GMD-A.2, GMD-B.4.
- Cylinder net: Aligns with GMD-B.4.

**Multiple Choice:**

CK-12: Faces, Edges, and Vertices of Solids: Practice questions with review content and video. Aligns with GMD-B.4.

Khan Academy: HSG-G-GMD includes HSG-GMD.B.4 Common Core aligned problems.

Also see practice questions with videos: Cross sections of 3D objects also includes rotating 2D objects in 3D: Aligns with MG-A.3 and GMD-B.4.

Test your spatial reasoning with the following free tests:

- Spatial Reasoning Test from 123test.com
- Spatial Reasoning Test from assessmentday.com

**Constructed-response:**

**Performance tasks:**

Illustrative Mathematics: Geometry:

- Global Positioning System 1: Students work with equations of spheres. Task aligns with standard GMD-B.4 and algebra standard CED-A.2.
- Global Positioning System 2: "This problem examines different scenarios for intersections of spheres from the point of view of the GPS device." Task aligns with standards GMD-B.4 and MG-A.1.
- Tennis Balls in a Can Task aligns with standards GMD-B.4 and MG-A.1.

Mathematics Assessment Project: High School: Geometry:

- Task 428: Modeling: Rolling Cups
- Task 439: Representing 3D Objects in 2D

Mathematics Vision Project, Secondary 3 Student Edition:

- Module 5: Modeling with Geometry: This module contains 8 classroom tasks. Multiple standards are addressed. Standard GMD-B.4 is found in tasks 1-3.

Common Core Math:
Intro | HS Geometry Domain: CO | SRT | C | GPE | **GMD** | MG |