 # Common Core High School Geometry Teaching and Learning Resources

## Domain: GMD: Geometric Measurement and Dimension Select the cluster for resources on this page:

• GMD-A: Explain volume formulas and use them to solve problems

• GMD-B: Visualize relationships between two-dimensional and three-dimensional objects ### GMD-A: Explain volume formulas and use them to solve problems

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: Geometry Formulas Explained: This is a series of four YouTube videos explaining formulas for the lateral area of a cone, volume of a cone, volume of a sphere, and surface area of a sphere.  Use with GMD-A.1.

Cut the Knot: Applet (Java required): Estimating the Circumference of a Circle: Use with GMD-A.1.

GeoGebra:

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

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

Mathematicas Visuales: Cavalieri's Principle: The Volume of a Sphere: Java applet.  Use with GMD-A.2.

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:

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:

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.

Khan Academy: Volume and Surface Area unit includes 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.

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:

Mathematics Vision Project, Secondary 2 Student Edition:

### GMD-B: Visualize relationships between two-dimensional and three-dimensional objects

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 objects, solids of revolution with this Java virtual manipulative, and rotate them 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:

Multiple Choice:

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

Khan Academy: 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 tests:

Constructed-response:

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:

Mathematics Vision Project, Secondary 3 Student Edition: 