**Math Topics**- Common Core
- Initiatives
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**Learning Support**- Standardized Test Prep
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Select the cluster for resources on this page:

**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: 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 |

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: Estimating the Circumference of a Circle: Use with GMD-A.1.

LearnZillion:

- 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.

NCTM Illuminations:
Pi Filling, Archimedes Style: In this unit, students inscribe
and circumscribe regular polygons in and around the unit circle,
which is known to have an area of π. Students then consider the area
of the polygons, using either an applet or a graphing calculator.
The area of the *n*-gons approaches π as *n*
increases. Similarly, students consider the perimeter of inscribed
and circumscribed regular polygons in and around a circle with unit
diameter. This exploration also leads to an approximation of π.
Also see NCTM's
Computing Pi applet.

Wolfram Math World: Circle

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

McDougal Littell ClassZone, Geometry, 2011: Select resources in Ch. 12 Surface Area and Volume of Solids for interactive practice: Animations (Virtual manipulatives, which also include Cavalieri's Principle), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. Use with GMD-A.

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.

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 one of the Interactives from Annenberg Media's Learner.org. Learn about three-dimensional shapes, calculate surface area and volume, and discover some of the mathematical properties of shapes. Use the tutorials and take the interactive multiple choice test.

Khan Academy: 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.

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.

McDougal Littell ClassZone, Geometry, 2011:

- Ch. 1.7 Quiz: Find Perimeter, Circumference, and Area
- Ch. 12.1 Quiz: Explore Solids
- Ch. 12.4 Quiz: Volumes of Prisms and Cylinders
- Ch. 12.5 Quiz: Volumes of Pyramids and Cones
- Ch. 12.6 Quiz: Surface Area and Volume of Spheres

OpenEd: Volume of a sphere: 5 questions combine multiple choice and free response. Includes related review resources. Aligns with GMD-A.2.

**Constructed-response:**

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.

OpenEd:

- Circles, cylinders and cones: 6 questions focus on free response. Includes related review resources. Aligns with GMD-A.1.
- Volume problems: 5 free response questions. Includes related review resources. Aligns with GMD-A.3.

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.

Wisc-Online, Learning Objects: In each of the following, learners review properties of the solid and use formulas to find volume and surface area. Examples are provided. Practice exercises are short answer:

**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: Standards: High School: Geometry: https://www.map.mathshell.org/stds.php?standardid=1367

- 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.

LearnZillion: 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.

NCTM Illuminations:
Conic Section Explorer: Use this virtual manipulative to
"Explore the different conic sections and their graphs. Use the **
Cone View** to manipulate the cone and the plane creating the
cross section, and then observe how the **Graph View** changes."
Use with GMD-B.4.

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:

- 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.

HMH School Publishers: Solid Figures: In this interactive learners identify the solid, given its views from top, bottom, sides. Answers are selected in multiple choice questions.

Khan Academy: Practice questions with videos: Cross sections of 3D objects: Aligns with GMD-B.4.

**Constructed-response:**

OpenEd: 2D Cross Sections of 3D Objects: 5 questions that focus on free response and related review resources. Aligns with GMD-B.4.

**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: Standards: High School: Geometry: https://www.map.mathshell.org/stds.php?standardid=1367

- 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.

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