Virginia SOL 8.MG.2.c
The Standard
Examine and explain the relationship between the volume of cones and cylinders, and the volume of rectangular prisms and square based pyramids.
Virginia Standards of Learning for Mathematics · The student will investigate and determine the surface area of square-based pyramids and the volume of cones and square-based pyramids.
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students match each cone to a cylinder and each square pyramid to a prism with the same base area and perpendicular height. They determine that each pointed solid has one-third of its partner's volume and explain the comparison with models or formulas.
What Mastery Looks Like
- Given dimensions or base area and height, a student correctly calculates both volumes and states the 1:3 ratio. The student explains that the comparison applies when paired solids have equal base areas and perpendicular heights.
Common Misconceptions
- Students often use one-half instead of one-third, or multiply by three when finding the smaller solid. They may compare solids with equal heights but unequal base areas. Some use slant height instead of perpendicular height, or forget to square the cone's radius.
How to Assess It
- Exit ticket: A cone and cylinder each have base area 30 cm² and height 8 cm; a square pyramid and prism each have base area 49 cm² and height 6 cm. Find all four volumes and explain each volume ratio.
Lesson moves
Ways to Teach It
Use rice to fill matched cone-cylinder and square pyramid-prism models, then count how many fills each larger solid holds.
Have students explain why a cone or square pyramid holds one-third as much as its matching prism, using base area and height.
Play a card match with solid diagrams, dimensions, volume expressions, and ratio cards, requiring students to justify every set.
Compare a waffle cone with a cylindrical cup sharing its radius and height, then calculate how much ice cream each holds.
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Related Standards
- 8.MG.2.b
Determine the volume of cones and square-based pyramids, using concrete objects, diagrams, and formulas.
- 5.MG.2.d
Investigate and develop a formula for determining the volume of rectangular prisms using concrete objects.
- 8.MG.2
The student will investigate and determine the surface area of square-based pyramids and the volume of cones and square-based pyramids.
- 8.MG.2.d
Solve problems in context involving volume of cones and square-based pyramids and the surface area of square-based pyramids.
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