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Instructor Overview

Lesson Overview

Students build prisms with fractional side lengths by using unit-fraction cubes (i.e., cubes with side lengths that are unit fractions, such as 13 unit or 14 unit). Students verify that the volume formula for rectangular prisms, V = lwh or V = bh, applies to prisms with side lengths that are not whole numbers.

Key Concepts

In fifth grade, students found volumes of prisms with whole-number dimensions by finding the number of unit cubes that fit inside the prisms. They found that the total number of unit cubes required is the number of unit cubes in one layer (which is the same as the area of the base) times the number of layers (which is the same as the height). This idea was generalized as V = lwh, where l, w, and h are the length, width, and height of the prism, or as V = Bh, where B is the area of the base of the prism and h is the height.

Unit cubes in each layer = 3 × 4

Number of layers = 5

Total number of unit cubes = 3 × 4 × 5 = 60

Volume = 60 cubic units

In this lesson, students extend this idea to prisms with fractional side lengths. They build prisms using unit-fraction cubes. The volume is the number of unit-fraction cubes in the prism times the volume of each unit-fraction cube. Students show that this result is the same as the volume found by using the formula.

For example, you can build a 45-unit by 35-unit by 25-unit prism using 15-unit cubes. This requires 4 × 3  × 2, or 24, 15-unit cubes. Each 15-unit cube has a volume of 1125 cubic unit, so the total volume is 24125 cubic units. This is the same volume obtained by using the formula V = lwh:V=lwh=45×35×25=24125.

15-unit cubes in each layer = 3 × 4

Number of layers = 2

Total number of 15-unit cubes = 3 × 4 × 2 =  24

Volume = 24 × 1125 = 24125 cubic units

 

Goals and Learning Objectives

  • Verify that the volume formula for rectangular prisms, V = lwh or V = Bh, applies to prisms with side lengths that are not whole numbers.