• MA.5.GR.3 Geometric Reasoning

    Solve problems involving the volume of right rectangular prisms. 



    Find the volume of a right rectangular prism with whole-number side lengths using a visual model and a formula.

    Benchmark Clarifications:
    Clarification 1: Instruction includes finding the volume of right rectangular prisms by packing the figure with unit cubes, using a visual model or applying a multiplication formula.
    Clarification 2: Right rectangular prisms cannot exceed two-digit edge lengths and responses include the appropriate units in word form.


    Purpose and Instructional Strategies
    The purpose of this benchmark is for students to make connections between packing a right rectangular prism with unit cubes to determine its volume and developing and applying a multiplication formula to calculate it more efficiently. Students have developed experience with area since Grade 3 (MA.3.GR.2.2). For volume, side lengths are limited to whole numbers in Grade 5, and problems extend to fraction and decimal side lengths in Grade 6 (MA.6.GR.2.3).

     Instruction should make connections between the exploration expected of MA.5.GR.3.1 and what is happening mathematically when calculating volume (MTR.2.1).

     Instruction should begin by connecting the measurement of a right rectangular prism to the calculation of a rectangle’s area. The bottom layer of the prism is packed with a number of rows with a number of cubes in each, like area of a rectangle is calculated with unit squares. From there, the third dimension (height) of the prism is calculated by the number of layers stacked atop one another.

     Having students explore how volume is calculated helps students see the patterns and develop a multiplication formula that will help them make sense of the two most
    common volume formulas, 𝑉 = 𝐵 × ℎ (where B represents the area of the rectangular prism’s base) and 𝑉 = 𝑙 × 𝑤 × ℎ. If students understand conceptually what the formulas mean, they are more likely to use them effectively and efficiently (MTR.5.1).

     When students use a multiplication formula, it is important for them to see that it is a matter of choice which dimensions of rectangular prisms are named length, width and height. This will help students understand that when calculating the volume of a rectangular prism, the three dimensions are multiplied together and that the order of
    factors does not matter (commutative property of multiplication).

    Common Misconceptions or Errors

     Students may confuse the difference between b in the area formula 𝐴 = 𝑏 × ℎ and B in the volume formula 𝑉 = 𝐵 × ℎ. When building understanding of the volume formula for right rectangular prisms, teachers and students should include a visual model to justify their calculations.

     Students may make computational errors when calculating volume. Encourage them to estimate reasonable solutions before calculating and justify their solutions after.
    Instruction can also encourage students to find efficient ways to use the formula. For example, when calculating the volume of a rectangular prism using the formula 𝑉 =
    45 × 12 × 2, students may find calculating easier if they multiply 45 x 2 (90) first, instead of 45 x 12. During class discussions, teachers should encourage students to share their strategies so they can build efficiency.

    Instructional Tasks
    Instructional Task 1
    The Great Graham Cracker Company is looking for a new package design for next year’s boxes. The boxes must be a right rectangular prism and measure 144 cubic centimeters.
    Part A. What are three package designs the company could use? Draw models and write equations to show their volumes.
    Part B. Dr. Cruz, the company’s founder, wants the height of the package to be exactly 8 centimeters. What are two package designs that the company can use? Draw models and write equations to show their volumes.

    Instructional Items
    Instructional Item 1
    Which of the following equations can be used to calculate the volume of the rectangular prism below?


    a. 𝑉 = 96 × 15
    b. 𝑉 = 15 × 8 × 12
    c. 𝑉 = 15 × 20
    d. 𝑉 = 27 × 8
    e. 𝑉 = 23 × 12

    Instructional Item 2
    A bedroom shaped like a rectangular prism is 15 feet wide, 32 feet long and measures 10 feet from the floor to the ceiling. What is the volume of the room?
    a. 57 cubic ft.
    b. 150 cubic ft.
    c. 4,500 cubic ft.
    d. 4,800 cubic ft.

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