• MA.5.GR.3 Geometric Reasoning

    Solve problems involving the volume of right rectangular prisms. 



    Solve real-world problems involving the volume of right rectangular prisms, including problems with an unknown edge length, with whole-number edge
    lengths using a visual model or a formula. Write an equation with a variable for the unknown to represent the problem.
    Example: A hydroponic box, which is a rectangular prism, is used to grow a garden in wastewater rather than soil. It has a base of 2 feet by 3 feet. If the volume of the box is 12 cubic feet, what would be the depth of the box?

    Benchmark Clarifications:
    Clarification 1: Instruction progresses from right rectangular prisms to composite figures composed of right rectangular prisms.
    Clarification 2: When finding the volume of composite figures composed of right rectangular prisms, recognize volume as additive by adding the volume of non-overlapping parts.
    Clarification 3: Responses include the appropriate units in word form.

    Purpose and Instructional Strategies
    The purpose of this benchmark is to solve real-world problems involving right rectangular prisms using a visual model or a formula. The real-world problems can require students to find an unknown side length or find the volume of a composite figure (MTR.7.1), if the figure can be decomposed into smaller right rectangular prisms. Students are expected to write an equation with a variable for the unknown to represent the problem. Similar expectations for area were developed in Grade 4 (MA.4.GR.2.1) and this work will be extended to include fraction and decimal side lengths in Grade 6 (MA.6.GR.2.3).

     Instruction of this benchmark can be combined with MA.5.GR.3.2 as students develop and apply understanding of calculating volume of right rectangular prisms using visual models and formulas (MTR.2.1).

     While finding volume, teachers should have students communicate and justify their decisions while solving problems (MTR.4.1).

     Instruction may include problems with the unknown side length being a fraction (MA.5.FR.1.1). For example, if a box has a base of 5in x 3in, and a volume of 20in3, what is the length of its missing side?

     During instruction teachers should allow students the flexibility to use different equations for the same problem. For example, to find the height of a rectangular prism with volume 120 and base dimensions 3 and 10, students can use the any of the follow equations: 120 = 3 x 10 x h or 120 = 30h or 120 ÷ 30 = h.

    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 use 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 first multiply 45 x 2 (90), 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 places packages of their graham crackers into a larger box for shipping to area grocery stores. Each package of graham crackers is a right rectangular prism that measures 18 cubic inches. The base of each package of graham crackers measures 2 inches by 3 inches. Packages are placed upright into the shipping box.

    Part A. If the larger shipping box is a cube with edges that are each 30 inches, how many layers of graham cracker packages can the shipping box hold? Show your
    thinking using a visual model and equation(s).
    Part B. Will the packages reach the top of the shipping box? If not, what will be the length of the gap from the top of the package to the top of the shipping box?
    Part C. How many graham cracker packages will fit in the shipping box?

    Instructional Items
    Instructional Item 1
    Select all of the following that could be the dimensions of the base of a rectangular box with height of 16in and volume of 128in3
    a. 2in x 4in
    b. 3in x 3in
    c. 1in x 8in
    d. 4in x 2in
    e. 56in x 56in 



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