5.MD.5

Standard

Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.


Student Language:

"I can find the volume of an object using the formulas V = l x w x h and V = b x h."

Explanation

About the Math, Learning Targets, and Rigor

5.MD.5a-b

These standards involve finding the volume of right rectangular prisms. Students should have experiences to describe and reason about why the formula is true. Specifically, that they are covering the bottom of a right rectangular prism (length x width) with multiple layers (height). Therefore, the formula (length x width x height) is an extension of the formula for the area of a rectangle.

For B x h: Ask students what strategies can be used to determine the volume of the prism based on the number of cubes in the bottom layer. Expect responses such as “adding the same number of cubes in each layer as were on the bottom layer” or multiply the number of cubes in one layer times the number of layers.

When given 24 cubes, students make as many rectangular prisms as possible with a volume of 24 cubic units. Students build the prisms and record possible dimensions.

5.MD.5c

This standard calls for students to extend their work with the area of composite figures into the context of volume. Students should be given concrete experiences of breaking apart (decomposing) 3-dimensional figures into right rectangular prisms in order to find the volume of the entire 3-dimensional figure.

Example 1: Students determine the amount of concrete needed to build the steps in the diagram below.

Example 2: A homeowner is building a swimming pool and needs to calculate the amount of water needed to fill the pool. The design of the pool is shown in the illustration below.

Resources

Videos


EngageNY Lessons


Extra Practice


Released Assessment Questions

PARCC

Common Core State Standards

5.MD.5b

Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

b. Apply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

      • Tasks are with and without contexts.
      • 50% of tasks involve use of V = l × w × h and 50% of tasks involve use of V = B × h.
      • Tasks may require students to measure to find edge lengths to the nearest cm, mm or in.


5.MD.5c

Relate the operations of multiplication and addition and solve real world and mathematical problems involving volume.

c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

      • Tasks require students to solve a contextual problem by applying the indicated concepts and skills.


Performance Indicators: 5.MD.5b, 5.MD.5c

Level 5: Exceeds Expectations

Solves real-world and mathematical problems by applying the formulas for volume, relating volume to the operations of multiplication and addition, and recognizing volume is additive by finding the volume of solid figures of two or more non-overlapping parts.


Level 4: Meets Expectations

Given a visual model, solves realworld and mathematical problems by applying the formulas for volume, relating volume to the operations of multiplication and addition, and recognizing volume is additive by finding the volume of solid figures of two non-overlapping parts.


Critiquing/Reasoning Standards

5.C.1-3

Base explanations/reasoning on the properties of operations.

(Content Scope: Knowledge and skills articulated in 5.MD.5a)

    • Students need not use technical terms such as commutative, associative, distributive, or property.


5.C.6

Base explanations/reasoning on concrete referents such as diagrams (whether provided in the prompt or constructed by the student in her response).

(Content Scope: Knowledge and skills articulated in 5.MD.C)


5.C.8-2

Present solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equals signs appropriately (for example, rubrics award less than full credit for the presence of nonsense statements such as 1 + 4 = 5 + 7 = 12, even if the final answer is correct), or identify or describe errors in solutions to multi-step problems and present corrected solutions.

(Content Scope: Knowledge and skills articulated in 5.MD.5c)

    • Multi-step problems must have at least 3 steps.


Performance Indicators: 5.C.1-3

Level 5: Exceeds Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student constructs and communicates a well-organized and complete written response based on explanations/reasoning using the:

    • properties of operations
    • relationship between addition and subtraction
    • relationship between multiplication and division

Response may include:

    • a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
    • an efficient and logical progression of steps with appropriate justification
    • precision of calculation
    • correct use of grade-level vocabulary, symbols and labels
    • justification of a conclusion
    • evaluation of whether an argument or conclusion is generalizable
    • evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate). Provides a counter-example where applicable.

Level 4: Meets Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student constructs and communicates a well-organized and complete written response based on explanations/reasoning using the:

    • properties of operations
    • relationship between addition and subtraction
    • relationship between multiplication and division

Response may include:

    • a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
    • a logical progression of steps
    • precision of calculation
    • correct use of grade-level vocabulary, symbols and labels
    • justification of a conclusion
    • evaluation of whether an argument or conclusion is generalizable
    • evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate).


Performance Indicators: 5.C.6

Level 5: Exceeds Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response based on operations using concrete referents such as diagrams--including number lines (whether provided in the prompt or constructed by the student) and connecting the diagrams to a written (symbolic) method, which may include:

    • a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
    • an efficient and logical progression of steps with appropriate justification
    • precision of calculation
    • correct use of grade-level vocabulary, symbols and labels
    • justification of a conclusion
    • evaluation of whether an argument or conclusion is generalizable
    • evaluating, interpreting, and critiquing the validity of other’s responses, approaches, and reasoning, and providing a counterexample where applicable

Level 4: Meets Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response based on operations using concrete referents such as diagrams--including number lines (whether provided in the prompt or constructed by the student) and connecting the diagrams to a written (symbolic) method, which may include:

    • a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
    • a logical progression of steps
    • precision of calculation
    • correct use of grade-level vocabulary, symbols and labels
    • justification of a conclusion
    • evaluation of whether an argument or conclusion is generalizable
    • evaluating, interpreting, and critiquing the validity of other’s responses, approaches, and reasoning.


Performance Indicators: 5.C.8-2

Level 5: Exceeds Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response by:

    • analyzing and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately
    • evaluating explanation/reasoning if there is a flaw in the argument
    • presenting and defending corrected reasoning

Response may include:

    • a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
    • an efficient and logical progression of steps with appropriate justification
    • precision of calculation
    • correct use of grade-level vocabulary, symbols and labels
    • justification of a conclusion
    • evaluation of whether an argument or conclusion is generalizable
    • evaluating, interpreting and critiquing the validity of other’s
    • responses, approaches and reasoning, and providing a counter-example where applicable

Level 4: Meets Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response by:

    • analyzing and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately
    • distinguishing correct explanation/reasoning from that which is flawed
    • identifying and describing the flaw in reasoning or describing errors in solutions to multi-step problems
    • presenting corrected reasoning

Response may include:

    • a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
    • a logical progression of steps
    • precision of calculation
    • correct use of grade-level vocabulary, symbols and labels
    • justification of a conclusion
    • evaluation of whether an argument or conclusion is generalizable
    • evaluating, interpreting and critiquing the validity of other’s responses, approaches and reasoning