# MA.5.GR.3.1

Explore volume as an attribute of three-dimensional figures by packing them with unit cubes without gaps. Find the volume of a right rectangular prism with whole-number side lengths by counting unit cubes.

### Clarifications

Clarification 1: Instruction emphasizes the conceptual understanding that volume is an attribute that can be measured for a three-dimensional figure. The measurement unit for volume is the volume of a unit cube, which is a cube with edge length of 1 unit.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved

## Benchmark Instructional Guide

• NA

### Vertical Alignment

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### Purpose and Instructional Strategies

• This benchmark introduces volume to students. Their prior experiences with volume were restricted to liquid volume (also called capacity). The concept of volume should be extended from the understanding of area starting in Grade 3 (MA.3.GR.2.1), with the idea that a layer (such as the bottom of cube) can be built up by adding more layers of unit cubes. In Grade 6, (MA.6.GR.2.3) students solve volume problems involving rectangular prisms with fraction and decimal side lengths.
• As students develop their understanding of volume, they recognize that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. This cube has a length of 1 unit, a width of 1 unit and a height of 1 unit and is called a cubic unit. This cubic unit is written with an exponent of 3 (e.g., in3, 33). Students connect this notation to their understanding of powers of 10 in our place value system (MTR.5.1).

### Common Misconceptions or Errors

• Students may incorrectly fill figures to find volume with cubes. Students need to ensure there is no empty space included and that unit cubes are equally-sized and packed tightly in without overlaps.

### Strategies to Support Tiered Instruction

• Instruction includes providing unit cubes and having students build rectangular prisms with specific dimensions and then calculating the volume.
• For example, the teacher provides students with unit cubes and the following dimensions: length is 8 units, width is 4 units, and height is 5 units. Students stack equally sized unit cubes and ensure that the cubes are packed tightly with no gaps or overlaps to create a solid three-dimensional figure. Students begin building the figure as shown below, continuing to fill it in until complete. Students calculate the volume by multiplying 8 × 4 × 5 and then decompose the figure and count the cubes to determine if their calculation is correct.
• Instruction includes providing rectangular prisms filled with cubes. Some are filled correctly with no gaps or overlaps, and others have the cubes filling the rectangular prism, but with gaps left between them. Students identify which are stacked correctly to find volume and which are not stacked correctly and record the dimensions of the number of cubes for the height, length, and width, counting the total to determine the volume.

Instructional Task 1 (MTR.6.1

Molly is putting her cube-shaped blocks into their storage container after she finishes playing with her sister. The storage container is shaped like a right rectangular prism and she has a total of 120 blocks. The bottom layer of her storage container holds exactly 6 rows of 4 blocks each with no gaps or overlaps. The storage container holds exactly 6 layers of blocks with no gaps or overlaps.
• Part A. Will all of Molly’s blocks fit in the storage container? Explain how you know using drawings and equations.
• Part B. If there is enough room, determine how many more blocks Molly could fit in the storage container. If there is not enough room, determine how many blocks will not fit be able to fit in the storage container.

### Instructional Items

Instructional Item 1

What is the volume of the right rectangular prism?

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
5012070: Grade Five Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7712060: Access Mathematics Grade 5 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))
5012065: Grade 4 Accelerated Mathematics (Specifically in versions: 2019 - 2022, 2022 and beyond (current))
5012015: Foundational Skills in Mathematics 3-5 (Specifically in versions: 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.5.GR.3.AP.1: Explore volume as an attribute of three-dimensional figures that can be measured by packing them with unit cubes without gaps.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Volume in Cubic Units:

Students are asked to determine the volume of a rectangular prism in cubic units.

Type: Formative Assessment

Volume With Improvised Units:

Students determine how many improvised units of volume will fill a large box.

Type: Formative Assessment

How Do We Determine Volume?:

Students are asked to determine how a unit cube can be used to measure the volume of a rectangular prism.

Type: Formative Assessment

Determining Volume:

Students analyze a rectangular prism that contains one layer of unit cubes and are asked to explain how to determine the volume of the entire prism using only the information given.

Type: Formative Assessment

How Do You Find the Volume?:

Students discuss the volume of a prism measured in cubic units with gaps between the unit cubes used to measure its volume.

Type: Formative Assessment

## Lesson Plans

Building Apartments: Connecting Volume of Centimeter Cubes to the Formula V = B x h:

Students will build "apartments" with centimeter cubes by packing boxes (template included).  In addition, they will use centimeter cubes to build a variety of rectangular prisms and record the area of the base (B) and height (h) on a worksheet.  They will use that information to complete the volume formula, V = B x h.  Students will think about how the volume changes as the height and base of rectangular prisms change.

Type: Lesson Plan

Volume: It's All About the Count:

In this lesson, students will learn the concept of volume as an attribute of solid figures, using unit cubes in various arrangements with a focus on rectangular prisms.

Type: Lesson Plan

Manipulating Cubic Units:

Students will recognize volume as an attribute of solid figures and understand concepts of volume measurement. They will measure volumes by counting unit cubes, using cubic centimeters and cubic inches.

Type: Lesson Plan

Finding Volume (Utah Education Network):

In this lesson students will learn how to calculate and compare volumes of rectangular prisms.

Type: Lesson Plan

Who Wants To Carry a Million?:

"Students calculate the volume of a million dollars in 1\$ bills and the dimensions of a box large enough to hold a million dollars" (from the Beacon Lesson Plan Library).

Type: Lesson Plan

Building Rectangular Prisms Part 1:

This is the first part of a two-part volume lesson. In this lesson, students will build foundational concepts for volume and count cubes to find volume. In the second part lesson Building Rectangular Prisms Part 2 (attached), students will discover the volume formulas length x width x height and base x height as they build rectangular prisms. They will use the formulas to find volume in real world situations.

Type: Lesson Plan

## Original Student Tutorials

Building Blocks of Volume :

Build on your previous knowledge of area and learn how to calculate volume in cubic units with this interactive tutorial.

Type: Original Student Tutorial

Working for Wonka:

Demonstrate how a rectangular prism can be carefully filled without gaps or overlaps using the same size unit cubes and then use this model to determine its volume, in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Expert

B.E.S.T. Journey:

What roles do exploration, procedural reliability, automaticity, and procedural fluency play in a student's journey through the B.E.S.T. benchmarks? Dr. Lawrence Gray explains the path through the B.E.S.T. maththematics benchmarks in this Expert Perspectives video.

Type: Perspectives Video: Expert

Computing Volume Progression 1:

Students are asked to determine the number of unit cubes needed to construct cubes with given dimensions.

Surface Area and Volume:

In this activity, students adjust the dimensions of either a rectangular or triangular prism and the surface area and volume are calculated for those dimensions. Students can also switch into compute mode where they are given a prism with certain dimensions and they must compute the surface area and volume. The application keeps score so students can track their progress. This application allows students to explore the surface area and volume of rectangular and triangular prisms and how changing dimensions affect these measurements. This activity also includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

## Teaching Idea

Volume of Rectangular Prisms:

This lesson is designed to introduce students to the concept of volume and how to find the volume of rectangular prisms. This lesson provides links to discussions and activities related to volume as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

Type: Teaching Idea

## Tutorials

Volume: Four Examples of Counting Unit Cubes:

This Khan Academy tutorial video illustrates measuring volume by counting unit cubes.  Models in this video include composite figure of rectangular prisms although only counting is used to find volume, not an equation.

Type: Tutorial

Volume: How to Measure It:

This Khan Academy tutorial video describes measurement in one, two, and three dimensions.

Type: Tutorial

## Unit/Lesson Sequence

Three Dimensional Shapes:

In this interactive, self-guided unit on 3-dimensional shape, students (and teachers) explore 3-dimensional shapes, determine surface area and volume, derive Euler's formula, and investigate Platonic solids. Interactive quizzes and animations are included throughout, including a 15 question quiz for student completion.

Type: Unit/Lesson Sequence

## MFAS Formative Assessments

Determining Volume:

Students analyze a rectangular prism that contains one layer of unit cubes and are asked to explain how to determine the volume of the entire prism using only the information given.

How Do We Determine Volume?:

Students are asked to determine how a unit cube can be used to measure the volume of a rectangular prism.

How Do You Find the Volume?:

Students discuss the volume of a prism measured in cubic units with gaps between the unit cubes used to measure its volume.

Volume in Cubic Units:

Students are asked to determine the volume of a rectangular prism in cubic units.

Volume With Improvised Units:

Students determine how many improvised units of volume will fill a large box.

## Original Student Tutorials Mathematics - Grades K-5

Building Blocks of Volume :

Build on your previous knowledge of area and learn how to calculate volume in cubic units with this interactive tutorial.

Working for Wonka:

Demonstrate how a rectangular prism can be carefully filled without gaps or overlaps using the same size unit cubes and then use this model to determine its volume, in this interactive tutorial.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorials

Building Blocks of Volume :

Build on your previous knowledge of area and learn how to calculate volume in cubic units with this interactive tutorial.

Type: Original Student Tutorial

Working for Wonka:

Demonstrate how a rectangular prism can be carefully filled without gaps or overlaps using the same size unit cubes and then use this model to determine its volume, in this interactive tutorial.

Type: Original Student Tutorial

Computing Volume Progression 1:

Students are asked to determine the number of unit cubes needed to construct cubes with given dimensions.

## Tutorials

Volume: Four Examples of Counting Unit Cubes:

This Khan Academy tutorial video illustrates measuring volume by counting unit cubes.  Models in this video include composite figure of rectangular prisms although only counting is used to find volume, not an equation.

Type: Tutorial

Volume: How to Measure It:

This Khan Academy tutorial video describes measurement in one, two, and three dimensions.

Type: Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.