# MA.5.GR.3.3

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.

### Examples

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?

### 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.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Composite Figure
• Rectangular Prism

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### 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 includes problems with the unknown side length being a fraction (MA.5.FR.1.1).
• For example, if a box has a base of 5 in × 3 in, and a volume of 20 in3 , 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 × 10 × 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 X 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 × 2 (which equals 90), instead of 45 × 12. During class discussions, teachers should encourage students to share their strategies so they can build efficiency.

### Strategies to Support Tiered Instruction

• Instruction includes the use of visual models to justify calculations when using the volume formula for right rectangular prisms.
• Instruction includes differentiating between base in the area formula, Area = b × h and base in the volume formula, Volume = B × h. Teacher provides students with models of two-dimensional figures, and three-dimensional figures, and has them identify which formula they will use and what the base in each image is. Students highlight the lines included in the base measurement for each figure and use the base to calculate the area or volume.
• For example, the teacher provides students with a set of models like the one shown below. The teacher asks students which image they would use the area formula for and which image they would use the volume formula for. Students then highlight the measurements used for the base in the formula. For the first figure, students would use volume and the formula B × h with B = 16 × 4. For the second figure, students would find area and use the formula b × h with b = 16.
• Instruction includes providing models of two-dimensional and three-dimensional figures with the area and volume formula labeled and color-coded with the measurements.
• For example, the teacher provides students with the following set of models and has students explain the difference in the base measurement in each formula. Students calculate the area or volume of each figure using the formula.
• Instruction includes providing a graphic organizer that requires students to estimate the volume of real-world examples provided and then solve using any strategy they would like. Students then compare their strategy to the strategies used by other students.
• For example, the teacher provides students with a graphic organizer similar to the one shown below. Students use it to find the volume of the given example and then compare their strategy to others.
• Instruction includes estimating reasonable solutions before calculating and justifying 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 × 2 (which equals 90), instead of 45 × 12. During class discussions, teachers should encourage students to share their strategies so they can build efficiency
• Instruction includes providing worked examples of volume and having students determine which strategy is the better strategy to use and why.
• For example, the teacher provides students with the following image and two examples of how students solved for volume. Student A solved the area of the base first using the Distributive Property to help with the multiplication. Student B used the Associative Property of Multiplication and multiplied 20 × 5 first. Students discuss both strategies and explain which would be easier and why.

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 128 in3
• a. 2in × 4in
• b. 3in × 3in
• c. 1in × 8in
• d. 4in × 2in
• e. 56in × 56in

*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.3: Solve real-world problems involving the volume of right rectangular prisms with given whole-number edge lengths using a visual model or formula.

## Related Resources

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

## Formative Assessments

Using Additive Reasoning When Finding Volume:

Students are asked to find the volumes of solids composed of rectangular prisms.

Type: Formative Assessment

Determining Dimensions:

Students are given the total volume of a rectangular prism and constraints on one dimension and asked to provide dimensions that would fit the constraints.

Type: Formative Assessment

## Lesson Plans

Lunchbox Volume:

This lesson focuses on the application of volume knowledge.  Students will need to add the volumes of individual right rectangular prisms to find total volumes.

Type: Lesson Plan

Getting the Top Mini-Fridge not a Small Deal:

In this MEA, students will create a procedure to rank five mini-refrigerators to determine which one should be purchased for the school by the PTA based on size, type, features, energy usage, and cost.  In the process, students will solve real-world problems involving the multiplication of multi-digit numbers with decimals to the hundredths, including using money.  Students will also determine the volume of a rectangular prism using a formula.

Type: Lesson Plan

Building Pools:

I

In this Model Eliciting Activity, MEA, students will work in teams to determine a the most suitable pool for a home construction company to build. Students will need to calculate the volume of the pool, make decisions based on a table of data, and write a letter to the customer providing evidence for their decisions.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

Type: Lesson Plan

Storage for Storage:

In this MEA, 5th Grade students will work in teams to determine a procedure for ranking Storage Sheds for a construction company that is moving to a new facility to purchase. Students will need to calculate the square feet and cubic feet of space for the Storage Shed, make decisions based on a table of data, and write a letter to the client providing evidence for their decisions.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

Type: Lesson Plan

Formulating Volume:

Students will use situational stories to help them apply the formulas V = l × w × h and V = B × h  to find the volumes of right rectangular prisms with whole-number edge lengths.

Type: Lesson Plan

Computing Volume Progression 2:

Students are asked to find the volume of water in a tank that is 3/4 of the way full.

Computing Volume Progression 3:

Students are asked to find the height of a rectangular prism when given the length, width and volume.

Computing Volume Progression 4:

Students are asked to apply knowledge of volume of rectangular prisms to find the volume of an irregularly shaped object using the principle of displacement.

## Tutorial

Volume: Decomposing a Solid Figure Example:

This Khan Academy tutorial video illustrates finding the volume of an irregular figure made up of unit cubes by separating the figure into two rectangular prisms and finding the volume of each part.

Type: Tutorial

## STEM Lessons - Model Eliciting Activity

Building Pools:

I

In this Model Eliciting Activity, MEA, students will work in teams to determine a the most suitable pool for a home construction company to build. Students will need to calculate the volume of the pool, make decisions based on a table of data, and write a letter to the customer providing evidence for their decisions.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

Getting the Top Mini-Fridge not a Small Deal:

In this MEA, students will create a procedure to rank five mini-refrigerators to determine which one should be purchased for the school by the PTA based on size, type, features, energy usage, and cost.  In the process, students will solve real-world problems involving the multiplication of multi-digit numbers with decimals to the hundredths, including using money.  Students will also determine the volume of a rectangular prism using a formula.

Storage for Storage:

In this MEA, 5th Grade students will work in teams to determine a procedure for ranking Storage Sheds for a construction company that is moving to a new facility to purchase. Students will need to calculate the square feet and cubic feet of space for the Storage Shed, make decisions based on a table of data, and write a letter to the client providing evidence for their decisions.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

## MFAS Formative Assessments

Determining Dimensions:

Students are given the total volume of a rectangular prism and constraints on one dimension and asked to provide dimensions that would fit the constraints.

Using Additive Reasoning When Finding Volume:

Students are asked to find the volumes of solids composed of rectangular prisms.

## Student Resources

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

Computing Volume Progression 2:

Students are asked to find the volume of water in a tank that is 3/4 of the way full.

Computing Volume Progression 3:

Students are asked to find the height of a rectangular prism when given the length, width and volume.

Computing Volume Progression 4:

Students are asked to apply knowledge of volume of rectangular prisms to find the volume of an irregularly shaped object using the principle of displacement.

## Tutorial

Volume: Decomposing a Solid Figure Example:

This Khan Academy tutorial video illustrates finding the volume of an irregular figure made up of unit cubes by separating the figure into two rectangular prisms and finding the volume of each part.

Type: Tutorial

## Parent Resources

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

Computing Volume Progression 2:

Students are asked to find the volume of water in a tank that is 3/4 of the way full.