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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.
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Rectangular Prism
Vertical Alignment
Previous Benchmarks
Next Benchmarks
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 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 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 × 2 (90) first, 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 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.
- For example, the students highlight the lines included in the base measurement for each figure. Then, they use the base to calculate the area or volume. The teacher provides students with a set of models like the one shown below asking 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 visual 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.
- 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 finding 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 × 2 (which equals 90) first, 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.
Instructional Tasks
Instructional Task 1 (MTR.2.1)
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.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorial
Problem-Solving Tasks
Teaching Idea
Tutorial
STEM Lessons - Model Eliciting Activity
Students will review rectangular prisms and the formula for finding the volume of rectangular prisms. Once students have determined the volume of a number of rectangular prisms (cereal boxes), the students will use that information to help a fictitious company in determining which cereal box they should use for their new product.
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
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.
Students will first review rectangular prisms and the formula for finding the volume of rectangular prisms. After students have determined the volume of a given set of rectangular prisms (aquariums), the students will use that information to help Seymour Phish in determining which aquarium he should purchase for his minnows.
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
This Model Eliciting Activity (MEA) asks students to develop a procedure for choosing a reptile or amphibian to place in a school reception area. Students will need to consider safety, price of animal, cost by week to feed animal, size and cost of the enclosure, and the life span of the animals they are considering. In the second portion of the problem statement, the students will need to prepare a budget and cost analysis for the year to consider if they have still made the correct choices while adding three more animals for consideration. The culminating activity for this MEA will have the student write a proposal for the Principal to state their choice of animal, give a year's budget for cost and care for the animal.
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
MFAS Formative Assessments
Students are asked to determine the volumes of two right rectangular prisms given the dimensions of one and the base area and height of the other.
Students are asked to count unit cubes to determine the volume of a right rectangular prism.
Students are asked to determine the volume of a rectangular prism by using both a formula and counting cubic units.
Students are asked to compare different strategies for finding the volume of a rectangular prism.
Original Student Tutorials Mathematics - Grades K-5
Help solve the problem of shipping video games and accessories to customers by calculating the volume of the containers needed in this interactive tutorial.
Student Resources
Original Student Tutorial
Help solve the problem of shipping video games and accessories to customers by calculating the volume of the containers needed in this interactive tutorial.
Type: Original Student Tutorial
Problem-Solving Tasks
Students are asked to determine the number of unit cubes needed to construct cubes with given dimensions.
Type: Problem-Solving Task
This purpose of this task is to help students understand what happens when you scale the dimensions of a right rectangular solid. This task provides an opportunity to compare the relative volumes of boxes in order to calculate the mass of clay required to fill them. These relative volumes can be calculated geometrically, filling the larger box with smaller boxes, or arithmetically using the given dimensions.
Type: Problem-Solving Task
Tutorial
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
Problem-Solving Tasks
Students are asked to determine the number of unit cubes needed to construct cubes with given dimensions.
Type: Problem-Solving Task
This purpose of this task is to help students understand what happens when you scale the dimensions of a right rectangular solid. This task provides an opportunity to compare the relative volumes of boxes in order to calculate the mass of clay required to fill them. These relative volumes can be calculated geometrically, filling the larger box with smaller boxes, or arithmetically using the given dimensions.
Type: Problem-Solving Task
