Author:
Chris Adcock
Subject:
Geometry
Material Type:
Lesson Plan
Level:
Middle School
Grade:
6
Provider:
Pearson
Tags:
6th Grade Mathematics, Cubes, Formulas, Rectangular Prisms, Volume
License:
Creative Commons Attribution Non-Commercial
Language:
English
Media Formats:
Interactive, Text/HTML

Education Standards

Volume Formula For Rectangular Prisms

Volume Formula For Rectangular Prisms

Overview

Lesson Overview

Students build prisms with fractional side lengths by using unit-fraction cubes (i.e., cubes with side lengths that are unit fractions, such as 13 unit or 14 unit). Students verify that the volume formula for rectangular prisms, V = lwh or V = bh, applies to prisms with side lengths that are not whole numbers.

Key Concepts

In fifth grade, students found volumes of prisms with whole-number dimensions by finding the number of unit cubes that fit inside the prisms. They found that the total number of unit cubes required is the number of unit cubes in one layer (which is the same as the area of the base) times the number of layers (which is the same as the height). This idea was generalized as V = lwh, where l, w, and h are the length, width, and height of the prism, or as V = Bh, where B is the area of the base of the prism and h is the height.

Unit cubes in each layer = 3 × 4

Number of layers = 5

Total number of unit cubes = 3 × 4 × 5 = 60

Volume = 60 cubic units

In this lesson, students extend this idea to prisms with fractional side lengths. They build prisms using unit-fraction cubes. The volume is the number of unit-fraction cubes in the prism times the volume of each unit-fraction cube. Students show that this result is the same as the volume found by using the formula.

For example, you can build a 45-unit by 35-unit by 25-unit prism using 15-unit cubes. This requires 4 × 3  × 2, or 24, 15-unit cubes. Each 15-unit cube has a volume of 1125 cubic unit, so the total volume is 24125 cubic units. This is the same volume obtained by using the formula V = lwh:V=lwh=45×35×25=24125.

15-unit cubes in each layer = 3 × 4

Number of layers = 2

Total number of 15-unit cubes = 3 × 4 × 2 =  24

Volume = 24 × 1125 = 24125 cubic units

 

Goals and Learning Objectives

  • Verify that the volume formula for rectangular prisms, V = lwh or V = Bh, applies to prisms with side lengths that are not whole numbers.

Volume of Cubes

Lesson Guide

Have students work in pairs to answer the questions, and then ask volunteers to share their answers and their reasoning.

Mathematics

Sixty-four smaller cubes fit inside the unit cube. To see this, students can visualize four 4-by-4 layers of small cubes fitting inside the unit cube. This is a total of 4 · 4 · 4, or 64 cubes.

Because the unit cube has a volume of 1 cubic unit, each 14-unit cube has a volume of 164 cubic unit.

[common error] Students may think that 4 of the small cubes fit inside the unit cube because the small cube is a 14-unit cube. Explain that the 14 unit refers to the edge length of the cube, not its volume.

Opening

Volume of Cubes

The cube at the left of the diagram has edge lengths of 1 unit. The cube at the right has edge lengths of 14 unit.

Volume is the amount of space inside a 3-D figure.

  • How many 14-unit cubes fit inside the 1-unit cube?
  • What is the volume of the 14-unit cube?

Math Mission

Lesson Guide

Discuss the Math Mission. Explain to students that they will be investigating whether the volume formula V = lwh works even if the dimensions of the prism are not whole numbers.

Opening

Build rectangular prisms with fractional edge lengths and find the volume of the prisms.

Find the Volume of Prisms

Lesson Guide

Introduce the Cube Builder interactive to students and make sure they know how to use it to make prisms with the unit-fraction cubes. Give students time to explore the Cube Builder interactive before they begin the problems.

Have students start the problems solo and then move into partner work after a few minutes.

For the first two problems, be sure students understand that they should first find the volume by reasoning (i.e., they should find the number of unit-fraction cubes in the prism and the volume of each unit-fraction cube). Then, they should apply the volume formula to see if they get the same result.

If students struggle to figure out how many 14-unit cubes to use in the first problem, suggest that they write all the dimensions as fractions with a denominator of 4. The dimensions are 34 unit by 54 unit by 24 unit, so the prism should be 3 cubes by 5 cubes by 2 cubes.

[common error] Some students may get the wrong answers when they use the formula because they do not remember how to multiply with fractions and mixed numbers. Remind students to write all the factors in fraction form and then simply multiply numerators and then multiply denominators.

ELL: When listening to students' responses, give students advance notice they will be presenting their work on a specific problem during the Ways of Thinking section. This will give them ample time to prepare a thoughtful response.

SWD: Struggling students may still need explicit instruction for multiplying fractions. Provide small group instruction to make sure all students can multiply fractions accurately to find the volume of a rectangular prism.

Interventions

Student has difficulty building the 114-unit by 34-unit by 12-unit prism.

  • Each cube has an edge length of 14 unit. So, to know how many cubes to use, you need to figure out how many 14-units are in each dimension.
  • Try writing the mixed numbers as fractions.
  • How many fourths are in 34? How many fourths are in 54? How many fourths are in 12?

Answers

  • There are 30 14-unit cubes in the prism.
  • V = 30 · 164 = 3064
    The volume of the prism is 3064 cubic unit, or 1532 cubic unit.
  • 54·34·24=3064

The volume is 3064 (or 1532) cubic unit, which is the same. Yes, the volume formula works for this prism.

Work Time

Find the Volume of Prisms

When a rectangular prism has whole-number edge dimensions, you can find its volume using the formula V = l ⋅ w ⋅ h, where l, w, and h represent the length, width, and height of the prism. Investigate whether this formula works even if the dimensions are not whole numbers.

Use the 14-unit cubes to build a prism with a length of 114 units, a width of 34units, and a height of 12 unit.

  • How many 14-unit cubes fit inside the prism?
  • What is the volume of the prism?
  • Does the volume formula
    V = l ⋅ w ⋅ h (or V = lwh) work for this prism?

INTERACTIVE: Cube Builder

Hint:

How many 14 -unit cubes fit inside a 1-unit cube?

The Formula

Lesson Guide

In this problem, students must recognize that the volume of a 13-unit cube is 127 cubic unit. Remind them of how they found the volume of a 14-unit cube in the Opening and suggest that they use similar reasoning for the 13-unit cube.

Interventions

Student gives the wrong volume for the prism made of 13-unit cubes.

  • Try using the Cube Builder interactive to build a unit cube from the 13-unit cubes.
  • How many cubes do you need?
  • If the unit cube has a volume of 1 cubic unit, what is the volume of each of the 13-unit cubes?

Answers

  • There are 40, 13-unit cubes in the prism.
  • V=13·13·13=127
    The volume of each 13-unit cube is 127 cubic unit.
  • V=40·127=4027=11327
    The volume of the prism is 11327 cubic units.
  • V=123·113·23=53·43·23=4027=11327
    The volume is 11327 cubic units, which is the same. Yes, the volume formula works for this prism.

Work Time

The Formula

Use the 13-unit cubes to build a prism with a length of 123 units, a width of 113 units, and a height of 23 unit.

  • How many 13-unit cubes fit inside the prism?
  • What is the volume of each 13-unit cube?
  • What is the volume of the prism?
  • Does the volume formula V = lwh work for this prism?

INTERACTIVE: Cube Builder

Hint:

How many 13-unit cubes fit inside a 1-unit cube?

Applying the Formula

Lesson Guide

[common error] Some students may get the wrong answers when they use the formula because they do not remember how to multiply with fractions and mixed numbers. Remind students to write all the factors in fraction form and then simply multiply numerators and then multiply denominators.

SWD:

Struggling students may still need explicit instruction for multiplying fractions. Provide small group instruction to make sure all students can multiply fractions accurately to find the volume of a rectangular prism.

Answers

  • V=453525=24125
    The volume is 24125 cubic unit.

Work Time

Applying the Formula

  • Use the volume formula to find the volume of a prism with a length of 45 unit, a width of 35 unit, and a height of 25 unit.

Using the Formula

Lesson Guide

Remind students to look at the dimensions of the paper prisms they made in the previous lesson—they will need those dimensions for this problem.

[common error] Some students may get the wrong answers when they use the formula because they do not remember how to multiply with fractions and mixed numbers. Remind students to write all the factors in fraction form and then simply multiply numerators and then multiply denominators.

Interventions

Student can't calculate the dimensions of the prisms.

  • Remember the dimensions of the paper you used to make the prisms, use those dimensions to help you find the dimensions of the prisms.
  • When you fold the paper to make the prism, each side of the prism is one forth of the paper.

Answers

  • Shorter prism: V=234·234·812=114·114·172=2,05732=64932
    The volume of the shorter prism is 64932in3.
    Taller prism: V=218·218·11=178·178·11=317964=494364
    The volume of the taller prism is 494364in3.
  • Answers will vary. The shorter prism has the greater volume.

Work Time

Using the Formula

  • Calculate the volume of each of the two prisms you made in Lesson 1.
  • Was your prediction about which prism has the greater volume correct?

Hint:

Use the formula to find the volume of the prisms.

Prepare a Presentation

Prepare For Ways of Thinking

Look for students who are reasoning correctly and incorrectly about building the prisms and computing the volumes. Select these students to share their work during Ways of Thinking.

Mathematical Practices

Mathematical Practice 1: Make sense of problems and persevere in solving them.

In the Challenge Problem, students need to determine how to use only what they see in the video to estimate the volume of the cube and then to compare the volume of the cube with the volume of the man's body. Look for students who approach this problem in interesting or unique ways. These students can share their methods in Ways of Thinking.

Interventions

Student has trouble starting the Challenge Problem.

  • To find the volume of a cube, what information do you need to know?
  • How can you estimate the length of each edge of the cube?
  • How does the man's height compare to the edge length of the cube?
  • What would you estimate is the average height of a man?

Challenge Problem

Answers

  • Answers will vary. Possible answer: I estimate the man is about 5 ft 8 in. tall. By comparing the edge length of the cube to the man's height, the edge length appears to be about 5 ft, so the volume is 5 · 5 · 5 = 125, or about 125 ft3.
  • Answers will vary. Possible answer: If I estimate that the man is a rectangular prism with dimensions 523 ft high by 112 ft wide by 23 ft thick, then his volume is 173·32·23=10218=523, or about 523 ft3. Therefore, his body would take up 523÷125=173÷125=173·1125=17375=0.0453, or about 5% of the cube.

Work Time

Prepare a Presentation

  • Explain how you found the volume of one of the cubes from the lesson.
  • Explain how you found the volume of one of the rectangular prisms from the lesson.
  • Support your explanations with diagrams.

Challenge Problem

This video shows an act in which a man does acrobatics inside of a cube.

  • Estimate the volume of the cube.
  • Estimate the percentage of the volume of the cube that is taken up by the man’s body.

VIDEO: Aerial Cube Act

Make Connections

Lesson Guide

Have students share their presentations. Work together as a class to correct any errors and misconceptions.

Have students who did the Challenge Problem share their work.

Mathematical Practices

Mathematical Practice 6: Attend to precision.

Ask students to explain, step by step, how they found their solutions. Encourage students to use clear, precise language.

Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.

Other students should critique the presenters' reasoning and make suggestions for how the solutions might be improved.

Mathematical Practice 4: Model with mathematics.

In the Challenge Problem, some students may have used one or more rectangular prisms to model the man and approximate his volume. Ask these students to describe how they reasoned about creating their models.

Performance Task

Ways of Thinking: Make Connections

  • Take notes about your classmates’ methods for finding the volume of a cube and a rectangular prism.

Hint:

As your classmates present, ask questions such as:

  • What problems did you encounter in finding the volume?
  • What knowledge of multiplying fractions did you use?
  • Did you make an estimate of the volume first? How did you make your estimate?
  • How did the Cube Builder help you?
  • How do you know your answer is correct?

Volume of Rectangular Prisms

A Possible Summary

You can build a prism with fractional side lengths using unit-fraction cubes. To find the volume of the prism, multiply the number of unit-fraction cubes times the volume of each cube. This gives you the same answer as you would get by using the volume formula V = lwh.

ELL: Give students plenty of wait time as some of them are learning and processing new mathematical information in a second language. Make a chart of the key points that students make during their presentations on volume of rectangular prisms.

Formative Assessment

Summary of the Math: Volume of Rectangular Prisms

  • Write a summary about how to build and find the volume of rectangular prisms.

Hint:

Check your summary.

  • Do you explain how to build a prism with cubes and find the volume of the prism?
  • Do you provide a formula for finding the volume of any rectangular prism?

Reflect On Your Work

Lesson Guide

Have each student write a brief reflection before the end of the class. Review the reflections to see what students like about using the volume formula.

Work Time

Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

I like using a formula to find volume because …