Volume of Compound Figures

Objectives

In this lesson, students will find the volume of compound figures by decomposing the figures into right-rectangular prisms. The lesson begins with a concrete representation of compound figures using cubic-centimeter blocks, and then moves to abstract pictures or sketches of compound figures. Students will:

compute the volumes of compound figures comprised of rectangular prisms.

Essential Questions

When is it appropriate to estimate versus calculate?
What makes a tool and/or strategy appropriate for a given task?
Why does what we measure influence how we measure?
In what ways are the mathematical attributes of objects or processes measured, calculated, and/or interpreted?
How precise do measurements and calculations need to be?

Vocabulary

Cubic Unit: A unit for measuring volume.
Customary System: A system of weights and measures frequently used in the United States. The basic unit of weight is the pound; the basic unit of capacity is the quart.
Measurement Unit: A specific quantity used as a standard of measurement.
Metric System: A system of measurements used throughout the world based on factors of 10. It includes measures of length, weight, and capacity.
Volume: The amount of space enclosed in a solid (3-dimensional) figure. Volume is measured in cubic units.

Duration

90–120 minutes

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Materials

cubic centimeter cubes
copies of Volume of Compound Figures practice worksheet (M-5-1-2_Volume of Compound Figures and KEY.docx)
copies of Compound Figures practice worksheet (M-5-1-2_Compound Figures Practice Worksheet and KEY.docx)
copies of the Lesson 2 Exit Ticket (M-5-1-2_Lesson 2 Exit Ticket and KEY.docx)

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

Volume of compound figures game:

http://www.sheppardsoftware.com/mathgames/geometry/shapeshoot/VolumeShapesShoot.htm

Formative Assessment

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- The Compound Figures practice worksheet may be used to determine how accurately students can calculate the volume of compound figures
- The Volume of Compound Figures practice worksheet may be used to gauge the level of student mastery.
- Use the Lesson 2 Exit Ticket (M-5-1-2_Lesson 2 Exit Ticket and KEY.docx) to quickly assess student understanding of computing the volume of a compound figure

Suggested Instructional Supports

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Scaffolding, Active Engagement, Modeling

W:	Students will learn to compute the volume of compound figures by decomposing the figure into multiple rectangular prisms. Students will also learn how to determine the dimensions of each rectangular prism after the decomposition. Students will also gain experience checking their answers for reasonableness.
H:	Hook students by showing a picture of a compound figure and asking them how to determine its volume. Students will likely suggest decomposing the figure into rectangular prisms.
E:	Engage students by asking them to build compound figures and then to decompose the figures to help compute their volumes. Ask students to work in pairs to complete the Volume of Compound Figures practice worksheet.
R:	Students will complete the Compound Figures practice worksheet in class or at home to review finding the volume of compound figures.
E:	Evaluate student understanding by observing while students complete the Compound Figures practice worksheet.
T:	Using suggestions in the Extension section, the lesson may be modified to meet the needs of students. Additional suggestions are provided both for students who may benefit from additional practice and those students who are ready for a challenge beyond the requirements of the standard. The Routine section gives suggestions for reviewing lesson concepts throughout the year.
O:	The focus of this lesson is to have students find the volume of compound figures by decomposing the figures into rectangular prisms. The lesson begins by having students build compound figures and compute their volume, a concrete representation. Students then use their understanding of the decomposition of these compound figures to move to the abstract, computing the volume of figures presented only as pictures.

Instructional Procedures

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Activity: Volume of Compound Figures

Ask students to work in groups of three to complete this activity. Also, distribute at least
150 cubic-centimeter cubes to each group of students.

Project the picture of the first compound figure from the Volume of Compound Figures worksheet. Explain to students that this is called a compound figure because it is made up of more than one rectangular prism. “Please use the cubic-centimeter blocks to build this compound figure.” When students are finished, ask them “What strategies can be used to find the volume of this compound figure?” Some students may suggest counting the number of cubes they used. This is a valid method. Others may suggest decomposing the compound figure into two rectangular prisms. This is the preferred method because it is more efficient. Engage students in using this method to find the volume of the compound figure.

Distribute the Volume of Compound Figures practice worksheet to each student (M-5-1-2_Volume of Compound Figures and KEY.docx). Now students can follow along with the steps, recording the dimensions and volume of each rectangular prism, and computing the total volume of the compound figure.

Instruct students to “Pull the compound figure apart to form two rectangular prisms.” Physically separate the figure into two rectangular prisms by pulling the blocks apart. Now ask students, “What are the dimensions of the two rectangular prisms?” Ask for volunteers, and record these measurements on the Volume of Compound Figures practice worksheet. Now ask students, “What is the volume of each rectangular prism?” Then give students time to compute the volume of each rectangular prism. Ask students to volunteer to write their answers on the board. Encourage them to show how they computed those values. Now ask, “What is the volume of the compound figure?” Students will likely suggest adding the volumes of the two rectangular prisms. Ask for a volunteer to give the sum.

Be sure to ask students which method they think is faster and more efficient—counting all the blocks or decomposing the figure into two rectangular prisms.

Now project the picture of the second compound figure from the Volumes of Compound Figures practice worksheet. Ask students to build the figure and compute its volume. Monitor students as they work. Be sure they have built the figure to the appropriate dimensions. Also, observe if they are able to separate the figure into three rectangular prisms to compute the volume. When students are finished working, ask for volunteers to show their work on the board.

Students should now transition to finding the volume without using blocks. Project the picture of the third compound figure. Ask students, “Where should we pull apart the figure to form two rectangular prisms? Can someone show us by drawing a line on the figure?” Likely a student will be able to draw the line, as shown on the KEY.

Ask students to work together to find the dimensions of the two rectangular prisms, and then use these prisms to find the volume of the compound figure. When students are finished, ask for volunteers to show their work on the board. (It is best if the compound figure is projected so students can use the figure to describe their thinking.)

Ask students to work together to find the volume of the fourth compound figure. Observe if they are able to separate the figure into three rectangular prisms to compute the volume. When students are finished working, ask for volunteers to show their work on the board.

Extension:
- Routine: Students may be given the Compound Figures practice worksheet (M-5-1-2_Compound Figures Practice Worksheet and KEY.docx) to review concepts from the lesson.
Students may also use the following Web-based game to practice calculating volumes of compound rectangular prisms:

http://www.sheppardsoftware.com/mathgames/geometry/shapeshoot/VolumeShapesShoot.htm
- Small Groups: Students who need additional practice may by pulled into small groups to work on creating compound figures with blocks. Ask one or two students to create a compound figure using blocks. Then support students in separating the figure into rectangular prisms and computing the volume of each. Transition to creating a compound figure and visually separating it. (Do not actually separate the figure into rectangular prisms.) Ask students to compute the volumes of each prism just by looking at the original figure. This prepares students to calculate volume in abstract settings, without the assistance of manipulatives.
- Expansion: Students who are ready for a challenge can draw nets on graph paper for the compound figures in this lesson’s activities. Color coding can be an option to show congruent faces. Students can cut out the nets and make three-dimensional models of these compound figures. Students can also create models of other compound figures of their choosing and then determine the volume.

Volume of Compound Figures

Lesson Plan