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

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

- 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.

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

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

Scaffolding, Active Engagement, and Modeling
W: Students will learn to calculate the volume of compound figures by decomposing them into rectangular prisms. Students will also learn how to calculate the dimensions of each rectangular prism following decomposition. Students will also acquire knowledge in evaluating their answers for reasonableness. 
H: Show a picture of a complex figure and ask students to determine its volume. Students will most likely suggest decomposing the figure into rectangular prisms. 
E: Encourage students to build and breakdown compound figures to calculate volumes. Ask students to work in pairs to complete the Volume of Compound Figures practice worksheet. 
R: Students will practice finding the volume of compound figures using the Compound Figures practice worksheet, either in class or at home. 
E: Evaluate student comprehension by observing them complete the Compound Figures practice worksheet. 
T: Modify the lesson based on student needs, as suggested in the Extension section. Additional suggestions are included for both students who could benefit from more practice and students who are ready for a challenge beyond the requirements of the standard. The Routine section includes strategies for reviewing lessons throughout the year. 
O: This lesson aims to teach students how to decompose compound figures into rectangular prisms and calculate their volume. The lesson begins with students constructing compound figures and calculating their volume, a concrete representation. Students then use their understanding of the decomposition of these compound figures to the abstract, calculating the volume of figures presented only as pictures.

Activity: Volume of Compound Figures

Ask students to do this activity in groups of three. Distribute at least 150 cubic centimeter cubes to each student group.

Project the image of the first compound figure from the Volume of Compound Figures worksheet. Explain to students that this is a compound figure because it consists of multiple rectangular prisms. "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 could recommend counting the amount of cubes they used. This is a valid way. Others might recommend dividing the composite figure into two rectangular prisms. This is the recommended strategy because to its higher efficiency. Encourage students to use this procedure to determine the volume of the compound figure.

Give each student a Volume of Compound Figures practice worksheet (M-5-1-2_Volume of Compound Figures with KEY). Students can now continue the instructions, recording the dimensions and volume of each rectangular prism and calculating the total volume of the compound figure.

Instruct students: "Pull the compound figure apart to form two rectangular prisms." Separate the figure into two rectangular prisms by dragging the blocks apart. Ask students: "What are the dimensions of the two rectangular prisms?" Ask volunteers and report their measurements on the Volume of Compound Figures practice worksheet. Ask students: "What is the volume of each rectangular prism?" Then allow them time to calculate the volume of each rectangular prism. Ask students to volunteer to write their answers on the whiteboard. Encourage them to demonstrate how they calculated those values. Ask the question, "What is the volume of the compound figure?" Students are likely to advise adding the volumes of the two rectangular prisms. Ask a volunteer to give the sum.

Make sure to ask students which option they believe is faster and more efficient: counting all the blocks or dividing the figure into two rectangular prisms.

Now, project the image of the second compound figure from the Volumes of Compound Figures Practice Worksheet. Ask students to construct the figure and calculate its volume. Keep an eye on students while they are working. Check that the figure is built to the correct dimensions. Also, see whether they can split the figure into three rectangular prisms to calculate the volume. When students have done working, ask for volunteers to display their work on the board.

Students should now switch to finding the volume without utilizing blocks. Project the image of the third compound figure. Ask students, "Where should we separate the figure to make two rectangular prisms? Can you show us by drawing a line on the figure?" It is likely that a student will be able to draw the line shown on the KEY. 

Students should work together to determine the dimensions of the two rectangular prisms, and then utilize these prisms to calculate the volume of the compound figure. When students have finished, ask for volunteers to present their work on the board. (It is best to project the compound figure so that students can utilize it to describe their thinking.)

Ask students to collaborate to determine the volume of the fourth compound figure. Determine whether they can divide the figure into three rectangular prisms to get the volume. When students have done working, ask for volunteers to display their work on the board.

Extension:

Routine: Students may be given the Compound Figures practice worksheet (M-5-1-2_Compound Figures Practice Worksheet with KEY) to review lesson contents. 
Students can also utilize the following Web-based game to learn calculating the volumes of compound rectangular prisms. 

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

Small Groups: Students who require further practice can be divided into small groups to concentrate on building compound figures with blocks. Ask one or two students to make a compound figure out of blocks. Then help students separate the figure into rectangular prisms and calculate the volume of each. Proceed to create a complex figure and visually separate it. (Do not actually cut the figure into rectangular prisms.) Ask students to calculate the volumes of each prism simply by glancing at the original figure. This teaches students to compute volume in abstract scenarios without the use of manipulatives. 

Expansion: Students who are ready for a challenge might draw nets on graph paper to represent the compound figures in this lesson's exercises. Color coding can be used to show congruent faces. Students can cut out the nets to create three-dimensional representations of the complex forms. Students can also build models of other compound figures of their choice and then calculate the volume.