• The Using Volume Formulas practice worksheet (M-5-1-1_Using Volume Formulas and KEY.docx) may be used to determine how well students apply the volume formulas.
• Use the Volume of Rectangular Prisms practice worksheet (M-5-1-1_Volume of Rectangular Prisms and KEY.docx) to determine how well students use the volume formulas to solve real-world problems.
• Use the Lesson 1 Exit Ticket (M-5-1-1_Lesson 1 Exit Ticket and KEY.docx) to get a quick assessment of students’ understanding of using both volume formulas for right-rectangular prisms.

Students should work in groups of two or three. Distribute at least one ruler with centimeter markings to each group. Also, distribute at least 100 cubic-centimeter blocks to each group. (The unit cubes in sets of base ten blocks are usually cubic centimeters.)

First, ask students to measure the dimensions of an individual cubic-centimeter block. “Using your ruler, measure the length, width, and height of one block. Measure to the nearest centimeter.” After students have had time to find these measures, ask them to report the measurements. Explain to students that these cubes represent cubic centimeters because the length, width, and height are all 1 centimeter. (Although grade 5 students learn about exponents, in this lesson, “cubic centimeters” is a better notation than cm³. It is important to refer to these as cubic centimeters or cubic cm to help students remember that volume can be thought of as how many cubes fit inside of a rectangular prism.)

Now distribute the Rectangular Prisms Nets A, B, C, D, and E (M-5-1-1_Rectangular Prism Nets.docx). Copy these nets onto cardstock or heavy paper. Nets F, G, H, and I will be used later in the lesson. Also, distribute scissors and tape for students to cut out the nets so they may fold and tape them to create rectangular prisms.

Ask students to cut out the nets. “Please cut out the nets. Begin by cutting out nets A, B, and C. Cut around the edge, not on each line shown. This is very important as the extra lines are fold lines to be used later.”

While students are cutting out the nets, distribute the Volume of Rectangular Prisms practice worksheet (M-5-1-1_Volume of Rectangular Prisms and KEY.docx). Write these directions for the activity on the board:

1. Fold and tape the net to create a rectangular prism.
2. Measure and record the length, width, and height of the prism. Measure to the nearest centimeter.
3. Fill the rectangular prism with cubic-centimeter blocks. Record how many cubic centimeters were needed to fill the prism (the volume).
4. Look for patterns in the numbers in the table on the worksheet.

When each group has cut out at least one net, use net A to demonstrate how to fold and tape the net to form a rectangular prism. (Remind students that each side is a rectangle, so this is called a rectangular prism.) “Please fold and tape net A. Now please measure the length, width, and height of rectangular prism A. Measure to the nearest centimeter.”

Ask students what the measurements were for prism A. Write these measurements on the board:

length = 8 cm                         width = 3 cm                           height = 1 cm

Often students struggle to identify which dimensions are the length, width, and height. Explain to students that often the length is considered the longest side, but this is not always the case. There is no right way to record these measurements. It is just important to measure all three dimensions. Since the rectangular prism has no top, most all students will agree that 1 cm is the height of the box when they position it with the open top upward.

Now show students to place cubic-centimeter blocks into prism A. Explain that the goal is to determine how many cubic-centimeter blocks are needed to fill prism A. It is not necessary to finish filling prism A during your demonstration. Instead, ask students to complete this task on their own, and record the count on the Volume of Rectangular Prisms sheet. Then students should continue working in their groups measuring the dimensions and filling each of the prisms B, C, D, and E.

When groups are finished with prisms A, B, C, D, and E, ask groups of students to volunteer to provide the dimensions and volume for each prism.  Ask students, “Who can explain a pattern you observed?” At least one group of students is likely to have identified that multiplying the dimensions length, width, and height results in the volume of the prism. If not, you may need to specifically ask students to examine the three dimensions and identify what operation on these three numbers would result in the volume of the prism.

Now write this pattern on the board and declare it to be a formula. Remind students that the formula summarizes the pattern so they do not need to place blocks into each prism to determine its volume.

Formula to find the Volume of a Rectangular Prism:

*Explain to students that  represents area of the base. In a rectangular prism, the shape of the base is a rectangle. The formula to find the area of a rectangle is length × width. Therefore,  is the same as .  Explain to students that the volume of any right prism can be found by multiplying the area of the base by the height, as long the correct formula is substituted for the area of the shape of the base. (For example, the volume formula for a right triangular prism is , because the base of a triangular prism is a triangle, and the formula to find the area of a triangle is .)

Distribute the Nets for Rectangular Prisms F, G, H, and I. Again, be sure students have tape and scissors. As students are cutting out the nets, write these directions on the board. Be sure all students understand the directions. Remind students that the last column is blank and should be used to compute the volume.

1. Measure and record the length, width, and height of the prism.
2. Compute the volume of the prism using the formula .
3. Fill the prism with cubic centimeter blocks to check your answer.

Monitor students as they are working. Observe whether or not students are filling the prism with cubic-centimeter blocks to verify their volume computations. Prism F requires 144 cubic centimeters to fill the prism. This was done intentionally so students may choose to start thinking of more efficient ways to determine how many blocks will be needed to fill the prism. Some groups may recognize that they need only form the bottom layer of blocks and then determine how many layers there would be. This is the formula  in action. This formula computes the volume of the prism by multiplying the number of blocks in one layer (or the area of the base B)  by the number of layers of blocks are needed to fill the prism (or the height of the prism h).

Once students are finished, ask them to share the measurements and volume for each prism and write them in the table on the board. Now ask students to return to prism F and determine how many cubic centimeters are needed to make one layer in the bottom of the prism.
(6 × 8 = 48) Then ask them how many layers of 48 blocks will be needed to fill the prism.(3)

Ask students, “For prisms G, H, and I, please determine how many cubic centimeters are in one layer and how many layers are needed to fill the prism.” Write this on the board, and ask students to record this information. (Answers are included in the table, but you shouldn’t write them on the board.)

Now ask students, “How can we use the number of cubic centimeters in each layer and the number of layers in the prism to compute the volume?” Students may suggest addition or multiplication. For example, students may suggest adding 48 + 48 + 48 for Prism F because there are 3 layers. Remind students that multiplying 48 and 3 does the repeated addition suggested here. Ask students to use the information in this table to compute the volume of each prism F, G, H, and I. Students may want to refer back to the Volume of Rectangular Prisms practice worksheet to determine if these are the same answers they found using the first volume formula.

The goal is to help students understand the two volume formulas and to realize that the second formula is a simplified version of the first. To do so, extend the table on the board as follows:

Help students understand that the number of blocks in one layer is the same as the area of the base and that the number of layers is the height of the prism. To do so, ask students “How could you use the measurements of the prism to quickly determine how many cubic
centimeter blocks will be needed to form one layer?” (Multiply the length by the width.) Then ask students, “What shape is the base of the prism?” (rectangle) Now ask “What do we compute by multiplying the length and width of a rectangle?” Students will likely remember that this computation yields the area of the rectangle.

Write this on the board:

Be sure to explain each statement. First, the formula for finding the volume of a rectangular prism is stated. Second, parentheses were inserted. Third, length multiplied by width is the area of the base so that substitution was made. Fourth, rewrite using variables instead of words.

Ask students to use the formula  to compute the volume of prisms A, B, C, D, and E. Again, they can write these computations in the last column. This gives them an opportunity to practice using this formula.

It will be difficult for students to understand why this second formula is needed, as the first formula seems sufficient. However, this second formula is very important as it can be used to find the volume of all right prisms and cylinders too. (It is also listed in the Grade 5 standards.) This second formula will be revisited in future grades when students compute the volume of other prisms, not limited to right-rectangular prisms.

Extension:

• Routine: As real-life situations arise during the school year, have students practice computing the volume of different containers. Students will also get additional practice with this skill in lesson 2 of this unit, when they find the volume of compound figures composed of multiple rectangular prisms.
• Small Groups: Students who need additional practice may by pulled into small groups to work on developing greater understanding of the volume formulas. Ask one student to build a rectangular prism using cubic-centimeter blocks. Using the dimensions of the prism, ask all students to compute the volume of the prism. Students should verify the computation by counting the cubic-centimeter blocks. Now, ask another student to build a rectangular prism and repeat the activity. When students tire of counting the individual blocks, encourage them to count the number of blocks in a layer and multiply by the number of layers. Help them understand the formulas are efficient ways to count the number of cubic-centimeter blocks.
• Expansion: Students who are ready for a challenge beyond the requirements of the standard can draw the rectangular prisms on isometric dot paper to create three-dimensional drawings, being sure to accurately label the dimensions. Also, students who are ready can estimate the volume of other three-dimensional figures. Many different food containers can be brought from home. Students can estimate the volume of these containers by filling them with centimeter cubes. Students can design ways to estimate volume because cubes do not always fit nicely into some three-dimensional figures.