• Use Life-Sized Net Worksheet to determine student progress.
• Use the Building Rectangular Prisms worksheet to help guide instruction.
• Observations while students complete the Eleven Nets worksheet will aid in deciding if they need more learning or practice opportunities.
• The presentation activity may also be used to assess student comprehension.

The presentation of materials and activities throughout the lesson affords opportunities for progressing from concrete representations to abstract thought. Students are not given formulas or answers but rather are required to discuss strategies with a partner. The inclusion of a variety of methods, tangible objects, investigations, and discussion fosters student engagement. Verbal checks for understanding, formative assessments, and a discursive review gauges understanding. The presentation task ties together the concepts of the lesson.

“When we look at the characteristics of solid objects such as cubes and rectangular prisms, we often draw them on the board or on paper, and we try to show a representation of a three-dimensional object in two dimensions. How is this possible, when a blackboard or a whiteboard is a flat surface and a cube is a solid object? Think about what you see when you look at a cube drawn in two dimensions. Why are you able to see it in three dimensions? The way you look at a two-dimensional drawing and see it as a solid object can help you think about the meaning of volume and surface area.”

In this lesson, students will become adept at using nets to model rectangular prisms and cubes. They will learn how to discuss the concept of surface area and volume in conceptual terms. They will also learn to describe the various strategies used to determine surface area and volume and relate the strategies to the formulas. Throughout the lesson, you will assess their successful modeling of solids with drawings and concrete models.

Start the lesson with a demonstration of the following geometric solids (from a set of manipulatives): cube, rectangular prism, triangular prism, hexagonal prism, square pyramid, triangular pyramid, and cylinder.

“How do these shapes differ from the shapes you have worked with before?” (They are three-dimensional.) “What do we call the sides of each shape?”(faces)

“How would we find the area of these shapes? With solids, the area is called surface area.”

To further demonstrate three-dimensional solids and their faces, introduce students to one of NCTM’s applets, Geometric Solids, at http://illuminations.nctm.org/ActivityDetail.aspx?ID=70.

Using the “data-show” projector connected to a computer, do a brief demonstration of how to manipulate the applet. Then allow students to explore further in groups on the group’s computer. Have students rotate the shape and color and count the faces. Experience with the applet will help students develop a conceptual understanding of three-dimensional solids. Allow students to experiment with all shapes, explaining that most of the shapes will be covered later on in their math career. Focus on the cube and pyramid solids within the applet for this lesson.

“We just examined a wide variety of solids. For the purpose of this lesson, which is surface area and volume, we are going to concentrate on cubes and rectangular prisms today. Let’s get started.”

Activity 1: Modeling with Nets

“We are going to create life-sized nets of some boxes.”

Have various boxes of different shapes and sizes available (rectangular prisms or square prisms, cubes). Examples can include any food containers (e.g., boxes containing cereal, crackers, ice cream cones, popcorn, etc.), a set of different-sized gift boxes, a large-sized box, boxes containing toys, etc.

Group students into pairs and give each pair scissors and four boxes, making certain to provide a variety of boxes to each pair.

“We are going to cut apart each box and create the box’s net. A net is the two-dimensional shape that can be folded to create the three-dimensional solid. Each box is either a rectangular prism or a cube. What is the difference between the two? Is a cube a prism?” (Students should answer that both are prisms. A cube simply has six square faces, whereas a rectangular prism has six rectangular faces.) “A prism is a three-dimensional solid that has two congruent and parallel faces that are polygons. The remaining faces are rectangles. Prisms are named by their bases.”

Using one of the boxes, demonstrate how to safely cut an edge. Remind students what congruent means.

“Now you may cut each box so that you get six faces by cutting along just enough edges to make the box flat. This will create your life-sized net. Next, color each pair of congruent faces a different color. Measure the sides of each face and label them in units.”

Once students have finished cutting apart each of the boxes and coloring each pair of corresponding congruent faces, provide time for students to walk around the room and view all of the different nets.

Activity 2: Understanding the Concept of Surface Area

Once students are back in their seats, remind them of the initial discussion of area of three-dimensional solids.

“Do you recall us using the term surface area a moment ago to find the area of a three-dimensional solid? Each pair of students will identify a couple of different strategies for determining surface area. You will use the Life-Sized Net Worksheet (see M-6-4-1_Life Sized Net Worksheet in the Resources folder) to record your item, strategy or formula, surface area, and any observations you make while working. Before getting started, we need an idea of what surface area means. Can anyone help us get an idea of the meaning of surface area?” Students should respond that surface area is the combined area of the outside faces of the solid.

“Let’s discuss the strategies you found for finding surface area of rectangular prisms or cubes.” Strategies include determining the area of each face and adding the areas together or determining the area of three faces and multiplying by 2.

“Can we write a general formula for finding surface area of a rectangular prism?”

Have students discuss this question with a partner, using the Life-Sized Net Worksheet as a reference.

Students should respond that the area of a rectangle can be found by multiplying length by width, . So, in order to find the surface area of a rectangular prism, you multiply the area of each face by 2 and add the products together. Lead students to write SA = 2lw + 2lh + 2wh.

“We know one face would have dimensions l and w, another would have dimensions l and h, and yet another would have dimensions w and h. We know that each face has an opposite face with the same dimensions. So, we need to multiply the area of each face by 2 and then sum the products.”

“Now, let’s find a formula for finding the surface area of a cube.” Students should respond that the area of a square can be found by multiplying one side by another side,  or . So, in order to find the surface area of a cube, you multiply the area of one side by 6. Lead students to write , or .

“We know the area of each face is equal to s² so we can multiply the area of one face by 6, since a cube has six square faces.”

Activity 3: Understanding the Concept of Volume

“Surface area tells us about the total area of the outside of the solid. The concept of volume gives us different information about the solid. What does it mean to examine the volume of a solid? If we ask, ‘What is the volume of this box?’ what are we really asking? These questions lay the foundation for our present discussion.” Students should respond that volume relates to the amount of space an object holds.

“We are going to examine volume using a twofold procedure.”

Activity 3—Part 1

“First of all, we will determine how many unit cubes it takes to fill up this empty box. We may have to provide an approximation. We will report our measurement using cubic units, or units³.” Choose any size of box and illustrate the process of filling up the box with rows and columns of unit cubes.

Activity 3—Part 2

“Second, we will use a set of colored cubes to build/create our own rectangular prism. We will examine the dimensions of the prism and report on the volume.

“Let’s first build a cube with sides of 3 units in length. We will build a cube like the one shown.” (Draw a cube, with labeled dimensions, on the board.)

“We will use 3 rows of 3 cubes (9 cubes) to form the bottom of the cube.” Model this process for the class, placing each cube on the projector. [Note: Transparent cubes work best here.]

“Here is an illustration of what our partially-filled cube looks like now.

“It will take 3 layers of 9 cubes (27 cubes) to complete the cube.” Model this process for the class.

“Notice the cube is a 3 unit by 3 unit by 3 unit cube, and . This can also be written as 3³ = 27. Here is an illustration of our completed cube.

“How many square units made up our cube?” (27)

“Yes, so our volume is 27 units³. Now, it’s time for you to explore!

“You will build rectangular prisms of varying dimensions, as listed in the table on the Building Rectangular Prisms worksheet (M-6-4-1_Building Rectangular Prisms and KEY.docx), using your set of cubes. After building each prism, you will fill in the table, recording the dimensions and volume for each solid. The goal of this activity is to make a connection between the dimensions of the prism and the procedure or formula used to find the volume.

“Let’s discuss the strategies you found for determining the volume of rectangular prisms or cubes.” Strategies include counting the number of cubes that fill the prism or cube; multiplying the number of cubes that cover the bottom (area of the base) by the number of layers of cubes; and finding the product of the dimensions.

“Can we write a formula for finding the volume of a rectangular prism?”

Have students discuss this question with a partner, using the table as a reference.

Students should reiterate the discussion above and state that volume can be found by finding the product of the dimensions, . Also, show students that  because we multiplied the number of cubes that covered the base (B represents the area of the base) by the number of layers or height of the cube.

“We can see from the table that volume can be found by multiplying together the dimensions. Now you have a conceptual understanding of volume and an algorithm for performing the calculation. This formula, along with the one for surface area, will prove helpful in the next lesson.

“Let’s take a look at another figure, a cylinder. The volume of a cylinder can be found using the same formula V = B · h.” Show students a cylinder such as a soup can or oatmeal container. “What is the shape of the base? How do we find the area of the base?” (The shape of the base is a circle. The formula to find the area of a circle is A = πr².) “Let’s determine the volume of a container with a radius of 3 inches and a height of 5 inches.”

At this point you could even introduce other figures (e.g., pentagonal prism) for students to get the idea that the volume is the area of the base times the height.

Lead a discussion about surface area and volume. The discussion will focus on prior misconceptions related to the two measurement types and on ways the lesson has remedied those faulty perceptions. Students will engage in conversation, related to jobs and/or real-world situations that involve surface area and volume problems on a daily basis. Finally, students will discuss their perceptions and understanding of nets, any difficulties they had with visualization, and ways they adapted to or overcame those difficulties.

To review nets once more and determine students’ abilities to draw nets for a given solid, hand out the Eleven Nets worksheet (M-6-4-1_Eleven Nets and KEY.docx). Students will be asked to illustrate the 11 nets that form a cube by coloring the appropriate six faces on each blank grid.

Presentation Activity

Ask students to prepare a demonstration of the concepts of net, surface area, volume, and prisms. They may use technology resources such as PowerPoint, construct drawings for display, and/or build models with readily available materials such as cardboard, and use objects they can collect from home such as empty boxes, cartons, or express delivery shipping containers. Students may work individually or in groups of two or three. For each demonstration, students must answer:

1. What can you tell about the object’s volume or surface area?

2. How does the object’s volume or surface area compare with a similar object?

Students will be required to incorporate real-world objects, discussing properties and measurements of the objects. Students will also be encouraged to compare real-world objects, prompting questions, such as, “Which can hold more?” and “Which has a larger surface area?” Students may use animation, video, and other engaging media forms to enhance their demonstration.

Extension:

Use the following strategies to tailor the lesson to individual needs and interests:

• Routine: Have students review the concept of nets periodically throughout the school year by playing the following online game:

http://www.harcourtschool.com/activity/mmath/mmath_dr_gee.html

• Small Group: Students who could benefit from additional instruction may find the following instructional video helpful:

http://www.onlinemathlearning.com/surface-area-prism.html

• Expansion: Have students draw a net for a cylinder and develop a strategy for determining the formulas for area and volume. Students could also use other three-dimensional figures such as pentagonal or hexagonal prisms, draw a net, and develop a strategy for determining the area and volume. For further expansion, introduce students to the concept of the volume of pyramids.