<p>The lesson encourages students to visualize solids as nets and to understand the concepts of surface area and volume. Students will:&nbsp;<br>- use drawings (nets), real models, and an applet to visualize solids in two-dimensional and three-dimensional forms.&nbsp;<br>- develop strategies for determining surface area.&nbsp;<br>- determine strategies for finding volume.&nbsp;<br>- discuss modeling and strategies.&nbsp;</p>

<p>- How may detecting repetition or regularity assist in solving problems more efficiently?&nbsp;<br>- How do spatial relationships, such as shape and dimension, help to create, construct, model, and represent real-world scenarios or solve problems?&nbsp;<br>- How may using geometric shape features help with mathematical reasoning and problem solving?&nbsp;<br>- How may geometric properties and theorems used to describe, model, and analyze problems?&nbsp;</p>

<p>- Congruent Figures: Figures that have the same size and shape. Congruent angles have the same measure; congruent segments have the same length.&nbsp;<br>- Cylinder: A solid that has two parallel, congruent bases (usually circular) connected with a curved side.&nbsp;<br>- Net: A two-dimensional shape that can be folded to create a three-dimensional figure.&nbsp;<br>- Prism: 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.&nbsp;<br>- Surface Area: The sum of the areas of all of the faces of a three-dimensional figure.&nbsp;<br>- Volume: The amount of space found within a solid.</p>

<p>- “data-show” projector connected to a computer&nbsp;<br>- NCTM’s applet, Geometric Solids, at <a href="http://illuminations.nctm.org/ActivityDetail.aspx?ID=70">http://illuminations.nctm.org/ActivityDetail.aspx?ID=70</a> &nbsp;<br>- boxes of various shapes and sizes (four for each pair of students)&nbsp;<br>- scissors&nbsp;<br>- crayons or colored pencils&nbsp;<br>- tape&nbsp;<br>- copies of the Life-Sized Net Worksheet (M-6-4-1_Life Sized Net Worksheet)&nbsp;<br>- copies of the Building Rectangular Prisms worksheet (M-6-4-1_Building Rectangular Prisms and KEY)&nbsp;<br>- copies of the Eleven Nets worksheet (M-6-4-1_Eleven Nets and KEY)&nbsp;<br>- unit cubes (transparent, if possible)</p>

<p>- Use the Life-Sized Net Worksheet to assess student progress.&nbsp;<br>- Use the Building Rectangular Prisms worksheet as a guide for instruction.&nbsp;<br>- Observations when students complete the Eleven Nets worksheet will help determine if they require additional learning or practice opportunities.&nbsp;<br>- The presentation activity can also be used to determine student comprehension.&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling and Formative Assessment&nbsp;<br>W: Introduce the use of nets to model rectangular prisms and cubes.&nbsp;<br>H: Review three-dimensional shapes and the related three-dimensional vocabulary. Show more three-dimensional shapes online. Ask students to rotate the shapes while counting the faces, edges, and vertices.&nbsp;<br>E: Give students boxes of different sizes to cut apart and make their own nets. Discuss options for determining the surface area and volume of their prism, using units rather than actual measurements.&nbsp;<br>R: Review lesson ideas by talking about real-world situations that include surface area and volume. Distribute the Eleven Nets worksheet so that students can discover the various nets for a cube.&nbsp;<br>E: Have students produce a presentation based on a prism model. Prepare to answer questions on its surface area and volume, as well as make comparisons between it and other similar prisms.&nbsp;<br>T: The Extension section allows you to personalize the lesson to the needs of your students. The Routine section includes strategies for reviewing lesson concepts throughout the school year. The Small Group section contains suggestions for students who could benefit from more teaching or practice. The Expansion section includes activity suggestions for students who are willing to go above and beyond the requirements of the standard.&nbsp;<br>O: This lesson aims to teach students how to calculate area and volume without formulas. By the end of the class, students will have a solid understanding of nets.&nbsp;</p>

<p>The presentation of materials and activities throughout the lesson provides opportunity to move from concrete representations to abstract thought. Students are not given formulas or answers, but are instead asked to discuss strategies with a partner. The use of various methods, tangible objects, studies, and discussions increases student participation. Verbal checks for understanding, formative assessments, and a discursive review gauges understanding. The presentation task links together the lesson's concepts.<br><br><strong>"When we study the properties of solid objects such as cubes and rectangular prisms, we frequently draw them on the board or on paper, we try to show a representation of a three-dimensional object in two dimensions. How is this possible, given that a blackboard or whiteboard is a flat surface and a cube is a solid object? Consider what you see when you look at a cube drawn in two dimensions. Why can you see it in three dimensions? Looking at a two-dimensional drawing as if it were a solid object can help you understand the concept of volume and surface area."</strong><br><br>This lesson will teach students how to use nets to model rectangular prisms and cubes. They will learn to describe the concept of surface area and volume in conceptual terms. They will also learn to describe the various strategies for calculating surface area and volume, as well as how to apply the strategies to the formulas. Throughout the lesson, you will examine their ability to model solids using drawings and concrete models.&nbsp;<br><br>Begin the lesson by demonstrating the following geometric solids (with manipulatives): cube, rectangular prism, triangular prism, hexagonal prism, square pyramid, triangular pyramid, and cylinder.<br><br><strong>"How do these shapes differ from the shapes you have worked with before?"</strong> (<i>They are three-dimensional.</i>) <strong>"What do we call the sides of each shape?"</strong>(<i>faces</i>)&nbsp;<br><br><strong>"How would we calculate the area of these shapes? The area of a solid is known as its </strong><i><strong>surface area</strong></i><strong>."</strong>&nbsp;<br><br>To further demonstrate three-dimensional solids and their faces, direct students to one of NCTM's applets, Geometric Solids, located at <a href="http://illuminations.nctm.org/ActivityDetail.aspx?ID=70">http://illuminations.nctm.org/ActivityDetail.aspx?ID=70</a>.<br><br>Use the "data-show" projector linked to a computer to give a brief demonstration of how to operate the applet. Allow students to explore further in groups using the group's computer. Ask students to rotate the shape and color and count the faces. Students' conceptual grasp of three-dimensional solids will improve as they use the applet. Allow students to play with all shapes while emphasizing that the majority of the shapes will be covered later in their math career. This lesson will focus on the cube and pyramid solids in the applet.&nbsp;<br><br><strong>"We have just analyzed a wide range of solids. Today's lesson on surface area and volume will focus on cubes and rectangular prisms. Let's get started.</strong><br><br><strong>Activity 1: Modeling with Nets</strong><br><br><strong>"We are going to create life-sized nets of some boxes."</strong>&nbsp;<br><br>Have a variety of boxes in different shapes and sizes (rectangular prisms, square prisms, and cubes). Examples include any food containers (e.g., boxes containing cereal, crackers, ice cream cones, popcorn, etc.), a set of various-sized gift boxes, a large-sized box, toy boxes, and so on.&nbsp;<br><br>Students should be divided into pairs and given scissors and four boxes each pair, with a variety of boxes provided to each pair.<br><br><strong>"We'll cut apart each box to make the net. The net is a two-dimensional shape that can be folded to form a three-dimensional solid. Each box is either a rectangular prism or a cube. What's the difference between them? Is the cube a prism?"</strong> (<i>Students should respond that both are prisms. A cube simply has six square faces, whereas a rectangular prism has six rectangular faces.</i>) <strong>"A prism is a three-dimensional solid made up of two congruent and parallel polygon faces. The remaining faces are rectangles. Prisms are named by their bases."</strong>&nbsp;<br><br>Use one of the boxes to demonstrate how to safely cut an edge. Remind students what "congruent" means.<br><br><strong>"Now, cut each box into six faces by cutting along just enough edges to make the box flat. This will build a life-sized net. Next, color each pair of congruent faces a different color. Measure and label the sides of each face in units."</strong><br><br>After students have done cutting apart each box and coloring each pair of related congruent faces, allow them to walk around the room and see all of the different nets.<br><br><strong>Activity 2: Understanding the Concept of Surface Area</strong><br><br>When students return to their seats, remind them of the opening discussion of the area of three-dimensional solids.&nbsp;<br><br><strong>"Do you recall us using the word </strong><i><strong>surface area</strong></i><strong> a moment ago to calculate the area of a three-dimensional solid? Each pair of students will identify a few different methods for calculating surface area. You will use the Life-Sized Net Worksheet </strong>(see M-6-4-1_Life Sized Net Worksheet in the Resources folder) <strong>to record your item, technique or formula, surface area, and any observations you make while working. Before we get started, we need to define surface area. Can somebody explain the definition of surface area?"</strong> Students should respond that surface area is the total area of the outside faces of the solid.<br><br><strong>"Let's discuss the strategies you found for finding surface area of rectangular prisms or cubes."</strong> Strategies include calculating the area of each face and adding the results together, or calculating the area of three faces and multiplying by 2.&nbsp;<br><br><strong>"Can we write a general formula for finding surface area of a rectangular prism?"</strong>&nbsp;<br><br>Encourage students to discuss this subject with a partner, using the Life-Sized Net Worksheet as a reference.<br><br>Students should respond that the area of a rectangle can be calculated by multiplying the length and width, <i>A = l × w</i>. So, to calculate the surface area of a rectangular prism, multiply the area of each face by 2 and sum the products. Instruct students to write <i>SA = 2lw + 2lh + 2wh</i>.<br><br><strong>"We know that one face would have dimensions </strong><i><strong>l</strong></i><strong> and </strong><i><strong>w</strong></i><strong>, another would have dimensions </strong><i><strong>l</strong></i><strong> and </strong><i><strong>h</strong></i><strong>, and yet another would have dimensions </strong><i><strong>w</strong></i><strong> and </strong><i><strong>h</strong></i><strong>. We know that each face has an opposite side with the same dimensions. So, double the area of each face by 2 and then sum the products."</strong>&nbsp;<br><br><strong>"Now, let's find a formula for finding the surface area of a cube."</strong> Students should respond that the area of a square can be calculated by multiplying one side by the other, <i>A = s × s</i> or <i>A = s²</i>. So, to get the surface area of a cube, multiply the area of one side by 6. Teach students to write <i>SA = 6 × s × s</i>, or <i>SA = 6s²</i>.<br><br><strong>"We know the area of each face is equal to </strong><i><strong>s²</strong></i><strong> so we can multiply the area of one face by 6, since a cube has six square faces."</strong><br><br><strong>Activity 3: Understanding the Concept of Volume</strong><br><br><strong>"The surface area indicates the whole area of the outside of the solid. The concept of volume provides us with varied information about the solid. What does it mean to investigate the volume of a solid? What exactly do we mean when we ask, 'What is the volume of this box?' These questions provide the basis for our current discussion."</strong> Students should respond that volume refers to the amount of space that an object occupies.&nbsp;<br><br><strong>"We are going to examine volume using a twofold procedure."</strong><br><br><strong>Activity 3 – Part 1</strong><br><br><strong>"First and foremost, we shall determine the number of unit cubes required to fill this empty box. We may have to provide an approximation. We will record our measurement using cubic units or units³."</strong> Choose any size box and demonstrate the process of filling it with rows and columns of unit cubes.<br><br><strong>Activity 3 – Part 2</strong><br><br><strong>"Second, we'll use a set of colored cubes to construct our own rectangular prism. We shall study the prism's dimensions and report on its volume.&nbsp;</strong><br><br><strong>"Let's first construct a cube with sides that are 3 units long. We'll make a cube similar to the one displayed."</strong> (Draw a cube with labeled dimensions on the board.)</p><figure class="image"><img style="aspect-ratio:193/118;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_9.png" width="193" height="118"></figure><p><strong>"We will use 3 rows of 3 cubes (9 cubes) to form the bottom of the cube."</strong> Model this procedure for the class by placing each cube on the projector. [Note: Transparent cubes work best here.]&nbsp;<br><br><strong>"Here's an example of how our partially-filled cube looks like now.</strong></p><figure class="image"><img style="aspect-ratio:222/127;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_10.png" width="222" height="127"></figure><p><strong>"It will take 3 layers of 9 cubes (27 cubes) to complete the cube."</strong> Model this procedure for the class.&nbsp;<br><br><strong>"Notice that the cube measures 3 units by 3 units by 3 units, and 3 • 3 • 3 = 27. This can also be expressed as 3³ = 27. Here's an illustration of our completed cube.&nbsp;</strong></p><figure class="image"><img style="aspect-ratio:222/224;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_11.png" width="222" height="224"></figure><p><strong>"How many square units made up our cube?"</strong> (<i>27</i>)<br><br><strong>"Yes, our volume is 27 units³. Now it's your turn to explore!</strong><br><br><strong>"You will use your cube set to build rectangular prisms of varied dimensions, as shown in the table on the Building Rectangular Prisms worksheet</strong> (M-6-4-1_Building Rectangular Prisms and KEY). <strong>After you've built each prism, fill in the table with the dimensions and volume of each solid. The purpose of this activity is to create a connection between the dimensions of the prism and the process or formula used to calculate the volume.</strong><br><br><strong>"Let's discuss the strategies you found for determining the volume of rectangular prisms or cubes."</strong> 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 calculating the product of the dimensions.&nbsp;<br><br><strong>"Can we write a formula for finding the volume of a rectangular prism?"&nbsp;</strong><br><br>Encourage students to discuss this question with a partner, using the table as a reference.<br><br>Students should repeat the preceding discussion and state that volume can be calculated by finding the product of the dimensions, <i>V = l × w × h</i> . Also, show them that <i>V = B • h</i> since we multiplied the number of cubes that covered the base (B symbolizes the base's area) by the number of layers or cube height.&nbsp;<br><br><strong>"We can see from the table that volume can be calculated by multiplying the dimensions together. You now have a conceptual grasp of volume and an algorithm for performing the calculation. This formula, along with the one for surface area, will be useful in the upcoming lesson.</strong><br><br><strong>"Let's look at another figure: a cylinder. The volume of a cylinder can be calculated using the formula </strong><i><strong>V = B • h</strong></i><strong>."</strong> Show students a cylinder, such as a soup can or oatmeal container. <strong>"What shape is the base? How can we calculate the area of the base?"</strong> (<i>The shape of the base is a circle. The formula to calculate the area of a circle is A = πr².</i>) <strong>“Let's determine the volume of a container with a radius of 3 inches and a height of 5 inches.”</strong></p><figure class="image"><img style="aspect-ratio:117/159;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_12.png" width="117" height="159"></figure><p>At this point, you may offer alternative figures (for example, a pentagonal prism) to help students understand that volume is the area of the base multiplied by the height.<br><br>Lead a discussion about surface area and volume. The discussion will focus on previous conceptions about the two measurement types and how the lesson has corrected those inaccurate perceptions. Students will engage in conversations related to jobs and/or real-world scenarios that involve surface area and volume problems on a daily basis. Finally, students will describe their perceptions and understanding of nets, any difficulties they encountered with visualization, and how they responded to or overcame those difficulties.&nbsp;<br><br>Distribute the Eleven Nets worksheet (M-6-4-1_Eleven Nets and KEY) to students to review nets and assess their ability to draw nets for a given solid. Students will be instructed to illustrate the 11 nets that form a cube by coloring the appropriate six faces on each blank grid.<br><br><strong>Presentation Activity</strong><br><br>Ask students to create a demonstration of the concepts of net, surface area, volume, and prisms. They may use technology resources such as PowerPoint, create display drawings, and/or make 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 can work individually or in groups of two or three. For each demonstration, students must respond:&nbsp;<br><br>1. What do you know about the object's volume or surface area?&nbsp;<br><br>2. How does the object's volume or surface area compare to another similar object?<br><br>Students will be required to incorporate real-world objects and discuss their features and measurements. Students will also be encouraged to compare real-world objects by asking questions like, "Which can hold more?" and "Which has a larger surface area?" Students can use animation, video, and other compelling material to enhance their demonstration.<br><br><strong>Extension:&nbsp;</strong><br><br>Use the following strategies to adapt the lesson to individual needs and interests:&nbsp;<br><br><strong>Routine:</strong> Throughout the school year, have students study the concept of nets by playing the following online game.<br><a href="http://www.harcourtschool.com/activity/mmath/mmath_dr_gee.html">http://www.harcourtschool.com/activity/mmath/mmath_dr_gee.html</a>&nbsp;<br><br><strong>Small Group:</strong> Students who could benefit from further teaching might find the following instructional video useful:&nbsp;<br><a href="http://www.onlinemathlearning.com/surface-area-prism.html">http://www.onlinemathlearning.com/surface-area-prism.html</a>&nbsp;<br><br><strong>Expansion:</strong> Ask students to draw a net for a cylinder and develop a strategy for calculating 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 calculating area and volume. To expand further, introduce students to the concept of the volume of pyramids.</p>