<p>In this lesson, students will visualize and work with nets. Students will:&nbsp;<br>- create nets for a specific solid.<br>- draw/name a solid for a certain net.<br>- make predictions.&nbsp;<br>- apply spatial estimation and reasoning.&nbsp;<br>- develop strategies for calculating surface area and volume.&nbsp;<br>- investigate relationships between solids.</p>

<p>- What are the different solid properties and their relationships?</p>

<p>- <strong>Net:</strong> A two-dimensional representation of a three-dimensional figure composed of polygons from which, by folding along certain edges and joining others, a polyhedron can be constructed.<br>- <strong>Oblique Polyhedron:</strong> A polyhedron whose longitudinal axis is not perpendicular to its base.<br>- <strong>Oblique Prism:</strong> A prism whose longitudinal axis is not perpendicular base.<br>- <strong>Polyhedron:</strong> Solid with polygons for faces.<br>- <strong>Prism:</strong> A polyhedron with two congruent parallel bases and lateral faces that are parallelograms.<br>- <strong>Pyramid:</strong> A polyhedron with a polygonal base and three or more triangular lateral faces.<br>- <strong>Right Prism:</strong> A prism whose longitudinal axis is perpendicular to its base.<br>- <strong>Solid:</strong> A three-dimensional figure.<br>- <strong>Surface Area:</strong> The total area surrounding a three-dimensional figure.<br>- <strong>Volume:</strong> The capacity of a solid; the amount a solid can hold.</p>

<p>- copies of Nets Matching (M-G-3-2_Nets Matching and KEY)<br>- manipulates for shapes<br>- copies of Net 1 (M-G-3-2_Net 1)<br>- copies of Net 2 (M-G-3-2_Net 2)<br>- copies of Net 3 (M-G-3-2_Net 3)<br>- copies of Surface Area and Volume (M-G-3-2_Surface Area and Volume.doc)<br>- copies of Relationships Between Quantities and KEY (M-G-3-2_Relationships Between Quantities and KEY)</p>

<p>- Matching nets to their solid representations is a visual/spatial skill students can develop with practice. Using the Nets Matching activity sheet (M-G-3-2_Nets Matching and KEY), students can discover errors in their own representations. Take note of how students are looking at the nets, turning the pages in different directions, and using their hands to assist with visualization.&nbsp;<br>- Predictions, conjectures, and chart completion reveal some gaps in students' grasp of the relationships between solid objects and their two-dimensional representations. Asking leading question, such as "Which edge matches this side of the triangle?" can assist students with visualization.</p>

<p>Scaffolding, Active Engagement, Modeling, Explicit Instruction<br>W: Hands-on learning with manipulatives and small graphical representations. Scaffolding, active engagement, and modeling are all featured in this lesson. Students are not provided any formulas or algorithms. Instead, the lesson is intended to encourage students' growth of each topic. This allows students to make critical connections between and within solids. Students are challenged to draw relationships and compare solids for both types of measurement. When simply looking at two solids, which one looks to hold more? Support your argument. Classroom discussion, active participation in making predictions, visualizing and drawing nets, making comparisons, and arguing one's point, as well as the culminating PowerPoint presentation activity and review at the end of the activity, are all ways to evaluate and assess students' learning.<br>H: This lesson engages students by challenging them to visualize nets without prior exposure. Students must also develop ways for determining surface area and volume on their own, with no explicit procedures provided. Finally, students must connect and compare solids in various ways. The use of manipulatives, small pictorial representations, and connections to the real world keeps students interested and challenged.&nbsp;<br>E: The lesson is divided into two parts. Part 1 introduces nets and their relationship to three-dimensional objects. Part 2 discusses geographical estimation, connections, comparisons, and relationships. The incorporation of visual, auditory, and kinesthetic representations accommodates a wide range of learning methods.&nbsp;<br>R: The review summarizes the lesson with a discussion and exercise. Students must create connections between the topics covered in the course and apply these topics to the real world. Most importantly, students will be given skills to conceptualize surface area and volume, as well as net visualization, without having to memorize formulas or diagrams. Students should have access to a formula sheet for solids.&nbsp;<br>E: Classroom discussions and group activities encourage self-evaluation of lesson comprehension. Students will explain how accurately a net represents its corresponding three-dimensional object, as well as how well the three-dimensional object represents the net.&nbsp;<br>T: Using diverse representations and learning styles throughout the class allows for differentiated instruction. The group activity throughout the lesson offers the necessary support for students who are having difficulties visualizing and/or understanding the material.&nbsp;<br>O: The lesson begins with physical manipulatives, progresses to visualization and abstract thinking, promotes generalizations, and culminates with a discussion. The teacher models and facilitates all of the activities in order to develop conceptual and procedural understanding of nets, surface area, and volume in the context of visualization.</p>

<p><strong>Part 1: Making Connections Between Nets and Solids</strong></p><p>Show students a variety of solids (manipulatives), including the following:</p><p>Cone</p><p>Cube</p><p>Right Cylinder</p><p>Right Hexagonal Prism</p><p>Oblique Rectangular Prism (parallelogram Lateral Faces).</p><p>Right Rectangular Prism.</p><p>Square Pyramid</p><p>Triangular Prism</p><p>Triangular Pyramid</p><p>Ask the following questions: <strong>"How can we unfold each of these solids? Is there more than one way to do this for each? Can you predict how the net will seem for each?"</strong></p><p>Review the meaning of net in the Tier III Vocabulary.</p><p><strong>"Let's have a look at some solids and nets together. We'll make a solid for a right rectangular prism and another for a right triangular pyramid. We will search the net for a cube, cylinder, and square pyramid. There are numerous nets that can be created for a right rectangular prism. I encourage you to discover every possible net."</strong></p><p>Show the rectangular prism and triangular pyramid manipulatives to the students. First, ask them to visualize what the drawings of each solid would look like. Have a class discussion on the attributes and various layouts for each. Then, have two class volunteers come to the front and draw the two solids. Next, ask students to guess what the nets of each could look like. What descriptions are possible? Which are impossible? Why? Create a list of all descriptions and possibilities. Then, bring two additional class volunteers to the front and design the possible nets. Do students concur? Do students disagree? Why?</p><p>Divide students into small groups. Allow students to investigate a rectangular prism and its nets using many cereal boxes. (Other relevant boxes are pasta boxes, butter boxes, and some types of right rectangular prisms that are congruent solids, allowing for a straightforward comparison of the many types of nets for a given prism.) Students should note the number of faces, edges, and vertices. Students should also cut each box in various ways to determine all possible nets for the solids.</p><p>The T-shape is one possible net configuration. Discuss the other possible nets. Have students share their nets with the class, recording which ones worked and which did not. During the discussion, create an answer table describing the possible cube nets (6).</p><p><strong>"Now draw the other solids and nets."</strong></p><p>Have students work in groups of three or four to draw the solid and possible nets. Students complete a chart like the one shown below. Encourage students to use as many nets as possible for each solid. (Students can utilize the previous discussion to complete the solids and nets for the rectangular prism and triangular pyramid). Choose amongst the following shapes: right rectangular prism, right hexagonal prism, cube, right triangular prism, square pyramid, triangular pyramid, cylinder, or cone.</p><p>After the groups have completed the chart, have the students display their drawings in the front of the room. Students come to the front, review the drawings, and discuss which ones appear to be correct.</p><p><strong>"Take the right triangular prism net, cut along the outer edges, and fold along the dotted lines to create the solid. Examine the shape to see if the drawn net creates the expected solid."</strong></p><p>Refer students to an appropriate formula for calculating the surface area of a rectangular prism. For example, 2(<i>lw + lh + wh</i>). Point out that a right rectangular prism has four rectangles for the lateral faces and two congruent bases. The right triangular prism has three lateral rectangular sides and two triangular bases.</p><p>The square pyramid consists of four triangular lateral faces and a square base.</p><p>Before estimating the surface area of each three-dimensional shape, have students provide the following verbal descriptions.</p><p>Finish with a matching activity (M-G-3-2_Nets Matching and KEY). Students match each solid to a possible net. There are polyhedra and non-polyhedra.</p><p><strong>Part 2: Estimations, Comparisons, and Calculations Using Nets</strong></p><p><strong>Activity 1: Using Nets to Find Surface Area and Volume</strong></p><p>To encourage students to use visualization to understand and calculate surface area and volume, as well as to look at a solid to determine the range of capacity and/or surface area, concentrate on spatial relationships within solids made by paper nets.</p><p><strong>"Let's first look at some solids and calculate surface areas and volumes for each."</strong></p><p>Explain to students that they will be using a paper net of a right rectangular prism to determine surface area calculation strategies. No formulas will be provided. Students develop strategies and associated formulas during the lesson. The lesson concludes with a review of formulas. Distribute copies of Net 1 (M-G-3-2_Net 1).</p><p>Use a large pair of scissors to cut around the shape. When the net is cut, fold along the line segments to transform the flat two-dimensional net into a three-dimensional solid. Then, fold the solid back into the two-dimensional net.</p><p>Note: Hold a copy of the net in your hand and cut it using scissors so that students may follow along. It's helpful to have a net already cut out so that students may see what it looks like before they start. Instead of cutting out the entire net, students frequently cut out each face separately. Even though this appears to be a simple direction, many students may have problems cutting the net in one piece. Keep in mind that certain students may have some difficult cutting out the net in one piece, therefore having numerous precut nets for student use is always beneficial.</p><p>Measure the dimensions of each rectangle face and mark them on the net. Measure each dimension to the nearest tenth of a centimeter. Find the area of each rectangle face to the nearest tenth of a square centimeter. Ask the following questions.</p><p><strong>"What does surface area mean?"</strong> (<i>the amount the solid covers or takes up; the overall area of the solid</i>)</p><p><strong>"How can the total surface area be found?"</strong> (<i>add up the areas of the six faces</i>)</p><p><strong>"What do you notice about the surface area of the faces?"</strong> (<i>there are three sets of computations, each with two faces with the same area.)</i></p><p>Fold the net into a solid.</p><p><strong>"What do you notice about the surface areas of opposite faces in parallel planes?"</strong> (<i>they are the same/equal.</i>)<br>Now, hold the solid so that its base is "A." Examine the lateral area.</p><p><strong>"Is there another way to calculate the lateral area other than combining the four lateral faces together? Is there a relationship in the model that might provide a formula for the surface area of any right prism?"</strong></p><p>Show that the four lateral faces make up a larger rectangle with a length equal to the perimeter of the base (the sum of the four lateral faces' lengths) and a width equal to the prism's height.</p><p><strong>"How can we write a simple formula for the surface area of the right rectangular prism?" </strong>(<i>SA</i> = <i>ph</i> + 2<i>B</i>)</p><p><strong>"How would we calculate the volume of the net if we transformed it into a right rectangular prism? How can our nets help with this calculation? How would we determine the volume? How can our nets help with this calculation? What would we do with the net? What units might be used to measure the solid? Consider the base to be a small sheet of paper, and stack numerous sheets of the same size on top of it until the stack equals the height of the solid. The space consumed is equal to the area of the base multiplied by the height of the solid. So the prism's volume is calculated using the formula </strong><i><strong>V = lwh</strong></i><strong> or </strong><i><strong>V = Bh</strong></i><strong>. Cubic units will be used to measure volume."</strong></p><p>Students should understand that the surface area of polyhedra can be computed by determining the area of each face and adding the areas of the faces. The volume of polyhedra varies depending on their type, and their formulas change. This is a good opportunity to introduce a generic formula sheet.</p><p><strong>"Now that we've looked at the surface area and volume of one right rectangular prism, I'd like you to look at the surface area and volume of a right triangular prism with a net. Cut the net out by following the solid lines. Fold along the dotted lines to create a solid. Measure each dimension to the nearest tenth of a centimeter. Calculate the base area, lateral area, total surface area, and volume of the right triangular and record the figures into the indicated spaces on the net." </strong>(After students finish this, you may want them to use a glue stick or clear tape to attach the tabs inside the solid figure. Students can keep the solids in net form inside their notebooks for future reference while working in class or with homework.)</p><p>Remind students that in a right rectangular prism, the lateral faces are always rectangular. If a prism has a polygonal face that is not a rectangle, that face is the base. This is an important step in ensuring that surface area and volume are computed accurately for non-rectangular prisms.</p><p><strong>"I'm going to distribute two additional nets: a regular square pyramid and a triangular prism. Cut the nets, measure their measurements (to the nearest tenth of a centimeter), and compute the surface area and volume of each. Refer to the formula sheet provided to help you calculate volume."</strong> Distribute copies of Nets 2 and 3 (M-G-3-2_Net 2 and M-G-3-2_Net 3).</p><p><strong>"We shall now compare solids. Let's consider: Which holds more? A cylinder having a radius of 3 inches and a height of 6 inches, OR a rectangular prism with dimensions of 3 inches by 3 inches by 6 inches?"</strong></p><p><strong>"How might we determine which has the greatest capacity?"</strong></p><p>Students are likely to use similar strategies to those stated previously.</p><p>Students should make each solid by measuring its exact dimensions and using an accurate scale. Students can then fill each solid with a variety of materials, such as grains, seeds, or cubic units. Even though the cylinder will have gaps when filled with cubic units, the experience allows for estimation and yields a good approximation.</p><p>Students should note that the supplied cylinder has more than three times the capacity of the rectangular prism described. In other words, the cylinder was filled with the same amount of material as the rectangular prism, but three times more. Liquid materials, such as water, are more exact measures, but they require waterproof structures to be functional. Fine-grain dry materials, such as rice, will yield reasonably precise results. It may be useful to emphasize to students that the finer the grain of material, the more accurate the estimate.</p><p>Encourage students to examine the actual resulting formulas as well.</p><p>The formula for the volume of a cylinder is <i>V = πr²h.</i></p><p>The formula for the volume of a rectangular prism is <i>V = lwh.</i></p><p><strong>"Based on the formulas alone, which solid would you predict to have the greatest volume, when considering corresponding dimensions?"</strong></p><p>Students should discuss how the square function of the radius represents how quickly the volume of a cylinder increases. Include the following activities:</p><p>Ask students to create five different cylinders and rectangular prisms with two of the same dimensions. Thus, students will determine the dimensions of five cylinders and five rectangular prisms. Consider the following chart:</p><figure class="image"><img style="aspect-ratio:618/503;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_12.png" width="618" height="503"></figure><p>Students study both the volumes and the ratios. Ask students, <strong>"Why would the cylinder volume expand so quickly? What is the ratio telling us? How do we interpret the ratio?"</strong> (<i>The answers will vary</i>.)</p><p>Returning to the initial problem, you can see that the cylinder holds around 169.65 cubic units, whereas the rectangular prism only holds 54 cubic units. See below.</p><p>Volume of Cylinder:</p><p><i>V = π · r² · h</i></p><p><i>= π · 9 · 6</i></p><p><i>= 54π</i></p><p><i>≈ 169.65</i><br>&nbsp;</p><p>The volume of the rectangular prism:</p><p><i>V = lwh</i></p><p><i>= 3 · 3 · 6</i></p><p><i>= 54</i><br>&nbsp;</p><p>Divide students into groups of three or four again. Have students think of another couple of solids to compare. This time, students will compare which has a larger surface area and which stores more/has a higher capacity.</p><p>Reconvene and discuss the solids studied by each group.</p><p>Finally, the class will go over the methods used to determine the surface area and volume of each solid, as well as generalizations for computations across related groups of solids. For example, what general process/formula may be used to calculate the surface area of prisms and cylinders? Pyramids and cones, etc.?</p><p>Distribute the Surface Area and Volume worksheet (M-G-3-2_Surface Area and Volume).</p><p><strong>Activity 2: Relationships Between Quantities</strong></p><p>In the previous activity, students compared similar measurements of various solids with similar dimensions. In this activity, students focus on similar solids using scale factors. Thus, proportionality comes into play. Share copies of Relationships Between Quantities and KEY (M-G-3-2_Relationships Between Quantities and KEY).</p><p>During this activity, students investigate relationships between quantities of measurement. For example, students compare the surface areas and volumes of different solids and calculate their ratios. What conjectures can be made? Are there any patterns to discuss? Which ones? What can we infer from these patterns?</p><p>Begin by telling students, <strong>"Let's first compare the surface areas and volumes of various solids."</strong></p><p>Each solid's measurements may differ since we are now comparing surface areas and volumes both across and within solids. The graphic clarifies any confusion. (Students should write the formulas as well.) Note. Students are investigating and comparing the surface area to volume ratios within and between solids. Also, before handing out the table, fill in column one with some dimensions for the students to compare.</p><p>In certain situations, students must convert measures to approximate decimals.</p><figure class="image"><img style="aspect-ratio:596/706;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_13.png" width="596" height="706"></figure><p>After finishing the Relationship Between Quantities table, ask the students, <strong>"What comparisons can you make for each solid between the two measurements? What comparisons can you draw between the two measurements across solids? Are there any patterns? What are the patterns? What generalizations can you make?"</strong></p><p>In the class discussion that follows this activity, talk about how to find the base of a right prism and how important it is to get it right. Compare and contrast the various solids, and discuss how the formulas differ in respect to the shapes of the faces that make the solid. For example: Right prisms have two bases and a lateral area, just like cylinders. However, the bases of a cylinder are circles, which must be used in area calculations, whereas the bases of a prism are polygons, and the area of the base is calculated by the polygon's area formula. A right prism's lateral area is always made up of rectangles (at least three in a right triangular prism), but a cylinder's lateral area is made up of one rectangle wrapped around its circular base. Volume formulas are similar in that the volume is calculated by multiplying the solid's base area by its height.</p><p>Another good comparison is the regular square pyramid against the right prism. The conventional square prism has a single square base, but the right prism has two bases. A right prism's lateral faces are rectangles, whereas a pyramid's lateral faces are triangular. Another distinction feature of a pyramid is the common intersection of the lateral faces, known as the pyramid's vertex. The volume calculations are similar in that they use the base area times the height of the solid, but the volume of a pyramid is one-third that of a prism with the same height and base dimensions.</p><p><strong>"Now, let's look at some similar solids and see how scale factors and proportions can help us solve problems."</strong> Utilize the following visuals.</p><figure class="image"><img style="aspect-ratio:545/455;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_14.png" width="545" height="455"></figure><p>Pyramid A has a surface area of 35 square meters and a volume of 24 cubic meters. The scale factor between Pyramid A and Pyramid B is 1:4. What is Pyramid B's surface area and volume?</p><p>Measurements of Pyramid B:</p><figure class="image"><img style="aspect-ratio:228/140;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_15.png" width="228" height="140"></figure><p><i>V</i> = 1,536 cubic meters.</p><p>Have students come up with another challenge involving two new similar solids. Ask them to draw conclusions based on these two examples. <strong>"What does this tell you? What can you do with this information? What relationships can you establish with what we did before?"</strong></p><p><strong>"Finally, we see how a change in one dimension affects the surface area and volume measurements. How does changing one dimension affect surface area and volume in different solids? For example, does changing the radius have a greater impact on the surface area and volume of a sphere, a cone, or a cylinder?"</strong></p><p>Guide students through an example with a cone with the following dimensions.</p><figure class="image"><img style="aspect-ratio:177/186;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_16.png" width="177" height="186"></figure><p><i>r = 6</i></p><p><i>h = 9</i></p><p><br><i>SA = B + (1/2)Cl</i>, where <i>B</i> represents the base area, <i>C</i> the circumference, and <i>l </i>the slant height.</p><figure class="image"><img style="aspect-ratio:186/143;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_17.png" width="186" height="143"></figure><p><i>V = (1/3)Bh</i>, where <i>B</i> represents the area of the base.</p><p><i>V = (1/3)π r² h</i></p><p><i>= (1/3) π 36 • 9</i></p><p><i>= (1/3) π 324</i></p><p><i>= 108π</i></p><p><i>≈ 339.29</i></p><p><strong>"Now, let's change one dimension of the cone and look at the new surface area and volume. Let's compare the results with a chart. Let's also utilize the chart to see how changes in one dimension affect surface area and volume in solids. We can answer our question, 'Does a change in radius more strongly effect the surface area and volume of a cone or a cylinder?'"</strong></p><p><strong>“We can see that changing one dimension of the cone had a greater impact on its volume. Why would that be? Please fill in the columns with the remaining solids.”</strong></p><figure class="image"><img style="aspect-ratio:606/393;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_18.png" width="606" height="393"></figure><p><strong>"Compare the rows of cones and cylinders. What did you determine? Which is most strongly influenced by a change in dimension, radius? Why?"</strong></p><p>Review the formulas for calculating surface area and volume. Use the chart below.</p><figure class="image"><img style="aspect-ratio:490/725;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_19.png" width="490" height="725"></figure><p>Finish with a discussion of relationships, patterns, similarity, and general concepts connected to solid comparisons.<strong> "How did this activity increase your overall understanding of relating surface areas and volumes for solids?"</strong></p><p><strong>Review Activity</strong></p><p>Students should prepare a brief presentation on the use of nets for visualization, spatial estimation, and comparisons of surface area and volume. Have students consider the following questions:</p><p>What relationships are important to note?</p><p>Where and how do such comparisons happen in the actual world?</p><p>Students select one main area of study, such as nets and pure visualization, estimation and comparison, or surface area and volume correlations. The ultimate goal is for students to create connections between all three activities, resulting in a strong foundation and understanding of nets, surface area, volume, and their interactions. These presentations should be about 5-10 minutes long. About 25 minutes of preparation time should be allowed. If you choose, you can assign this as homework and have students bring it to class the following day. This would be a good method to review the ideas covered in this lesson before moving on to the next.</p><p><strong>Extension:</strong></p><p>Expand the discussion to Platonic Solids. Students may visualize and draw nets for all five Platonic Solids, as well as estimate their surface area and volume. The discussion should not focus on actual calculations, but rather on visualization, estimation, and reasoning abilities. Students should develop connections inside and between solids using measurements and changes in solid dimensions, as well as their impact on measurements.</p><p>Furthermore, students can apply these concepts to everyday situations. Where do you see platonic solids? When might Platonic Solids measurements be useful in the actual world? Could you give an example or examples?</p><p>Use the applet at <a href="http://www.cs.mcgill.ca/~sqrt/unfold/unfolding.html">http://www.cs.mcgill.ca/~sqrt/unfold/unfolding.html</a> to teach students about Platonic solids and their nets, as well as pentominoes.</p>