<p>In this lesson, students will interact with, build, and visualize solids and their properties. Students will:&nbsp;<br>- examine and determine the properties of solids, including polyhedra and non-polyhedra.&nbsp;<br>- use properties to determine relationships between and within solids.&nbsp;<br>- create solids.<br>- classify and sort solids.&nbsp;<br>- participate in higher-level visualization using cross sections.</p>

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

<p>- <strong>Cone:</strong> A solid with a circular base.<br>- <strong>Cross Section:</strong> The area formed by the intersection of the plane and the solid.<br>- <strong>Cylinder:</strong> A solid with two congruent and parallel circular bases.<br>- <strong>Edge:</strong> The line segment that adjoins two faces of a solid; the intersection of two faces of a polyhedron.<br>- <strong>Face:</strong> The side of a polyhedron.<br>- <strong>Non-polyhedron:</strong> Solid with curved surfaces or combination of curved and flat surfaces.<br>- <strong>Plane:</strong> A two-dimensional surface that extends without end in all directions.<br>- <strong>Platonic Solid:</strong> A polyhedron with faces that are congruent regular polygons.<br>- <strong>Polyhedron:</strong> Solid with polygons for faces; edges of each face are line segments, i.e., flat.<br>- <strong>Prism:</strong> A polyhedron with two congruent parallel bases.<br>- <strong>Pyramid:</strong> A polyhedron with a polygonal base and three or more triangular faces.<br>- <strong>Solid:</strong> A three-dimensional figure.<br>- <strong>Sphere:</strong> A solid that is formed by points that are an equal distance from the center.<br>- <strong>Vertex:</strong> The point on a solid where edges meet; the edges of three or more faces of a polyhedron.</p>

<p>- copies of Classifying Solids (M-G-3-1_Classifying Solids and KEY)<br>- copies of Cross Sections (M-G-3-1_Cross Sections and KEY)<br>- copies of Lesson 1 Exit Ticket (M-G-3-1_Lesson 1 Exit Ticket and KEY)<br>- Student access to computers during class time</p>

<p>- Classroom discussions can help you find common challenges among several students, such as visualizing two-dimensional representations of solids.&nbsp;<br>- Observe the applet explorations to see how well students advance from visualizing to manipulating the three-dimensional object.&nbsp;<br>- Lesson 1 Exit Ticket (M-G-3-1_Lesson 1 Exit Ticket and KEY) asks students to explain their grasp of specific characteristics of three-dimensional object, represent cross sections, and categorize solids based on their attributes.</p>

<p>Active Engagement, Modeling<br>W: Exploratory activities and modeling will show students' conceptual comprehension and knowledge throughout the the lesson. An active-engagement strategy does not only provide students with strong definitions, classification criteria, and explicit cross-section directions. Instead, students play an active role in self-reflection and meaning development. Real-world connections are interspersed to help concentrate the lesson and demonstrate the need of such knowledge. Students will be evaluated utilizing a variety of formative assessment strategies. Students will be evaluated based on their active participation, classroom discussions, classification and cross section activities, and the creativity and mathematical accuracy of their sorting solids presentation.&nbsp;<br>H: The overall design of the lesson, which is exploratory in nature, puts the student at the forefront of his/her learning. This will captivate the student. The frequent discussions, higher-level requirements, connection to the real world, and enthusiasm surrounding the concluding sorting presentation provide a way to keep the student engaged throughout the lesson.&nbsp;<br>E: The lesson is divided into two parts. Part 1 explores solid and their properties, relationships, and classifications. Part 2 focuses on a more detailed property (i.e., cross sections). The use of actual objects, real-world connections, investigation, and self-reflective visualization all contribute to a more realistic knowledge. The lesson aims to assist all learners achieve by using visual, auditory, and kinesthetic representations.&nbsp;<br>R: Revisiting the original problems raised during the exploration of the set of solids and requiring students to construct their own definitions, students will need to rethink how to describe three-dimensional objects in two-dimension space. Students are not provided answers to any of the questions, or given direct definitions for any terms. Students, on the other hand, are responsible for developing their own concepts, definitions, and generalizations.<br>E: The lesson's discursive and exploratory requirements encourage students to evaluate two-dimensional representations of solids.&nbsp;<br>T: The use of kinesthetic and virtual manipulatives, graphic organizers, representations, and explorations can differentiate instruction for students with varied levels of understanding. The ability to collaborate with partners and groups benefits students who value social interaction.&nbsp;<br>O: The lesson is structured to include tangible explorations followed by abstract generalizations. The teacher models and facilitates the explorations, which increase overall comprehension of solids.<br>&nbsp;</p>

<p><strong>Part 1: Initial Exploration</strong></p><p>Begin the lesson by exploring a set of solids that are commonly available from providers such as ETA/Cuisenaire, EAI, and Nasco. The set should comprise a pyramid, sphere, cylinder, cone, and prism. Ideally, a set should comprise all of these solids, as well as a cube, rectangular prism, square pyramid, triangular pyramid, triangular prism, and potentially a hemisphere and hexagonal prism. It is critical to include the oblique solids so that students may make immediate and one-on-one visual comparisons to the right solids. Include other commonly used solids that are not part of formal teaching materials, such as cereal boxes, pasta boxes, soup cans, and tennis ball tubes. Companies such as FedEx, UPS, and USPS offer shipping tubes shaped like triangular prisms.</p><p>Allow students plenty of opportunity to investigate the properties of each. Divide the class into pairs. Each student should work in pairs to create a sketch that contains an illustration, name, and full explanation of each solid. Students are expected to use prior knowledge and vocabulary such as faces, edges, and vertices.</p><p>Following this activity, the class will gather for a whole-class discussion to confirm the information provided in the chart.</p><p>Ask the following questions:</p><p><strong>"How are the prisms related?"</strong> (<i>lateral faces and parallel bases</i>)</p><p><strong>"How are the pyramids related?"</strong> (<i>one base, one vertex, polygon base</i>)</p><p><strong>"How can you differentiate between a prism and pyramid?"</strong> (<i>The pyramid has one base, while the prism has two.</i>)</p><p><strong>"Which ones were polyhedra? Which ones were not?" </strong>(<i>polyhedral: pyramid, prism; non-polyhedral: sphere, cone, cylinder</i>)</p><p><strong>"Which objects in the room are prisms? Pyramids? Spheres? etc. What other real-world solids are there?"</strong> (<i>The answers vary: the room itself is a rectangular prism, as are file cabinets, tissue boxes, doors, and textbooks are all rectangular prisms.</i>)</p><p>Students should discuss polyhedral and non-polyhedral, using terms like "flat" and "curved surfaces." <i>Polyhedra </i>are solids having polygonal faces. <i>Non-polyhedra</i> solids include circles, cones, and cylinders, which have curved surfaces. Toblerone candy boxes, Laughing Cow cheese packages (triangular prisms), tea infusers (which could also be triangular prisms), basketball (sphere), silo, and tennis ball container (cylinders) are examples of real-world solids. Simple observations of packing materials in a supermarket can provide other instances.</p><p>Students should review their responses to the preceding questions in the lesson's review section.</p><p><strong>Drawing Solids</strong></p><p>After raising students' interest in solids and answering general questions about them, give them the opportunity to draw solids themselves.</p><p>Have students collaborate with a partner. One partner is given a list of solids to pick from, while the other is tested on his or her ability to draw that solid. The student has one minute to draw each solid. Ask students to sketch rapidly and observe the whole object rather than the details. Allowing students to practice, discard, and try again is worth the effort and time. If the solids are not finished within one minute, ask the student to try again. The partners will then change roles. The triangular pyramid, pentagonal prism, hexagonal prism, and oblique prism are some of the most complex illustrations.</p><p>At the end of this lesson, a representative from each group will present a sketch to the class. (The drawings may be repeated if needed.)</p><p><strong>Applet Exploration</strong></p><p>After students have drawn the solids by hand, use NLVM's Isometric 3-D applet for students, which is available at <a href="http://nlvm.usu.edu/en/nav/frames_asid_129_g_4_t_3.html?open=activities&amp;from=category_g_4_t_3">http://nlvm.usu.edu/en/nav/frames_asid_129_g_4_t_3.html?open=activities&amp;from=category_g_4_t_3</a></p><p>Have them use the applet to make various solids. This activity is advanced in terms of visualizing and understanding each type of solid.</p><p>Make certain that students thoroughly comprehend the instructions, particularly those that instruct them to begin with the hidden view and finish with the sides in plain view. Show one of these drawings on the board if necessary. If available, isometric paper might be used for this task.</p><p>Please see the sample Rectangular Prism.</p><figure class="image"><img style="aspect-ratio:1281/801;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_2.png" width="1281" height="801"></figure><p>Ask students to discuss any problems they encountered when creating the various solids.</p><p>Consider the following questions:&nbsp;</p><p><strong>"Which solids were the most difficult or easiest to create? Why?"</strong></p><p><strong>"What did you learn about three-dimensional shapes from exploration with the applet?"</strong></p><p><strong>"Were your visualization abilities improved? How?"</strong></p><p><strong>"Did this activity provide any insight into the previous questions' answers? For example, does it assist you grasp the concepts of surface area and volume? Can you provide an example?"</strong></p><p><strong>Classification Activity</strong></p><p>Give students copies of the Classifying Solids activity sheet (M-G-3-1_Classifying Solids and KEY). Students should classify shapes as polyhedrons or non-polyhedrons, with polyhedra being classified as prisms or pyramids and non-polyhedra as cylinders, cones, or spheres. The sheet is set up as a graphic organizer. The purpose of this activity is for students to summarize and evaluate the knowledge they obtained while drawing, categorizing, and discussing.</p><p><strong>Faces, Edges, and Vertices: Applet Exploration #1</strong></p><p>At the end of the classification activity, give students another opportunity to rotate a solid in space, manually color and count the faces, edges, and vertices, and discover a possible relationship between these characteristics. Students calculate Euler's formula.</p><p>Show students how to use NCTM's Geometric Solids applet, which can be found at <a href="http://illuminations.nctm.org/ActivityDetail.aspx?ID=70">http://illuminations.nctm.org/ActivityDetail.aspx?ID=70</a>. The main challenge for students is to discover whether there is a relationship between the number of faces, edges, and vertices. Is there a pattern? If so, what's the pattern? See the chart below.</p><figure class="image"><img style="aspect-ratio:599/292;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_3.png" width="599" height="292"></figure><p>Relationship: _____________________. <i>(F + V = E + 2)</i></p><p><strong>Faces, edges, and vertices: Applet Exploration #2</strong></p><p>To visually see Euler's formula realized for solids, use NLVM's Platonic Solids applet, which may be found at: <a href="http://nlvm.usu.edu/en/nav/frames_asid_128_g_4_t_3.html?open=instructions&amp;from=category_g_4_t_3.html.">http://nlvm.usu.edu/en/nav/frames_asid_128_g_4_t_3.html?open=instructions&amp;from=category_g_4_t_3.html.</a></p><p>The focus of this lesson is not on platonic solids, but the applet illustrates the relationship between attributes and displays the number of faces, edges, and vertices when each property is selected. Furthermore, after clicking all of the components, the applet displays Euler's formula. A platonic solid is a polyhedron whose faces are congruent regular polygons. The classical Platonic solids include the tetrahedron (4 faces), cube (6 faces), octahedron (8 faces), dodecahedron (12 faces), and icosahedrons (20 faces). Students also examined the five platonic solids using the NCTM Geometric Solids applet. That applet simply had one more solid. The Wikipedia page on Platonic Solids (http://en.wikipedia.org/wiki/Platonic_solids) includes animation sequences that rotate each solid to highlight the relationships between its faces.</p><p>The purpose of including both applets is twofold: 1) to allow students to count and record the number of faces, edges, and vertices without the applet's assistance, as well as detect the pattern on their own; and 2) to confirm and verify their records and conclusions.</p><p>Encourage students to investigate each of the five Platonic Solids (Tetrahedron, Cube, Octahedron, Dodecahedron, and Icosahedron).</p><p>See the Tetrahedron below.</p><figure class="image"><img style="aspect-ratio:1288/809;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_4.png" width="1288" height="809"></figure><p><strong>Final Sorting Activity</strong></p><p>Divide students into groups of three or four and have them sort solids based on similarities and differences. Students can sort solids based on the number of faces, edges, and vertices; face kinds; polyhedra/non-polyhedra; prisms/pyramids; shape of base; shape of possible cross-sections, and so on.</p><p>Make this an engaging activity by including students in the process of identifying particular categories that highlight similarities and differences. The presentation on solid sorting is open-ended. Encourage your students to be as creative as possible.</p><p>Ask students to use real-world objects in their presentations as well. In addition, ask them to explain why sorting is important, as well as the real-world implications. Other examples of real-world things are Hershey kisses, tea bags, beverage cans, cone cups, water towers, pizza boxes, balls, butter quarters, and shipping tubes.</p><p>Example questions: What are the similarities and differences between a cylinder and a right rectangular prism? What is the difference between a pyramid and a right rectangular prism. What are the similarities and differences between cylinders and cones?</p><p><strong>Part 2: Cross Sections</strong></p><p>Introduce planes and cross sections after students have learned about qualities, their relationships, and classification. A plane is two-dimensional and extends indefinitely in all directions. A cross section is the area created by the intersection of a plane and a solid.</p><p><strong>Cross Section Applet Exploration</strong></p><p>Introduce students to NLVM's Platonic Solids-Slicing applet, which is available at <a href="http://nlvm.usu.edu/en/nav/frames_asid_126_g_4_t_3.html?open=instructions&amp;from=category_g_4_t_3.">http://nlvm.usu.edu/en/nav/frames_asid_126_g_4_t_3.html?open=instructions&amp;from=category_g_4_t_3.</a></p><p>Guide students through the demonstration, demonstrating how solids can have multiple cross sections. Allow students adequate time to explore. Understanding cross sections and calculating the resultant shape is the pinnacle of visualization.</p><p>Here is a possible cross section of a cube.</p><figure class="image"><img style="aspect-ratio:1289/808;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_5.png" width="1289" height="808"></figure><p>In this graphic, a plane was sliced through one corner of the cube.</p><p><strong>Review</strong></p><p>Review the questions posed in the initial exploration,</p><p>How are the prisms related? (<i>for example, the number of faces, edges, and vertices</i>)</p><p>How are the Pyramids related? (<i>shape of base</i>)</p><p>How do you distinguish between a prism and a pyramid? (<i>the common vertex of a pyramid is not coplanar with the base</i>)</p><p>Which ones were polyhedra? Which ones were not? (<i>polyhedrons have polygonal faces</i>)</p><p>Which objects in the room are prisms? Pyramids? Spheres? etc. What additional real-world solids exist?</p><p>Students should discuss the following ideas:</p><p>Prisms are solids with two parallel, congruent bases and parallelogram-shaped lateral faces. In a right prism, the lateral faces are rectangular. The bases can take numerous shapes. Pyramids are made up of three triangular lateral faces and one polygonal base. Triangular pyramids and rectangular pyramids differ in that triangular pyramids have three triangular lateral faces whereas rectangular pyramids have four.</p><p>Pyramids and prisms were the only polyhedra among the solids examined. The cylinders, cones, and spheres were not polyhedra.</p><p>Possible objects for every solid:</p><p>Prisms: boxes, the room itself, books, a storage closet, and so on.</p><p>Pyramids: food pyramid.</p><p>Spheres: globe, soccer ball, basketball, baseball, etc..</p><p>Cylinders: any can, cylindrical waste basket, candle holder, and so on.</p><p>Cones: ice cream cones, pencil tip, etc.</p><p>Have things readily available that can be cut in cross section to demonstrate the relationship between the plane and solid. Place the cut cross section on paper and trace it with a pencil to reveal its shape. Ask students to find solid models that can be easily brought to the classroom, such as toilet paper and paper towel rolls.</p><p>Discuss additional real-world solids.</p><p>Have students talk about the most fascinating and useful aspect of the lesson. How has their perception of solids changed? Do they have a greater comprehension of cross sections? Will students become more aware of solids in the real world?</p><p>Finally, ask students to develop definitions for the different solids shown in the chart below.</p><figure class="image"><img style="aspect-ratio:598/322;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_167.png" width="598" height="322"></figure><p>A whole-class discussion can be used to review definitions.</p><p><strong>Review Activity</strong></p><p>Examples of individual and group presentations that students could give include:</p><p>PowerPoint</p><p>descriptive and illustrative document, such as a text.</p><p>radio announcement</p><p>newspaper article</p><p>multimedia presentation.</p><p>tour of real-world objects and interactive synthesis of classmates' ideas (directed and facilitated by the group)</p><p><strong>Finding Cross Sections</strong></p><p>Before asking students to draw planes and locate cross sections of solids, present a few solids, ask where a possible plane might intersect, and then examine the resulting cross section with students.</p><p>Draw cross sections for a cube, a rectangular pyramid, and a sphere. See below.</p><p>Note: A cube can have seven different cross sections. However, in addition to the one shown in the applet, we will illustrate two more of them here. Encourage students to identify all possible cross sections. See the illustrations below.</p><figure class="image"><img style="aspect-ratio:231/297;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_7.png" width="231" height="297"></figure><p>A cube having an intersecting plane parallel to one of its sides. This shows a square cross section.</p><figure class="image"><img style="aspect-ratio:457/280;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_8.png" width="457" height="280"></figure><p>A cube with an intersecting plane that is perpendicular to one of the faces but not to its edges. This shows a rectangular cross section.</p><figure class="image"><img style="aspect-ratio:214/496;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_9.png" width="214" height="496"></figure><p>The intersecting plane of a rectangular pyramid is perpendicular.</p><figure class="image"><img style="aspect-ratio:472/316;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_10.png" width="472" height="316"></figure><p>A sphere intersecting a plane results in a circle's cross section.</p><p>*Include a real-world comparison of cross sections. For example, the cross section of a tree stump contains concentric rings. However, the cross section of a floor board shows longitudinal rectangles as the pattern.</p><figure class="image"><img style="aspect-ratio:383/329;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_11.png" width="383" height="329"></figure><p><strong>Drawing Cross Sections for Solids</strong></p><p>Provide the Cross Sections Activity Sheet (M-G-3-1_Cross Sections and KEY). Students demonstrate higher-level thinking by drawing several solid intersections and then disclosing the cross section's shape.</p><p>The activity includes five solids: triangular prisms, rectangular prisms, triangular pyramids, cones, and cylinders.</p><p>Close the lesson with the Lesson 1 Exit Ticket (M-G-3-1_Lesson 1 Exit Ticket and KEY).</p><p><strong>Extension:</strong></p><p>Assign students to investigate the cross sections of the platonic solids.</p><p>Have students investigate the shadows of prisms, pyramids, spheres, cones, and cylinders.</p><p>In the faces, edges, and vertices Applet Exploration #1, include real paper solid objects. Some students may have a better understanding by grasping a physical thing rather than the technology-driven representation. The following website provides templates for similar uses:</p><p><a href="http://www.enchantedlearning.com/math/geometry/solids">http://www.enchantedlearning.com/math/geometry/solids</a></p>