Part 1: Initial Exploration

Start the lesson with an exploration of a set of solids, such as are widely available through suppliers such as ETA/Cuisenaire, EAI, and Nasco. [IS.5 - All Students] The set should include at a minimum a sphere, cylinder, cone, prism, and pyramid. Optimally, you want a set that includes all of these solids, plus a cube, rectangular prism, square pyramid, triangular pyramid, triangular prism, and possibly a hemisphere and hexagonal prism. It is important to include the oblique solids so that students can make immediate and one-to-one visual comparisons to the right solids. Include other familiar solids that are not part of formal instructional materials, such as cereal boxes, pasta boxes, soup cans, and tennis ball tubes. Shipping tubes in the shape of triangular prisms are available from companies such as FedEx, UPS, and USPS.

Allow students ample time to examine the properties of each. Divide class into pairs. Have each student in pairs make a sketch that includes an illustration, name, and detailed description of each solid. Students are to pull from prior knowledge and use such terms as faces, edges, vertices, etc. [IS.6 - Struggling Learners]

Following this activity, the class will come together for a whole class discussion and verify the information provided in the chart.

Ask the following questions: [IS.7 - Struggling Learners]

• “How are the prisms related?” (lateral faces, parallel bases)

• “How are the pyramids related?” (one base, one vertex, polygon base)

• “How can you differentiate between a prism and pyramid?”

  (prism has two bases, pyramid one)

• “Which were polyhedra? Which were not?”

  (polyhedral: pyramid, prism; not polyhedral: sphere, cone, cylinder)

• “What objects in the room are prisms? Pyramids? Spheres? etc. What other real-world solids are there?” (Answers vary: the room itself is a rectangular prism, as are file cabinets, tissue boxes, doors, and textbooks.)

Students should discuss polyhedra and non-polyhedra, using such descriptors as “flat” and “curved surfaces.” Polyhedra are solids with polygons for faces. Non-polyhedra, such as circles, cones, and cylinders, are also solids, but have curved surfaces. Other real-world solid objects include Toblerone candy boxes, Laughing Cow cheese packets (triangular prisms), tea infusers (may be triangular prisms), basketball (sphere), silo, tennis ball container (cylinders). Simple observation of packaging materials in a supermarket can yield other useful examples.

Students should revisit the answers to the questions above in the review section of the lesson.

Drawing Solids [IS.8 - Struggling Learners]

After raising students’ interest regarding solids and general questions pertaining to solids, offer students an opportunity for actually drawing solids.

Have students work with a partner. One partner is given a list of solids to choose from, and the other partner is tested on his/her ability to draw that solid. The student is given one minute to draw each solid. Ask students to sketch quickly and observe the whole object, rather than the details. [IS.9 - Struggling Learners] Allowing time for them to practice, discard, and retry is worth the effort and time. For those solids not finished within one minute, ask the student to try again. The partners will then switch roles. The more difficult drawings include the triangular pyramid, pentagonal prism, hexagonal prism, and oblique prisms.

At the close of this activity, a representative from each group presents a drawing to the class. (The drawings may repeat, if necessary.) [IS.10 - Struggling Learners]

Applet Exploration

After students draw the solids by hand, model NLVM’s Isometric 3-D applet for students available at http://nlvm.usu.edu/en/nav/frames_asid_129_g_4_t_3.html?open=activities&from=category_g_4_t_3.html.

Have them create various solids using the applet. Such an activity is high-level in terms of visualization and understanding of each type of solid. [IS.11 - All Students]

Make sure students fully understand the directions, particularly those advising them to start with the hidden view and finish with the sides in plain view. Demonstrate one of these drawings on the board if necessary. If available, isometric paper is useful for this activity.

See the sample Rectangular Prism.

Ask students to discuss any difficulties they had with the creation of the various solids.

Ask the following questions: [IS.12 - Struggling Learners]

• “Which solids were the most difficult/easiest to create? Why?”

• “What did you learn about three-dimensional shapes from exploration with the applet?”

• “Were your visualization abilities improved? How?”

• “Did this activity provide insight into the answers of the questions from before? For example, did it help you understand the concepts of surface area and volume? Can you give an example?”

Classification Activity

Distribute to students copies of the Classifying Solids activity sheet (M-G-3-1_Classifying Solids.doc). Have students classify shapes as polyhedron or non-polyhedron and further classify polyhedra as prisms or pyramids and classify non-polyhedra as cylinders, cones, or spheres. The sheet is set up as a graphic organizer. The goal of this activity is for students to summarize and review the knowledge they gained in doing the drawing, classification, and discussion.

Faces, Edges, and Vertices: Applet Exploration #1

At the close of the classification activity, provide yet another exploration that allows students to rotate a solid in space; manually color and count the faces, edges, and vertices; and determine a possible relationship between these properties. Students determine Euler’s formula.

Model for students the usage of NCTM’s Geometric Solids applet available at http://illuminations.nctm.org/ActivityDetail.aspx?ID=70. The primary task is for students to determine whether a relationship exists between the number of faces, edges, and vertices. Is there a pattern? If so, what is the pattern? See chart below.

Relationship: _____________________ (F + V = E + 2)

Faces, Edges, and Vertices: Applet Exploration #2

In order to visually see Euler’s formula realized for the solids, model the usage of NLVM’s Platonic Solids applet, available at: http://nlvm.usu.edu/en/nav/frames_asid_128_g_4_t_3.html?open=instructions&from=category_g_4_t_3.html.

The focus of this lesson is not on platonic solids, but the applet provides a nice illustration of the relationship between properties and provides the number of faces, edges, and vertices as each property is clicked. In addition, the applet shows Euler’s formula, once all components have been clicked. A platonic solid is a polyhedron with faces that are congruent regular polygons. The classical Platonic solids are the tetrahedron (4 faces), cube (6 faces), octahedron (8 faces), dodecahedron (12 faces), and icosahedrons (20 faces). Note: Students explored the five platonic solids with the NCTM Geometric Solids applet as well. That applet simply had one additional solid. The Wikipedia page on Platonic Solids: http://en.wikipedia.org/wiki/Platonic_solids contains animation sequences that rotate each solid to show the relationships of their faces.

The point of including both applets is twofold: 1) to allow students to count and record the number of faces, edges, and vertices on their own without the applet doing it for them, as well as to find the pattern on their own; and 2) to confirm and verify their records and findings.

Ask students to explore each of the five Platonic Solids (Tetrahedron, Cube, Octahedron, Dodecahedron, and Icosahedron).

See the Tetrahedron below.

Final Sorting Activity

Divide students into groups of three to four and have students sort solids according to similarities and differences. For example, students can sort solids by the number of faces, edges, and vertices; types of faces; polyhedra/non-polyhedra; prisms/pyramids; shape of base; shape of possible cross-sections; etc.

Make this activity engaging and involve students in the process of determining the specific categories, representative of similarities and differences. The presentation of sorting of solids is open-ended. Encourage students to be as creative as possible.

Ask students to include real-world objects in the presentation as well. In addition, probe them to discuss the reasoning behind the importance of sorting, as well as real-world implications. Further examples of real-world objects include Hershey kisses, tea bags, beverage cans, cone cups, water towers, pizza boxes, balls, butter quarters, and shipping tubes.

Sample 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 difference between a cylinder and a cone?

Part 2: Cross Sections

After students have an understanding of properties, relationships among properties, and classification, introduce planes and cross sections. A plane is two-dimensional and extends without end in all directions. A cross section is the area formed by the intersection of the plane and the solid.

Cross Section Applet Exploration

Introduce students to NLVM’s Platonic Solids-Slicing applet available at http://nlvm.usu.edu/en/nav/frames_asid_126_g_4_t_3.html?open=instructions&from=category_g_4_t_3.html.

Guide students through the demonstration, illustrating the fact that solids can have more than one cross section. Provide ample time for students to explore. Understanding cross sections and determining the resulting shape is at the height of visualization.

Here is a possible cross section of a cube:

In this illustration, a plane was sliced through one corner of the cube.

Review

Review the questions posed in the initial exploration.

• How are the prisms related? (e.g., number of faces, edges, vertices)

• How are the pyramids related? (shape of base)

• How can you differentiate between a prism and pyramid? (pyramid has a common vertex that is not coplanar with the base)

• Which were polyhedra? Which were not? (polyhedrons have polygon faces)

• What objects in the room are prisms? Pyramids? Spheres? Etc. What other real-world solids are there?

Students should touch on the following ideas:

• Prisms are solids that have two parallel, congruent bases, and lateral faces that are parallelograms. In a right prism, the lateral faces are rectangles. The bases can be various shapes. Pyramids have three triangular lateral faces and one base that is a polygon. Triangular pyramids and rectangular pyramids can be differentiated from each other because triangular pyramids have three triangular lateral faces and rectangular pyramids have four triangular lateral faces.

• Of the solids examined, the pyramids and prisms were polyhedra. The cylinders, cones, and spheres were non-polyhedra.

• Possible objects for each solid:

• Prisms: boxes, the room itself, books, storage closet, etc.

• Pyramids: food pyramid.

• Spheres: globe, soccer ball, basketball, baseball, etc.

• Cylinders: any can, cylindrical waste basket, candle holder, etc.

• Cones: ice cream cones, pencil tip, etc.

Have objects on hand and readily available that may be cut in cross section as a demonstration of the relationship between the plane and solid. Position the cut cross section on paper and trace it with a pencil to show the shape. Ask students to find solid models that can be easily brought to the classroom, e.g., toilet paper and paper towel rollers.

Discuss other real-world solids.

Have students discuss the most interesting and helpful part of the lesson. How has their view of solids changed? Do they have a better understanding of cross sections? Will they be more aware of solids in the real world?

Finally, have students write definitions for the various solids provided in the chart below.

Review definitions in a whole-class discussion.

Review Activity

Some examples of individual and group presentations students could do include:

• PowerPoint

• descriptive and illustrative document, as in the form of a text

• radio announcement

• newspaper article

• multimedia presentation

• tour of real-world objects and interactive synthesis of ideas from classmates (guided and facilitated by group)

  Finding Cross Sections

Prior to asking students to draw planes and find cross sections of solids, introduce a few solids, query as to where a possible plane might intersect, and then explore the resulting cross section with students.

Illustrate cross sections of a cube, rectangular pyramid, and sphere. See below.

Note: There are seven cross sections possible with a cube. However, we will simply illustrate two more of those here in addition to the one illustrated with the applet. Encourage students to find all possible cross sections. See illustrations below.

Cube with an intersecting plane parallel to one of the sides. This reveals a square cross section.

Cube with an intersecting plane perpendicular to one of the faces but not perpendicular to the edges of that particular face. This reveals a rectangular cross section.

Rectangular pyramid with intersecting plane perpendicular.

Sphere with intersecting plane with resulting cross section of a circle.

*Include a real-world comparison of cross sections. For example, the cross section of a tree stump has concentric rings. However, the cross section of a floor board has longitudinal rectangles shown as the pattern.

Drawing Cross Sections for Solids

Provide the Cross Sections activity sheet (M-G-3-1_Cross Sections and KEY.doc). Students reveal higher-level thinking by drawing various intersections of the solid and then revealing the shape of the cross section.

Included in the activity are five solids: triangular prism, rectangular prism, triangular pyramid, cone, and cylinder.

Close the lesson with Lesson 1 Exit Ticket (M-G-3-1_Lesson 1 Exit Ticket and KEY.doc).

Extension:

• Assign students to explore cross sections of the platonic solids.

• Have students explore shadows of prisms, pyramids, spheres, cones, and cylinders.

• During the faces, edges, and vertices Applet Exploration #1, include actual paper solid objects. Some students may have an improved understanding by holding a physical object rather than its technology-driven image. The following Web site provides templates for such uses:

  http://www.enchantedlearning.com/math/geometry/solids