• Matching nets to their corresponding solid representation is a visual/spatial skill that students can improve with practice. Using the Nets Matching activity sheet (M-G-3-2_Nets Matching and KEY.doc) can help students identify where they are making 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 aid their visualization.

• Predictions, conjectures, and completion of charts identify some of the weaknesses in students’ understanding of the relationships between solid objects and their two-dimensional representations. Asking leading questions such as, “Which edge matches this side of the triangle?” can help them in visualization.

Part 1: Making Connections Between Nets and Solids [IS.5 - Struggling Learners]

Show students various solids (manipulatives), including the following: [IS.6 - All Students]

• Cone

• Cube

• Right Cylinder

• Right Hexagonal Prism

• Oblique Rectangular Prism (parallelogram lateral faces)

• Right Rectangular Prism

• Square Pyramid

• Triangular Prism

• Triangular Pyramid

Pose the following questions: “How can we unfold each of these solids? Is there more than one way to do so for each? Can you predict what the net will look like for each?”

Review the definition of net in the Tier III Vocabulary.

“Let’s first look at a couple of solids and nets together. We will create a solid for a right rectangular prism and one for a right triangular pyramid. We will explore the net for a cube, cylinder, and a square pyramid. There are several nets we can create for a right rectangular prism. I encourage you to find all of the possible nets.”

Show students the manipulatives of the rectangular prism and triangular pyramid. First ask students to imagine what the drawings of each solid look like. Have a class discussion of the properties and possible layout of each. Then have two class volunteers come to the front and draw the two solids. Next ask students to predict what the nets of each might look like. Which descriptions are possible? Which are not possible? Why? Make a list of all descriptions and possibilities. Then, have two more class volunteers come to the front and draw the possible nets. Do students concur? Do students disagree? Why?

Divide students into small groups. Have students explore a rectangular prism and its nets using several cereal boxes. (Other useful boxes include pasta boxes, butter boxes, some types of right rectangular prisms that are congruent solids so that a clear comparison can be made between the different types of nets for a particular type of prism.) Students should record information on the number of faces, edges, and vertices. Students should also cut each box in different ways, determining all of the possible nets for the solids.

One possible net is the T-shape. Discuss the other possible nets. Have students share their nets with the class, recording the ones that worked and the ones that didn’t work. During the discussion, build an answer table that describes the possible nets of a cube (6).

“Now draw the other solids and nets.”

Have students work in groups of three or four and draw the solid and possible nets. Students complete a chart similar to the one shown below. Encourage students to draw as many nets as possible for each solid. (Students can use the prior discussion to fill in the solid and nets for the rectangular prism and triangular pyramid.). Select from the following shapes: right rectangular prism, right hexagonal prism, cube, right triangular prism, square pyramid, triangular pyramid, cylinder, and cone.

After the groups complete the chart, have students post their drawings at the front of the room. Students come to the front, review the drawings, and discuss the ones that look correct.

“Take the right triangular prism net, cut along the outer edges, and fold along the dotted lines, thus creating the solid. Examine the shape to determine whether or not the net drawn is actually a net that creates the intended solid.”

Refer students to a suitable formula for the surface area of a rectangular prism. For example, 2(lw + lh + wh). Point out that in a right rectangular prism, there are four rectangles that are the lateral faces, and two congruent bases. For the right triangular prism, there are three lateral rectangular faces and two triangular bases.

For the square pyramid, there are four triangular lateral faces and one square base.

Have students complete these verbal descriptions of the surface area for each three-dimensional shape before calculating its surface area.

End with a matching activity (M-G-3-2_Nets Matching and KEY.doc). Students match each solid with a possible net. Included are polyhedra and non-polyhedra.

Part 2: Estimations, Comparisons, and Calculations Using Nets

Activity 1: Using Nets to Find Surface Area and Volume [IS.7 - All Students]

To encourage students to employ visualization in understanding and calculating surface area and volume, as well as in looking at a solid to determine the range of capacity and/or surface area, focus on spatial relationships within solids formed by paper nets.

“Let’s first look at some solids and calculate surface areas and volumes for each.”

Explain to students that a paper net of a right rectangular prism will be used to determine strategies for calculating surface area. No formulas will be given. Students determine strategies and accompanying formulas within the lesson. Review formulas at the end of the lesson. Distribute copies of Net 1 (M-G-3-2_Net 1.doc).

Using a large pair of scissors, cut around the shape. When the net is cut out, fold along the line segments so that the flat two-dimensional net can be manipulated into the three dimensional solid. Then unfold the solid back into the two-dimensional net.

Note: You may want to hold a copy of the net in your hand and cut with scissors so students can follow. It’s useful to have a net already cut out to show students what the net looks like before beginning. Often students will cut out each face rather than the net as one whole. Even though this seems like a simple direction, many students can have some difficulty cutting out the net in one piece. Keep in mind that some students may have less developed fine motor skills, and it’s always useful to have several precut nets for student use.

Measure the dimensions of each rectangular face, marking the dimensions on the net. Measure each dimension to the nearest tenth of a centimeter. Calculate the area of each rectangular face to the nearest tenth of a square centimeter. Ask the following questions. [IS.8 - Struggling Learners]

“What does surface area mean?” (the amount the solid covers or takes up; the total area of the solid)

“How can the total surface area be found?” (add up the areas of the six faces)

“What do you notice about the surface area of the faces?” (there are three sets of calculations, each having two faces with the same area)

Fold the net into a solid.

“What do you notice about the surface areas of opposite faces in parallel planes?” (They are the same/equal.)

Now hold the solid so that the base is “A.” Examine the lateral area.

“Is there another way the lateral area could be found besides adding the four lateral faces together? Is there some relationship that you can see in the model that would give a formula for the surface area of any right prism?”

Show that the four lateral faces make up a larger rectangle whose length is the perimeter of the base, that is the sum of the lengths of the four lateral faces, and whose width is the height of the prism.

“How can we write a simple formula for the surface area of the right rectangular prism?” (SA = ph + 2B)

“Looking at the net made into a right rectangular prism, how would we find the volume? How can our nets help with that calculation? How would we find the volume? How can our nets help with that calculation? What would we do with the net? What units would we use to measure the solid? Think about the base being a small sheet of paper and having multiple sheets of same-sized paper stacked on top of it, until the stack is the same as the height of the solid. The space consumed is equal to the area of the base times the height of the solid. So the formula for the volume of the prism is V = lwh or V = Bh. Cubic units will be used as the measure of volume.”

Students should realize that the surface area of polyhedra can be calculated by finding the area of each face and finding the sum of the areas of the faces. Volume is dependent on the type of polyhedra and the formulas are different from each other. This is a good point to introduce a generic formula sheet.

“Now that we have examined the surface area and volume of one right rectangular prism I would like you to examine the surface area and volume of a right triangular prism using a net. Cut the net out by cutting along the solid lines. Fold along the dotted lines to form the 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 values in the indicated spaces on the net.” (Once students have completed this, you may want them to use a glue stick or clear tape to attach the tabs inside the solid figure. It is useful for students to leave the solids in net form so they can carry it around inside their notebooks for future reference while working in class or with homework.)

Note: Remind students that in a right rectangular prism, the lateral faces are always rectangles. If a prism has a polygonal face that is not a rectangle, that face is the base. This is a crucial point in making sure surface area and volume are calculated correctly for non-rectangular prisms.

“I am going to hand out two additional nets, a regular square pyramid and a triangular prism. Cut out the nets, measure the dimensions (to the nearest tenth of a centimeter), and calculate the surface area and volume of each. Refer to the formula sheet given to aid in your calculations of volume.” Distribute copies of Net 2 and Net 3 (M-G-3-2_Net 2.doc and M-G-3-2_Net 3.doc).

“We will now make comparisons between solids. Let’s consider: Which holds more: A cylinder with dimensions of radius = 3 inches and height = 6 inches OR a rectangular prism with dimensions of 3 inches by 3 inches by 6 inches?”

“How might we determine which has the greatest capacity?”

Students will likely come up with similar strategies mentioned previously.

Students should create each solid, measuring the actual dimensions and creating a solid with an accurate scale. Students can then fill each solid with various materials, including rice, seeds, and cubic units. Even though the cylinder will have gaps when filled with cubic units, the experience taps into estimation and provides a nice approximation.

Students should notice that the given cylinder holds over three times the amount of the rectangular prism described. In other words, the cylinder was filled up using the same measured amount of material over three times that of the rectangular prism. Liquid material such as water is a more accurate measure, though it requires waterproof structures to be of practical use. Fine-grain dry materials such as rice will produce reasonably accurate approximations. It may be useful to point out to students that the finer the grain of material, the more accurate the estimate,

Encourage students to look at the actual resulting formulas as well.

Formula for volume of a cylinder: V = πr²h

Formula for volume of a rectangular prism:     V = lwh

“Based on the formulas alone, which solid would you predict to have the greatest volume, when considering corresponding dimensions?”

Students should discuss the fact that the square function of the radius shows how quickly the volume increases with a cylinder. Include the following activity:

Ask students to come up with five different cylinders and rectangular prisms with two of the same dimensions. Thus, students will determine different dimensions for five cylinders and five rectangular prisms. Consider the chart below:

Students examine the volumes as well as the ratios. Ask students: “Why would the cylinder volume increase more quickly? What does the ratio tell us? How can we interpret the ratio?” (Answers will vary.)

In revisiting the original problem, you see that the cylinder holds approximately 169.65 cubic units, whereas the rectangular prism only holds 54 cubic units. See below.

Volume of the Cylinder: [IS.9 - All Students]

V = π r² h

= π · 9 · 6

= 54π

≈ 169.65

Volume of the Rectangular Prism:

V = lwh

= 3 · 3 · 6

= 54

Break students into groups of three or four again. Have students brainstorm another couple of solids to compare. This time students will determine which has a greater surface area as well as which holds more/has greater capacity.

Reconvene and discuss the solids explored by each group.

Finally, the class will discuss the strategies used to calculate surface area and volume of each solid, as well as generalizations made for calculations across similar groups of solids. For example, what general process/formula can be used when finding surface area of prisms and cylinders? Pyramids and cones, etc.?

Distribute M-G-3-2_Surface Area and Volume (M-G-3-2_Surface Area and Volume.doc).

Activity 2: Relationships Between Quantities

In the previous activity students compared same measurements of different solids with similar dimensions. In this activity students focus on similar solids by way of use of scale factors. Thus, proportionality comes into play. Distribute copies of Relationships Between Quantities (M-G-3-2_Relationships Between Quantities.doc and M-G-3-2_Relationships Between Quantities KEY.docx).

In completing this activity, students visit relationships between quantities of measurement. For example, students compare surface areas and volumes of various solids and examine the ratios of each. What conjectures can be made? Are there any patterns to discuss? Which ones? What can we determine from these patterns?

Start by telling students, “Let’s first make some comparisons of the surface areas and volumes of different solids.”

The measurements of each solid can be different, as we are now not only comparing the surface areas and volumes across solids, but within as well. The chart clarifies any misunderstandings here. (Students should write the formulas as well.) Note. Students are examining the ratio of surface area to volume within each solid and comparing the ratios across solids. Also, before handing out the table, fill in column one with some dimensions you wish students to compare.

Students need to convert measurements to approximate decimals in some cases.

After completing the Relationship Between Quantities table, ask the class, “What comparisons can you make for each solid across the two measurements? What comparisons can you make for the two measurements across solids? Are there any patterns? What are the patterns? What generalizations can you make?” [IS.10 - Struggling Learners]

In the class discussion following this activity, discuss how to determine the base of a right prism and the importance of identifying the base correctly. Compare and contrast the various solids and talk about how the formulas vary in relationship to the shapes of the faces making the solid. For example: Cylinders have two bases and a lateral area as do right prisms. However, the bases in a cylinder are circles, necessitating the use of in calculations of area, whereas the bases of a prism are polygons and the area of the base is determined by the area formula for that polygon. The faces of the lateral area of a right prism are always rectangles (and at least three as in a right triangular prism) whereas the lateral area of a cylinder is one rectangle wrapped around the circular base. Formulas for volume are alike in that the volume is the base area times the height of the solid.

Another good comparison is the regular square pyramid and the right prism. The regular square prism has one square base in contrast to the right prism’s two bases. The lateral faces of a right prism are rectangles unlike the triangular lateral faces of a pyramid. Another distinction of a pyramid is the common intersecting point of the lateral faces called the vertex of the pyramid. The volume formulas are similar in that base area times height of the solid is used in both, and yet the volume of a pyramid is one-third the volume of a prism with the same height and base dimensions.

“Now, let’s look at some similar solids and see how scale factors and proportions can help us solve problems.” Use the following visuals.

Pyramid A has a surface area of 35 square meters and a volume of 24 cubic meters. The scale factor of Pyramid A to Pyramid B is 1:4. What is the surface area and volume of Pyramid B?

Measurements of Pyramid B:

 

V = 1,536 cubic meters

Have students come up with another problem, with two new similar solids. Ask them to make some conclusions based on these two examples. “What does this tell you? How can you use this information? What connections can you make to what we did previously?”

“Finally, we see how change in one dimension affects the other measurements of surface area and volume. How does the effect of change in one dimension on surface area and volume vary across solids? For example, does a change in radius more heavily impact the surface area and volume of a sphere, a cone, or a cylinder?”

Guide students through an example using a cone with the following dimensions:

r = 6

h = 9

SA = B + (1/2)Cl, where B is the area of the base, C is the circumference, and l is the slant height.

V = (1/3)Bh , where B is the area of the base. [IS.11 - All Students]

V = (1/3)π r² h

= (1/3) π 36 • 9

= (1/3) π 324

= 108π

≈ 339.29

“Now, let’s change one dimension on the cone and examine the new surface area and volume. Let’s compare the results using a chart. Let’s also use the chart to examine how changes in one dimension impact surface area and volume across solids. We can answer our question, ‘Does a change in radius more heavily impact the surface area and volume of a cone or a cylinder?’”

“We can see that a change in one dimension on the cone more heavily impacted the volume. Why would that be? Please fill in the columns for the remaining solids.”

“Compare the rows for cone and cylinder. What did you determine? Which is the most heavily influenced by a change in the dimension, radius? Why?”

Review the formulas for finding surface area and volume. Use the chart below.

Close with a discussion on relationships, patterns, similarity, and overall ideas related to comparisons within and between solids. “How did this activity increase your overall understanding of relating surface areas and volumes for solids?”

Review Activity

Have students create a brief presentation on the use of nets for the purpose of visualization, spatial estimation, and comparisons, involving surface area and volume. Have students consider these questions:

• What relationships are important to note?

• Where and how are such comparisons made in the real world?

Students choose one key area to focus on, such as nets and pure visualization, estimation and comparisons, or relationships involving surface area and volume. The utmost goal is for students to make connections across all three activities, thus walking away with a strong foundation and understanding of nets, surface area, volume, and relationships thereof. These presentations should be approximately 5–10 minutes in length. Approximately 25 minutes of preparation time should be given. If you wish, you can assign this as homework and have students bring this to class the next day. This would be an excellent way to review the topics of this lesson before proceeding with the next lesson.

Extension:

• Expand the discussion into the realm of Platonic Solids. Students can visualize and draw nets for the five Platonic Solids, as well as estimate surface area and volume for each. The discussion should not focus on the actual calculations, but instead, focus on visualization, estimation, and reasoning skills. Students should make connections within and across the solids, by way of measurements and changes in dimensions of solids and the impact on measurements.

  In addition, students can relate these solids to everyday occurrences. Where do you see Platonic Solids? When might Platonic Solids measurements be needed in the real world? Can you provide an example or examples?

• Provide students with an opportunity to explore Platonic solids and their nets, as well as pentominoes and their nets, using an applet found at: http://www.cs.mcgill.ca/~sqrt/unfold/unfolding.html .