Packaging Engineering

Project Packaging Engineering Objectives To use nets to represent and calculate the surface area of 3-dimensional shapes; and to apply concepts of area, surface area, and volume to design a package for a pair of shoes. www.everydaymathonline.com etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Doing the Project Recommended Use During or after Unit 9 Key Concepts and Skills Apply place-value concepts to convert among units of measurement. [Number and Numeration Goal 1] Use nets and formulas to find the surface area of 3-dimensional shapes. [Measurement and Reference Frames Goal 2] Find the volume of 3-dimensional figures. [Measurement and Reference Frames Goal 2] Represent 3-dimensional shapes with nets. [Geometry Goal 2] Key Activities Students discuss issues related to the design of packages, including area, surface area, and volume. They draw nets to represent 3-dimensional figures and to find surface area. They create original designs for shoe packages. Key Vocabulary packaging engineer Extending the Project Mathematical Practices SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP8 Content Standards 6.G.1, 6.G.2, 6.G.4 Materials Math Masters, pp. 402 402C and 408 Student Reference Book, pp. 214 226 packages of various shapes and sizes base-10 blocks (1 cube, 1 long, 1 flat, and 1 large cube for demonstration purposes; optional) per partnership: 120 cm cubes scissors ruler tape Students use the Internet to research interesting or unique package designs. Materials computer with Internet access Additional Information A packaging engineer designs and selects packaging materials and containers such as boxes, bottles, and cartons for companies involved in manufacturing or shipping goods. People who work in this industry must be problem solvers prepared to tackle social, economical, and environmental challenges. The packages they create must appeal to the consumer, but also protect the product and be cost-effective. Packaging engineers must consider the shapes of 3-dimensional objects as well as the surface area and volume of each container in order to construct a package that is both practical and economical. Advance Preparation Encourage students to bring in packages from different items or advertisements that depict different packaging designs. Try to collect a variety of packages, including cylindrical canisters (such as soup cans), rectangular prisms (such as cereal boxes), other prisms (such as candy packages), and irregularly-shaped packages (such as bags or liquid cleaning product containers). You might also display pictures of interesting packages from Web sites such as the following: www.puma.com/cleverlittlebag www.toblerone.co.uk blog.pentawards.org www.thedieline.com www.packagedesignmag.com www.gdusa.com/issue_2010/03_mar/feature/winners/index.php Depending on the needs of your students, consider making this a two-day project. Use the first day to review the concepts of area, surface area, and volume. On the second day, have students create designs for the shoe packages. Consider having students bring in an additional pair of shoes to use for reference as they design their shoe packages. 958 Project 9 Packaging Engineering

1 Doing the Project Introducing Packaging Engineering WHOLE-CLASS DISCUSSION Ask: What are some everyday items that are sold in packages? Sample answers: eggs, juice, cereal, shoes, paper towels, shampoo When you think of packages, which shapes come to mind? Sample answers: rectangular prisms, cylinders Why do you think this is? Answers vary. Display several different packages for students to examine, or display pictures of packages from printed advertisements or the Internet. Include a variety of shapes and sizes. Ask students to comment on the designs of the different packages. Use questions like the following to guide the discussion: How would you describe the design of this package? What item is this package designed to hold? Could there be another package design that would hold the same item? What do you notice about the size of this package? Why do you think the package is this size? Display two different packages for the same product, such as a bag of cereal and a box of cereal. Ask: What are the advantages and disadvantages of each package design? Explain that a packaging engineer is someone who designs and selects containers and packaging materials for companies involved in manufacturing or shipping goods. Tell students that in this project, they will examine some of the issues that a packaging engineer must consider, and then they will have the opportunity to design their own packages. For the purposes of these activities, packages can be thought of as containers in which an item or items are packed for transportation, storage, and sales. Project 9 958A

Name Date Time 9 Polyhedron Net Project Master Surface Area and Package Design (Math Masters, p. 402) WHOLE-CLASS ACTIVITY Math Masters, p. 402 One issue a packaging engineer must consider is the surface area of the container. Ask students why surface area might be important to a packaging engineer. Sample answer: Surface area determines how much material the engineer must use to create the package. A package with less surface area will use fewer materials and be more cost efficient than one with more surface area. Tell students that one step a packaging engineer might take when developing a design is to draw a net of the package. Drawing a net can help the engineer visualize a design and calculate the amount of material needed without actually creating the package. Distribute a copy of Math Masters, page 402 to each student. Have students predict the shape that will be made when the net is assembled and check their predictions by cutting out and assembling the net. Square pyramid Ask students to use the assembled net and a ruler to find the surface area of the pyramid in square inches. Surface area = 21 in.² Ask students to share their strategies for finding the surface area. NOTE The prisms and cylinders that students have encountered in Everyday Mathematics are right prisms and right cylinders, meaning that the bases are perpendicular to the other faces of the prisms. Students should assume that the solids pictured on Math Masters, page 402A are right prisms and a right cylinder. Project Master Name Date Time 9 Using Nets to Find Package Surface Area A company has given you sketches and dimensions of several packages it wants to create. The company has asked you to determine how much material will be needed to create each package. Draw and label a net of each package using the sketches below. Then use the nets to calculate the surface area of each package. Use 3.14 for π. Sample nets are given. Package Rectangular prism: 5.5 in. Triangular prism: 2 in. 4 in. 12 in. 5.5 in. 8 in. Net and Surface Area 4 in. 2 in. 4 in. 2 in. Surface area: 82 in. 2 12 in. 2 in. 2 in. 10 in. 8 in. Using Nets to Find Package Surface Area (Math Masters, p. 402A) PARTNER ACTIVITY Have students work in partnerships to complete Math Masters, page 402A. Encourage students to handle and disassemble the sample packages from the initial discussion if it helps them to visualize the nets. When most students are finished, discuss their strategies and solutions. Note that there are several possible nets that can be drawn for each 3-dimensional figure. Have students compare some of the different possible nets. For Problem 3, encourage students to discuss the relationship between the circumference of the circular base and the length of the rectangular section of the net. Students may need to remove a label from a soup can to help them visualize the net of the cylinder. NOTE As students work, encourage them to think about how they can use easy area formulas to help them remember how to find the area of other shapes. For example, the area of a triangle can be found by composing two congruent triangles into a parallelogram and taking half of that area: A = 1_ 2 b h. 10 in. 8 in. 12 in. Surface area: 408 in. 2 Cylinder: 5 in. 2.6 in. 5 in. 2.6 in. 31.4 in. 5 in. 238.64 Surface area: in. 2 Math Masters, p. 402A 958B Project 9 Packaging Engineering

Volume and Package Design (Math Masters, p. 402B) PARTNER ACTIVITY Project Master Name Date Time 9 Open Rectangular Prism Another issue that a packaging engineer must consider when designing a package is the volume of the package. Ask students why they think volume is an important consideration for a packaging engineer. Sample answer: The volume of a package determines how much space is available for a particular product. A package must have enough volume to hold the item or items without damaging them. Distribute one copy of Math Masters, page 402B to each partnership. Have partners cut out the net and assemble the open rectangular prism. Tell the class that there are many ways to find the volume of a container. One way is to pack the container with unit cubes. Tell students that they will pack this prism with centimeter cubes to find the volume in cubic decimeters. Ask: What fraction of a decimeter is a centimeter? 1_ dm, or 0.1 dm, 10 because there are 10 centimeters in 1 decimeter. What fraction of a square decimeter is a square centimeter? 1_ 100 dm2, or 0.01 dm 2, because there are 100 square centimeters in 1 square decimeter (10 cm 10 cm = 1 dm 1 dm). What fraction of a cubic decimeter is a cubic centimeter? 1_ dm³, or 0.001 dm³, because there are 1,000 cubic 1,000 centimeters in 1 cubic decimeter (10 cm 10 cm 10 cm = 1 dm 1 dm 1 dm). Math Masters, p. 402B Adjusting the Activity If students have difficulty determining the relationship between square centimeters and square decimeters or cubic centimeters and cubic decimeters, have them use base-10 blocks to model the relationships. 1 cm = length of cube 1 dm = length of long 1 cm 2 = area of cube face 1 dm 2 = area of flat face 1 cm 3 = volume of cube 1 dm 3 = volume of big cube A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Have students find the volume of the prism by packing the prism with centimeter cubes. Ask them to give their answer in cubic decimeters. 0.12 dm 3 Ask students to explain how they found the volume. Sample answer: 120 centimeter cubes fit in the prism. Each cubic centimeter is 0.001 dm 3. 120 0.001 is 0.12 dm 3. Next, ask students to find the dimensions of the prism in decimeters. 0.6 dm by 0.4 dm by 0.5 dm Project 9 958C

Project Master Name Date Time 9 Designing a Shoe Package You will be designing three packages that could hold a pair of shoes that might fit a sixth-grader. Each design will be a different shape: one will be a rectangular prism, one will be a triangular prism, and one will be a cylinder. All three designs must be able to easily hold the shoes. You might want to start by making estimates for the minimum dimensions and volume of each of your package designs. Consider using a pair of shoes to help you make your estimates. Then carefully draw a net for each design on a separate piece of grid paper. 1 unit on the grid should represent 2 inches. 1. Calculate the exact surface area and volume of each design and record them in the chart below. Design Shape Surface Area (in. 2 ) Volume (in. 3 ) Rectangular prism Answers vary. Triangular prism Cylinder 2. Which of your designs do you think is the most economical? Explain your answer. Sample answer: I think the triangular prism is the most economical because it has the smallest surface area and will use the fewest materials. 3. Which of your designs do you think will be the best for a store display? Explain your answer. Sample answer: I think the rectangular prism would be the best for a store display because the packages would stack easily. Have them calculate the volume of the prism in cubic decimeters using the formula for volume of a rectangular prism (V = l w h). 0.6 dm 0.4 dm 0.5 dm = 0.12 dm 3 Ask students to explain why the results from counting centimeter cubes and the results from using a formula are the same. Sample answer: They are both different strategies for finding the volume of a rectangular prism. Since we used the same prism for both strategies, the volume should remain the same. Ask students whether they think packing containers with unit cubes would be a good way for a packaging engineer to find the volume of containers. Sample answer: No. Many shapes cannot be completely filled with whole unit cubes. Using formulas is a better strategy. 4. Which of your designs would you recommend to a shoe company to use for a new pair of shoes that they are marketing? Use surface area, volume, and the overall package design to defend your choice. Answers vary. Designing a Shoe Package (Math Masters, pp. 402C and 408; Student Reference Book, pp. 214 226) PARTNER ACTIVITY Math Masters, p. 402C Ask: What issues should packaging engineers keep in mind when creating packages for specific products? Be sure to address the following points in the discussion: Packaging engineers must think about surface area when they create packages. The larger the surface area, the more materials must be used to make the package. Packaging engineers must think about volume when they create packages. The package must be able to actually hold the product they are packaging. Packaging engineers must think about the shape and size of the product that will be packaged. The container must be suitable for the product itself. Packaging engineers must think about the shape of their packages because they must consider how the packages will be transported, stored, and placed on shelves or displays. Packaging engineers must think about the overall design of their packages. The packages should be practical, but should also stand out to a consumer. Tell students that they should keep these points in mind as they design three possible packages that could be used to hold a pair of shoes. Distribute at least three sheets of centimeter grid paper (Math Masters, page 408) to each partnership and review the directions for the design project on Math Masters, page 402C. Have students refer to Student Reference Book, pages 214 226 for area, surface area, and volume formulas. 958D Project 9 Packaging Engineering

2 Extending the Project Researching Unique Package Designs INDEPENDENT ACTIVITY Students use the Internet to find an interesting or unique package design. They share their thoughts about why the packaging engineer might have designed that particular package. If possible, have them find information about the package design, or similar designs, on the Internet. Project 9 958E

Name Date Time 9 Polyhedron Net Copyright Wright Group/McGraw-Hill 402

Name Date Time 9 Using Nets to Find Package Surface Area A company has given you sketches and dimensions of several packages it wants to create. The company has asked you to determine how much material will be needed to create each package. Draw and label a net of each package using the sketches below. Then use the nets to calculate the surface area of each package. Use 3.14 for π. Package Net and Surface Area Rectangular prism: 2 in. 4 in. 5.5 in. Surface area: in. 2 Triangular prism: 12 in. Copyright Wright Group/McGraw-Hill Cylinder: 10 in. 2.6 in. 8 in. 5 in. Surface area: in. 2 Surface area: in. 2 402A

Name Date Time 9 Open Rectangular Prism Copyright Wright Group/McGraw-Hill 402B

Name Date Time 9 Designing a Shoe Package You will be designing three packages that could hold a pair of shoes that might fit a sixth-grader. Each design will be a different shape: one will be a rectangular prism, one will be a triangular prism, and one will be a cylinder. All three designs must be able to easily hold the shoes. You might want to start by making estimates for the minimum dimensions and volume of each of your package designs. Consider using a pair of shoes to help you make your estimates. Then carefully draw a net for each design on a separate piece of grid paper. 1 unit on the grid should represent 2 inches. 1. Calculate the exact surface area and volume of each design and record them in the chart below. Design Shape Surface Area (in. 2 ) Volume (in. 3 ) Rectangular prism Triangular prism Cylinder 2. Which of your designs do you think is the most economical? Explain your answer. 3. Which of your designs do you think will be the best for a store display? Explain your answer. Copyright Wright Group/McGraw-Hill 4. Which of your designs would you recommend to a shoe company to use for a new pair of shoes that they are marketing? Use surface area, volume, and the overall package design to defend your choice. 402C