Summary
Students find the volume and surface area of a rectangular cardboard box (such as a cereal box), and then figure out how to convert that box into a new, cubical box having the same volume as the original. As students construct new, cubeshaped boxes from the original box material, they discover that the cubical box has less surface area than the original, and thus, a cube is a more efficient way to package things.Engineering Connection
Engineers design product packaging to protect the products and efficiently utilize materials. Students think like engineers when they determine that a cube is a more efficient use of material.
Learning Objectives
 Students will be able to determine the dimensions of a cube when given its volume.
 Students will be able to assert that a cube has less surface area than a rectangular prism of the same volume, and then prove this assertion with examples.
Educational Standards
Each TeachEngineering lesson or activity is correlated to one or more K12 science,
technology, engineering or math (STEM) educational standards.
All 100,000+ K12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN),
a project of D2L (www.achievementstandards.org).
In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics;
within type by subtype, then by grade, etc.
Each TeachEngineering lesson or activity is correlated to one or more K12 science, technology, engineering or math (STEM) educational standards.
All 100,000+ K12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).
In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.
NGSS: Next Generation Science Standards  Science

Evaluate competing design solutions using a systematic process to determine how well they meet the criteria and constraints of the problem.
(Grades 6  8 )
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This Performance Expectation focuses on the following Three Dimensional Learning aspects of NGSS:Science & Engineering Practices Disciplinary Core Ideas Crosscutting Concepts Evaluate competing design solutions based on jointly developed and agreedupon design criteria. Alignment agreement:
There are systematic processes for evaluating solutions with respect to how well they meet the criteria and constraints of a problem. Alignment agreement:
Common Core State Standards  Math

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.
(Grade 6 )
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Fluently divide multidigit numbers using the standard algorithm.
(Grade 6 )
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Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.
(Grade 6 )
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Solve realworld and mathematical problems involving area, surface area, and volume.
(Grade 6 )
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Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
(Grade 7 )
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Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
(Grade 7 )
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International Technology and Engineering Educators Association  Technology

Use tools, materials, and machines safely to diagnose, adjust, and repair systems.
(Grades 6  8 )
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Interpret and evaluate the accuracy of the information obtained and determine if it is useful.
(Grades 6  8 )
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The selection of designs for structures is based on factors such as building laws and codes, style, convenience, cost, climate, and function.
(Grades 6  8 )
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State Standards
North Carolina  Math

Solve realworld and mathematical problems involving area, surface area, and volume.
(Grade
6 )
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Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.
(Grade
6 )
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Fluently divide multidigit numbers using the standard algorithm.
(Grade
6 )
More Details
Do you agree with this alignment?

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.
(Grade
6 )
More Details
Do you agree with this alignment?

Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
(Grade
7 )
More Details
Do you agree with this alignment?

Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
(Grade
7 )
More Details
Do you agree with this alignment?
Materials List
Each student needs:
 a pair of identical, rectangular cardboard boxes; ask students to bring these from home, thus it is unlikely that any students will bring in the exact same box types; boxes that work well are small or mediumsized product packaging boxes from cereal, crackers and pasta, macaroni and cheese, pudding, cake mix, hot chocolate, tea, instant oatmeal packets, computer diskettes, contact lens solutions, toothpaste
 sturdy scissors
 metric ruler
 large envelope or zipperstyle plastic bag (such as quart or gallon sizes)
 large paper clip
 Instructions for Students Handout
To share with the entire class:
 masking tape, several rolls
 transparent tape, several rolls
Worksheets and Attachments
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PreReq Knowledge
 Students should know how to determine both the surface area and volume of a rectangular prism.
 Students should be able to measure lengths accurately to the nearest millimeter.
Introduction/Motivation
[Show students this video: http://pbskids.org/designsquad/video/packagedesign/.]
What is a packaging engineer? [Answer: An engineer who has to design packages for products, keeping designs marketable yet costeffective.] What sorts of things do packaging engineers need to consider? [Answer: keeping solutions that look good to the consumer, not waste materials, use cheap materials, pick dimensions that keep the solution costeffective, i.e. how many will fit in each shipping box, etc.]
Today, you are going to take one of your two, identical boxes, and cut it up and tape it back together to make a cubeshaped box that has the same volume as the original, rectangular box. In order to create a cubeshaped box from a rectangular box, you will have to work backwards. For a cube of known volume, you need to be able to figure out how to find its dimensions. If the volume of a cube is equal to its length cubed, the length of any side of a cube is equal to the cube root of its volume. (Work through some simple examples to help illustrate this.) For example, what would the dimensions be for a box with a volume of 8 cubic cm, or for a box with a volume of 27 cubic inches?
(Move on to some harder examples.) What if the volume of a box was 21 cubic inches? (If students have graphing calculators such as TI82 or TI83, they can find cube roots easily using the MATH function key. If not, they will get some good practice with estimation and trialanderror as they determine that the cube root of 21 is about 2.76. When students make their own cubeshaped boxes, they will work in cm and mm, so cube roots need not be taken out beyond the nearest hundredth. Once students are clear on how to work these types of surface area and volume problems, they are ready to conduct the activity.)
Procedure
 Gather materials and make copies of the Instructions for Students Handout.
 Give each student a copy of the handout and direct them to read all the instructions before beginning.
 Hand out the materials and have students begin, guided by the handout.
 Measure the dimensions of your box to the nearest millimeter: length, width and height.
 Calculate the box surface area.
 Calculate the box volume. Remember to include units in your answers.
 Your box was probably made from one flat piece of cardboard, so carefully open the glued edges to deconstruct your box. Trim off any parts or flaps that were hidden from view when the box was intact. Look at the identical other box to make sure before cutting.
 Calculate what the length of any side of your new cubeshaped box should be.
 Trade papers and boxes with another student in class to check each others' measurements and calculations. Correct as necessary.
 On the inside of the openedout box, efficiently draw the six identical squares you need to make your cubeshaped box. You may need to tape together some of the oddshaped leftover scraps in order to make the last one or two cube sides.
 Cut out the six cube sides, saving all leftover scraps and putting them in an envelope or bag.
 Neatly tape the six sides together to form the cubeshaped box. Tip: Use masking tape on the inside so the cube is sturdy and looks good. Use clear tape on the outside for additional strength.
 Calculate the surface area of the cubeshaped box.
 Compare this number to the surface area of the original box. Same? Different? By how much?
 Find the total area of all the scraps. For the oddly shaped ones, divide them into squares and rectangles to make it easier to measure and calculate area.
 Compare this number to the number you figured was the difference in the surface areas between the original and cubeshaped boxes. Same? Should they be the same? Why or why not?
Assessment
Activity Calculations: Ask students to determine the difference in surface areas of a rectangular box of given dimensions, and a cubeshaped box having the same volume as the rectangular box.
Quiz: At activity end, provide a sketch of a consumer product packaged in a rectangular box, and include the dimensions of the box in the sketch. Ask students to sketch a cubeshaped box, including its dimensions, that has the same volume as the rectangular box.
Investigating Questions
 How do the volumes of your two boxes compare? (Answer: They are identical.)
 How do the surface areas of your two boxes compare? (Answer: The cubeshaped box has less surface area than the original, rectangular box.)
 Which method of packaging consumer goods uses less cardboard—rectangular or cubical boxes—assuming equal volumes? (Answer: Cubical boxes have less surface area than rectangular boxes of the same volume, so the cubical box use less cardboard.)
 If cubeshaped boxes use less material, why do most consumer goods come packaged in rectangular boxes? (See the Lesson Closure section.)
 How does the total area of the leftover scraps from your original box compare to the difference in surface area between your original and cubeshaped box? (Answer: The areas should be the same, but for most students their total scrap area will be at least several square centimeters less than the difference in box surface areas. See the Lesson Closure section.)
Troubleshooting Tips
Remind students to keep all of the scraps from their original box as they cut the cube sides from it. If some students have the same boxes as others, suggest they write their initials on the backs of any scraps.
Most students will not be able to complete the construction of their cubeshaped boxes during one class period, so use paper clips to keep the parts of boxesinprogress together overnight. Have students store their scraps in plastic bags or envelopes, which can be clipped to the partially assembled boxes.
Activity Extensions
Give the students a problem that will force them to evaluate competing design solutions while considering the constraints of a design problem.
Question:
Katie is a packaging engineer for a new company that is producing candy! She has been asked to determine the optimal shape and size of the candy packages, and must follow these constraints, or guidelines:
 The company would like to fit as many candy packages as possible into shipping boxes that are 18" x 18" x 18".
 The company needs a design solution that ensures the candy packages won't shift around inside the shipping boxes during transit.
Katie has done most of the calculations already and has narrowed down her choices to:
 A circular design with a radius of 3".
 A rectangular design with dimensions of 3" x 4.5" x 1.5".
 A cube design with sides that are 3" each.
Which option will allow the company to ship the most packages per box?
Solution:
 The students can first eliminate the circular design. Based on the constraints given by the company, they do not want any spaces that may result in loose packing. If they packed the boxes with interlocking circles, they wouldn't shift, however you would fit less boxes than the cubed design.
 The cubed design would allow for 216 candy boxes to be packed per shipping box.
 The rectangular design would allow for 288 candy boxes to be packed per shipping box.
Discussion:
Therefore, the rectangular design is the best based on the constraints provided by the company! What if the company added that they also wanted a novel design solution so that it would be highly marketable to the consume? Well, that would make it more complicated. A circular package, while less packable than the rectangular, might actually look really cool to a consumer! At this point, the company would need to evaluate the tradeoffs between multiple designs. Sometimes finding one design that abides by all engineering constraints can be tricky!
Contributors
Mary R. Hebrank, project writer and consultantCopyright
© 2004 by Regents of the University of Colorado; original © 2004 Duke UniversitySupporting Program
Engineering KPhD Program, Pratt School of Engineering, Duke UniversityAcknowledgements
This content was developed by the MUSIC (Math Understanding through Science Integrated with Curriculum) Program in the Pratt School of Engineering at Duke University under National Science Foundation GK12 grant no. DGE 0338262. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.
This lesson and its associated activities were originally published, in slightly modified form, by Duke University's Center for Inquiry Based Learning (CIBL). Please visit http://ciblearning.org/ for information about CIBL and other resources for K12 science and math teachers.
Last modified: August 22, 2018
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