## THE MAX AND MIN OF A CAPT'N CRUNCH BOX Grades 8-10

Students make models to explore the process of keeping one factor (either area or volume) constant while minimizing or maximizing the other.
Mathematical Eye II Program 4 "Paper Engineering"
Mathmedia #10 "Area and Volume"
The Riddle of Wizard's Oak #3 "From 3D to 2D
Students will use knowledge of area and volume to explore minimum and maximum. Curriculum standard 13 refers to students study of the underpinnings of calculus. This lesson involves every student in hands on activity making models. The students will determine a "best" shape for getting the maximum volume or minimum area.
For the opening activity each pair of students will need:
• a pair of scissors
• a roll of tape
• one sheet of regular typing paper.
To create their own box each pair will need
• a used cereal box
• scissors
• tape
• poster board
Discuss the construction of a topless box from the sheet of typing paper. Have students work with a partner. I have found assigning each pair a different measure for the tabs by increments of quarter or half inch (depending on the number of students) works well. Discuss reasonable measurements. Less than a half inch is too small and over two and a half is probably too much. Have each pair create their box and find the volume. This activity keeps the area relatively constant and changes the volume. Find the box with the greatest volume. This activity can be extended with the use of equations and examination with a graphing utility to find the maximum volume. Where x is the measure of one side of a square tab: A=(8.5 )(11)-4x2 and Volume y=(8.5-2x)(11-2x)x. The best screen for viewing is x between .5 and 2.5 and y between 60 and 70 cu. in. This activity allows less aggressive students to work with a partner and allows for many extensions. (The Mathematics Teacher, "A Core Curriculum in Geometry", April, 1992, Page 300-302 , M. Tietze). For the video and the cereal box the problem is reversed where the volume is to remain about the same and the area will change.
START Mathmedia Program 10 "Area and Volume" and tell students to answer the following: "Look for and be able to give two reasons you might want to change the design of a cereal box." PAUSE when she says, "Okay is there anything else we should know?" Check for understanding with answers that they want to save material and a unique shape would sell. Review that they will compute the volume and area as on the video. RESUME where left off and play until she talks about BB players getting \$100,000 for their picture on a box of cereal. Ask, "What rectangular prism gives the most volume for the least area?" Check for understanding with the answer cube. Then ask, "Why did they not choose the cube?" Discuss the reasons given about shelf space. Ask what other shapes are possible. Go to the beginning of Mathematical Eye II #4 "Paper Engineering". PLAY to where they show the Easter egg packages. FAST FORWARD through the paper folding and Eulers Formula (the cartoon goat and Swiss mountains). RESUME the second time you see the green triangular cake. Play through the end of the program.

The students are asked to make a new design for a cereal box. I make this more opened ended than the video Students may use any justification for the new design and are not limited to rectangular prisms. They have had work with area and volume formulas for prisms, pyramids, cylinder, cones, and spheres. As a project some have explored geodesic domes. Their assignment is to give the area and volume of the old box. The students are to state a reason for a new design. This is to be substantiated by calculations for area and volume of the new design. Experimentation reveals that the best design for volume is a sphere but it is impractical as a design. In the case of a cylinder, when the diameter and height are close to being equal the volume is maximized.

Students use poster board and make the model. The area and volume of the original box is compared with the new one. Encourage creativity. Students will want to design the box, put a toy inside, make a spout to open and other gimmicks.
The students present their findings in the form of a letter to the cereal company. They need to give the old area and volume with the improved. They need to express why they picked the shape they used. The letter should be addressed to the company (it is on the cereal box) and may be mailed. The assessment for the activity is done with this letter and the model they made.
Have students pick a product they see in the store. After recording the shape of at least three different packages, they discuss both the advantages and disadvantages for each. Ideas about convenience, attractiveness, economy, and ecology should be considered..
Have students contact a manufacturer of packaging materials and see if there is a design department. Ask about what considerations they use in creating a new package for a product. Ask if they would speak to the class and possibly bring samples of package.. Maybe even samples of a product!

### Master Teacher: Martha Tietze

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