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Open-Ended Template

 

 

Names:  Vivian Loewenstern, Merle Reinford, Frank Rudnik, Karen Tucker

 

Grade Level:  9 – 12 

Content Area:  Measurement/Estimation

 

PA Standards Addressed:

2.1, 2.2, 2.3, 2.8, 2.9, 2.11

NCTM standards addressed:

Measurement, Number and Operations,

Algebra, Geometry, Problem Solving,

Reasoning and Proof, Communication,

Connections, and Representation

 

 

 

Problem Name:  Cereal Box Design

 

Problem:

 

        A cereal company wants to minimize the cost of packaging their cereal.  The cereal is packaged in a box with dimensions of 1 inch by 4 inches by 16 inches (pictured below).  Find the integral dimensions of the box which would require the least material to produce, but would hold the same volume.

 

       

 

 

 

                                                     4”                                                       (Figure may not be drawn to scale.)

 

 

                                                                                             1”

                                                                       16”

 

Directions:

 

Show or describe each step of your work, even if you did it in your head or used a calculator,

 

                                                                                                 AND

 

write an explanation stating the reason(s) why you chose your solution. Consider the relationship between the dimensions of any box and the minimum material required to produce that box.

 

 

 

 

 

Problem Name:  Cereal Box Design

 

Grade Level:  9 – 12                                           Content Area:  Measurement/Estimation

 

Problem Solution:

 

Find the volume of the given box:  V = lwh

                                                        V = 16 x 4 x 1 = 64 cubic inches

 

Find the surface area of the given figure:  SA = 2lw + 2wh + 2lh

                                                                         SA = 2(16)(4) + 2(4)(1) + 2(16)(1) = 168 sq in

 

Find the factors of 64 to keep the volume constant: 

 

a)    1 x 1 x 64

d)    4 x 4 x 4

b)   1 x 8 x 8

e)    16 x 2 x 2

c)   1 x 32 x 2

f)     8 x 4 x 2

 

Find the surface areas of the other possible boxes:  

a)       SA =  2(1)(1) + 2(1)(64) + 2(1)(64) = 258 sq in

b)       SA =  2(1)(8) + 2(1)(8) + 2(8)(8) = 160 sq in

c)       SA =  2(1)(32) + 2(32)(2) + 2(1)(2) =  196 sq in

d)       SA =  2(4)(4) + 2(4)(4) + 2(4)(4) = 96 sq in

e)       SA =  2(16)(2) + 2(16)(2) + 2(2)(2) =  136 sq in

f)        SA =  2(8)(4) + 2(4)(2) + 2(8)(2) = 112 sq in

 

The box with the minimum surface area is a box with dimensions of 4 inches by 4 inches by 4 inches.  The box is a cube.

 

The surface area is minimized as the dimensions approach the same number.

 

 

 

Problem Name: Cereal Box Design

 

Grade Level:  9 –12                                                   Content Area: Measurement and Estimation

 

SPECIFIC RUBRIC

 

5

Advanced Understanding – Excellent

Ø       Correct numerical answer – 96 sq. in.

Ø       All work shown and explained – (However, not all surface areas need to be calculated)

Ø       Explanation includes why steps were performed

Ø       Nothing incorrect – (All quantities have appropriate labels)

Ø       Why the solution chosen was a minimum – (e.g. The surface area is minimized when the dimensions approach the same number)

4

Satisfactory Understanding

Ø       Correct numerical answer – 96 (Labels can be missing)

Ø       All work shown and explained

Ø       Explanation includes why steps were performed

Ø       Nothing incorrect

Ø      Generalization is missing – Student did not state the reason why the chosen solution produces the minimum

3

Almost Satisfactory Understanding

A)

Ø       Correct answer with/without labels

Ø       All work shown with no explanation

Ø       Work may contain one or two computational errors

B)

Ø       Correct answer with/without labels

Ø       Adequate work shown or explained – (Some steps are missing but you can follow what is being done)

Ø       All appropriate mathematical procedures are used

C)

Ø       Incorrect numerical answer due to one calculation or copying error

Ø       All procedures correct and adequate work shown

2

Partial Understanding

A)

Ø       Correct answer with minimal work or some explanation

B)

Ø       Incorrect numerical answer due to more than one calculation or copying error

Ø       All procedures are correct

C)

Ø       Incorrect numerical answer due to one procedural error (e.g. When calculating surface area all 6 surfaces were not included)

D)

Ø       Incorrect or no answer with some understanding of procedures

Ø       Must have: 1) volume of original box   2) surface area of original box   3) an understanding that to keep volume constant, factors of the original volume must be used

 

1

Minimal Understanding

A)

Ø       Correct solution with no work or explanation OR the work and / or explanation is illegible

B)

Ø       Incorrect numerical answer due to more than one procedural error

C)

Ø       Incorrect or no answer with minimal understanding of procedures.

Ø       Must have 2 of the following: 1) volume of original box   2) surface area of original box   3) an understanding that to keep volume constant, factors of the original volume must be used

0

No Understanding

Ø      Correct answer - however work doesn’t support the answer

Ø      “I don’t know” – “I don’t understand”

Ø      No response

 

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