Inquiry-based Lesson Project
Stark_Team 4
Hahn, Kosenski, & Presutto
1. Lesson Title: What’s “Width” the Cereal? (Package Optimization)
2. Lesson Summary: Students will examine the packaging of several types of
cereal boxes, make generalizations about their observations. Students will
minimize the packaging.
3. Key Words:
 Surface Area - refers to the total area of the exposed surface of a 3dimensional solid, such as the sum of the areas of the exposed sides of
a polyhedron.
 Volume - The amount of space occupied by a three-dimensional
object or region of space, expressed in cubic units.
 Optimization - Calculating the independent variables in a function so
as to generate the best function value for a given set of conditions.
Optimization usually involves maximizing or minimizing a function.
4. Background Knowledge:
This activity will have students exploring relationships between surface area
and volume of rectangular prisms to determine the most economical
packaging. Students should have a basic knowledge of how to use a
TI-Nspire calculator, as well as the following grade level indicators:
- Measurement Grade 7 Indicator 4:
Solve problems involving proportional relationships and scale factors;
e.g., scale models that require unit conversions within the same
measurement system.
-Measurement Grade 7 Indicator 5:
Analyze problem situations involving measurement concepts, select
appropriate strategies, and use an organized approach to solve narrative
and increasingly complex problems.
-Measurement Grade 7 Indicator 8:
Understand the difference between surface area and volume and
demonstrate that two objects may have the same surface area, but
different volumes or may have the same volume, but different surface
areas.
-Measurement Grade 7 Indicator 9:
Describe what happens to the surface area and volume of a threedimensional object when the measurements of the object are changed;
e.g., length of sides are doubled.
-Patterns Function and Algebra Grade 7 Indicator 8:
Use formulas in problem-solving situations.
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Inquiry-based Lesson Project
Stark_Team 4
Hahn, Kosenski, & Presutto
-Patterns Function and Algebra Grade 7 Indicator 11
Use graphing calculators or computers to analyze change;
e.g., distance-time relationships
5. State Standard and Benchmark:
-Patterns Function and Algebra Grade 8 Indicator 16:
Use graphing calculators or computers to analyze change;
e.g., interest compounded over time as a nonlinear growth pattern
-Measurement Grade 8 Indicator 3:
Use appropriate levels of precision when calculating with
measurements.
-Patterns Function and Algebra Grade 9 Indicator 3:
Describe problem situations (linear, quadratic and exponential) by using
tabular, graphical and symbolic representations.
-Measurement Grade 9 Indicator 5:
Solve problems involving unit conversion for situations involving
distances, areas, volumes and rates within the same measurement
system.
- Patterns Functions and Algebra Grade 10 Indicator 10:
Solve real-world problems that can be modeled using linear, quadratic,
exponential or square root functions.
6. Learning Objectives:
Student Will Be Able To:
a. Efficiently gather data by calculating the surface area and volume of
several different size cereal boxes to test for correlations between
dimensions, surface area and volume.
b. Make conjectures on class data.
c. Determine the best dimensions for minimal packaging.
7. Materials:
a. Several cereal boxes.
b. Standard rulers
c. TI-Nspire graphing calculator
8. Procedures:
1. Students are placed into teacher selected groups.
2. Each group receives a cereal box.
3. Groups will measure the length, width, and height of their cereal box.
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Inquiry-based Lesson Project
Stark_Team 4
Hahn, Kosenski, & Presutto
4. Record data on student record sheet. (Record Sheet 1)
5. Calculate the surface area and volume of original dimensions.
(Record Sheet 1)
6. Students will be completing three options and recording data.
(Record Sheet 1)
a. Option 1: Find dimensions using same width but different height
and length that produce the same volume as original
dimensions.
b. Option 2: Find dimensions using three different dimensions that
produce the same volume as original dimensions but
generate a smaller surface area.
c. Option 3: Change all three dimensions from option 2 that produces
the same volume as original dimensions but generate a smaller
surface area.
d. Option 4: Change all three dimensions from option 2 & 3 that
produces the same volume as original dimensions but generate
a smaller surface area.
7. Analyzing the Data
Enter your data in the calculator.
a. Turn the calculator on.
b. Press Home; 3:Add Lists & Spreadsheets
c. Move the cursor to the cell next to the letter A and type in
“Length”.
d. Press enter.
e. Move the cursor to the cell next to the letter B and type in “Surface
Area”.
f. Press enter.
g. Students should have four points of data entered. (Original and
three options)
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Inquiry-based Lesson Project
Stark_Team 4
Hahn, Kosenski, & Presutto
Open a new page.
a. Ctrl + I; 5: Add Data & Statistics. A scatter plot is created.
b. Move the cursor to the y-axis until you see “Click to add variable”.
Press Enter. Highlight “Surface Area” press Enter.
c. Move the cursor to the left of the x-axis until you see “Click to add
variable”. Press Enter. Highlight “Length” and press Enter.
9. Students will complete Analysis Sheet 1 and Extension 1.
9. Assessment:
1. Check students progress in class.
2. Have students submit completed activities. (Record Sheet 1,
Analysis Sheet 1 & Extension 1)
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Inquiry-based Lesson Project
Stark_Team 4
Hahn, Kosenski, & Presutto
Student Handout
What’s “Width” the Cereal?
GOAL: Efficiently gather data by calculating the surface area and volume of
several different size cereal boxes to test for correlations between surface
area and volume and determine the best packaging for optimization.
What you need to complete:
1. Your group will receive a cereal box.
2. Your group will measure the length, width, and height of their cereal box.
3. Record data on student record sheet. (Record Sheet 1)
4. Calculate the surface area and volume of the cereal box. (Record Sheet 1)
5. Complete the following three options:
a. Option 1: Find dimensions using same width but different height
and length that produce the same volume as original
dimensions but generate a smaller surface area.
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Inquiry-based Lesson Project
Stark_Team 4
Hahn, Kosenski, & Presutto
b. Option 2: Find dimensions using three different dimensions that
produce the same volume as original dimensions but
generate a smaller surface area.
c. Option 3: Change all three dimensions from option 2 that produces the
same volume as original dimensions but generate a smaller
surface area.
d. Option 4: Change all three dimensions from option 2 & 3 that
produces the same volume as original dimensions but generate
a smaller surface area.
6. Analyze the Data
Enter your data in the calculator.
a. Turn the calculator on.
b. Press Home; 3:Add Lists & Spreadsheets
c. Move the cursor to the cell next to the letter A and type in
“Length”.
d. Press enter.
e. Move the cursor to the cell next to the letter B and type in “Surface
Area”.
f. Press enter.
g. Students should have four points of data entered. (Original and
three options)
Open a new page.
a. Ctrl + I; 5: Add Data & Statistics. A scatter plot is created.
b. Move the cursor to the y-axis until you see “Click to add variable”.
Press Enter. Highlight “length” and press Enter.
c. Move the cursor to the left of the x-axis until you see “Click to add
variable”. Press Enter. Highlight “Surface Area” and press Enter
7. Answer Analysis questions and Extension 1.
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Inquiry-based Lesson Project
Stark_Team 4
Hahn, Kosenski, & Presutto
Student Record Sheet 1
NAMES: ____________________________________________________
DATE: ___________________
Cereal Box Data
Height
(in)
Length (in)
Original
Dimensions
Option 1
(same width)
Option 2
(change all
dimensions)
Option 3
(change all
dimensions)
Option 4
(change all
dimensions)
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Width (in)
Surface
Area (in2)
Volume
(in3)
Inquiry-based Lesson Project
Stark_Team 4
Hahn, Kosenski, & Presutto
(Student Analysis Sheet 1)
Name:____________________________________
Date:_________________
1. Compare the surface areas of the three options with the original surface area.
What do you notice?
2. Which length offers the smallest surface area? How does this length compare to
its width and height?
3. Compare your answers from question two with other classmate’s answers. What
do you notice?
4. Can you make a generalization about the best size for all cereal boxes based on
your data?
5. What is the ideal shape of packaging for any cereal box?
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Inquiry-based Lesson Project
Stark_Team 4
Hahn, Kosenski, & Presutto
Extension 1:
Based on your response to problem 5, answer the following questions for a cereal
box with a volume of 300 in3 …
a. What would be the appropriate dimensions for optimal packaging?
b. Now draw a sketch of the cereal box with dimensions labeled.
c. If a typical kitchen cabinet has dimensions of 15″ x 12″x 36″, how many
cereals boxes with the above dimensions would fit in the kitchen cabinet?
Extension 2:
a. Since optimal packaging of a cereal box is a specific shape, how does this
ideal “shape” affect the surface area?
b. Why is it important for the cereal box manufacturer to minimize surface
area?
c. Optimal packaging of a cereal box is a specific shape, why do you think
manufacturers do not use this design? Support your opinion with valid
reasons.
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Inquiry-based Lesson Project
Stark_Team 4
Hahn, Kosenski, & Presutto
(Analysis Sheet 1 Solutions Page)
Name:____________________________________
Date:_________________
1. Compare the surface areas of the three options with the original surface area.
What do you notice?
They should notice the surface areas are different than the original.
2. Which length offers the smallest surface area? How does this length compare to
its width and height?
Answers will vary but they should pick the length that is closest in value to the
width and the height. (Measurements will be closest to a cube)
3. Compare your answers from question two with other classmate’s answers. What
do you notice?
They should notice that other groups’ measurements will be close to a cube.
4. Can you make a generalization about the best size for all cereal boxes based on
your data?
They should say that measurements will be close to a cube.
5. What is the ideal shape of packaging for any cereal box?
They should have cube for the answer.
Extension 1:
Based on your response to problem 5, answer the following questions for a cereal
box with a volume of 300 in3 …
a. What would be the appropriate dimensions for optimal packaging?
Length ≈ 6.69 in
Width ≈ 6.69 in
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Height ≈ 6.69 in,
Inquiry-based Lesson Project
Stark_Team 4
Hahn, Kosenski, & Presutto
b. Now draw a sketch of the cereal box with dimensions labeled.
c. If a typical kitchen cabinet has dimensions of 15″ x 12″x 36″, how many
cereals boxes with the above dimensions would fit in the kitchen cabinet?
You can fit 10 cereal boxes with the above dimensions.
Extension 2:
a. Since optimal packaging of a cereal box is a specific shape, how does this
ideal “shape” affect the surface area?
The ideal shape produces the smallest surface area.
b. Why is it important for the cereal box manufacturer to minimize surface
area?
Less packaging costs less.
c. Optimal packaging of a cereal box is a specific shape, why do you think
manufacturers do not use this design? Support your opinion with valid
reasons.
Answers will vary.
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