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Applying Functions to the Real World of Business:

A Mathematical Modeling Approach

A Tech-Math Module

Tisha Jones, TeacherSouth Central High School, Winterville, NC

Shavonte Mills, StudentTanese Newton-Love, Student

Andrew Wilson, Senior Technical EngineerCMI Plastics

http://www.cmiplastics.com/index.htm

PrefaceThe purpose of this module is to expose high school algebra students to types of algebra used in the world of business. The math topic that will be addressed is linear functions and decision-making (maximizing space). This module is most appropriate for students taking 8th grade mathematics, a high school Introduction to Math or an Algebra 1 class. It could also be used as in introduction to an Algebra II or Advanced Functions and Modeling Class. The module consists of five-90 minute class sessions, with the 4th day used to prepare student presentations and the 5th day used for the actual student presentations. Before this module is presented to the class, students should already know and understand how to solve equations, find slope and write an equation of a line in slope intercept form. This module leads the way to the topic of Scatter plots and Lines of Best Fit.

This module was specifically developed with the assistance of Andrew Wilson, Senior Technical Engineer of CMI Plastics, Consolidated Models, Inc. in Ayden, North Carolina. The context of the module is that students use functions of plastic trays and height, and then plastic trays and cost to determine the total cost to fill a specific box to its capacity.

CMI Plastics specializes in custom thermoforming. Thermoforming is a manufacturing process where plastic sheet is heated to a pliable forming temperature, formed to a specific part shape in a mold, and trimmed to create a usable product. The plastic sheet is heated in an oven to a high enough temperature that it can be stretched into or onto a mold and cooled to a finished shape. CMI serves numerous industries including: medical, aerospace, cosmetic, transcript, food, industrial and retail. Their products include for example clamshells cosmetic discs and medical trays.

CMI was created during WWII in Bronx, NY. The founder Arthur Hasselbach, Sr. took his passion for airplanes and the popularity of models and used them to fuel his desire to pursue making model airplanes. After the attack on Pearl Harbor the company began production for the US Military, fabricating model planes for Naval Intelligence, to be used for training purposes. In the early 1950’s, CMI began working with the heating and cooling of plastics. Since 75% of the total annual sales came from plastics and packaging CMI transformed into a forming and contract packaging company and began to phase out its model kits.

Since then CMI has worked on the Apollo Space Program, forming the battery housings for Lunar Lander Module, which sit on the moon today to working on medical packaging focusing on projects for Johnson and Johnson. Through the 80’s the company expanded service towards the cosmetic and beauty industry, working with companies such as L’Oreal. When CMI, now owned and managed by Stephen Hasselbach, CEO and his sons, relocated to Pitt County North Carolina in 2007, it has allowed for an expansion and continued success of the family business.

This module was developed during the spring and summer of 2009. Initial piloting in the classroom occurred during the Fall of 2009.

A Tech-Math Module | Applying Functions to the real World of Business 2

Table of Contents

Preface pg. 2

Learning Objectives Pg. 4-5

Materials List pg. 6

Module Time Frame pg. 7

Activity #1 Packing Boxes

Teacher Notes pg. 8-9 Student pages pg. 10-11 Solutions pg. 12 Rubric pg. 13

Activity #2 Function, Function, What’s your Function?

Teacher Notes pg. 14 Student pages pg. 15-17 Solutions pg. 18-20

Activity #3 The Price is Right

Teacher Notes pg. 21 Student pages pg. 22-23 Solutions pg. 24-26

Appendix pg. 27-30


Learning Objectives

North Carolina Standard Course of Study

Introduction to Math and 8th Grade Mathematics

Number and Operations Competency Goal 1 The learner will understand and compute with real numbers. 1.02 Develop flexibility in solving problems by selecting strategies and using mental computation, estimation, calculators or computers, and paper and pencil.

Data Analysis and Probability Competency Goal 3 The learner will understand and use graphs and data analysis. 3.01 Collect, organize, analyze, and display data (including scatter plots) to solve problems. 3.02 Approximate a line of best fit for a given scatter plot; explain the meaning of the line as it relates to the problem and make predictions.

Algebra Competency Goal 4 The learner will understand and use linear relations and functions. 4.03 Solve problems using linear equations and inequalities; justify symbolically and graphically.

Algebra 1

Data Analysis and Probability Competency Goal 3 The learner will collect, organize, and interpret data with matrices and linear models to solve problems. 3.03 Create linear models for sets of data to solve problems. Interpret constants and coefficients in the context of the data.

Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.

Algebra Competency Goal 4 The learner will use relations and functions to solve problems.4.01 Use linear functions or inequalities to model and solve problems; justify results.

NCTM PSSM – National Council for the Teachers of Mathematics Principles and Standards for School Mathematics

Algebra Standard- Understand patterns, relations and function- Represent and analyze mathematical situations and structures using algebraic symbols- Analyze change in various context

Problem-Solving Standard


- Build new mathematical knowledge through problem solving Communication

- Communicate mathematical thinking coherently and clearly to peers, teachers, and others

- Analyze and evaluate the mathematical thinking and strategies of others Connections

- Recognize and apply mathematics in contexts outside of mathematics Representation

- Use representations to model and interpret physical, social and mathematical phenomena

Reasoning and Proof- Make and investigate mathematical conjectures- Select and use various types of reasoning and methods of proof.

Lesson Objectives

Calculate the rate of change in one variable as another variable increases. Describe relationship among the graph, symbol rule (equation), table of values and related

situation for a linear function. Interpret meaning of the slope and y-intercept of the graph of a linear function in a context. Write a rule (equation) for a linear function given its graph or table of sample values. Use linear functions to answer questions about the situations that they describe. Use a linear model to predict the value of one variable given the value of the other and

describe the rate of change in one variable as the other increases in a meaningful way. Use a graphing calculator to find the linear regression model for a set of data.


Materials List

Activity #1

2 yard sticks for each group Taped pallet with bottom dimensions of box 3 to 4 Plastic trays for each group Black line Master “Packing Boxes” Blackline Master “Packing Boxes Activity Sheet” Paper Pencil

Activity #2

Graph paper Ruler Pencils Black line Masters “Function, Function, What’s your Function?” Graphing Calculator

Activity #3

Pencil Black line Master “The Price is Right!” Graphing Calculator


Module Time Frame

Day #1 Introduction of Unit of Study

Activity #1 – 90 minutes

Day #2 Activity #2 – 60 minutes

Day #3 Activity #3 – 60 minutes


Packing Boxes Teacher Notes

Introduce unit of study“In this unit of study, we are going to investigate how a particular business uses mathematics in their daily work routine”.

Introduce business partner“(Show slide of company, see Appendix #1) CMI Plastics, short for Consolidated Models Incorporated is in Ayden, North Carolina. The company specializes in custom thermoforming. (Show slide of machine that heats the plastic, see Appendix #2) Thermoforming is a manufacturing process where a plastic sheet is heated in an oven to a high enough temperature that it can be stretcher into on onto a mold and cooled to a finished shape. Some products include cosmetic discs (Show slide of cosmetic discs, see Appendix #3), medical trays and the product me are going to use in our activity today, perfume trays. (Show example of model used in activity, See Appendix #4. Most of these materials are packed in special cardboard boxes called Gaylord boxes with specific dimensions. (Show slide of opened empty box, see Appendix #5). Once the plastics are packed the box is closed and then sealed in shrink wrap, then plastics are ready to be shipped to the company of choice. (Show slide of box being sealed and stack for shipping, See Appendix #6).” See Appendix for pictures of slides.

Introduce scenario (ESSENTIAL QUESTION)“Let’s pretend you are the senior technical engineer at CMI Plastics and are in charge of determining the cost involved in packing a standard sized Gaylord box, with plastic trays to capacity. How can we begin to answer this question?” (Listen to suggestions and then list for class to display.)

Introduce lesson 1 activity“In order to determine the cost you must first design the best way to pack as many trays as possible into a standard size Gaylord box. The box has a length of 44 inches, a width of 39 inches, and a height of 46 inches. One major concern is that the trays cannot be stacked vertically; they must lie on their sides. When the trays are stacked vertically, pressure causes the nesting of the trays to become so tight that it cannot be pulled apart. (Give example of when plastic cup are pushed tightly above one another and how hard it is to pry them apart).”

Explain student expectations


“By the end of the activity, students need to provide a diagram of how the first (bottom) layer of the box will look. Include dimensions of box, dimensions of the stacked trays, and dimensions of any gaps or extra space, if any. Provide a clear explanation of how you determined how your design will look. Be sure to discuss and include any necessary calculations, etc. Make sure this explanation includes the final amount of trays needed to fill the entire box.”

Allow Students to work in groups.Group students in groups of 4 or 5. Give ample time to work.

List and discuss student strategiesAfter the students have completed the activity, while they are giving their explanations to the class, make a largely displayed list of the different strategies used.

ClosureDiscuss the listed strategies, show similarities and differences.“Tomorrow we will continue to discuss strategies that can be used to help pack this box with trays to its capacity but in a lot less time.”


Packing Boxes You are the senior technical engineer at CMI Plastics and you are in charge of determining the cost involved in packing a standard sized Gaylord box, with plastic trays to capacity. How are you going to begin this task? What information do you need to know before the cost can be figured in?

Explore

Design the best way to pack as many trays as possible into a standard size Gaylord box.

Dimensions of the box: Length = 44 inchesWidth = 39 inchesHeight = 46 inches

Major concernso The trays cannot be stacked vertically; they must lie on its

sides. When the trays are stacked vertically, pressure causes the nesting of the trays to become so tight that it cannot be pulled apart.

o All layers in the box must be the same.o All trays in the box must be the same.o All trays must lie on the same side.

Expectations

Items to be turned in by end of class period:

Provide a diagram of how the first (bottom) layer will look on the provided chart paper. Include dimensions of the box, dimensions of the stacked trays and dimensions of any gaps, if any. Include the number of trays needed to fill the first layer of the diagram.

Provide a clear explanation of how you determined how your design will look. Be sure to discuss and include any necessary calculations etc. Make sure this explanation includes the final number of trays needed to fill the entire box.

Prepare to present this information in written form and in a brief 2 to 3 minute presentation.


Name _____________________ Group Color ___________ Period _______

Using the rectangle below, draw a sketch of how the first (bottom) layer of the boxed looks. Include dimensions of the box, dimensions of the stacked trays and dimensions of any gaps, if any. Include the number of trays needed to fill the first layer of the diagram.

The original box is 40 by 44 inches. The scale of the drawing is 1 inches to 8 inches.

Explain how you determined how your design will look. Be sure to discuss and include any necessary calculations etc. Make sure this explanation includes the final number of trays needed to fill the entire box.


Packing Boxes Solutions

Depending on the size of the plastic tray will actually determine the number of trays that will fit in the box. Make sure that students are nesting the trays within each other when figuring out how many trays fit in a row. For a tray that is 8 in x 10 in x 2 in and a ½ nestle, the Gaylord box can fit 1,544 trays. If the tray is smaller then it will hold more and if it is larger then it will fit less.


Packing Boxes Grading Rubric

Math - Problem Solving : Packing Boxes Activity

Teacher Name: Tisha Jones

Student Name: ________________________________________

CATEGORY 4 3 2 1Diagrams and Sketches

Diagrams and/or sketches are clear and greatly add

to the reader's understanding of the

procedure(s). Diagram includes specific

dimensions needed to understand it.

Diagrams and/or sketches are clear and

easy to understand, but some dimensions

are inaccurate.

Diagrams and/or sketches are

somewhat difficult to understand, because some dimensions are

missing.

Diagrams and/or sketches are difficult to understand or are not used, because all

dimensions are missing.

Strategy/Procedures Typically, uses an efficient and effective strategy to

solve the problem. Includes necessary

calculations and includes final amount needed to fill

the entire box.

Typically, uses an effective strategy to solve the problem.

Includes some calculations and final amount needed to fill

the entire box.

Sometimes uses an effective strategy to solve problems, but

does not do it consistently, missing

several necessary calculations.

Rarely uses an effective strategy to

solve the problem, has no calculations and no final amount needed to fill the entire box.

Working with Others Student was an engaged partner, listening to

suggestions of others and working cooperatively

throughout lesson.

Student was an engaged partner but

had trouble listening to others and/or working

cooperatively.

Student cooperated with others, but

needed prompting to stay on-task.

Student did not work effectively with others.

Mathematical Reasoning

Uses complex and refined mathematical reasoning.

Uses effective mathematical

reasoning

Some evidence of mathematical

reasoning.

Little evidence of mathematical

reasoning.


Function, Function… Teacher Notes

Introduce Mathematical Connection“In industrial, engineering and business applications it is sometimes necessary to develop a mathematical model to predict how something will perform. The mathematical model is based on a set of sample data and the model that is developed is then used to predict behavior in new situations. Using the information gathered in activity one, this activity will help you develop a mathematical mode (an equation) to help fill the same box to its capacity.”

Introduce scenario“The owner of CMI Plastics has come to you and complained about the time it has taken you to determine how many trays will fit in one box. This is taking too long and time is money. You need a quick way to determine this information.”

Introduce Activity #2“In order to do this you need to develop a mathematical model (an equation) to describe the amount of plastic trays needed to gain a specific height, width, or length to fill the box to its capacity. Follow the direction on the worksheet, to help guide you to a quick way to solve this business’ problem.”

Explain Student Expectations“At the end of the activity, students need to have a completed worksheet using their assigned plastic tray. Include any and all calculations needed to find requested information.”

ClosureDiscuss the equations developed and reasoning how they got their answers. “Tomorrow we will continue to discuss linear regression, but we are going to use it to decide how it can be used to answer our essential question.”


Function, Function, What’s Your Function?

Introduction

The owner of CMI Plastics has come to you and commented about the time it has taken you to determine how many plastic trays will fit in one box.

“The process has taken too long to complete and time is money.” The owner of CMI Plastics enlists their employees help in figuring out a way to reduce the amount of time.

In this activity you will need to develop a mathematical model (an equation) to describe the height, width, or length of a varying number of plastic trays. You will be provided with the same plastic trays used for the Packing Boxes Activity, rulers and graphing calculators. Your goal is to develop an equation that can be used to predict the number of plastic trays needed to fill the Gaylord box to its capacity.

Collecting Data

Part 1 To get started; first measure the height of the trays starting with one, then two trays stacked etc. Record the collected information in the table below.

Does the height appear to be a linear function of the number of plastic trays? Explain.

Developing Your Model

Part 2 After you have measured the heights of several plastic trays stacked loosely; graph the information from the table on a coordinate plane with the number of plastic trays on the x-axis and the height of the trays on the y-axis. Draw a line through the points on the graph.


Number of Plastic Trays

Total Height

123456

Part 3 Using the table and or graph you developed to determine the equation in slope-intercept form for calculating the height of the trays (h) in inches if the number of plastic trays (t) is stacked. Explain the thinking you used to write the equation.

Exploring Your ModelPart 4 Use the graph to find the slope and y-intercept. Explain what these values tell about the number of trays and their heights.

Testing Your ModelPart 5 Use your equation to find the height for 10 trays and the height of 95 trays.

Use your equation to find the number of trays that would have a height of 85 inches.

Using TechnologyPart 6 Using the graphing calculator enter the data points into the list menu. Use the calculator to determine the equation that will best describe your data. Compare it to the equation you developed earlier.

Using the table set and table key options on the calculator, Find the height of 25 trays and 235 trays.

Using the table key option on the calculator, find how many trays would have a height of 150 inches.


Final AnswerHow many plastic trays are needed to fill the bottom layer of the box to its capacity?

How many plastic trays are needed to fill the entire box to its capacity?


Function, Function… Solutions

Collecting Data (Sample Answers)

Part 1 To get started; first measure the height of the trays starting with one, then two trays stacked etc. Record the collected information in the table below.

Does the height appear to be a linear function of the number of plastic trays? Explain. Yes. The total height is increasing by the same amount as the number of plastic trays increase.

Developing Your Model

Part 2 After you have measured the heights of several plastic trays stacked loosely; graph the information from the table on a coordinate plane with the number of plastic trays on the x-axis and the height of the trays on the y-axis. Draw a line through the points on the graph.


Number of Plastic Trays

Total Height

1 32 3.53 44 4.55 56 5.5

Part 3 Using the table and or graph you developed to determine the equation in slope-intercept form for calculating the height of the trays (h) in inches if the number of plastic trays (t) is stacked. Explain the thinking you used to write the equation.

h=.5 t+2.5

Exploring Your ModelPart 4 Use the graph to find the slope and y-intercept. Explain what these values tell about the number of trays and their heights.The slope is 0.5 and the y-intercept is (0, 2.5). The slope tells me how much the height increase with the addition of each tray. The y-intercept tells me what the height would be if you had 0 trays, which does not make sense in this real world situation.

Testing Your ModelPart 5 Use your equation to find the height for 10 trays and the height of 95 trays.

10 trays = 7.5 inches

95 trays = 50 inches

Use your equation to find the number of trays that would have a height of 85 inches.

85 inches = 165 trays

Using TechnologyPart 6 Using the graphing calculator enter the data points into the list menu. Use the calculator to determine the equation that will best describe your data. Compare it to the equation you developed earlier.

Using the table set and table key options on the calculator, Find the height of 25 trays and 235 trays.25 trays = 15 inches235 trays = 120 inches


Using the table key option on the calculator, find how many trays would have a height of 150 inches.150 inches = 295 trays

Final AnswerHow many plastic trays are needed to fill the bottom layer of the box to its capacity?

Sample Answer 427

How many plastic trays are needed to fill the entire box to its capacity?

Sample Answer 2135


The Price is Right Teacher Notes

Introduce Mathematical Connection.“Now that we have learned how to develop a mathematical model and use that to help us make a decision or help us solve a real world business problem, we are going to use that same concept to help you answer the ultimate question, What is the cost involved in packing a standard sized Gaylord box, with plastic trays, to capacity?” We will now focus on cost.

Introduce Business Connection.These plastic trays are formed from large rolls of plastic that are perforated into smaller plastic sheets. The trays are then cut from the large sheets of plastic. Depending on the size of the tray determines how many trays can be made from that sheet of plastic. The leftover plastic is saved and then recycled.

Introduce Scenario.“The owner of CMI Plastics is now pleased with your work and determination of finding a quick way to maximize space inside a Gaylord box.” The owner now says, “I knew you could do it. Now let’s discuss cost. I need to know how much we should charge Company A to produce a box full of these trays”.

Introduce Activity #3Ask leading questions:

What new information do we need to know in order to determine the cost of the contents of this packed Gaylord box? (Lead them to say, we need to know how many molds per sheet and then how many sheets need to fill the box and cost per mold.)

Explain student expectations.“At the end of the activity, students need to have a completed worksheet using their assigned plastic tray. Include any and all calculations needed to find requested information.”

Allow students to work in pairs.

Closure


The Price is Right Part A. CMI Plastics cuts the plastic trays out of large plastic sheets. Depending on the type of tray, 4 to 6 trays can be cut out of one sheet of plastic.

1. If CMI Plastics is making plastic trays for company A that fit ______ (number) trays per plastic sheet. Complete a table of values for the first 10 sheets of plastic used. Label the first column with the number of sheets and the second column with the number of plastic trays.

2. What is the rate of change in the number of plastic trays produced as the number of sheets increases?

3. Write an equation that tells the number of plastic trays (t) produced as the number of plastic sheets (s) increases.

4. Use your rule to find the number of trays for 20 sheets and the number of trays for 125 sheets.

5. Use your rule to find the number of sheets needed to produce 5,000 plastic trays.


Part B. The CMI Plastics engineering team has determined the number of trays to fit in one Gaylord box. Now they must determine the cost of the contents in the box.

1. If the amount to produce one tray is ______ (cost), and the cost of the one Gaylord box is 29.95, complete a table of values for the first 10 plastic trays. The first column is the number of plastic trays and the second column is the total cost.

2. What is the rate of change in the total cost for trays as the number of trays increases?

3. Write an equation that tells the cost of the plastic (C) trays as the number of trays (t) produced increases?

4. Use your rule to find the cost for 20 plastic trays and the cost for 125 plastic trays.

5. Use your rule to find the number of trays that can be produced for $200. Does this answer make sense? Why or why not? Use a complete sentence to describe the amount of trays can be produced for $200.


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The Price is Right SolutionsPart A. CMI Plastics cuts the plastic trays out of large plastic sheets. Depending on the type of tray, 4 to 6 trays can be cut out of one sheet of plastic.

1. If CMI Plastics is making plastic trays for company A that fit 4 to 6 (number) trays per plastic sheet. Complete a table of values for the first 10 sheets of plastic used. Label the first column with the number of sheets and the second column with the number of plastic trays.

Sample answers:

2. What is the rate of change in the number of plastic trays produced as the number of sheets increases?

4 or 6 (the number of trays per plastic sheet)

3. Write an equation that tells the number of plastic trays (t) produced as the number of plastic sheets(s) increases.

t = 4s or t = 6s

4. Use your rule to find the number of trays for 20 sheets and the number of trays for 125 sheets.

t = 4 (20) = 80 t = 6 (20) = 120

t = 4 (125) = 500 t = 6 (125) = 750

5. Use your rule to find the number of sheets needed to produce 5,000 plastic trays.

t = 4s 5000 = 4s s = 1250 t = 6s 5000 = 6s s = 833


Number of sheets

Number of plastic trays

1 42 83 124 165 206 247 288 329 36

10 40

Number of sheets

Number of plastic trays

1 62 123 184 245 306 367 428 489 54

10 60

Part B. The CMI Plastics engineering team has determined the number of trays to fit in one Gaylord box. Now they must determine the cost of the contents in the box.

1. If the amount to produce one tray is $0.10 (cost), the cost of the one Gaylord box is 29.95, complete a table of values for the first 10 plastic trays. The first column is the number of plastic trays and the second column is the total cost.

Number of plastic trays Cost ($)1 30.052 30.153 30.254 30.355 30.456 30.557 30.658 30.759 30.85

10 30.95

2. What is the rate of change in the total cost for trays as the number of trays increases?

$0.10

3. Write an equation that tells the cost of the plastic (C) trays as the number of trays (t) produced increases?

C = .10 t + 29.95

4. Use your rule to find the cost for 20 plastic trays and the cost for 125 plastic trays.

C = .10(20) + 29.95 = $31.95 C = .10 (125) +29.95 = $42.45

5. Use your rule to find the number of trays that can be produced for $200. Does this answer make sense? Why or why not? Use a complete sentence to describe the amount of trays can be produced for $200.


C = .10 t + 29.95

200 = .10 t +29.95

t = 1700.5

This does not make sense because you can not make a half of a tray. 1700 trays can be produced for $200.


AppendixActivity 1 Slides

#1

#2


#3


#4

#5


#6


€¦ · web viewthermoforming is a manufacturing process where plastic sheet is heated to a...

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