A Modeling Experience
A TechMath Module
Ernest A Hoke Jr , Teacher
EB Aycock , Greenville, NC
Joanna Paul, Student
Butch Allen , Business Partner
Domtar, Plymouth, NC
July 2008
Preface
Within this module, students will explore various Algebra and Geometry topics. Topics include: scale drawings, surface area and volume of cylinders and prisms, equations and inequalities including systems, and the idea of maximum capacity. This module is most appropriate for students in 8th Grade Mathematics, Algebra 1, and Geometry, but could be used in any mathematics classroom from 7th grade to Calculus depending on the specific outcome expectations of the teacher. Students will work in collaborative groups of 5 to 7 throughout the module. The module completion time is approximately five, ninety-minute sessions, while a sixth day could be added for student presentations. The culminating project of this module is for the students to design a warehouse model to maximize the capacity of two different products within the safety parameters set by the warehouse.
The module begins by requiring the student group to construct scale drawings of the product that will be stored in the warehouse. Students then construct 3D models of the products out of plastic sheets to calculate surface area and volume. Students must then calculate a system of equations and/or inequalities for the profit of the two products in order to fill their warehouse with the appropriate numbers of each product. Using the warehouse’s safety parameters, students will then design the interior of their warehouse to maximize capacity.
This module can be used with any warehouse and any product. This particular module was specifically developed with the assistance of Butch Allen, transportation manager of Domtar in Plymouth, North Carolina. The context of the unit is that students use calculations of product volumes as well as profit expectations to design how a warehouse should be designed to maximize capacity of the two products. Actual Domtar warehouse and product dimensions were used within this module to aid in connections between classroom and business settings.
According to their website, Domtaris the largest manufacturer in North America and second largest in the world of uncoated free-sheet, based on production capacity. Uncoated free-sheet papers are used for office and business printing, forms, envelopes, text and trade books, commercial printing, and stationary. The Domtar facility in Plymouth, North Carolina, was opened in 1937 by the North Carolina Pulp Company. The first paper machine began production in 1947, while fine paper production did not start until 1965. In 2007, Domtar acquired this facility from Weyerhaeuser which occupied the facility since 1957. Approximately 489,000 tons of paper are produced from this facility each year, with an additional 486,000 tons of pulp production. The facility houses two pulp lines and two paper machines.
This module was developed during the Spring and Summer of 2008. Initial piloting in the classroom occurred during the Fall of 2008. With further editing, the module was piloted in additional classrooms in the Spring of 2009.
Table of Contents
Learning Objectives1
- North Carolina Standards
- Principles and Standards for School Mathematics
List of Materials2
Module Timeframe3
Teacher Notes4
- Bubble Activity
- Warehouse Products Activity
- Warehouse Design Activity
Student Tasks5
- Bubble Activity
- Warehouse Products Activity
- Warehouse Design Activity
Sample Solutions6
- Bubble Activity
- Warehouse Products Activity
- Warehouse Design Activity
Rubric for Evaluation7
- Warehouse Design
- Presentation
Appendix8
Learning Objectives
North Carolina Standard Course of Study
Seventh Grade Mathematics
2.01Draw objects to scale and use scale drawings to solve problems.
2.02Solve problems involving volume and surface area of cylinders, prisms, and composite shapes.
3.03Use scaling and proportional reasoning to solve problems related to similar and congruent polygons.
Eighth Grade Mathematics
1.02Develop flexibility in solving problems by selecting strategies and using mental computation, estimation, calculators or computers, and paper and pencil.
3.01Represent problem situations with geometric models.
4.01Collect, organize, analyze, and display data (including scatterplots) to solve problems.
5.03Solve problems using linear equations and inequalities; justify symbolically and graphically.
Algebra 1
4.01Use linear functions or inequalities to model and solve problems; justify results.
4.03Use systems of linear equations or inequalities in two variables to model and solve problems. Solve using tables, graphs, and algebraic properties; justify results.
Geometry
1.02Use length, area, and volume of geometric figures to solve problems. Include arc length, area of sectors of circles; lateral area, surface area, and volume of three-dimensional figures; and perimeter, area, and volume of composite figures.
2.01Use logic and deductive reasoning to draw conclusions and solve problems.
Principles and Standards for School Mathematics
National Council of Teachers of Mathematics Standards
Numbers and Operations
Algebra
Geometry
Measurement
Problem Solving
Reasoning and Proof
Communications
Connections
Representation
List of Materials
Bubble Activity
For each student:
Bubble Activity Worksheet
For each student group:
1 Roll of Clear Painter’s Plastic
Several Pairs of Scissors
1 or 2 Rolls of Clear Packing Tape
Rulers, Yard Sticks or Measuring Tapes
Box Fan
Graph Paper
Warehouse Products
For each student:
Warehouse Products Worksheet
Graph Paper
Graphing Calculators
Computer with Graphing Software
Warehouse Design Modeling
For each student:
Warehouse Design Worksheet
For each student group:
Access to computer
Module Timeframe
Day One:“The Bubble Activity”
Creation of Nets
Calculations of Surface Area
Calculations of Volume
Scale Drawings
Simple Measurement
Day Two:“The Bubble Activity”
Simple Measurement
Construction of Prism / Cylinder
“Warehouse Products”
Equations
Day Three:“Warehouse Products”
Equations
“Warehouse Design”
Modeling
Equations / Systems
Day Four:“Warehouse Design”
Modeling / Systems
Day Five:“Warehouse Design”
Modeling / Systems
OPTIONAL:
Day Six:“Warehouse Design Presentations”
Presentations of Designs
Bubble Activity
Objectives:
Create scale nets of rectangular prisms and cylinders with given dimensions
Calculate surface area and volume of rectangular prisms and cylinders
Use of simple measuring devices (yard sticks, rulers, measuring tapes)
Materials:
Groups of 6-7 students will need the following materials:
1 roll clear painter’s plastic
Several pairs of scissors
1 or 2 rolls clear packing tape
Yard Sticks or Measuring Tapes
Box Fans
Graph Paper
Procedure:
- Instruct groups to cut out the pieces necessary to build a specific rectangular prism and cylinder. The dimensions of the rectangular prism is 3’ x 4’ x 5’, while the cylinder has a diameter of 6’ and a height of 5’. Tell them that they must create a net for a rectangular prism and cylinder to scale on the graph paper before they proceed. Once their scale nets are approved, students can advance to the cutting of the painter’s plastic. Students will cut out each piece of the net and tape it together so that there are no holes (an airtight bubble). As they are constructing the prism and cylinder, instruct them to write in permanent marker the dimensions of each piece of plastic.
- Once students have built their prisms and cylinders, cut a hole in one of the sides approximately the size of a box fan. Tape the plastic to the box fan so that it is air-tight. Turn the fan on low speed. Hold the fan at first, so it does not tip over. As the “bubble” starts to inflate, take a pair of scissors and cut a slit about 3 feet long in another side of the “bubble”. Avoid cutting directly across from the box fan. This will prevent the bubble from lifting off the floor. It also serves as a door to get inside if students wish to explore.
- Students should complete the activity sheet. They will record the dimensions of their “bubbles and make additional calculations. **Use 3.14 for π **
Bubble Activity Sheet
Prism: (3’ x 4’ x 5’)
Dimensions for Net:
Length ____3 ft_____ Width____4 ft_____ How Many?__2__
Length ____3 ft_____ Width____5 ft_____ How Many?__2__
Length ____4 ft_____ Width____5 ft_____ How Many?__2__
Sketch the Net Here:
3 ft
3 ft
Total Surface Area:____94 ft2______
Show Work Here:
= 2(3 x 4) + 2(3 x 5) + 2(4 x 5) = 2(12) + 2(15) + 2(20) = 24 + 30 + 40 = 94
Total Volume:_____60 ft3______Show Work Here:
= (3 x 4) x 5 = (12) x 5 = 60
Cylinder: (Diameter of 6’, Height of 5’)
Dimensions for Net:
Length _~18 feet 10 inches__ Width___5 feet___ How Many?__1___
Radius Length _3 feet_ Circumference_~18 feet 10 inches_ How Many?__2___
Sketch the Net Here:
3 ft
5 ft5 ft
~18.84 ft
Total Surface Area:___150.72 ft2____
Show Work Here:
= 2 (32π) + 5(18.84) = 2(9π) + 94.2 = 18π + 94.2 = 56.52 + 94.2 = 150.72
Total Volume:___141.3 ft3_____
Show Work Here:
= (32π) x 5 = 9π x 5 = 45π = 141.3
Warehouse Products
Objectives:
Create equations with a real world application
Using technology to solve problems
Materials:
Graph paper
Graphing Calculators
Computer with a graphing utility (Excel, TI Interactive)
Background Information:
In the previous “Bubble Activity” we investigated the attributes of two different three-dimensional figures, a rectangular prism and cylinder. These two figures represent two different products that Domtar stores in their warehouse. The rectangular prism represents a pallet of boxed paper, while the cylinder represents a large roll of paper that Domtar manufactures.
Procedure:
The “Warehouse Products Worksheet” poses a scenario to students regarding the products at Domtar.
Each product is given a manufacture price and selling price.
No matter how many of each product they produce, they will always have a standard cost of $100 for the rolls of paper and $175 for the pallets of paper.
The manufacturing price for each pallet of paper is $250, and they sell them for $600.
The manufacturing price for the rolls of paper is $125, and they sell them for $575.
Calculate a Profit Equation for each of the two products.
Graph the equations using a graphing utility (Excel, TI-Interactive)
Pallet of PaperRolls of Paper.
$175Operating Costs$100
$600Selling Price$575
$250Production Costs$125
$350Profit on Each Item$450
y = 350x - 175Profit Equationy = 450x – 100
- Below is an example of a graph students may create with Excel.
Warehouse Design
The culminating project of this module is for students to design a warehouse to maximize its profit. Students must make numerous assumptions to complete this assignment. A rubric and outline are provided to guide students through this process.
One of the most important parts to this modeling assignment is for students to understand that there is not one correct answer. Defense of their decisions is vital. Reminding the students of the outline and rubric is a key to student success.
Students must remember and follow all safety guidelines.
- Products can only be stacked 3 high.
- Only same product can be stacked on each other.
- Every walkway throughout the warehouse must be at least 10 feet wide.
Students must also keep in mind that there is a maximum and minimum amount of product to be held in the warehouse.
- To keep up with demand, Domtar must keep in stock at least 100 pallets of paper and 75 rolls of paper.
- The floor space of the warehouse totals 16,905 ft2. So by figuring that the paper pallets use 12 ft2 of floor space using the entire area, a total of 1408 pallets could be placed, and if we pile them 3 high, there is a total of 4224 pallets of paper that could be in the warehouse. Using the same processes, the rolls of paper take up approximately 28.3 ft2 for 598 rolls, stacking them 3 high allows for a total of 1794 rolls of paper.
Bubble Activity
Objectives:
Create scale nets of rectangular prisms and cylinders with given dimensions
Calculate surface area and volume of rectangular prisms and cylinders
Use of simple measuring devices (yard sticks, rulers, measuring tapes)
Materials:
Your group will need:
1 roll clear painter’s plastic
Several pairs of scissors
1 or 2 rolls clear packing tape
Yard Sticks or Measuring Tapes
Box Fans
Graph Paper
Procedure:
- You will be building two three-dimension figures, a rectangular prism and a cylinder.
- Rectangular Prism : 3’ x 4’ x 5’
- Cylinder: Diameter of 6’, Height of 5’
- Create scale drawings of the nets on graph paper.
- Once finished get teacher permission to proceed.
- Use the materials provided to you to construct the two objects.
- Cut out each piece of the net and tape it together so that there are no holes (an airtight bubble).
- Write in permanent marker the dimensions of each piece of plastic.
- Once complete, you are ready for your teacher to “blow up your bubble”.
- Complete the Bubble Activity Sheet. **Use 3.14 for π**
Bubble Activity Sheet
Prism: (3’ x 4’ x 5’)
Dimensions for Net:
Length ______Width______How Many?_____
Length ______Width______How Many?_____
Length ______Width______How Many?_____
Sketch the Net Here:
Total Surface Area:______Show Work Here:
Total Volume:______Show Work Here:
Cylinder: (Diameter of 6’, Height of 5’)
Dimensions for Net:
Length ______Width______How Many?_____
Radius Length ______Circumference______How Many?_____
Sketch the Net Here:
Total Surface Area:______Show Work Here:
Total Volume:______Show Work Here:
Warehouse Products Activity
Objectives:
Create equations with a real world application
Using technology to solve problems
Materials:
Graph paper
Graphing Calculators
Computer with a graphing utility (Excel, TI-Interactive)
Background Information:
Domtar manufactures various paper products at their Paper and Pulp Mill in Plymouth, North Carolina. We will be looking at two of their products, pallets of boxed paper and large rolls of paper.
At the mill, copier paper is made, cut, packaged, stored, and shipped. Each pallet of paper costs $250 to manufacture, while they sell them for $600. Operating costs for the production line is $175 no matter how many pallets of paper are produced. Boxed paper is placed on pallets that are 3 feet wide, 4 feet long, and 5 feet high. Pallets are then shrink wrapped with plastic and stored in the warehouse. Domtar must keep on stock in their warehouse at least 100 of these pallets at all times.
Large paper rolls are also manufactured at Domtar. The paper is 5 feet wide and rolled into a cylinder that has a diameter of 6 foot. Each roll costs $125 to produce along with a $100 operating cost for the production line to start. Domtar sells these large rolls of paper for $575. Due to demand, they must keep at least 75 rolls in the warehouse.
Procedure:
Calculate a profit equation for each of these two products. Remember to explain each part of the equation. Then graph these two equations using a graphing utility (Excel, TI-Interactive).
Warehouse Design
A new warehouse is being built specifically for pallets of paper and rolls of paper at Domtar’s Plymouth, NC mill. Your team will be responsible for designing the interior of the warehouse, including where the rows of product will be placed.
Things to remember:
- Attached you will find the diagrams of the new warehouse.
- Walkways throughout the warehouse are at least 10 feet wide.
- To best ensure safety in the warehouse, pallets and the rolls of paper can only be stacked 3 high.
When completing this modeling assignment, you must remember that there is not one correct answer. The key is for your group to make decisions and support them. Your warehouse design assignment includes both a diagram as well as a written portion. You may use any of your available resources to make your scale diagram of the warehouse. The written portion contains four major components: Statement of the Problem, Statement and Defense of Assumptions, Development of the Mathematical Model, and Strengths and Weaknesses. Follow the Warehouse Design Outline Write-Up as a guide to complete the written portion.
Warehouse Design Outline Write-Up
Student Name:______Class:______Date:______
Category / 3: Exceeds Standards / 2: Meets Standards / 1: Needs Revisions / 0: Not Included / Comments