Mathematics Of Doing, Understanding, Learning, and Educating for Secondary Schools

MODULE(S2):Mathematical Modeling

Adapted for 2021-2022 Pilots

Version 2.2 Summer 2021

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The Mathematics Of Doing, Understanding, Learning, and Educating Secondary Schools (MODULE(S2)) project ispartially supported by funding from a collaborative grant of the National Science Foundation under GrantNos. DUE-1726707,1726804, 1726252, 1726723, 1726744, and 1726098. Any opinions, findings, and conclusions orrecommendations expressed in this material are those of the authors and do not necessarily reflect the views of theNational Science Foundation.

WRITING TEAM

Lead Authors

Cynthia Anhalt University of Arizona Mathematics Education Researcher

Ricardo Cortez Tulane University Applied Mathematician

Brynja Kohler Utah State University Mathematics Teacher Educator

Contributing Authors

Jacy Beck Utah State University Graduate Student MS

Karolyn Christopher Treasure Mountain Junior High School Mathematics Teacher

Gretchen Knight Utah State University Preservice Secondary Teacher

Alyssa Noble Utah State University Preservice Secondary Teacher

Patrick Seegmiller Utah State University Graduate Student PhD

Will Tidwell Utah State University Graduate Student PhD

John Womack Sky View High School Mathematics Teacher

This course is designed as a semester-long course on mathematical modeling for futuresecondary school mathematics teachers. While several assumptions have molded the structureand design of these materials, we are aware that mathematics teacher preparation programs varywidely, and thus lessons are intended to be adapted to fit the needs of many instructors at variousinstitutions. We have assumed a 15-week semester, and created these materials in 3 Modulessuch that each one takes approximately 5 weeks. We have further assumed that the course meetstwice per week in 75-minute class sessions, but adaptation into other formats is expected. Thetitles and aims of each module are summarized in the table below. Each module contains sevenlessons, and is further described in an introductory section prior to the table of contents.

Our general philosophy is that future mathematics teachers will learn the appropriateknowledge, skills, and dispositions for teaching mathematical modeling by engaging inmathematical modeling activities themselves, and reflecting on the process and skills required asthey develop their own modeling competencies.

Module 1 The Mathematical Modeling Process and Purpose

• Create models with attention to units, dependent and independent variables, and informallyanalyze parameter sensitivity.

• Analyze the cyclical process of mathematical modeling and tasks from the K-12 curriculum.• Value mathematical modeling as an approach to gaining understanding of real world issues,

current events, problems and questions of all kinds.• Develop strategies for selecting tasks/topics with attention to student motivation, opportunities

to address mathematics in the K-12 curriculum, and potential for addressing importantscientific and social issues.

Module 2 Incorporating Real Data in Mathematical Modeling

• Recognize that the modeling process requires careful analysis of model assumptions andrevision.

• Derive, solve, and interpret first order differential equations.• Collect and use real data for model parameterization and validation, and develop a deep

understanding of parameter fitting algorithms.• Analyze classic models including Newton’s Law of Cooling, the Torricelli Model for fluid flow,

and the pendulum equation.

Module 3 Diverse Perspectives in Mathematical Modeling

• Use mathematical modeling to address social justice and environmental issues.• Appreciate how models have evolved over time with contributions from diverse cultures and

individuals.• Study compartment models including SIR models of disease transmission.• Conduct an independent investigation through a self-chosen course project.

Equitable Teaching Practices in MODULE(S2)Curriculum

The curriculum produced by the Mathematics of Doing, Understanding, Learning and Educatingfor Secondary Schools (MODULE(S2)) Project is an outgrowth of the work of the MathematicsTeacher Education Partnership (MTE-Partnership), a national collaborative working towardsimproving the number and quality of secondary mathematics teachers prepared in institutions ofhigher education. The MTE-Partnership is founded on a set of Guiding Principles(MTE-Partnership, 2014) which include a focus on transforming preparation programs so thatprospective teachers develop and convey views that “mathematics is a living and evolvinghuman endeavor” (p. 4), and develop teaching practices that “demonstrate a dedication toequitable pedagogy” (p. 5).

As such, the MODULE(S2) curriculum materials include opportunities to engage in the use ofteaching practices that support developing content knowledge in equitable ways as well aspractices that support developing productive dispositions and identities in mathematics. TheMathematics Teaching Practices (NCTM, 2014) describe quality instructional practices with afocus on developing content knowledge of learners. This core set of eight research-basedteaching practices support equitable teaching and are shown in the framework below (Figure 1).When implemented intentionally as interconnected practices in mathematics teaching, thesepractices provide space for the instructor to view all students as mathematical thinkers and todevelop agency among the learners in the classroom (Berry, 2019).

Figure 1. The Mathematics Teaching Framework demonstrating the interconnected nature ofteaching practices that support equitable teaching. (Boston, Dillon, Smith, & Miller, 2017, p. 215).

In addition to implementing practices which focus on building content knowledge, theMODULE(S2) materials are intended to “strengthen mathematical learning and cultivate positivestudent mathematical identity” (Aguirre, Mayfield-Ingram, & Martin, 2013, p. 43). Throughoutthe MODULE(S2) materials, instructors will find opportunities to enact the practices of:

• Going deep with mathematics

• Leveraging multiple mathematical competencies

• Affirming mathematics learners’ identities

• Challenging spaces of marginality

• Drawing on multiple resources of knowledge (Aguirre et al., 2013).

As instructors engage with the MODULE(S2) curriculum materials, efforts should be made toutilize equitable teaching practices in order to engage prospective teachers in learning aboutmathematics and learning about teaching in equitable ways. Through the activities and practicescontained in the MODULE(S2) materials, instructors and students will have opportunities toreflect on the power dynamics inherent in the teaching and learning of mathematics and considerhow that reflection might inform their practice. Instructors can find more detailed descriptions ofthese practices in the first three references below.

Aguirre, J., Mayfield-Ingram, K., & Martin, D. B. (2013). The impact of identity in K8 mathematics:Rethinking equity-based practices. Reston, VA: National Council of Teachers of Mathematics.

Berry, R. Q. (2019, May). President’s message: Examining equitable teaching using the mathematicsteaching framework. National Council of Teachers of Mathematics. Retrieved from www.nctm.org.

Boston, M. D., Dillon, F., Smith, M. S., & Miller, S. (2017). Taking action: Implementing effectivemathematics teaching practices in grades 912. Reston, VA: National Council of Teachers ofMathematics.

Mathematics Teacher Education Partnership. (2014). Guiding principles for secondary mathematicsteacher preparation. Washington, DC: Association of Public Land Grant Universities. Retrievedfrom www.aplu.org.

National Council of Teachers of Mathematics (2014). Principles to actions: Ensuring mathematicalsuccess for all. Reston, VA: Author.

Module I

The Mathematical Modeling Process and Purpose

The central goal of Module 1 is to lead learners to understand that mathematical modeling (MM)problems are developed in many ways – from observations of the world around us, claims madeby authorities, scientific discoveries, desire for predictions, etc. Every mathematical modelshould have a purpose. In contrast with prepackaged math “application” problems from typicalschool textbooks, observations from the real world (referred to as situations) must betransformed into a math problem (problem posing), solved, and then reconnected with theoriginal situation to determine if the purpose has been achieved. This module will lead learnersto address the roles of variables and parameters in the mathematical models they develop.Through metacognitive reflection on their own work on MM tasks, they will gain experiencewith the elements of MM. MM practices include becoming comfortable making assumptions thatare known to be false but can help gain initial insight into the situation knowing that the falseassumption will need to be revised and the modeling process iterated.

This module aims to expose participants to the process of mathematical modeling as a way todescribe, explain, understand, or predict situations arising in everyday life; connect everydayexperiences with classroom mathematics; explore several frameworks used to illustrate themathematical modeling process; develop an understanding of modeling goals in the secondarymathematics curriculum; and help prepare mathematics teachers to incorporate mathematicalmodeling activities into their curriculum.

In this module, course participants will:

• Create models with attention to units, dependent and independent variables, andinformally analyze parameter sensitivity.

• Analyze the cyclical process of mathematical modeling and tasks from the K-12 curriculum.

• Gain appreciation for mathematical modeling as an approach to understanding real worldissues, current events, problems and answering questions of all kinds.

• Develop strategies for selecting tasks/topics with attention to student motivation,opportunities to address mathematics in the K-12 curriculum, and potential for addressingimportant scientific and social issues.

Overview of course content and assignments:

• Lesson 1 Developing Ways of Thinking for Mathematical Modeling

◦ Video Introduction

◦ Current Event with Mathematical Analysis

• Lesson 2 Fighting Floods with Sandbags

◦ Sandbags Problem Reflection

• Lesson 3 Elements of the Mathematical Modeling Process

◦ Readings About Mathematical Modeling

• Lesson 4 Predicting the Evolution of STDs in the USA

◦ STD Revised Model or STD Model Draft

◦ STD Modeling Report

• Lesson 5 Modeling for Water Conservation

◦ Water Conservation Report

◦ Simulation of Practice - Written

• Lesson 6 Analyzing Modeling Tasks: Rolling Cups

◦ Rolling Cups Reflection

• Lesson 7 Critical Reading of Mathematical Models: Muffin Sale Task

◦ Simulation of Practice - Video

◦ Critical Reading - Baseball Article

These lessons are each described briefly in the table below, along with the emphasizedmathematical knowledge for teaching (MKT) or the main mathematical ideas and teachingpractices that arise in the lesson.

Module 1Lesson

Description Essential Mathematics

1 DevelopingWays ofThinking forMathematicalModeling

Introduction to several mathematicalmodeling examples from thenewspaper and everyday life. Reallife situations can be more deeplyexplored through the application ofmathematics.

The essential mathematicalknowledge for teaching (MKT) hereis to raise awareness of the manysituations all around that can beviewed mathematically,quantitatively, systematically,logically. A specific sub-focus is topay attention to the variables andunits that arise inequations/relations representingaspects of these situations. What arethe quantities, and what are theirunits?

2 FightingFloods withSandbags

This lesson stems from an authenticsituation, and leads to developingfunctions from number sequences toexplain a discrepancy in publisheddocuments regarding the number ofsandbags required to build a levee.Spoiler alert: a difference inparameter values can explain theresults.

Derive the formula for the triangularnumbers, and use it to create otherrelated formulas. Distinguish anddiscuss parameters versus variablesin a model. How do model resultsdepend on parameters?

3 Elements oftheMathematicalModelingProcess

Learners reflect on the process theywent through to solve the sandbagsproblem, and describe their work interms of the mathematical modelingcycle in CCSSM. They are introducedto modeling standards across thevarious conceptual categories of highschool mathematics, and theexpectation that teachers integratemodeling experiences into thesubject-matter they teach.

The essential MKT in this lesson is tolearn that modeling in schoolmathematics is pervasive acrosscontent strands and grade levels. Payattention to the expectations forstudents to develop modelingproficiency in the various branchesof mathematics content.

Module 1Lesson

Description Essential Mathematics

4 Predictingthe Evolutionof STDs in theUSA

Learners are prompted to develop amodel and predict the futureregarding infections of sexuallytransmitted diseases based on anauthentic published report. In thesecond day of the lesson they extendand revise their models and discussessential components of reportsabout mathematical models.

Learners develop both discrete andcontinuous functions from numbersequences. Continue analysis ofparameters by comparing modelpredictions based on differentstrategies for reducing disease.

5 Modeling forWaterConservation

Learners explore the question ofwhether a shower or a bath is moreefficient for conservation of waterresources.

Learners utilize variable ranges tomake reasonable predictions,working with linear functions andinequalities. This lesson requiresstudents to consider flow rates andvolumes and conduct a comparativeanalysis.

6 AnalyzingModelingTasks: RollingCups

Learners review and apply theirknowledge about the mathematicalmodeling process as they analyze theRolling Cups task from the MARS(Mathematics Assessment ResourceService) project. This allows them toreflect on curricular materials in lightof the modeling process, and applygeometry to develop a function formaking a prediction. The samegeometry could be extended orapplied to determine turning radiion skateboards (and other vehicles)or eggs (which may have biologicalrelevance).

Learners analyze the mathematicalwork of secondary school students,identify errors, and suggest ways toimprove work. Similar triangles andproportional reasoning are sufficientfor determining the maximum rollradius of a given cup (frustum of acone). Learners often start withtrigonometry and other mathematicsmore complicated than necessary, sosimplifying the problem is anadvantage.

7 CriticalReading ofMathematicalModels:Muffin SaleTask

Here learners are prompted tocritically review materials preparedfor teachers that deal withmathematical modeling. The muffinstask is an engaging problem thatprompts school students to analyzedata that is quadratic in nature andfind a maximum.

Learners will critically read themathematical presentation forunderstanding - noting subtletiesthat arise with discrete versuscontinuous functions. The discretevertex differs from the continuousvertex of the quadratic.

Contents

I The Mathematical Modeling Process and Purpose

1 Developing Ways of Thinking for Mathematical Modeling 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Crowd-size Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Crowd-size Estimation Historical Version . . . . . . . . . . . . . . . . . . . . . 71.2.3 Water Crisis in Flint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 In-Class Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.1 Assignment: Video Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 Handout 1: Mathematical Modeling Written Survey . . . . . . . . . . . . . . . 141.3.3 Handout 2: Crowd Size Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.4 Assignment: Current Event with Mathematical Analysis . . . . . . . . . . . . 181.3.5 Handout 3: Flint Water Crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.6 Handout 4: Standards and their Connections to this Lesson . . . . . . . . . . 20

2 Fighting Floods with Sandbags 212.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Sample Approach & Possible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 In-Class Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.1 Handout 1: US Army Corps of Engineers Sandbag Article . . . . . . . . . . . 302.4.2 Handout 2: Missouri Department of Natural Resources Sandbag Article . . . 312.4.3 Handout 3: Fighting Floods with Sandbags . . . . . . . . . . . . . . . . . . . . 322.4.4 Handout 4: Possible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4.5 Handout 5: Standards and their Connections to this Lesson . . . . . . . . . . 372.4.6 Assignment: Sandbags Problem Reflection . . . . . . . . . . . . . . . . . . . . 38

3 Elements of the Mathematical Modeling Process 393.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 General Approach for Mathematical Modeling . . . . . . . . . . . . . . . . . . 453.3 In-Class Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.1 Handout 1: Modeling Cycle - Bare . . . . . . . . . . . . . . . . . . . . . . . . . 473.3.2 Handout 2: Modeling Cycle - Described . . . . . . . . . . . . . . . . . . . . . . 483.3.3 Handout 3: CCSSM Modeling Standards - Portioned for Group Work . . . . 493.3.4 Handout 4: Standards and their Connections to this Lesson . . . . . . . . . . 543.3.5 Assignment: Readings about Mathematical Modeling. . . . . . . . . . . . . . 55

4 Predicting the Evolution of STDs in the USA 564.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3 Commentary, Sample Approach & Possible Model . . . . . . . . . . . . . . . . . . . . 624.4 In-Class Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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4.4.1 Handout 1: Live Science Article and Data . . . . . . . . . . . . . . . . . . . . . 734.4.2 Handout 2: Predicting the Evolution of STDs in the USA . . . . . . . . . . . . 744.4.3 Handout 3: US Census data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4.4 Handout 4: Guidelines for Preparing a Mathematical Modeling Report . . . . 764.4.5 Handout 5: Standards and their Connections to this Lesson . . . . . . . . . . 774.4.6 STD Revised Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.4.7 STD Modeling Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.5 Rubrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Water Conservation: Shower v. Bath 835.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.3 Commentary, Sample Approach & Possible Model . . . . . . . . . . . . . . . . . . . . 865.4 In-Class Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4.1 Handout 1: USEPA Water Conservation Plan Guidelines . . . . . . . . . . . . 905.4.2 Handout 2: Indoor Water Efficiency Fact Sheet - Portland Water Bureau . . . 915.4.3 Handout 3: Water Conservation Problem and Guidelines for Preparing a

Mathematical Modeling Report . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.4.4 Handout 4: Standards and their Connections to this Lesson . . . . . . . . . . 935.4.5 Water Conservation Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.5 Rubrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.6 MODULE 1 Simulation of Practice (SoP) - Written . . . . . . . . . . . . . . . . . . . . 98

6 Analyzing Modeling Tasks: Rolling Cups 1006.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.3 Commentary, Sample Approach & Possible Model . . . . . . . . . . . . . . . . . . . . 1046.4 In-Class Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4.1 Handout 1: Student Work from MARS Resource . . . . . . . . . . . . . . . . . 1066.4.2 Handout 2: Standards and their Connections to this Lesson . . . . . . . . . . 1086.4.3 Rolling Cups Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7 Critical Reading of Mathematical Models: Muffin Sale Task 1117.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.3 Commentary, Sample Approach & Possible Model . . . . . . . . . . . . . . . . . . . . 1147.4 In-Class Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.4.1 Handout 1: Muffins Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.4.2 Handout 2: Student Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.4.3 Handout 3: Baseball Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.4.4 Handout 4: Standards and their Connections to this Lesson . . . . . . . . . . 122

7.5 MODULE 1 Simulation of Practice (SoP) - Video . . . . . . . . . . . . . . . . . . . . . 1247.5.1 Critical Reading - Baseball Article . . . . . . . . . . . . . . . . . . . . . . . . . 126

A Module 1 Assessment and Synthesis 127A.1 Reflection on the Mathematical Modeling Process . . . . . . . . . . . . . . . . . . . . 127

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2 Fighting Floods with Sandbags

Length: 1 Class Meeting, ˜ 75 minutes

Overview

Summary This lesson has been prepared as an initial modeling task before mathematical modeling has been defined or introduced. The general goal is for students to gain experience using mathematics to analyze a real situation and “discover” mathematical modeling first through this task and formalize the process at a later lesson. The specific goal of the lesson is to determine a plausible reason for a discrepancy in two official documents for the number of sandbags needed to build a wall of specified dimensions.

Goals • Extract information from documents, identify variables, and generate functions to describe

the number of sandbags required to build a levee of specified dimensions.

• Apply problem solving strategies.

Materials • Handout 1: US Army Corps of Engineers Sandbag Article

• Handout 2: Missouri Department of Natural Resources Sandbag Article

• Handout 3: Fighting Floods with Sandbags

• Handout 4: Possible Model

• Modeling Course Module 1 slides

• Homework Assignment: Sandbags Problem Reflection

Pedagogical Note. Handout 4 should be printed ahead of time and distributed to PSTs only after they go through the modeling process on their own. It may be useful for PSTs to be able to analyze, view, and discuss a model as created by an instructor. Additionally, they will be able to use it as a reference through the remainder of the course.

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CCSSM Standard Connection to Lesson

MP4: Model with mathematics This is a real-life situation (not contrived or concocted) that arose from noticing a discrepancy in published documents, leading to the question of what may have caused the discrepancy.

6-EE.9: Use variables to represent twoquantities in a real-world problem thatchange in relationship to one another asdependent and independent variables.Analyze the relationship between thedependent and independent variables usinggraphs and tables.

As the thickness of the bags and height of the wall change, the number of bags needed to construct a 100 feet sandbag wall will change in relationship to that. PSTs will use variables in an equation to show how these input changes effect the output.

7-G.6: Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

The sandbag walls are roughly triangular prisms built of rectangles. Using these shapes to construct the larger wall allows predictions of size, total sand needed, etc.

A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.*

To find an answer to the sandbags problem, creating a equation and graphing it will help students visualize a solution.

F-BF.1: Write a function that describes a relationship between two quantities.*

The number of sandbags needed is a quadratic function of the desired height of the wall.

G-MG.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost).*

Recognizing that the sandbag wall is a triangular prism, participants can use properties of triangles and rectangles to accurately describe the wall. Additionally, they can use properties of numbers and shapes to properly model the structure.

Concepts Beyond CCSSM Connection to the Lesson

Triangular numbers The sandbags are arranged in triangles. Using properties of triangular numbers, including the closed formula for finding them, a function can be created to model building a wall of various heights.

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Lesson Overview

Description

Briefly re-introduce the Sandbags task (since it was launched at the end of Developing Ways of Thinking for Mathematical Modeling) and pose the problem as described below. Prompt participants to work with a small group. Encourage participants to keep track of their ideas and to think outside the box while creating their model. Participants can use the articles they read for homework as a reference, and should begin with sharing ideas and important information they read.

1. Call on specific groups to share their results and approaches to the sandbags problem.

2. Introduce Sandbags Problem Reflection. Instructions: Write brief answers to the followingquestions using a different sheet of paper for each item:

(a) What information did you consider necessary to answer the question in the problem? Wheredid you find all this information? Did you make any assumptions or choices in the solution?

(b) What mathematics did you use to solve the problem? How did you select those mathematicalconcepts?

(c) Do you feel that the process of solving this problem was linear (step-by-step) or did you findyourself backing up, making changes to your previous decisions and then continuing with thesolution? If so, give an example of this.

(d) Did your solution make sense to you? Discuss the reasonableness of your solution and whatyou did to improve it if it was not reasonable.

3. Distribute, consider, and discuss Handout 3 with the mathematical model provided by theinstructor. Note that the estimate for the number of sandbags needed depends sensitivelyon the model parameter for the thickness of the bags. How does this approach differ fromthe group presentations?

4. As appropriate during the lesson, include some remarks about triangular numbers, andmathematician Johann Carl Friedrich Gauss and his work with triangular numbers. Checkthat PSTs are comfortable explaining the derivation of a formula for the sum of nconsecutive integers.

Pedagogical Note. It is helpful to organize small groups prior to class and facilitate a quick transition into these groups. We recommend changing groups periodically throughout the course so PSTs develop collaboration skills with a variety of their colleagues. Monitor the groups as they work and make selections of groups with various approaches. Urge groups to keep good records of their work and record not only their calculations, but their thinking so they can better reflect on the process.

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Introduce the Task Burlap sacks filled with sand are used to prevent or reduce floodwater damage. Properly filled and placed sandbags can act as a barrier to divert moving water. Sandbags are also used successfully to prevent overtopping of streams with levees. According to the U.S. Army Corps of Engineers, filling sandbags is a two-person operation. Bags should be filled between one-third to one-half of their capacity. This allows the bags to be stacked with a good seal. The bags should be placed as a pyramid, staggering the position for multiple layers. Read the articles in Handout 1 and Handout 2

Pose the Problem Present the task as described in Handout 3: Fighting Floods with Sandbags.

Height of Sandbag Wall

Your Estimate Army Corps of Engineers Estimate

Missouri Dept. of Natural Resources Estimate

1 foot 600 bags 500 bags

2 feet 2,100 bags 1,000 bags

3 feet 4,500 bags 2,100 bags

4 feet 7,800 bags 3,600 bags

5 feet - 5,500 bags

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Sample Approach & Possible Model

Looking ahead at the lesson on the typical elements of the modeling process, sometimes it is difficult to get started on an open-ended problem like this one, especially without prior experience. It helps to start by asking questions like:

1. What information do we need to estimate the number of sandbags needed per 100 feet?

2. Where do we find this information?

3. What do we do with information we need but we can’t find?

4. How do we develop a procedure to estimate the number of sandbags?

5. What mathematics might be useful?

6. How do we present this procedure?

7. How do we know how accurate our estimate is?

The figure below shows a typical bag viewed from the side (left), an example of the pyramid stacking using 3 layers seen from downstream (center) and a top view (right).

Pedagogical Note. Students can work in small groups, and the instructor can walk around listening to the small group discussions and answering questions. Anticipate questions regarding how to get started, making assumptions and other choices. Encourage students to be creative, as this is likely to be an unusual problem for them.

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The answers likely to come up are that we need to know the dimensions of each filled sandbag, especially the length (to make sure we cover the 100 feet) and the thickness (to make sure we cover the height of the wall). Also needed is the proper way to stack the sandbags. Most of this information is in the handouts.

After reading the handouts and possibly looking up other sources, we will need to make decisions in order to proceed. Decisions and choices that are stated here as assumptions. Assumptions for the model:

• Each sandbag is 12 inches long when filled.

• The thickness of each sandbag is no more than 6 inches depending on how much the bag isfilled.

• The staggering of the bags in the downstream direction does not affect the total number ofbags need (see figure below). This is because the staggering results in a half bag stickingout at one end of the wall and a half bag short at the other end, so they compensate for oneanother.

• The bags are stacked in a triangular pyramid form with 1 bag at the top, 2 bags in the nextlayer down, and so on (like the figure above). If the pyramid has 5 layers, then the bottomlayer will have 5 bags. The figure below shows an example of a pyramid with 3 layers each.

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A General Mathematical Model:

The number of bags needed for a 100-foot long wall is equal to the number of layers in the pyramid times 100 (since the bags are 1 foot long). Let’s define:

• T to be the thickness (in inches) of the bags,

• NL to be the number of layers in each pyramid, and

• NB to be the number of bags required to build a pyramid with NL layers.

Then the total number of sandbags needed for a 100-foot long wall is 100NB. To compute NB, the number of bags required to build a pyramid with NL layers, we sum that number of bags in each layer:

NB = 1 + 2 + 3 + . . . + NL = 1/2(NL)(NL + 1) [connection to triangular numbers]

Pedagogical Note. PSTs can start with specific numbers and generalize as a later step.

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Therefore, the total number of bags for the wall as a function of the number of layers is

total number of bags = 100NB = 50(NL)(NL + 1)

We need to know how many layers are needed. This depends on the thickness of the bags (how much sand is placed in the bags) and the height of the wall we want to build. If the bag thickness is T = 6 inches, then we need NL = 2 layers for every foot of wall height. If the bag thickness is T = 4 inches, then we need NL = 3 layers for every foot of wall height. In general, if the bag thickness is T inches, then we need NL = 12/T layers for every foot of wall height. So, the complete mathematical model for a wall of height H feet is: given a sandbag thickness T (in inches), compute the number of layers with NL = (12/T)H. Then the total number of bags needed is 50(NL)(NL + 1). The table below is a summary for different bag thicknesses.

Wall Height,

N feet

Number of Layers Needed,

NL = (12/T)H

Total Number of Bags,

50(NL)(NL + 1)

Wall Height,

Number of Layers Needed,

NL = (12/T)H

Total Number of Bags,

50(NL)(NL + 1) 6-inch thick bags (T = 6) 4-inch thick bags (T = 4)

1 foot 2 300 1 foot 3 600

2 feet 4 1,000 2 feet 6 2,100

3 feet 6 2,100 3 feet 9 4,500

4 feet 8 3,600 4 feet 12 7,800

5 feet 10 5,500 5 feet 15 12,000

. . . . . .

H feet 2H 100(H)(2H + 1) H feet 3H 150(H)(3H + 1)

Wall

Height,

N feet

Number of

Layers Needed,

NL = (12/T)H

Total Number

of Bags,

50(NL)(NL + 1)

Wall

Height,

Number of

Layers Needed,

NL = (12/T)H

Total Number

of Bags,

50(NL)(NL + 1) 3-inch thick bags (T = 3) 2-inch thick bags (T = 2)

1 foot 4 1,000 1 foot 6 2,100

2 feet 8 3,600 2 feet 12 7,800

3 feet 12 7,800 3 feet 18 17,100

4 feet 16 13,600 4 feet 24 30,000

5 feet 20 21,000 5 feet 30 46,500

. . . . . .

H feet 4H 200(H)(4H + 1) H feet 6H 300(H)(6H + 1)

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Conclusion: The number of sand bags needed for the wall depends sensitively on the thickness of the bags once filled with sand. If we assume they are 4 inches thick (green table column) then the numbers match those of the Army Corps of Engineers. If we assume the bags are 6 inches thick (orange table column), then we would need fewer bags and the numbers match those of the Missouri document. The only exception is the 1-foot tall wall. It may be that their estimate uses different assumptions.

Is our model acceptable?

Consider evaluating your model critically. It helps to ask questions like:

• Are the results from my model reasonable?

• Are there other variables that should be taken into account?

• Are the assumptions made acceptable or should they be changed?

• What if the bag thickness in 5-inches? Then NL = 12/5 = 2.4 layers!

Since both estimates, from the Army Corps of Engineers and from the Missouri document, are obtained from our model by just changing one parameter (the thickness of the filled bags), we can conclude that we answered the question satisfactorily. The fourth bullet is something we didn’t think about before, but it can be accounted for by simply redefining NL to be “the smallest integer greater or equal to 12/T.” Symbolically, this is written 𝑁" = $%&

'(.For instance, $%&

'( =

⌈2.4⌉ = 3

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In-Class Resources

HANDOUT 1: US ARMY CORPS OF ENGINEERS SANDBAG ARTICLE

Sandbagging techniques Brochure from US Army Corps of Engineers, Northwestern Division, 2004

Citation: U.S. Army Corps of Engineers, Northwestern Division. (2004). Sandbagging techniques [Brochure]. Retrieved from usace.contentdm.oclc.org

HANDOUT 2: MISSOURI DEPARTMENT OF NATURAL RESOURCES SANDBAG ARTICLE

How to construct a sandbag emergency levee Article from Missouri Department of Natural Resources by Carol S. Comer, March 2014

Citation: Comer, C. S. (2014, March). How to construct a sandbag emergency levee. Missouri Department of Natural Resources, retrieved from dnr.mo.gov/pubs/pub2217.htm.

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SANDBAGS PROBLEM REFLECTION

Write brief answers to the following questions using a different sheet of paper for each item:

1. What information did you consider necessary to answer the question in the problem?Where did you find all this information? Did you make any assumptions or choices in thesolution?

2. What mathematics did you use to solve the problem? How did you select thosemathematical concepts?

3. Do you feel that the process of solving this problem was linear (step-by-step) or did youfind yourself backing up, making changes to your previous decisions and then continuingwith the solution? If so, give an example of this.

4. Did your solution make sense to you? Discuss the reasonableness of your solution andwhat you did to improve it if it was not reasonable