adapted for afﬁliation and course version term taught draft · mathematical modeling adapted for...

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

MODULE(S2):Mathematical Modeling

Adapted for Affiliation and Course

Version Term Taught

This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License. acknowledgmentsThe Mathematics Of Doing, Understand, 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.

Draft

To Do List

Overview Document . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CCSSM Standards and Connections to Lessons for ALL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

instructor vs. student view for ALL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

L4: Change this format to match more like the other lessons, clarify and revise HW #5 draft and HW #6 final . . . 51

L5: Sensitivities Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

L6: Do we want to include this pdf as a handout attached at the end of this lesson? . . . . . . . . . . . . . . . . . . 73

L7: do we also assume that the patient has an odd sleeping schedule? . . . . . . . . . . . . . . . . . . . . . . . . . . 84

L8: Simulation of Practice needs beefing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

L9: concepts beyond CCSSM? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Contents

I The Mathematical Modeling Process

1 Introductory Examples 1

Length: 2 Class Meetings, ˜ 150 minutes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Lesson Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Water Crisis in Flint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Introduce the Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Pose the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Sample Approach & Possible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Crowd-size Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6


Pose the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Crowd-size Estimation Historical Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7


Pose the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Sample Approach & Possible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Preparation for next lesson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 In-Class Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Handout 1: Water Donations in Flint, Michigan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Amid Flint crisis, Walmart, Coca-Cola, Nestle and PepsiCo to donate millions of water bottles 9

1.3.2 Handout 2: Crowd-size Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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How do the media and police estimate crowd sizes? . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.3 Handout 3 - Guided Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Flint Water Crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Crowd Size Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Crowd-size Estimation Historical Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Fighting Floods with Sandbags 15

Length: 1 Class Meeting, ˜ 75 minutes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Lesson Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16


Pose the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Possible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Preparation for Next Lesson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20


2.4.1 Handout 1: Sandbagging Techniques, US Army of Engineers Northwestern Division . . . . . . . . 22

2.4.2 Handout 2: Missouri Department of Natural Resources, Natural Disaster Assistance for MissouriCitizens - How to Construct a Sandbag Emergency Levee . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.3 Handout 3: Possible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Elements of the Mathematical Modeling Process 37

Length: 1.5 Class Meetings, ˜ 110 minutes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Lesson Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38


3.3 General Approach for Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41


3.4.1 Handout 1: Modeling Cycle - Bare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.2 Handout 2: Modeling Cycle - Described . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.3 Handout 3: CCSSM Modeling Standards - Portioned for Group Work . . . . . . . . . . . . . . . . 44

Group 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Group 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Group 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Group 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.5 Sample Questions from the Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Predicting the Evolution of STDs in the USA 51


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4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Summary and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Lesson Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


Pose the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Commentary, Sample Approach & Possible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

End of Day 1 and preparation for next lesson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62


4.4.1 Handout 1: Live Science Article and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4.2 Handout 2: Predicting the evolution of STDs in the USA . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4.3 Handout 3: US Census data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4.4 Handout 4: Guidelines for Preparing a Mathematical Modeling Report . . . . . . . . . . . . . . . . 65

5 Analyzing Modeling Tasks: Rolling Cups 66


5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Introduce the Task: Egg Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Introduce the Task: Longboard Turning Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Pose the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Lesson Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68


Possible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69



5.5 Written Simulation of Practice: Reflection on Rolling Cups . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6 Critical Reading: Muffin Task 72


6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Lesson Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73


Pose the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73


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Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

A Variation on the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75



6.4.1 Handout 1: “An Authentic Modeling Task That Models Quadratics” . . . . . . . . . . . . . . . . . 78

6.4.2 Handout 2: “Model Eliciting Activities: A Home Run” . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.5 Video Simulation of Practice: Analyze the Muffin Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Examine and Analyze Student Work and an Article . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7 Modeling Pain Medication 81


7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Lesson Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83


Pose the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83


A Possible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Other Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90



7.4.1 Handout 1 - Pain Medication Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8 Water Conservation, Shower v. Bath 93


8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

8.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


Pose the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Reporting Out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95




8.4.1 Handout 1: USEPA Water Conservation Plan Guidelines . . . . . . . . . . . . . . . . . . . . . . . . 98

8.4.2 Handout 2: Indoor Water Conservation Fact Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8.4.3 Handout 3: Water Conservation Problem and Guidelines for Preparing a Mathematical ModelingReport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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Citations: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

8.5 Video Simulation of Practice: Reflection on Teaching the Mathematical Modeling Process . . . . . . . . . 108

9 The Lost Cell Phone 109


9.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

9.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110


Pose the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110


Initial Model: 2D - Two-dimensional Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

A Revised Model: 3D- Three Dimensional Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113



9.4.1 Handout 1: The Lost Cell Phone Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

9.4.2 Guidelines for Preparing a Mathematical Modeling Report . . . . . . . . . . . . . . . . . . . . . . . 116

9.5 Written Simulation of Practice: Student Work on the Lost Cell Phone Task . . . . . . . . . . . . . . . . . . 117

9.6 Additional Resources for Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

9.6.1 “Mathematical Modeling: A Structured Process” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

10 The Modeling Process Revisited 119


10.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

10.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Lesson Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119


Pose the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Commentary, Sample Approach & Possible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120



10.3.1 Handout 1 - Questionnaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

10.4 Additional Articles and Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

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1 Introductory Examples

Length: 2 Class Meetings, ˜ 150 minutes

Overview

SummaryThis lesson sets the stage for the class. The problems may be different from what students are used to encountering.Students will raise mathematical questions with connections to several current events, serving as an introduction to anunderstanding of mathematical modeling.

Goals• Generate student motivation for seeing mathematics in everyday life.

• Orient students to an applied mathematicians’ view of the world.

• Close the gap between the outside world and school math.

• Help students realize that mathematics is so much more than what they have been learning in their classes andthat they can address social, political, or scientific issues using mathematical ideas to understand, analyze orpredict happenings in the world around them.

CCSSM Standard Connection to Lesson

MP4: Model with Mathematics PSMTs participate in an introductory modelingexperience. This will help prepare them for the restof the class. PSMTs use mathematics to estimate dailywater needs of school children in Flint, Michigan andto approximate how many people attend an eventbased on photographs or other observations.

A-CED.3 Represent constraints by systems ofequations, and interpret solutions as viable ornon-viable options in a modeling context.?

PSMTs use two linear equations to represent thenumber of sales and rentals of “The Interview” basedoff data from a New York Times article. Additionally,PSMTs interpret these results and briefly discuss whythey are useful.

A-REI.6 Solve systems of linear equations exactly andapproximately

Using the system of two linear equations to representhow many copies of the “The Interview” aredistributed and how much money is made, PSMTssolve for the number of sales and rentals.

7-RP.1 Compute unit rates associated with ratios offractions including quantities measured in like ordifferent units.

Students use units, rates, ratios, scaling, etc. to be ableto convert between the water needs in Flint, Michiganand how much water is donated. Additionally,students may use scaling and ratios in the crowd-sizeestimation problem.

N-Q.1 Use units as a way to understand problems andto guide the solution of multi-step problems.?

To find a solution to the Flint, Michigan Problem,students determine what units they want an answer in,and use conversions to solve for that.

Concepts Beyond CCSSM Connection to Lesson

Solving systems of equations with linear algebra Comes up in discussion of “The Interview”

Materials• Handout 1 - Water Donations in Flint, Michigan

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• Handout 2 - Crowd-size Estimation

• Handout 3 - Guided Worksheet

• Video - Introduction to Water Crisis

• Modeling Course Module 1 slides

Description

Lesson OverviewNote. Most of this information is also included in the Google slides presentation. Feel free to reference the slidesto guide your class as you teach.

Day 1: (One full day)

1. Begin by introducing Homework 1

(a) Introductory survey completed via video recording. This assignment will be completed outside of class.Instructions: Create a short video of yourself (2-5 minutes) as you might introduce yourself to a class. Pleaseaddress the following questions with as much (or as little) detail as you’d like. This information will also be used toselect topics of interest to you in this course, and as a test of the technology for making a video of your teachingpractice.

i. Your name, what you’d like to be called by your studentsii. A brief statement of your interests in and outside of school

iii. Where you’re from, languages you speak, places you’ve travelediv. Something about your family: large/small, occupations of parents or grandparents or siblings [only

select topics that are relevant to you]

Tool Tip. This would be a good time to teach students how to use specific the specific videotechnologies you want them to use when creating videos. It would also be a good time to discussassignment submission procedures and policies.

(b) Initial survey on mathematical modeling. This assignment will be a written assignment. Let studentsbegin and complete the survey in class to stimulate thoughts and conversations about modeling. Theywill submit the assignment at the end of today’s class. Instructions: Answer the following questions withsufficient detail.

i. What does mathematical modeling mean to you?ii. Describe your experiences with courses and topics most relevant to mathematical modeling.

iii. Describe any additional interests or information about you that you did not include in the video.

2. Raise mathematical questions with connections to current events. Included is an example about “TheInterview.” View the slides for specific details regarding this example.

(a) Introduce “The Interview” article regarding film sales. This film is a fictional comedy about Americanentertainers of which North Korea’s leader Kim Jong-Un is supposed to be a fan. The CIA then contactsthem to carry out an assassination plot. The film was condemned by North Korea, and generated somecontroversy when released. You may have seen this in the New York Times at the end of 2014(Cieply, Michael. ’The Interview’ Brings In $15 Million on Web. The New York Times, The New YorkTimes, 28 Dec. 2014,www.nytimes.com/2014/12/29/business/media/the-interview-comes-to-itunes-store.html.)

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http://abcnews.go.com/WNT/video/lead-found-tap-water-flint-michigan-36277516

(b) This problem serves as an example to how math comes up in real life. It is almost a textbook wordproblem, but the writer didn’t notice.

(c) As a class, discuss the mathematical parts of this article, and solve the system of linear equations.

• (definitions) R = Rentals, S = Sales• (transactions) R + S = 2, 000, 000• (dollars) 6 · R + 15 · S = 15, 000, 000• Solution: R = 1.67 million, S = 0.33 million

3. Discuss/establish class norms

Pedagogical Note. I recommend beginning with the list in the slides and discussing the rationale forthese norms. Then ask students for additional input on the norms. For example, a student might ask “Canwe have some freedom to select topics that interest us and pursue our own ideas and approaches?” Theanswer should be “Yes, of course.” And the instructor can add something to the class norm reflecting thisidea.

4. Discuss another introductory example. As you conduct your class, you can use these examples or substitutethem with more current examples. Included later in this document are in-depth descriptions of the tasks andproblems associated with both the water crisis issue and the crowd-size estimation issue.

(a) Water Crisis in Flint (A social political crisis.)

Note. Included in most lessons in these modules is a “Sample Approach and Possible Model” section.These are not intended to be used as a lecture. Rather, they are meant to be background knowledge andcontext for the instructor. Introduce the task, pose the problem, and let students begin their work. Afterstudents work and report on these problems, you could highlight certain parts from the Possible Model.In the duration of this course, students should feel freedom to try new things, create different models, andengage in the modeling process. As the instructor, you are responsible for helping them succeed in this,not simply showing them possible models.

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Pedagogical Note. Some freedom is given to you as the instructor as to how you want to run thesediscussions. Perhaps you could have students work in pairs for this portion of the class in contrast to theprevious example you did as a whole class. While they are working, move through the classroom to getan idea about what different students are working on. Their discussions will help you know what to focuson when you bring the class back together to dissect their work.

5. Introduce Homework 2 - Current Event with Mathematical Analysis. Instructions: For this assignment, locate anarticle from a news outlet and begin thinking and working quantitatively with the article content. Your write-up should be5 paragraphs in all. Answer the following questions:

(a) What is the article and source?

(b) Why did you choose this article? What is the main issue? 1 paragraph

(c) What kind of mathematics is included by the author? 1 paragraph

(d) Critique the quantitative presentation. What questions do you still have? 1 paragraph

(e) Be creative and expand. What mathematics might further enhance the news story? Develop some ideasand present your thinking. 2 paragraphs

Day 2: (One full day)

1. Begin class by reviewing the current events and analyses from Homework 2. Have several students share whatarticles they found, the mathematics that goes along with it, and what (if any) research they did to learn moreabout the specific situation.

2. Discuss solutions, implications, ideas, etc. from the Flint Water Crisis Problem.

3. Launch the Crowd-size Estimation problem (or something else current in the news - something regarding whowas telling the truth). Give students adequate time to discuss and work toward finding a solution. Again, moreinformation is found for this below as an attachment.

4. Briefly introduce the task from Lesson 2 – Sandbags. Burlap sacks filled with sand are used to prevent or reducefloodwater damage. Properly filled and placed sandbags can act as a barrier to divert moving water around. Sandbags arealso 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 allowsthe bags to be stacked with a good seal. The bags should be placed as a pyramid, staggering the position for multiple layers.

5. Introduce Homework 3: Read Handout 1 “Sandbagging Techniques” and Handout 2 “How to Construct a SandbagEmergency Levee” from Lesson 2 about the Sandbags problem. Familiarize yourself with the data from both informationsheets. Look for sandbagging techniques, discrepancies in estimates, and any other useful data from the articles.

Note. The purpose of Lesson 1 is to lead students to realize the quantitative nature of every day news. Anintended side effect of the lesson is to raise awareness about what’s going on in the world. We are not trying tojust use real world contexts to bring up mathematics problems, rather the other way around. We are raisingrelevant questions to real world issues and then seeing the mathematics.

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WATER CRISIS IN FLINT

Introduce the TaskThe Washington Post reported that there is a water emergency in the city of Flint, Michigan. Scientists found elevatedlevels of lead (which is poison) in Flint’s water supply. Blood tests revealed that lead contamination had nearlydoubled and tripled in children younger than 5 who were exposed to the highest lead levels. The water isundrinkable.

Watch the following video further describing the problem.http://abcnews.go.com/WNT/video/lead-found-tap-water-flint-michigan-36277516

Pose the ProblemRead the article Water Donations in Flint from Handout 1 and keep in mind the following questions:

• What can city officials do in this situation?

• What should residents do to get water?

• How should the community help?

Walmart, Coca Cola, Nestle, PepsiCo said that they will donate bottles of water for school children in Flint, Michiganto help with the city’s public health crisis over lead contaminated water. On January 26 the companies said that theyare planning to “collectively donate water to meet the daily needs of over 10,000 school children for the balance of thecalendar year.” To do so, the companies will send 176 truck loads of bottled water, up to 6.5 million bottles, to Flint.

Consider the following questions in regards to this situation. Work with a partner to understand and dissect the issue.

• How do we know how much water is enough to meet the “daily needs” of Flint school children until December31, 2016?

• Is the companies’ plan a good one?

• What information do we need to determine the daily needs of the Flint school children?

• Where do we find this information?

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

• How do we develop a procedure to determine how much water is actually needed in comparison with thedonation?

• How do we present this procedure?

• What mathematics might be relevant to use?

• How do we know how accurate our estimate is?

Sample Approach & Possible ModelTo begin, I remembered hearing that a good measure for adults to know how much water they need to drink daily isto convert their weight into ounces and divide by 2. Consider the following example. For someone who weights 150pounds, perform the conversion as follows. ( 150 lbs

1 ) · ( 1 oz1 lb ) · (

12 ) = 75 oz. That person should aim for drinking 75 oz of

water daily.

I wanted to validate that information on the Internet, as well as research what the recommendations are for children.Smart Sips for Healthy Kids gives the number of cups of water that children should drink according to their age.

As a general rule, here’s how much water kids should drink every day:

Ages 4 - 8 years: 5 cups = 40 oz.

Ages 9 - 13 years: 7 to 8 cups = 56 - 64 oz.

Ages 14 and up: 8 to 11 cups = 64 - 88 oz

Beyond just having safe drinking water, children need to have lead-free water for washing dishes, preparing food, andbrushing their teeth. The Center for Disease Control and Prevention says that bathing and showering in lead water isokay, as long as children do not swallow the water. I estimated that 2 gallons of water (256 ounces) would be sufficientfor dishes, food prep, and hygiene for one day.

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http://abcnews.go.com/WNT/video/lead-found-tap-water-flint-michigan-36277516

https://www.washingtonpost.com/news/post-nation/wp/2016/01/26/in-flint-walmart-coca-cola-nestle-and-pepsico-will-donate-millions-of-water-bottles-to-schoolchildren/

https://www.webmd.com/parenting/guide/kids-healthy-hydration#1

https://www.cdc.gov/nceh/lead/tips/water.htm

There are several assumptions we have to make in this process. We need to assume that within an age group, allchildren require the same amount of water - disregarding gender, weight, activity level, and the weather. Additionally,we will assume that the water bottles being donated are all plastic water bottles that carry 16 ounces of water asshown on the introductory video.

Additionally, recall that the donations were going to be delivered on January 26, 2016. However, 2016 is a leap year, sothere are 366 days. 366− 26 = 340. The number of days remaining in 2016 are 340. Consider the following calculationsregarding how much water was donated.

(6.5× 106 water bottles)·(16 oz per bottle) = 10.4× 107 oz

( 6.5× 106 bottles104 children ) · ( 1

340 days ) ≈ 1.91 bottles per child per day.

( 10.4× 107 oz104 children ) · ( 1

340 days ) ≈ 30.59 oz per child per day.

A child who is between ages 4 - 8 needs about 40 oz of drinking water each day. The donations do not adequatelyprovide enough water for the youngest children. For children who are 14+ years old, 30 oz is approximately only 1

3 ofthe total amount of water needed.

The average water needed across all age groups of children is 64 oz or four 16 oz water bottles. Calculate how manywater bottles would be needed for all children through the remainder of the school year.

4 bottles · 340 days · 104 children = 13.6× 106 bottles.

The donations from Walmart, Coca Cola, Nestle, and PepsiCo were very generous but will only provide slightly lessthan half of the necessary drinking water for school-aged children in Flint, Michigan. This conclusion is based on thefact that we are not considering how much more water is needed for safe food preparation and personal hygiene ontop of drinking water.

CROWD-SIZE ESTIMATION

Introduce the TaskHow can we estimate the size of a crowd? We usually hear in the news estimates of crowds at political rallies, outdoorconcerts or other events. For example, on January 18, 2017 the New York Times ran the article From Lincoln to Obama,How Crowds at the Capitol Have Been Counted with a history of how this has been done over the years.

Pose the ProblemThe picture below shows a crowd at an event. Discuss the following questions. Develop a procedure that can be usedto estimate the number of people in the picture.

• What information do we need to estimate the size of a crowd from an image like the picture?



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https://www.nytimes.com/interactive/2017/01/18/us/politics/How-Crowds-at-the-Capitol-Have-Been-Counted.html

https://www.nytimes.com/interactive/2017/01/18/us/politics/How-Crowds-at-the-Capitol-Have-Been-Counted.html

• How do we develop a general procedure to estimate the crowd size from a picture?

• How do we describe this procedure?



CROWD-SIZE ESTIMATION HISTORICAL VERSION

Introduce the TaskOnly three people have a US national holiday observed in their honor: Christopher Columbus, George Washington,and the Rev. Martin Luther King, Jr. He was a leader in the movement to end racial segregation in the United States.He was an advocate of non-violent protest and became the youngest man to be awarded the Nobel Peace Prize. Hismemorial was created in 2011.

Pose the ProblemThe “March on Washington,” when Martin Luther King gave his world-wide famous speech, “I Have a Dream,” washeld in August 28, 1963. The pictures show the crowd at the event.

Discuss the following questions. Develop a procedure that can be used to estimate the number of people that attendedthe event.


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CommentaryFollowing the students’ discussion, read and comment on the following article, How Do the Media and PoliceEstimate Crowd Sizes?. Discuss how their approaches were similar and different from the estimation techniquespresented in the article.

Sample Approach & Possible ModelThis sample approach includes the thought-process behind one way to estimate crowd size. The questions from aboveare listed with responses written in italics.

• What information do we need to estimate the size of a crowd from an image like the picture? We need to knowthe general size of the area the people are standing on. Is the ground flat, or are there bumps and hills?

• Where do we find this information? It may be hard to find the information for this particular picture. However, if weare dealing with a photograph from a recognizable area, you can use Google Maps or Google Earth to figure out the area.

• What do we do with information we need but we can’t find? If there is information we can’t find, we can makeassumptions or guesses. We do need to clearly state those assumptions, however, so other people can understand the workthat we did on the problem.

• How do we develop a general procedure to estimate the crowd size from a picture? One idea is to count and drawapproximately 10 people by 10 people squares. These squares will be larger near the bottom of the picture and will getsmaller as you move toward the top of the picture.

• How do we describe this procedure? Sketching these 10 by 10 squares onto the picture helps us make these estimations.

• What mathematics might be relevant to use? Another idea is to use some form of Riemann sums or trapezoidal sums.This is more of a calculus approach rather than a geometry approach.

• How do we know how accurate our estimate is? To figure out the accuracy of our estimate, we can compare ourresults with those of other people. Additionally, if an event has ticket sales, you can compare your results to that of theticket sales. Another good idea is to check if your estimation seems valid. If the number seems too high or too low for thesituation, keep trying and developing your procedure.

Preparation for next lesson• Students:

◦ Do Homework 1 - Introductory Video Survey and Written Survey

◦ Do Homework 2 - Current Event with Mathematical Analysis

◦ Do Homework 3 - Read Sandbag Articles

• Instructor:

◦ Print copies of the Possible Model Handout for Lesson 2 Sandbags

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http://www.todayifoundout.com/index.php/2014/11/news-police-like-estimate-crowd-size-parade-protest-like/


In-Class Resources

HANDOUT 1: WATER DONATIONS IN FLINT, MICHIGAN

(Citation: Bever, Lindsey. Amid Flint Crisis, Walmart, Coca-Cola, Nestl and PepsiCo to Donate Millions of WaterBottles. The Washington Post, WP Company, 26 Jan. 2016,www.washingtonpost.com/news/post-nation/wp/2016/01/26/in-flint-walmart-coca-cola-nestle-and-pepsico-will-donate-millions-of-water-bottles-to-schoolchildren/)

Amid Flint crisis, Walmart, Coca-Cola, Nestle and PepsiCo to donate millions of waterbottlesArticle from the Washington Post By Lindsey Bever January 26, 2016https://www.washingtonpost.com/news/post-nation/wp/2016/01/26/in-flint-walmart-coca-cola-nestle-and-pepsico-will-donate-millions-of-water-bottles-to-schoolchildren/

Walmart, Coca-Cola, Nestle and PepsiCo said Tuesday that they will donate up to 6.5 million bottles of water toschoolchildren in Flint, Mich., to help with the city’s public health crisis over lead-contaminated water.

The companies announced in a news release that they are planning to “collectively donate water to meet the dailyneeds of over 10,000 school children for the balance of the calendar year.” To do so, the companies will send 176truckloads of bottled water up to 6.5 million bottles to Flint.

“We are grateful for Walmart and their suppliers’ support during this crisis,” Bilal Tawwab, Flint Community SchoolsSuperintendent, said in a statement. “With their generous support, District students will have access to clean drinkingwater, and more importantly, the ability to focus on their education.”

In April 2014, Flint stopped drawing its water from Detroit and began pulling water from the Flint River. Residentssoon began complaining that the water smelled or was discolored. In October, Flint again began getting water fromDetroit but, by that time, some residents had been drinking the water for 19 months.

Researchers found elevated levels of lead in Flint’s water supply and reported that blood tests revealed that leadcontamination had nearly doubled and tripled in children younger than 5 who were exposed to the highest lead levels.

The city and state have since declared a public-health emergency.

Officials at Walmart said the water crisis has become personal.

“Those affected include our own associates, customers and their families,” said Beth Harris, store manager at a FlintWalmart, said in the statement. “Our associates are proud to be a part of the effort to help our friends and neighbors.”

Business and community leaders will announce the donation at the Walmart Supercenter in Flint.

“Access to safe water is a basic human right,” Tony West, executive vice president of government affairs at PepsiCo,said in the statement, adding that the donation announced Tuesday “will allow Flint school children and their parentsto focus on their education rather than where they can find clean water.”

Walmart said in the news release that it has “already donated 14 trucks of water, or 504,000 bottles, and 1,792 waterfilters to the Flint community” since July. It noted that Coca-Cola has donated nearly 80,000 bottles of Dasani and thatNeste Waters North America has donated more than 190,000 bottles to Flint community organizations.

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https://www.washingtonpost.com/news/post-nation/wp/2016/01/26/in-flint-walmart-coca-cola-nestle-and-pepsico-will-donate-millions-of-water-bottles-to-schoolchildren/

HANDOUT 2: CROWD-SIZE ESTIMATION

How do the media and police estimate crowd sizes?(Citation: Smallwood, Karl. How Do the Media and Police Estimate Crowd Sizes? Today I Found Out, 6 Nov. 2014,www.todayifoundout.com/index.php/2014/11/news-police-like-estimate-crowd-size-parade-protest-like/.)Article from Today I Found Out by Karl Smallwood November 6, 2014http://www.todayifoundout.com/index.php/2014/11/news-police-like-estimate-crowd-size-parade-protest-like/

Keith H. asks: When the police and media report crowd sizes of a parade or something, what do they base theirnumbers on?

Although the task of determining how many people attend something as large as say, a political rally or a protest mayseem like a daunting, almost impossible undertaking to do with any accuracy, with some basic information, it’sactually not that difficult to get reasonably accurate results. The most well-known method of estimating the size of agiven crowd is simply called “The Jacobs’ Method” as an ode to its inventor, Herbert Jacobs. Jacobs spent a fewdecades working for the Milwaukee Journal before retiring into teaching journalism at the University of California,Berkeley in the 1960s. He thought up his very simple crowd size estimate method after observing numerous VietnamWar protests outside of his office window.

Jacobs noticed that the area the students stood on had a repeating grid-like pattern, meaning he could very easilycount how many students occupied a certain amount of space by counting how many students on average seemed tobe able to stand inside a section of the grid. By doing this, he soon noticed some patterns.

For example, Jacobs found that in the most densely packed crowds, each person took up approximately 2.5 squarefeet. We should note that this is the absolute upper limit of a how dense a crowd can safely get, as in, you simplycouldn’t fit more people into a crowd this dense without someone being trampled or worse, which is probably whymost, including some scholarly articles on the subject we read, simply refer to it as “mosh-pit density”. In a dense, butmore manageable crowd, Jacobs observed that participants had a comparably more roomy 4.5 square feet whilst thosein a “light” crowd had a positively breezy 10 square feet to themselves.

In any event, once he had the approximate average number of students in each grid, he could then easily calculate thenumber of grids in an area occupied at a given density, and quite quickly come up with a very good estimate of howmany people were in a given crowd. Thus, the now Gold Standard, and remarkably simple, “Jacobs’ Method” wasborn.

This may sound like an overly simple solution but the truth is, it’s strikingly accurate when done by non-biasedobservers, and modern technology has only made it easier. For instance, tools like Google Earth have made learningthe exact size and area of a location, as well as dividing an area into grids, an almost trivial feat for just about anyone.And thanks to ubiquitous media coverage, any large gathering of people is going to have video or photographicfootage (if not just scanning the Tweetosphere for people in the crowd who may have gotten a good shot and posted itonline). So breaking things down from there is relatively trivial. Of course, one could get really fancy and take a photoof an entire crowd and use a bit of custom designed image processing software to programmatically count the peoplein a crowd for a more exact number, but the extra level of accuracy here over the properly executed Jacobs’ Methodisn’t really typically that much, nor all that necessary.

Of course, when giving estimations, sometimes the news media or the organizers of an event do like to fudge thenumbers a bit. Perhaps the most famous example is that of the Million Man March - a mass gathering of AfricanAmericans (mostly men) that took place in 1995. As you can probably guess from the name of the march, eventorganizers afterwards were very insistent that at least a million men had attended, with estimates going as high as twomillion. However, the National Parks Service disagreed and offered up a much lower, but still extremely significantfigure of around 400,000 individuals. But when something is called the Million Man March, 400,000 seems a bit of aletdown, even though it’s logically very much not; getting 400,000 people (about 1.2% of all African Americans in theUnited States at the time) to show up at such an event in Washington DC is really quite a feat.

Nevertheless, the NPS’s estimate incensed a key player behind the march, Louis Farrakhan, so much so that hethreatened to sue the NPS. As a direct result of the brouhaha that followed, the NPS is now banned by congress fromestimating the size of crowds in Washington, at least publicly. As they noted, if the President asks them for how big acrowd was, they’re happy to crunch the numbers given footage of the crowd. They just aren’t technically supposed touse tax payer dollars in this way anymore, so wouldn’t share that information with the media who, of course, couldquite easily come up with their own estimates.

So how many people actually attended the Million Man March? While an exact figure is impossible to discern, most

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researchers are in agreement that the original estimation of the NPS is pretty accurate. For example, in 2004 a pair ofresearchers, Clark McPhail and John D. McCarthy, worked out that in the location of the gathering there would havebeen space for a maximum of 1,048,206 people assuming that every inch of the crowd was as densely packed as safelypossible at 2.5 square feet per person. In the end, from the pictures available of the gathering, they determined that theNPS’s estimate of about 400,000 was quite accurate.

This isn’t a one-off case either; research has shown that the estimates of event organizers are consistently higher thanthose of the police, who tend to give more accurate predictions given the events generally take place in gatheringspaces that are well documented in case of emergency, in terms of how many people they can safely hold. Of course,event organizers (and sometimes the media) can have something to gain by overstating how large a crowd is, whilethe police and other official agencies generally do not.

That said, there are certainly examples out there of official agencies intentionally adjusting announced crowd sizes forone reason or another just like organizers love to do. Luckily, there’s a simple method of accurately estimating the sizeof a crowd that’s free from bias, and these days can be easily done even just by some guy sitting at home in his PJssurfing the web half way across the world from where the event is actually happening, which is more than a littleamazing. Don’t you think?

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HANDOUT 3 - GUIDED WORKSHEET

Flint Water CrisisWalmart, Coca Cola, Nestle, PepsiCo said that they will donate bottles of water for school children in Flint, Michiganto help with the city’s public health crisis over lead contaminated water. On January 26 the companies said that theyare planning to “collectively donate water to meet the daily needs of over 10,000 school children for the balance of thecalendar year.” To do so, the companies will send 176 truck loads of bottled water up to 6.5 million bottles to Flint.

Consider the following questions in regards to this situation. Work with a partner to understand and dissect the issue.

• How do we know how much water is enough to meet the “daily needs” of Flint school children until December31, 2016?

• Is the companies’ plan a good one?

• What information do we need to determine the daily needs of the Flint school children?



• How do we develop a procedure to determine how much water is actually needed in comparison with thedonation?

• How do we present this procedure?



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Crowd Size EstimationThe picture below shows a crowd at an event. Discuss the following questions. Develop a procedure that can be usedto estimate the number of people in the picture.








Following your discussion, we will read and comment on the following article, How Do the Media and PoliceEstimate Crowd Sizes?. During our discussion, write down how your approaches were similar and different from theestimation techniques presented in the article.

• Similarities

• Differences

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Crowd-size Estimation Historical VersionThe pictures show the crowd at the “March on Washington” event led by Martin Luther King, Jr. Discuss thefollowing questions. Develop a procedure that can be used to estimate the number of people that attended the event.








Following your discussion, we will read and comment on the following article, How Do the Media and PoliceEstimate Crowd Sizes?. During our discussion, write down how your approaches were similar and different from theestimation techniques presented in the article.

• Similarities

• Differences

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