Mathematical Modeling: Examples and Opportunities Holly Zullo and Allison Carson Westminster College

Overview MCM Alli share MCM work Contests for high school students Preparation Benefits Resources

Mathematical Contest in Modeling (MCM) 96-hour competition in January Team of 3 students Designed for undergraduates, but high school students can participate Students pick one of 6 open-ended applied problems Cost: $100 per team http://www.comap.com/undergraduate/contests/

MCM Problems A: continuous B: discrete C: data insights Hot bath B: discrete Space junk C: data insights The Goodgrant challenge D: operations research/network science Measuring the evolution and influence in a society’s information E: environmental science Are we heading towards a thirsty planet F: policy Modeling refugee immigration policies

Bathtub Problem After you sit in a bath for a while, the temperature cools. If you turn on the faucet again, what is the best method to keep the bath water at constant temperature and as close to the original temperature as possible?

Assumptions The bathtub is of a rectangular shape. The room temperature is constant and there is no outside airflow. The faucet delivers immediate hot water to the bath. Evaporation is constant over time. Mixing the water will assure a perfect mix throughout the tub. Heat moves through a liquid similar to how it moves through a solid

Model Theory Discretize the heat equation: Newton’s Law of Cooling 𝑇 𝑖+1,𝑗 = 𝑇 𝑖,𝑗 + 𝜅Δ𝐴 2 𝑇 𝑖,𝑗 − 𝑇 𝑖,𝑗−1 − 𝑇 𝑖,𝑗+1 Δ 𝑥 2 Newton’s Law of Cooling

Model Theory Differential Equations for perfect mixing *assumptions!

Experiment!

Simulations

Simulations

Results

HiMCM High school version of MCM 36-hour contest Time window in November (this year is Nov 4 – 17) Teams of 4 high school students Students pick one of two problems Cost: $75 per team http://www.comap.com/highschool/contests/himcm/index.html

Problem: City Crime and Safety What can we make of the massive amount of crime statistics collected in major cities? Beyond just reporting numbers, how can we use these data to determine the safeness of a city? Assume that you and your modeling team live in My City, a large international hub of commerce, technology, finance and travel, with a current population of 2.8 million people impacted by a metropolitan area of an additional approximately 6 million people. The data set provided (My_City_Crime_Data.xlsx) shows two weeks from police reports in My City and includes crimes listed by case number, date of occurrence, primary and secondary crime descriptions, crime location, whether an arrest was made, whether or not this was domestic crime, and the beat number of the police route. Part I: Using mathematical modeling, analyze the data. Create a safety rating for My City. Use your safety rating to specify a measure of how safe My City is. Part II: In addition to the HiMCM contest format, prepare a 1-2 page non-technical report for the Mayor of My City to describe your findings.

International Mathematical Modeling Challenge Up to two teams per country can compete All Finalist, National Finalist, and Outstanding teams from HiMCM invited (about top 10%) 5-day contest in April Top two papers forwarded for international judging Teams and advisors ranked Outstanding invited to awards ceremony Hope to grow to same prestige as International Mathematics Olympiad In 2nd year, had 40 teams from 23 countries Problem: insurance for athletic world-records

Moody’s Mega Math Challenge 14-hour contest Registration open in November; contest February 24-27, 2017 3 to 5 students per team; juniors and seniors only Limit of two teams per high school One problem 2016: Share and (car) share alike Cost: free! Scholarship prizes total $150,000 https://m3challenge.siam.org/challenge/about

Preparing Students Brainstorm on old problems Simplifying the problem Writing the paper Equation editor Winning papers freely available from IMMC and Moody’s Study for common sections Writing the summary Group work Respect Keeping everyone busy Time management

Benefits Sense of accomplishment Increased confidence Solved big problem Immersion Learned new math & better understand old concepts Worked without assistance from teacher Increased confidence Worked with applied math instead of just algebra and computations Better idea of how math might be used in careers Something to discuss on college/scholarship applications

Resources GAIMME (free download): http://www.siam.org/reports/gaimme-full_color_for_online_viewing.pdf Modeling Handbook (free download): https://m3challenge.siam.org/sites/default/files/uploads/siam-guidebook-final-press.pdf Math Models (membership $29/year): http://mathmodels.org/ Has solutions to MCM, HiMCM, and IMMC contests Holly Zullo hzullo@westminstercollege.edu Alli Carson ac0521@westminstercollege.edu