2.1: Mathematical Models–What’s Best?

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INTRODUCTION

Media outlets regularly publish rankings of cell phone carriers, travel destinations, colleges, TVs and credit cards, just to mention a few.

Here’s an example of one such ranking – the “Best Countries for Retirement,” according to investopedia.com. To determine which countries are the best for retirees, we use a scoring system that measures a variety of factors, including:

Ease of buying and owning property
Cost of renting
Benefits and discounts on things such as healthcare and entertainment
Visa and residency requirements

Healthy living
Development and infrastructure
Climate
Stability of the country’s political situation
Cost of living

The top 5 countries with the highest cumulative average score across all those categories are Panama, Costa Rica, Mexico, Ecuador, Malaysia.

When creating rankings like the one above, we first need to decide what factors (variables) will contribute to the overall ranking. Then, we need to decide how those factors can be measured (quantified). Finally, we need a formula that combines all of them into one overall score.

In your group, agree on a topic that interests everyone (or most of you!), and try to come up with a strategy to create a ranking system related to that topic. For example, you could think about how you’d rank the restaurants in your area.

After agreeing on a topic, think about:

What variables to use;
How to measure those variables; and
How to combine all of them into one overall score.

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

quantifying variables is sometimes difficult and there may be multiple methods to do so.
models can be used to help make real-life decisions, but it is likely that they do not take into account every possible variable.
mathematical models provide a method to estimate values that are difficult to measure directly.
mathematical models can be equations, graphs, tables, or described in words.

By the end of this collaboration, you should be able to

define what is meant by the term “variable.”
describe what a mathematical model is.
identify possible variables which may be used to model a given scenario.
use a given mathematical model to make comparisons.
use a multivariable mathematical model to estimate a value.

PROBLEM SITUATION 1: THE BEST COLLEGES

You read about the yearly U.S. News ranking of best colleges in Preparation M.1. Today, we are going to think more about this ranking system and then review another ranking system for the “best” community colleges. The media commonly uses “best” and “worst” rankings when they report on colleges. Often, these rankings are used to make decisions about what colleges are better than others.

(1) Using what you know about how U.S. News ranks colleges, discuss what things you might include in your decision in choosing a college. That is, what factors would be important to your decision?

Important factors for a decision or situation are called variables. You can remember this term because variables are the things that vary, or are different, across the people or things we are measuring. For example, first-year student retention and graduation rates are two factors that vary from school to school. These factors are also important in determining how good a school is. These factors are variables used in the U.S. News ranking system.

Variables, like first-year student retention or graduation rates, often have values that are numbers. These are called quantitative variables. Numbers can allow for more meaningful comparisons. For example, if the variable is first-year student retention for each college, we can use a number to give the rate at each college, such as the percentage of first-year students that go on to the second year of college. Then we can easily compare a college with an 80% retention rate to a college with a 40% retention rate. The comparison would not be as meaningful if only descriptive words had been used, such as “high” retention rate for the first college and “low” retention rate for the second.

When variables sort data into a limited number of values [such as a screening test being either positive or negative, or sizes of small, medium, and large), they are called categorical variables.

$Image showing two different types of variables: Quantitative and Categorical. Quantitative variables are numbers. For example, first-year retention rate, graduation rate, number of pets, annual income. Categorical variables are words. For example, eye color, blood type, mood.$

It can be difficult to decide what numeric value to use for certain variables. For example, U.S. News also uses a variable called faculty resources. This could be measured by average faculty salary (in dollars), money available for classroom equipment (also in dollars), or the number of hours of support faculty get from a mentor, or a combination of multiple things.

(2) Refer back to the list of variables (factors) you discussed in Question 1, and brainstorm how numeric values could be used to quantify them (express with numbers). Include the units of measurement.

A mathematical model is a description of a system, or situation, using mathematical concepts, language, and numbers. People often use variables to help create mathematical models; however, variables are not always needed for models. Like functions, mathematical models can be equations, tables, graphs, or descriptions in words.

A mathematical model may be created to make predictions about a situation or simply represent and explain a situation and its different components. Mathematical models are often not perfectly accurate, but they are useful ways to measure values. Many models have been shown to produce values that closely match real data. For example, meteorologists use information on wind speed, temperature, and humidity to predict the chance of rain. While meteorologists are not perfectly accurate, their models produce far more accurate predictions than guessing.

Most decisions using models are based on multiple variables. One way to compare options is to combine the models’ variables in a meaningful way. The combining of variables (usually by adding, subtracting, multiplying, dividing, or averaging) into one number is called building a mathematical model equation.

Now, we’ll study another ranking system, this time for community colleges. We will look at a model that combines the variables in a way that gives us one number for each community college. The resulting composite score will allow us to first compare and then rank a set of community colleges.

The Aspen Institute awards the Aspen Prize every other year. The award “honors those institutions that strive for and achieve exceptional levels of success for all students, while they are in college and after they graduate.”³ In addition to the prestige of winning this award, the winning college also receives a $1 million cash prize. To determine the 150 finalists for the award, the Aspen Institute collects information from 1000 community colleges related to the following variables (remember that we call them variables because their values may vary between colleges):

1. Performance – The performance score measures student success in persistence, degrees awarded, completion, and transfer;

2. Change – The change score measures consistent improvement in these areas over time; and

3. Minority Achievement and Family Income – The minority achievement score and family income score measure equitable outcomes for students of all racial/ethnic and socioeconomic backgrounds.

The values of these variables are given in the table below for four community colleges. Note that the table itself is a model for this situation.

College	State	Performance Score	Change Score	Minority Achievement Score	Family Income Score
Bergen Community College	NJ	299	800	178	70
Miami Dade College	FL	323	400	288	191
Northwest Iowa Community College	IA	503	200	437	99
Pearl River Community College	MS	444	600	275	118

Each of these scores is created from combining a great deal of information about many factors. For example, the performance score is created by using both the first-year student retention rate and the school’s graduation rate. There is no easy way to guess how these scores were calculated. In order to find out, you would need to look carefully at the details of the research plan. One must be careful when combining numbers with different units.

Later in this module, you will look specifically at how to build mathematical models. For this collaboration, you should concentrate on understanding the following:

the model that is given
what the model is used for
what the variables in the model measure

(3) Look at the scores in the table above. Which school do you think is the “best?” Why? (Hint: What data are you focusing on in the table?)

(4) The Aspen institute used the following mathematical model equation to calculate a score for each school. In this case, the model is represented by this equation:

R = 0.33*P + 0.33*C + 0.27*M + 0.07*F

where:

R = Overall ranking score (higher scores are best)

P = Performance score

C = Change score

M = Minority score

F = Family income score

Use this equation to calculate the overall score for each of the schools and enter it into the table below. According to the overall score you calculated, which school is best?

College	R
Bergen Community College	(i)
Miami Dade College	(ii)
Northwest Iowa Community College	(iii)
Pearl River Community College	(iv)

Here are visual representations of the models that we’ve discussed so far in this collaboration:

$Image representing the Aspen Institute model of college ranking. On the left are the variables: P = Performance score, C = Change score, M = Minority score, and F = Family income score. These variables converge into an equation in the middle of the image, and then on the right we have R = Overall ranking score (higher scores are best).$

$Image representing the course grade model. On the left are the variables: H = average Homework score, T = average Test score, A = Attendance. These variables converge into an equation in the middle of the image, and then on the right we have C = overall Course grade.$

$Image of mathematical weather model. Temperature, pressure, and other variables are highlighted on the left and point to the model. The model's output is the hurrican's path.$

PROBLEM SITUATION 2: WEIGHING THE MODELS

Throughout this module, we will investigate several mathematical models. Some build on the linear and exponential models that you may have seen in previous courses, while others are completely different types of models, such as polynomial models or trigonometric models. We next turn our attention to a mathematical model that estimates body size.

(5) According to the RAND study described in the Science Daily article⁴ that you read in the preparation, the proportion of Americans who are severely obese continues to increase. The study used Body Mass Index, or BMI, to classify individuals as obese. We will now consider the formula for BMI, as well as its relationship to other measurements of a person’s size.

(a) What variables are used to calculate BMI?

(b) How do you think the researchers in the RAND study obtained information about these variables? Imagine yourselves as the researchers. You will want a large, varied sample, and you have limited funds. What would be a good way to get people’s height and weight from diverse U.S. locations?

(c) If you want to determine whether someone is obese, are there any other variables that might be important to consider?

(6) The article defines Body Mass Index as weight in kilograms divided by the square of height in meters. In terms of a mathematical model equation, we can write this as

\[ BMI = \dfrac{weight\;(in\;kilograms)}{[height\;(in\;meters)]^2} \nonumber \]

The height and weight of athletes is often publicly available. The table below lists known heights and weights of several current and former (at the time their height and weight were measured) athletes. Calculate the BMI for each of the athletes. Use the fact that one inch is 0.0254 meters and one pound is about 0.454 kilograms.

(a) Discuss in your group before completing the entries for Rory McIlroy first. Round answers to two decimal places.

(b) Complete the rest of the table for the other athletes. Round answers to two decimal places.

Athlete	Height		Weight		Body Mass Index
Athlete	Feet (’) Inches (”)	Meters (m)	Pounds (lbs)	Kilograms (kg)	kg/m²
Rory McIlroy (Golf)	5’ 10”		160 lbs
Julianna Pena (UFC)	5’ 7”		135 lbs
Cam Gallagher (Baseball)	6’ 3”		230 lbs
Breanna Stewart (WNBA)	6’ 4”		170 lbs
Shaquille O’Neal (NBA)	7’ 1”		325 lbs
Maria Sharapova (Tennis)	6’ 2”		130 lbs

(7) The Centers for Disease Control and Prevention (commonly referred to as the CDC) suggests that

BMI below 18.5 is Underweight;
BMI between 18.5 and 24.9 is Normal;
BMI between 25 and 29.9 is Overweight;
and BMI of 30 or above is Obese.

What are the athletes’ classifications based on their BMIs? Write "Underweight", "Normal", "Overweight", or "Obese", based on your calculated BMI.

Rory McIlroy:

Julianna Pena:

Cam Gallagher:

Breanna Stewart:

Shaquille O'Neal:

Maria Sharapova:

(8) Recall the description of the BMI calculation: “weight in kilograms divided by the square of height in meters.” Let’s examine this mathematical model more closely.

(a) Consider the following two scenarios. In each scenario, which person has the lower BMI?

(i) Julie and Sarah have the same weight, but Julie is taller. Which of them has a lower BMI?

(ii) Mike and Jim have the same height, but Mike weighs more. Which of them has a lower BMI?

(b) If you wanted to lower your BMI, which variable could you reasonably change?

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.

________________________________________

³ http://highered.aspeninstitute.org/projects/

⁴ http://www.sciencedaily.com/releases/2012/10/121001132146.htm

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