INTRODUCTION - What is Modeling?

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A Path Through the Course
Student Materials
	Preparation Reading
	Activity 1-The Modeling Process
	Homework 1-Killer Bug Ate My Face
Teacher Notes
	Preparation Reading
	Activity 1-The Modeling Process
	Homework 1-Killer Bug Ate My Face
	Transparency 1-The Modeling Process

A Path Through the Course

This flowchart describes a recommended path through the Science and Engineering section of Mathematical Models with Applications course. The goal for this part of the course is for you to teach five units: Three are recommended, and two can be selected based on student career and subject interests, access to technology, and mathematics level. To help you determine which of these units might be appropriate for your classroom, here is some information on the unit choices.

After teaching Prediction and Rain you should teach one of the following units.

After teaching Oscillation, you should teach one of the following units.

Important note: These are recommendations! Every class is different, and teachers should use materials in a manner that considers the unique abilities and interests of their students.

Student Materials

Preparation Reading

Have you ever wondered how some people seem to know so much about our world and our selves? For example, how do doctors know the correct dosage for a particular medicine: twice a day, three times a day? Or how can a wildlife biologist be sure that an animal is endangered or overpopulated?

And how about the people who create things: the animators who draw moving pictures that appear natural and life-like, and the engineers who design and build things that move and perform complex tasks? Are the people that know and build these things all very smart? Do they have special abilities beyond the rest of us?

The thing that all of these people have in common is not supernatural intelligence or abilities. They have learned to apply a specific skill to the real issues they encounter in their jobs and lives. They have learned how to use mathematical modeling.

In this course you will be using mathematics to solve real problems. But you will also be using science, writing, research and your imagination to put the mathematics to work in the right way, at the right time. Let's look at a problem that affects schools that has been in the news lately.

Activity 1-The Modeling Process

Newspapers and television often relate stories of students carrying weapons to school, brandishing them at teachers and their peers. This is not merely a big-city problem. School crime and violence affect rural, suburban, and urban communities. Consider the following information:

• The Department of Justice recently reported that 100,000 guns are brought to school each day.
• The National Crime Survey, released in May 1991, found that about 50% of all violent crimes against youths aged 12-19 occur in school buildings, on school property, or on adjacent streets. Twelve percent of the violent crimes in school buildings involve an offender with a weapon.
• The Centers for Disease Control state that 1 in 5 school children reports carrying a weapon of some type. About 1 in 20 (5.3%) said they had brought a gun to school.
• Almost 1 in 3 students in 31 Illinois high schools said they had brought a weapon to school for self-protection at some time during their high-school career.
• Sixty percent of all administrators, nationwide, maintain that guns have never been involved in incidents at their schools.

You would like to know what all this means for your school.

Pose a specific question concerning these data to which you would like an answer.

What additional information would you like to have in order to 3

Newsweek 119: 10 (March 9, 1992): 25-8;
USA Today 122 (January 1994): 29-34.
Department of Education, School Violence and Discipline Report, Aug. 1996

Homework 1-Killer Bug Ate My Face

In June 1994, British tabloids fanned fears in England, with headlines such as "Eaten Alive," and "Killer Bug Ate my Face." The stories tell of a "deadly flesh-eating bacteria" that had killed 11 people in England and Wales in 1994. These fears spread to the United States when it was reported that a handful of cases, and at least two deaths, had occurred in the U.S. that same year.

The disease is passed by human-to-human contact, and its favorite route of entry is through an open wound. While the streptococcus bacterium is very common (it causes "strep throat"), the publicized cases involve a rare variant of the germ, called strep-A. If caught early, the infection is treated easily with antibiotics. The bug is so virulent, however, that it cannot be taken lightly.

Strep-A may now afflict 10,000-5,000 Americans per year; perhaps 3,000 die. However, this microbe has been around for years, and no one thinks it will suddenly cause an epidemic. Your chances of getting a life-threatening strep-A infection are roughly comparable to those of being hit by lightning.

Pose a specific question concerning these data to which you would like an answer.

What additional information would you like to have in order to answer your question?

Newsweek 123: 25 (June 20,1994): 32-33;
Time 143, (June 20, 1994): 54.

Teacher Notes

The two main goals of this introductory unit are to explain the process of modeling as represented in Transparency 1; and to have students begin the process of formulating the key issues that they will model. On a block schedule this is a one day unit.

Class Discussion (Day One)

After the introductory reading, break the students into groups and do activity one. Have the students present their questions to the class. Since the focus of this first activity is on the introduction to the modeling process, including the formulation of assumptions and their use in building their model, the activity and class discussions should be fairly open-ended.

The reading and questions are designed to give the students practice in analyzing a situation, deciding on questions that need to be asked, and determining what information is needed to help answer the questions.

Discussion can take place among the entire class, or in small groups.

Assign students Homework 1.

The Modeling Process (Day Two)

Use the examples from the activity and the homework and others that students might suggest to go through the modeling process (Transparency 1).

Annotation of Transparency 1

Define a problem.

Consider all the known factors, decide which are most relevant to the problem, and make assumptions.

Develop the model (relationships between the included factors).

Solve the model.

Does the solution make sense?

Test it with real data.

Is it user-friendly? Is it practical?
Can we use this model in other ways (e.g Other diseases, other parts of the country)? Does it always provide us with a realistic answer.

Maintain the model.

Has anything changed?

At any time in the modeling process, you may need to make adjustments.

In this introductory unit, it is important to reinforce the reasons for studying modeling. Discuss the modeling process, and the idea that various groups maybe affected differently by decisions that are made using these models. A helpful resource that discusses the modeling process is A First Course in Mathematical Modeling by Frank R. Giordano and Maurice D. Weir, published by Brooks/Cole, Monterey, CA, 1985. See Chapter 2, pages 29-40.

Preparation Reading

Have you ever wondered how some people seem to know so much about our world and our selves? For example, how do doctors know the correct dosage for a particular medicine: twice a day, three times a day? Or how can a wildlife biologist be sure that an animal is endangered or overpopulated?

And how about the people who create things: the animators who draw moving pictures that appear natural and life-like, and the engineers who design and build things that move and perform complex tasks? Are the people that know and build these things all very smart? Do they have special abilities beyond the rest of us?

The thing that all of these people have in common is not supernatural intelligence or abilities. They have learned to apply a specific skill to the real issues they encounter in their jobs and lives. They have learned how to use mathematical modeling.

In this course you will be using mathematics to solve real problems. But you will also be using science, writing, research and your imagination to put the mathematics to work in the right way, at the right time. Let's look at a problem that affects schools that has been in the news lately.

Activity 1-The Modeling Process

Newspapers and television often relate stories of students carrying weapons to school, brandishing them at teachers and their peers. This is not merely a big-city problem. School crime and violence affect rural, suburban, and urban communities. Consider the following information:

• The Department of Justice recently reported that 100,000 guns are brought to school each day.
• The National Crime Survey, released in May 1991, found that about 50% of all violent crimes against youths aged 12-19 occur in school buildings, on school property, or on adjacent streets. Twelve percent of the violent crimes in school buildings involve an offender with a weapon.
• The Centers for Disease Control state that 1 in 5 school children reports carrying a weapon of some type. About 1 in 20 (5.3%) said they had brought a gun to school.
• Almost 1 in 3 students in 31 Illinois high schools said they had brought a weapon to school for self-protection at some time during their high-school career.
• Sixty percent of all administrators, nationwide, maintain that guns have never been involved in incidents at their schools.

You would like to know what all this means for your school.

Pose a specific question concerning these data to which you would like an answer.

What additional information would you like to have in order to answer your question?

Newsweek 119: 10 (March 9, 1992): 25-8;
USA Today 122 (January 1994): 29-34.
Department of Education, School Violence and Discipline Report, Aug. 1996

Homework 1-Killer Bug Ate My Face

In June 1994, British tabloids fanned fears in England, with headlines such as "Eaten Alive," and "Killer Bug Ate my Face." The stories tell of a "deadly flesh-eating bacteria" that had killed 11 people in England and Wales in 1994. These fears spread to the United States when it was reported that a handful of cases, and at least two deaths, had occurred in the U.S. that same year.

The disease is passed by human-to-human contact, and its favorite route of entry is through an open wound. While the streptococcus bacterium is very common (it causes "strep throat"), the publicized cases involve a rare variant of the germ, called strep-A. If caught early, the infection is treated easily with antibiotics. The bug is so virulent, however, that it cannot be taken lightly.

Strep-A may now afflict 10,000-5,000 Americans per year; perhaps 3,000 die. However, this microbe has been around for years, and no one thinks it will suddenly cause an epidemic. Your chances of getting a life-threatening strep-A infection are roughly comparable to those of being hit by lightning.

Pose a specific question concerning these data to which you would like an answer.

What additional information would you like to have in order to answer your question?

Newsweek 123: 25 (June 20,1994): 32-33;
Time 143, (June 20, 1994): 54.

Transparency 1-The Modeling Process

• Define a problem.
• Consider all the known factors, decide which are most relevant to the problem, and make assumptions.
• Develop the model (relationships between the included factors).
• Solve the model.
• Does the solution make sense?
  Test it with real data.
  Is it user-friendly?
  Is it practical?
• Maintain the model.
  Has anything changed?