Mathematics, like English or French, is a language -- a language that we use to express ourselves, to communicate with others, and, perhaps most importantly, to think -- to manipulate ideas. Mathematical modeling is about how we use the language of mathematics to express, communicate, and think about the real world.

Learning mathematics by itself, is like learning the words and grammar of a language like English or French without ever talking, reading, or writing. Of course the grammar and vocabulary of mathematics are important but the real point of mathematics is using it -- mathematical modeling.

We begin our study of mathematical modeling with an example of mathematics in use -- a good example to use in your own classroom.

The thermometer stands at 10 degrees Fahrenheit and the weatherpeople are talking about a wind chill of twenty degrees below zero. Your students come into class shivering and stamping snow from their feet. This is a great opportunity to talk about mathematical modeling -- what does the wind chill mean? -- will the temperature of the water or a person drop below the temperature indicated by a thermometer when the wind is blowing? -- or will it just cool to the same temperature faster? At the front of your room you have equipment set up to explore the effects of temperature and wind.

For this exploration you need a TI CBL, three temperature probes, two containers for water, a fan, and some way to heat the water to boiling. Because you will be working with boiling water and electricity you must observe the appropriate safety precautions.

This exploration also requires one of the TI graphing calculators -- The TI-82, TI-83, TI-85, TI-86, or TI-92 -- and a program, TEMP, for these calculators that sets-up the TI-CBL to record temperature data and then recovers the data from the TI-CBL. Click on the appropriate icon below to get the program for your graphing calculator and for instructions about how to use your graphing calculator with the TI-CBL for this exploration.

This module is a good exercise in classroom management as well as an introduction to modeling. We want to collect data for analysis. The data collection takes the better part of an hour. So we must start collecting data at the very beginning of the period and then use the time while data is being recorded for other things. We will collect two sets of data simultaneously to study the effects of wind on cooling water.

• Before the class begins:

  Boil some water. This can be done using a microwave or a hot plate. When the class begins we will pour the boiling water into two identical shallow bowls.

  Place one bowl in the path of air from a fan and the other bowl in calm air. Place one temperature probe in each of the bowls and let a third temprature probe be loose in calm air. Connect the three probes to the CH1, CH2, and CH3 ports on the TI-CBL. Notice which probes are connected to which ports. Turn the fan off.

  Follow the instructions for your graphing calculator to download the program needed to set-up the TI-CBL and to recover the data from the TI-CBL.

  Connect the TI-CBL to your graphing calculator using the TI linking cable. Make sure the two ends of the linking cable are firmly inserted into the ports on the TI-CBL and the graphing calculator.

  Follow the instructions for your graphing calculator to run the program that sets up the TI-CBL to record data.

  You should see the word READY on the TI-CBL screen. You can disconnect your graphing calculator from the TI-CBL now if you want to use it while the experiment is running.

  You are ready for the class to begin.

• As soon as the class begins pour the same amount of boiling water into each of the two identical shallow bowls. Make sure that the temperature probes in the two bowls are immersed in the water, turn the fan on and immediately press the TRIGGER key on the TI-CBL.

  The TI-CBL will collect data while you work with the class.

• Just before the end of the class you will reconnect the TI-CBL to your graphing calculator and run a program that will recover the data recorded by the TI-CBL. This data can then be used by your students in the remainder of this module.

Discussion

When a hot object, like a cup of coffee, is placed in a cooler environment its temperature drops. The same thing happens to a person outside on a cold day. We know from personal experience that windy days feel colder than calm days. In fact, during winter weather reports often include the "wind chill factor" in addition to the temperature. In this module we study cooling and the effect that wind has on cooling. We begin the classroom discussion by considering an experiment like the one shown at the right. A cup of water was heated to boiling and then its temperature was recorded as it cooled. In class we are running a more complicated experiment with two containers of water cooling -- one in calm air and and one in windy air -- while we speak.

Very roughly a graph of the cooling water will look something like the sketch at the right. Notice that in this particular graph -- The initial temperature is about 190 degrees Fahrenheit. The water cools to room temperature in about 30 minutes. It takes the same amount of time to cool from 190 degrees to 150 degrees as it does to cool from 150 degrees to 110 degrees. The water temperature appears to drop steadily to room temperature and then stop dropping.

Before looking at actual data what do you think a graph of actual data will look like?

• Will the water cool faster when it is hot or when it is merely warm? Or will it cool steadily?

• Will the temperature of the water drop to room temperature and then stop dropping?

• How long will it take the water to cool?

Before going on sketch a graph showing the way that you think the water will cool.

Notice there is no link to click for an answer. The most important point we want to make in this module is that in modeling answers are judged not by looking in the back of the book or clicking on an "answer" link but by comparing the model with the real world.

After you have discussed these questions look at the data in the graph at the right. This data was collected with a simple experiment -- one bowl of cooling water. Click here to open a new window with a table showing the same data. Arrange these two windows so that they overlap and you can move easily back-and-forth between them by clicking on the inactive window to make it active. When you are done looking at this data close its window. The same data appears in your CAS window.

Notice that the temperature measurements are in degrees Celsius. Water boils (at sea level) at 100 degrees Celsius and it freezes at zero degrees Celsius.

Compare this graph with your discussion above.

The first model we look at as one possibility for describing cooling is called Newton's Law of Cooling or, better yet, Newton's Model of Cooling to emphasize that it is not enacted by Congress or handed down on stone tablets but is just one possible model and may or may not be a good model.

Newton's Model of Cooling says that the amount by which the temperature of an object cools in a set time interval is proportional to the difference between its temperature and the ambient or room temperature.

Thus, if we make a series of measurements

at equally spaced times we would expect that

where R is a constant and A is the ambient temperature. Thus,

or

where k is the constant R + 1. The value of R and, hence, k depends on the physical composition of the cooling object, its size, and how well insulated it is. If the object is small and poorly insulated then k will be relatively large but if the object is more massive and well insulated then k will be relatively small. If you are unfamiliar with models like this click here.

If we know the ambient temperature A then we can determine the value of the constant k using any two data points by noticing that

In a well-designed experiment, we would know the ambient temperature but all experiments are not well-designed and we often need to work with data that was collected by someone else and is incomplete.

If we don't know the ambient temperature then we can determine the values of both constants, A and k, using any three equally spaced data points as shown below.

Notice that this calculation is a good exercise in high school algebra.

Once we have determined the values of the constants A and k using these three data points we have a model given by

where S_i is the temperature predicted by the model at the time of the i-th data point.

Now you can use your CAS window to compare the data with various models based on Newton's Model of Cooling using different points to determine the values of the constants. Notice that each model matches the data exactly at the three points used to determine the values of the constants A and k.

You should experiment with Newton's Model of Cooling. Use this data and other data that you collect yourself, using your CAS window and curve fitting using various data points to see how well Newton's Model of Cooling describes the temperature a cooling object. Later you should see how well Newton's Model of Cooling describes the data for water cooling in calm air and for water cooling in windy air. The graph at the right shows the results from one experiment we did. Notice that the ambient temperature varied quite a bit, apparently because the furnace cycled on and off several times during data collection.

The purpose of this exploration is to discuss mathematical modeling -- the use of mathematics to describe and predict real world phenomenon. Newton's Model of Cooling is a mathematical description that we hope describes some of the phenomena we see when an object is cooling. Your students will be able to collect real world data using the TI-CBL and see how successfully Newton's Model of Cooling describes their data.

The most important lesson here is that whether a model is successful or not doesn't depend on looking up an answer in the back of the book or getting a good grade on homework or an exam but on how well it describes actual data.

Models like this one are successful in some circumstances but may fail in others. One way to test how well Newton's Model of Cooling describes a particular set of data is by computing the ratios

If Newton's Model of Cooling was completely successful then all these ratios would be the same.

We ran a number of different experiments measuring the temperature of containers of water as they cooled and we tried to describe the resulting data using Newton's Model of Cooling. All our results were the same. The ratios k_i were larger for high temperatures than for low temperatures. The same mechanism, modeled by Newton's Model of Cooling, cannot explain cooling at high temperatures and cooling at low temperatures.

The graph at the right shows the data we collected comparing water cooling in calm air with water cooling in windy air. Notice that not only does the water in windy air cool faster but that its temperature actually drops lower than the water in calm air. This effect cannot be described by Newton's Model of Cooling. It is due in part to a mechanism called evaporative cooling. Evaporative cooling is very useful. It is particularly effective in hot dry climates. In the southwestern states many homes are cooled by evaporative coolers, or swamp coolers, that use this phenomenon as a cheap source of air conditioning.

It is important for you and your students to do lots of experiments using the basic set ups described above and to analyze your results and compare them with the predictions made by Newton's Model of Cooling. Newton's Model of Cooling generally does a good job for water that is cooling in calm air and that is not too close to boiling or to the ambient temperature.

The main point of this module is that models like this one generally work in some situations but not in all situations. We need other models to describe what happens when the water is close to boiling. When the temperature of the cooling water is close to the ambient temperature then small errors in measurement obscure the situation.

There are several possible explanations for the difference between water cooling in calm air and water cooling in windy air.

• In calm air a layer of warm air forms just above the surface of the water. This layer acts as insulation and slows the cooling. In windy air no such insulating layer forms.

• In windy air evaporative cooling becomes more important. As a result not only does the water cool faster in windy air but it reaches a lower temperature.