Among the first customers for a new car are engineers from competing companies. Because they want their own cars to compare favorably with their competition, they need to know what their competition is doing and how. One of the most important tools for figuring out how a competitor's product works is reverse engineering. Engineers for one company have a general idea of the options available to their competitors and using this general idea they try to reconstruct what competing engineers have done. We are all reverse engineers -- for example, medical scientists try to reverse engineer the human body and football coaches try to reverse engineer the opposing team.
This course is about modeling, or reverse engineering -- not just reverse engineering a car or a piece of software but reverse engineering the world in which we live -- determining the how behind the what of the spread of an epidemic, the motion of the planets, the start of a war, or inflation.
Building a mathematical model is like building other kinds of models -- for example, visual models (pictures), verbal models (speech, articles, or books), and three dimensional models (sculpture). When you watch an artist work, you see his attention move back-and-forth from the scene in front of him to his canvas. This is the modeling cycle. We begin by looking at the real world and then we make a rough sketch. Then we go back to the real world, comparing what we see with our picture so far. We may erase a few lines on the picture, or change a color, or fill in some more details. In any case we build up the model step-by-step, starting with the real world and repeatedly comparing our model to the real world. In the process we often learn more about the real world. Features of the real world that don't immediately draw our attention often become more noticeable as we draw our picture.
We begin this course with a simple example of reverse engineering. We want to capture our students' interest and show them the power of mathematics to help them do things that they want to do, so we choose an example involving cars. One of the most important factors in the performance of a car is its transmission and drive train -- the mechanisms that convert the circular motion produced by the engine into the forward or backward motion of the car. The circular motion of the engine is measured in RPMs, or revolutions per minute and is indicated on some cars by a tachometer. In the United States the motion of a car is measured in MPH, or miles per hour and is indicated by a speedometer.
You can get an idea of the general principles involved in the drive train and transmission of a car by looking at a bicycle. The gears, chain, and tires of a bicycle convert the circular motion of the pedals into the forward motion of the bicycle. There is usually a cluster of several different gears attached to the pedals. The different gears have different numbers of teeth -- for example many bikes have a gear with 52 teeth. A front derailler positions the chain on one of these gears. If the chain is on the 52 tooth gear then each revolution of the pedals moves the chain a distance of 52 links.
The rear wheel also has a cluster of several gears and each gear has a different number of teeth -- for example, many bicycles have a 14 tooth gear attached to the rear wheel. A rear derailler positions the chain on one of these gears. If the chain is on the 14 tooth gear than each time the chain moves a distance of 14 links then the rear wheel goes through one complete revolution. If the real wheel has a diameter of 27" then the bicycle moves pi * 27 inches or roughly 84.82 inches.
Notice the way that the links on the chain fit onto the teeth on the gears and link the motion of the pedals to the motion of the wheels. Each time that the front gear revolves by one tooth the rear gear also revolves by one tooth.
If the front derailler is on the 52 tooth gear and the rear derailler is on the 14 tooth gear then each revolution of the pedals causes the chain to move by 52 teeth. Thus, the rear gear revolves by 52 teeth. Since the rear gear has 14 teeth this produces 52/14 revolutions of the rear wheel. If the diameter of the rear wheel is 27" then each revolution of the pedals moves the bicycle 315.06 inches = 26.25 feet = 0.0050 miles. Thus, if a bicyclist pedals at 90 revolutions per minute her speed will be 26.85 MPH with this combination of gears.
Bicyclists use the term cadence for the rate at which they are "spinning" the pedals. We are interested in the relationship between the cadence measured in RPM and the speed of the bicycle measured in MPH.
This can be a good way to involve your own students. You might ask a student to bring a bicycle to class and show the class how the chain, gears, and deraillers work. If none of the students in the class has a bicycle with different gears then an older sibling or parent might bring one in.
You could have the class make tables showing the conversion factors from cadence (measured in revolutions per minute) to speed (measured in miles per hour) for the different gear combinations on this bicycle or their own bicycles. This would be an interesting application in middle school of multiplication and division and the relationship between the diameter of a circle and its circumference.
Your computer algebra system window has examples showing how you can make computations like the ones above keeping track of units and even with automatic unit conversion. Work through the examples in your CAS window. Throughout this course you will be expected to use your CAS window on your own initiative whenever you think it might be helpful.
Draw a graph, "by hand," showing the relationship between the cadence (measured in revolutions per minute) of the pedals and the speed (measured in miles per hour) of the bicycle for the situation described above. Use the horizontal axis for the cadence of the pedals measured in RPM and the vertical axis for the speed of the bicycle measured in MPH. This graph is useful if you know how fast you can pedal and you want to know how fast you can ride the bicycle.
Make another graph showing the same relationship but this time use the horizontal axis for the speed of the bicycle measured in MPH and use the vertical axis for the cadence of the pedals measured in RPM. This graph is especially useful if you know how fast you want to go and want to determine how fast you must pedal. The two graphs show exactly the same information from two different perspectives.
This is a good example of functions and inverses. It can be used even before the formal terminology is introduced. Another way to see the same idea is by looking at tables of values. For example, suppose that we are looking at a bicycle with 24" diameter wheels, a 44 tooth front gear, and a 22 tooth rear gear. The relationship between speed (measured in miles per hour) and cadence (measured in revolutions per minute) can be written
or
We can express this with the following two tables
-------------------------------- -----------------------------
cadence (RPM) Speed (MPH) Speed (MPH) Cadence (RPM)
-------------------------------- -----------------------------
10 1.43 1.43 10
20 2.86 2.86 20
30 4.28 4.28 30
40 5.71 5.71 40
50 7.14 7.14 50
60 8.57 8.57 60
70 10.00 10.00 70
80 11.42 11.42 80
90 12.85 12.85 90
100 14.28 14.28 100
-------------------------------- -----------------------------
Notice the two tables are exactly the same except that the columns of the lefthand table were switched to obtain the righthand table. The lefthand table is more useful when we know cadence and want to determine speed. The righthand table is more useful when we know speed and want to determine cadence.
In practice, we would rarely use a table like the righthand table. Although it is mathematically correct, it is not very useful because people are more likely to ask questions like -- What cadence is necessary to achieve a speed of 5 MPH? -- than -- What cadence is necessary to achieve a speed of 5.71 MPH?
Although the transmission and drive train on a modern car is somewhat more sophisticated than the gears and chain on a bicycle the basic operation is the same. Using various combinations of gears, the engine's speed of R revolutions per minute is converted into the car's speed of kR miles per hour. The value of the constant k is determined by the combination of gears and the diameter of the tires. Manual transmissions on cars usually offer from three to five different forward gear combinations. Large trucks often have many more combinations. This is another chance to involve your students. If a student or her family has a car with a standard transmission and a tachometer as well as the usual speedometer ask her to measure the conversion factor, k, for each of the forward gears and for the reverse gear. We discuss collecting data with a real car below. If one of your students knows something about trucks or tractors, ask him or her to talk about gears and transmissions.
We are interested in automatic transmissions. When a car starts at rest and then accelerates the transmission starts with a low gear because this gives lots of "power" and high acceleration. As the speed of the car increases and the crankshaft in the engine is spinning fast the transmission shifts to a higher gear. This conserves fuel and saves wear-and-tear on the engine. But the acceleration suffers. You can usually feel this gear change as a slight jerk and hear the change in the engine noise. As the car continues to accelerate the transmission shifts to another higher gear for the same reasons as the first gear shift.
Click here to open a new window with a Java applet. Arrange your two windows so that they overlap and it is easy to go back-and-forth between them by clicking on the exposed portion of the inactive window to make it active. When you are done with this module remember to close the window with the Java applet.
The graph on the left in the Java applet shows data that you might obtain by watching the tachometer and speedometer of a car as you accelerate from a standing start. This applet can be used to investigate questions like -- At what speeds did the automatic transmission shift gears? -- What is the ratio between MPH and RPM in first gear? in second gear? in third gear?
You can change any of the settings.
Congratulations! -- You have just completed your first modeling cycle.
Compare these three steps to the figure below. The first step is indicated in this figure by the top arrow going from the real world to the picture. An artist looks at the real world and based on his knowledge of lighting, color, paint, and canvas makes a rough sketch on his canvas. The second step corresponds to the lower arrow going from the picture back to the real world. The artist compares his picture with the real world. The third step corresponds to the top arrow again. This time the artist modifies his picture based on the comparison between what he has drawn so far and the picture.
The last two steps are often repeated several times. We often begin with a simple model that works in simple situations and eventually build a more complex model for more realistic situations. For example, the automatic transmission in most cars is controlled by a computer that calculates shift points on the basis of many factors; it might use higher shift points when the driver accelerates rapidly to provide more pep and use lower shift points when the driver accelerates more slowly to improve gas mileage. A realistic model of a modern automatic trasnsmission would have to include all the factors that the transmission uses.
You can practice the modeling cycle using this Java applet. Click on the long blue bar on the left side at the bottom of the applet. Each click on this bar will generate a new set of data for a different car and transmission. Go through the modeling cycle for this car and transmission. You can return to the original car and transmission by clicking on the shorter green bar on the right side at the bottom of the applet.
Even better, you and your students can practice the modeling cycle with a real car and transmission. This requires a car with an automatic transmission and a tachometer as well as a speedometer. As the driver accelerates smoothly and slowly, have passengers record speedometer readings and tachometeter readings. One way to do this is with two passengers. The first passenger calls out when the speedometer reaches 5 MPH, 10 MPH, 15 MPH, ... and the other passenger records the corresponding tachometer readings. This is another chance to make connections between the classroom and the outside world. It is also a good chance to discuss the process of building ever more sophisticated models. Modern cars use fairly sophisticated computers to control the shift points based on many factors -- how quickly the car is accelerating, whether the road is level, going uphill, or downhill, the temperature of the engine, and so forth. All this is very important to the performance of the car -- how peppy it feels, what kind of gas mileage it gets, and so forth.
With a real car and real transmission the engine is "idling" even when the car is standing still. The graphs in the applet above are not completely correct because they don't show this idling.