In teaching calculus the "Rule of Four" is recognized as sound pedagogical approach that has spurred the development of innovative materials. The "Rule of Four" philosophy develops topics graphically, numerically, analytically, and verbally. With the increasing use of technology for mathematics instruction there is another component that can help students, the use of instructional animations. Ideally such animations should include interactive components to change parameters and thus provide a richer environment for learning. However, even animations that illustrate action and results for a fixed choice of parameters are powerful tools for instructors. This demo includes a gallery of such animations and some interactive demos that an instructor can use to strengthen students' skills for optimization problems. Prior to using the materials in this demo, the demo Constructing Equations from Word Problems can be used. This more general demo provides a toolbox of visual aids for geometrically oriented word problems. These visual tools are designed to help students to develop equations that provide an algebraic model for the problem. A number of the gallery items are connected to max-min problems. Many students have difficulty developing a mathematical model for an optimization problem. Some of the difficulty stems from a lack geometric visualization skills so that the written description of how things change can be translated into an algebraic form. An animation illustrating changes can help students focus on salient features of the geometric objects involved and hence provide an opportunity to construct algebraic equations that provide a model for the situation. With this in mind we have developed a set of animations that illustrate a variety of standard geometric optimization problems. We start with an outline of the general approach to setting up optimization problems. (The steps list below are really interrelated and are ordered merely as a guide so students have a starting point for aspects of the solution process.) The steps involved are illustrated with several examples. We then provide a collection of statements of optimization problems together with visual demos that can be used within a lecture or assigned for students to use for practice. We expect the user to use the animation tools and then supply the algebra and calculus to accompany the situation. Outline of steps in a optimization problem.
2. Read the problem: The reading of course must be accompanied by understanding. For beginning students one reading is rarely sufficient. The first reading can be used to get familiar with the general situation (the "character") of the problem.
For this reading identification of what is going on is a primary goal. A second (or later) reading can be used to focus on a geometric model of the general situation. Here is where an accompanying animation as part of a lecture can provide practice with the visualization of components that change. It is at this point that we usually tell the student to draw a diagram that is a geometric embodiment of the process described in the problem statement. This is a key interpretive step and we need to devise ways for students to practice this step. (See the gallery of animations below.) 3. Construct a diagram: Having seen an accompanying animation and discussed the situation to be modeled students are often better able to draw a general diagram and label it with appropriate terms. With a diagram that captures the theme of the problem students can more readily traverse the bridge to the algebraic expressions required for the mathematical model. 4. Equation construction: Using the diagram in conjunction with the problem's verbal description we need to develop the equations to model the situation. Most of the max-min problems will involve two equations in two unknowns. Here is where the geometric nature of the problem can provide clues to the equation construction. The terms length, perimeter, area, and volume suggest the style of equations to construct. One of the equations will be equal to a constant. We use that equation to solve for one of the unknowns and then substitute for it in the other equation to obtain an expression in terms of a single unknown. The equations derived are completely dependent on the "character" of the problem. Certain fundamentals arise repeatedly involving right triangles, rectangles, circles, cones, and spheres. 5. Differentiation: Before proceeding to take a derivative, it is recommended that the equation in the single unknown derived from Step 4 be inspected to see if there are any restrictions on the variable involved. (For example values at which it is not defined.) For ease of reference, let's denote the expression as y = f(x). Compute the derivative
Next we set f '(x) = 0 and solve for x. The values determined are candidates for the value of x that will produce the optimal solution. You may also need to include in the list of candidates endpoints of an interval to which f(x) is restricted due to the description of the problem. You then use standard calculus procedures to determine the value of x that gives the optimal value of y. __________________________________________________________ Example 1. A rectangular animal pen is to be constructed so that one wall is against an existing stone wall and the other three sides are to be fence. If 500 feet of fence is to be used, determine the dimensions and area of the pen with maximal area. From the reading/recognition phase we have
The text book approach is to now draw a static figure and algebraically encode this information. However using an animation we provide further visual stimulus that can help students construct the figure and identity the features listed. It is the use of the animation that provides another bridge from the verbal description to the algebra model. Ultimately we need to write equations, perform algebra, and then the calculus associated with max-min problems. Consider features of an animation to help visualize the situation.
In much the same way a cartoonist plans scenes we can imagine a "storyboard" that shows successive images like those below.
In this "story" we change the width, redraw the rectangle, and compute the area. A slider is used to change the width. In the figure below, note the restrictions on the width, based on constraints in the the problem.
The correspondence between the width and the area defines a function. If we had an algebraic expression for this function we could use calculus to determine it maximum. However only discrete data is available from the "story" so we can only plot this function. Moving the slider to many different positions generates a set of ordered pairs (width, area) whose graph we generate. See the next figure.
The preceding components can be seen in the animation shown next. In addition we pose leading questions to guide the development of the algebraic equations.
The animation shown above can be used as part of the process in Step 2 and be a valuable aid in Step 3. Suggestions for the use of this animation follow:
The way instructors use an animation such as this will vary depending upon the level of the class, the goals of the course, and other local factors. It is adaptable to a variety of situations. Using the animation with some of the suggestions above provides a nice visualization of the pen construction process. __________________________________________________________ Example 2. A person in a row boat at point P is a distance S miles from a straight shore line. The point A on the shore is directly opposite the boat. The objective is to travel from point P to point B on the shore a distance D miles from A in a minimum amount of time. If the person can row at R miles per hour and walk at W miles per hour where should the person land the boat between A and B? (Let X mark the spot where the boat is landed.)
This type of problem appears in many books with specific information included. The situation can be changed to pipeline construction or medical emergency transportation, but the basic setup is the same as depicted in the preceding figure. From the reading/recognition phase we are led to consider a picture right a way so that we keep things focused. Something as crude as the one above is fine. Such a figure really supplies valuable information. If we aren't careful we fall into the trap of looking at distances, but the objective is to minimize time. The fact that we have two different rates, walking and rowing, is a key feature. Other items that are evident once we have the figure in combination with the verbal description include the following.
All these aspects of the problem can be difficult to merge together to get a good feeling for the situation. An animation in which we can change the landing spot can aid in organizing this information in order to develop the algebraic model. We proceed as we did in Example 1 by specifying items as in developing a "story board".
In this case the "story board" can be constructed as follows.
In this "story" we change the landing spot X, redraw the triangle, and compute the travel times. A slider is used to change the landing spot. In the figure below, note the restrictions on the landing spot, based on constraints in the the problem.
The correspondence between the landing spot X and the total travel time defines a function. If we had an algebraic expression for this function we could use calculus to determine it maximum. However only discrete data is available from the "story" so we can only plot this function. Moving the slider to many different positions generates a set of ordered pairs (X, Time) whose graph we generate. See the next figure.
The preceding components can be seen in the animation shown next. In addition we pose leading questions to guide the development of the algebraic equations.
The animation can be used as part of the process in Step 2 and be a valuable aid in Step 3. Features of the animation and suggestions for using it follow:
The way instructors use an animation such as this will vary depending upon the level of the class, the goals of the course, and other local factors. It is adaptable to a variety of situations. Using the animation with the outline above provides a nice visualization of the general situation. Example 3. This example illustrates uses 3-dimensional geometry to visualize an optimization problem. The general problem is easy to state:
The development of the algebraic model is a bit more subtle here. To help students with this process we have included an auxiliary construction to help bridge the gap to the equation development step. The following animation tells a "story" as in Examples 1 and 2 and was constructed in a similar manner. The comments in Examples 1 and 2 can be easily modified to describe the setting for this optimization problem.
____________________________________________________________ A Gallery of Visualization DEMOS for Optimization Problems The following is a gallery of demos for visualizing common max-min problems. These animations can be used by instructors in a classroom setting or by students to aid in acquiring a visualization background relating to the steps for solving max-min problems. Two file formats, gif and mov are available.
For a detailed description click in the "General problem description" region.
David
R. Hill
and is included in Demos with Positive Impact with his permission. |
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DRH 9/16/02
last updated 6/1/2007 DRH
Since 10/18/2002