Constructing Equations from Word Problems

Demos with Positive Impact NSF DUE 9952306

 
Objective: This demo provides a toolbox of visual aids for geometrically oriented word problems.  These visual tools are designed to help students develop equations that provide an algebraic model for the problem. The visual aids are presented in a gallery .

Level: Algebra, Precalculus, or Calculus courses in high school or college.

Prerequisites: Students should be familiar with fundamental relationships between components of geometric figures like triangles, rectangles, circles,  rectangular solids, cylinders, cones, and spheres. The properties of similar triangles are also needed.

Platform: No particular software package is required. Support for a viewer of gif or mov files is required. Viewers within a browser, Windows media player, Quicktime, or a commercial program can be used. It is recommended that a viewer with a stop/start feature be used when incorporating the animations in a lecture format or when students view the animations on an individual basis. There are MATLAB files to accompany each demo and Java applets for selected demos.

Instructor's Notes: (We restrict our demos to measurement geometric word problems. A successful web based assignment that uses some of the visualization tools in the gallery of demos is available as a pdf file by clicking here.)

Many students have difficulty developing an algebraic equation from the description provided in a word problem. Some of the difficulty stems from a lack of the geometric visualization skills required for the written description to be translated into a figure. An animation of the geometric construction process helps students focus on salient features involved, providing an opportunity to algebraically develop an equation describing the physical situation. With this in mind we have constructed a set of animations that illustrate several standard geometric constructions that commonly occur in textbook word problems.

Many word problems that employ the use of geometric figures make use of common shapes like circles, triangles, rectangles, rectangular boxes, cylinders, and cones. Such word problems frequently involve perimeter, area, surface area, and/or volume. Many texts suggest that these concepts be reviewed to remind students of their meaning and to reacquaint students with appropriate formulas for computation. (Often a table of appropriate formulas is included, possibly in an appendix.) In addition, units of measurement should be reviewed. Understanding units of measurement assist students in developing appropriate equations from verbal descriptions in which units appear.

The skill of translating from verbal descriptions to algebraic equations is a prerequisite for success with optimization and related rate problems in calculus. Hence this demo can be used at various places in calculus courses prior to sections involving optimization and related rates. In some cases it may be appropriate to include practice formulating equations from verbal descriptions in early sections that deal with functions and graphs.

We start with a general outline of steps that apply to many verbal problems. (The steps listed 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 geometrically oriented word 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 supply the algebra to accompany the situation. The animations can also be used as 'preview' material for optimization problems.

Outline of steps in a measurement geometric word problem.

1. Recognition: Each word problem has a "character" of its own. The diversity of situations to be modeled is an issue that tends to inhibit students from seeing a pattern for the solution process. Since we are dealing with a word problem the algebraic aspects are imbedded within verbal descriptions that connect various components of the problem. Students need some guide posts to assist with recognition. Several suggested guide posts that can useful are:
  • Determine the geometric figure involved. Is it a plane figure or a solid?
  • Is more than one shape or figure involved?

Certainly the recognition process depends on "reading the problem", which is often given as Step 1 in text books.

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

  • What type of measurement is involved? Perimeter, area, surface area, or volume?

A second (or later) reading can be used to focus on a formula that will be needed to perform the measurement involved. Here is where an accompanying animation as part of a lecture can provide practice with the visualization of measurement components. It is at this point that we usually tell the student to draw a diagram that is a geometric embodiment of the situation 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. Label the diagram: Here the components of the diagram are to be identified as described in the verbal description. It important that the student correctly interpret which parts of the figure are known and which should be labeled with an unknown. In this step, using an animation as part of an example can aid in distinguishing between these two aspects of the problem. 

4. Find an equation linking the quantities: The result of this step is completely dependent on the "character" of the problem. It is possible that two or more equations must be combined to get a single equation in a single variable that relates the quantities involving measurements. This is where the basic knowledge of perimeter, area, surface area, and/or volume formulas must be used. Ultimately we want one equation in a single unknown that models the measurement requested in the word problem.


Example 1.  A large cylindrical can is to be designed from a rectangular piece of aluminum that is 25 inches long and 10 inches high by rolling the metal horizontally. Determine the circumference of the cylinder, the radius of the cylinder, the height of the cylinder, and the volume of the cylinder.

Many text books  include a figure that indicates the rolling process as shown below.

A simple animation of the process gives a nice visual component that helps students determine relationships between parts of the rectangle and aspects of the solid. Incorporated as a demo within a lecture, the animation enhances the discussion of the necessary geometric relationships and facilitates development of the appropriate measurements. Click here to see the animation. Rolling a sheet of paper is also a very effective prop in this case.

A related exercise that appears in optimization problems is often phrased as follows. 

A large can is to be designed to have volume 

The lateral side of the can is formed by rolling a rectangle of

Find the values of x and h for which the smallest amount of material is needed. Include the top and bottom of the can.

In order to set up the equations for this optimization problem we can draw upon the experience of the measurement word problem and its accompanying animation. Having laid a foundation by the measurement problem, the equations for the optimization problem are more readily constructed.


Example 2.  A square sheet of material is to be used to construct a box
with no top by cutting a square from each corner. If the material is 16 inches on a side, and a 3 inch square is cut from each corner,  what is the volume of the box?

Many text books  include a figure that indicates the process and the "finished box" as shown in the figure below.

         

A simple animation of the process gives a nice visual component that helps students determine relationships between the original square of material, the corner cut-outs, and the finished box. Incorporated as a demo within a lecture the animation enhances the discussion of the necessary relationships and facilitates development of the appropriate measurements. It is easy to generalize the specific case to where a square of x inches is cut from each corner. Click here to see the animation.

A related exercise that appears in optimization problems is often phrased as follows. 

A square sheet of material is to be used to construct a box with no top by cutting a square from each corner. If the material is s units on a side, determine size of a square to be cut from each corner so that the box has maximum volume.

In order to setup the equations for this optimization problem we can draw upon the experience of the measurement word problem and its accompanying animation. Having laid a foundation by the measurement problem, the equations for the optimization problem are more readily constructed.

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A Gallery of Visualization DEMOS for Word Problems

The following is a gallery of demos for visualizing common geometric word problems that involve a measurement. These animations can be used by instructors in a classroom setting or by students to aid in acquiring a visualization background relating to constructing algebraic equations that model a word problem. Two file formats, gif and mov are available. It is recommended that a viewer with a stop/start feature be used when incorporating the animations in a lecture format or when students view the animations on an individual basis. There are MATLAB files to accompany each demo and Java applets for selected demos.

  1. The gif animations should run on most systems and the file sizes are relatively small. 

  2. The mov animations require the QuickTime Player (version 5) which is a free download available by clicking here; these file are also small. (The mov files may not execute properly in old versions of QuickTime.) 

  3. The collection of animations in gif and mov format can be downloaded; see the 'bulk' zipped download category at the bottom of the following table.

  4. The MATLAB files that accompany each demo are available for downloading; see the 'bulk' zipped download category at the bottom of the following table.

  5. The Java applets that are available for selected demos can be downloaded with each applet in a folder; see the 'bulk' zipped download category at the bottom of the following table.

General problem description. Animation Sample Click to view the gif file. Click to view the mov file. MATLAB file available.  Run JAVA applet.
Cylinder formed by rolling a rectangle

Click to see gif animation Click to see mov animation

not available
Box formed by cutting out corners

Click to see gif animation Click to see mov animation

buildabox applet
Cone formed by cutting out a sector of a circle

Click to see gif animation Click to see mov animation

not available
A gutter built by folding a rectangle

Click to see gif animation Click to see mov animation

buildagutter applet
Surface area of a box

Click to see gif animation Click to see mov animation

not available
Cross-sectional area of an artery; fixed change

Click to see gif animation Click to see mov animation

artery applet
Cross-sectional area of an artery; percent change

Click to see gif animation Click to see mov animation

artery1 applet
Cubic crystal growth Click to see gif animation Click to see mov animation

crystal applet
 

Zipped  downloads ==>

Click here to download the set of animated gifs. Click here to download the set of animated movs. Click here to download the set MATLAB files. Click here to download the Java applets.


Credits:  This demo was developed by 

David R. Hill
Department of Mathematics 
Temple University

and is included in Demos with Positive Impact with his permission.


5/20/02  DRH      last updated 2/2/2005

 

Since 7/04/2002