# Human-Computer Interaction: Communication, Cooperation, and Application Design

By Hans-Jörg Bullinger; Jürgen Ziegler | Go to book overview

measures proficiency in the following NCTM standards: spatial sense, patterns, computation, statistics, and communication. Integral assessment components are designed into the multimedia project that enables teachers to quickly evaluate the students and their mathematical progress.

The mathematics involved in Wyzt's Playground are significant and have extensions in school mathematics beyond the fourth grade. The mathematical concept underlying Wyzt's Playground is a linear programming problem with constraint inequalities about space, time and money. The optimization function maximizes the number of kids who can play on the playground. Moreover, the answers must be integers. Thus, the mathematical problem, and the related assessment, can extend beyond fourth grade through high school algebra.

3 The Mathematical Model

The model is formulated in the following way: maximize z=ax + by + c where a, b, c are positive integers chosen for the number of kids using the equipment, choices represented by x, y and a required basketball court, and

dx + ey ≤ f (Space) gx + hy ≤ i (Cost) where d,e,f,g,h,i are all appropriate numbers selected on the basis of reasonableness of the solution, and x and y are the numbers of each type of equipment as above.

The solution then, is an integer choice of z representing the highest number of children on the playground constructed with pieces of x pieces of one equipment and y pieces of another, each of which will occupy space and cost money. As mentioned earlier, a third piece of equipment (a basketball court) was required for all playgrounds.

To illustrate the model and the solution, let us consider one of many playgrounds we could build. Suppose that we choose swings and slides as the type of equipment we will use along with the basketball court. An amount of \$10,000 is available for the playground. We also know that swings cost \$1000 each and can hold 8 kids, slides cost \$500 and can hold 4 kids, and the basketball court cost \$2500 and can hold 10 kids (at a time). Furthermore, the space available for the playground is 4900 square feet. The basketball court uses 1500 square feet of space, swings occupy 300 square feet of space and slides occupy 150 square feet of space. Our problem is now:

maximize z = 8x + 4y + 10 such that 300x + 150y + 1500 ≤ 4900 (Space) 1000x + 500Y + 2500 ≤ 10000 (Cost) where x, y and z must all be positive integers.

The solutions to some of our playground problems are not unique. For instance, one of several solutions to the example is x = 1, y = 13 and z = 70. Another is x = 7,y = 1 and z=70.

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