I have to say that Fred holds a very special place in my heart, since this is the model that got me involved in this project. In the spring of 1995, I was taking multivariable calculus with Prof. Klotz. After a particularly bloody quiz, he challenged the class to make a model of y = x * z to prove that we knew how to use level surfaces. I took him up on the offer, and thus was born Fred.
The instructions for this model are not a precise as some of the others. This really is something that you have to do by feel. I will outline the principles and techniques, you fill in the details.
- Materials you need include a cardboard box, string, scissors, tape, and a ruler. Some scratch paper would also be helpful. There are optional embellishments, such as paper to cover the box or markers to draw the axes.
- The first thing to note is that for any constant value of z (say, z = c), the equation becomes y = c * x, which is simply a line through the origin [ahhh, think back to the days of the slope-intercept form of a line...].
- Now it is time to get a little bit creative. Measure your box and decide on some appropriate scale. If you are lucky, your box will have dimensions in even numbers of inches or centimeters. If you are not lucky, you will just have to make do. Draw the axes with scales on the sides of the box. On two sides you will have the z and y axes, on the other sides you will have the z and x axes. If you draw the x and y axes on the bottom of the box, it will help you remember which way is positive and which way is negative.
- As you can see from the diagram, each side of the box is at a constant for one of the coordinates. To make the surface, you have to use the strings as the level curves. For example, say your box is 6 inches in the y direction. If you are making the level curve for z = c = 2, the string will intersect the side of the box that is at the top of the diagram at y = 6 / 2 = 3 and z = 2. Going back to the original equation, this makes x = 1.5. Punch a hole in the box at this point. (Pay close attention to which x direction is positive. In this case, it is left!) Then figure out where the string will intersect the bottom, punch a hole there, and tape a piece of string through the two holes. Repeat this for all of the level curves that you wish to make.
- An extension for really enthusiastic builders: You could make this surface in a soda bottle, using method similar to that outlined in the instructions for Leslie by using cylindrical coordinates. If you try this, let us know how it turns out!
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8 November 1995
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