Michael Mays and Laura Pyzdrowski
West Virginia University, Institute for Math Learning
The applet below is designed to guide students through "the gutter problem," an exploration that models the rectangular cross sectional area of a gutter as a function of the gutter height when the dimensions of the rectangle are varied. It would fit naturally in an algebra course as a supplemental activity to a unit on quadratic functions, or in a more advanced course as an illustration of mathematical modeling. The applet was developed using JBuilder and the Mathematical Java Toolkit of Joe Yanik. Source code is available under the GPL. This work was supported by the NSF, CCLI project number 0339117.
There are many ways to build a guided exploration using the applet. For example, the gutter applet has been embedded into a more detailed quadradic function exploration. A guide containing student questions and activities appropriate for an algebra course is provided, along with a key for teacher use. Some background teacher information is also available, justifying the laboratory in terms of algebra content standards and objectives. The student questions can be adapted for use with a graphing calculator, or used with an on line quadratic function graphing utility.
The applet is built using Java 1.4.2, and an appropriate browser plugin (the Runtime Environment is fine) might have to be installed before the applet will work.
The applet has several areas of interaction. The upper left panel is a graphical representation of the sheet of metal being creased to form the gutter, with a draggable point (in blue) that changes the shape of the gutter. Once the gutter is drawn, the upper right area lets you adjust the dimensions and the units to be used. Default dimensions are in pixels, but inches or centimeters might be more appropriate. If the dimensions are changed, the scale of the graphical representation and the graphing utility also change. As you drag the point to alter the gutter shape, there is a "Capture Data Point" button to add the x value (gutter height) and associated area to the table of values. After some values have been captured, "Plot Data Points" shows them on the graphing utility in the lower right. Then entering a function in the "A(x) =" text field will exhibit the graph to see if there is a good fit. The function should be entered with graphing calculator syntax, using x for the variable, * for multiplication, and grouping factors with parentheses. As long as there are at least three table values plotted, if the plotted equation passes through them an encouraging message is displayed. There is a navigational grid to scroll and zoom the graph, and trace enabled to locate points of interest. If at any time the figure is covered or obscured, "Refresh" will redraw it.
A good way to start is to click on "Draw Gutter," drag the point in the lower right hand corner of the gutter picture to make several different rectangles, and for each rectangle click on "Capture Data Point" to plot the point on the graph.