This quick tutorial shows you how to mathematically model data using Mathscribe 2.5. In this example, we assume you have been rolling balls down an inclined (sloped) plane for various distances d, and recording the time t to reach the bottom. (If your name is Galileo, then you are also ushering in the scientific revolution.) Begin by double-clicking on the Mathscribe program to start it, and choose to create a New file. Then modify the Independent and Dependent Variable tables to contain the following variables and limits, using the mouse or the <TAB> key to move between fields:
Now click on the Table link, and then type in the data below, separated by <TAB>s and <RETURN>s. (If you want to see this data plotted as points while you type, then click on Graphs before typing in these numbers.)
Here d is measured in meters, and t is measured in seconds, presumably using a stopwatch or computer-based lab equipment. Also note that if you had already entered your lab data into a spreadsheet or other program, you could just copy and paste it into a Mathscribe Table, without having to re-type it.
Now go back to your original window, select "y=" under Constraints, and type in an equation for t in terms of d that will hopefully model (approximate) this data. Of course, we're not sure exactly what equation to use, so we introduce another couple of parameters c and k, and guess (hope) that some equation of the form t=c*d^k will do the job:
After you've typed in this equation, click on the Graphs link to graph it. This will also automatically create sliders for the parameters c and k, and give them initial values of 0. You should therefore get the graph shown below. Notice that your data values are plotted on the graph, along with the curve t=c*d^k, assuming c=0 and k=0.
In addition, each variable's value is shown, along with t's root-mean-square (square root of average of square of) error, using these data values and approximating formula.
Your goal at this point is to choose better c and k values so that your formula more closely models your data. To see how the graph depends on c and k, try clicking or dragging at various places inside the sliders (under Parameters in the first window). To focus your efforts, you might want to edit the slider limits, changing them from -10 and 10 to 0 and 2 on each slider. Also, after clicking inside a slider, use the right-arrow or left-arrow keys to modify your selection by a small amount. Finally, try shift-clicking inside a slider to select a range of values. You'll get three curves in the Graphs window, corresponding to the left edge, right edge, and middle of your selection. Here is an example where k=0.5 and c=0.6,0.7,0.8:
We believe that all this clicking and dragging is very educational, as it really illustrates the algebraic model in an intuitive and hands-on way.
Notice that as you slide through different c and k values, the ~t column in the Table window is also updated. After you have found c and k values that appear to give you a good approximating graph, with a small root-mean-square error, you can use the Table window's ~t column to fine-tune your parameters. You should be able to get computed values that closely match your observed t values, as in the following table:
You can compare the columns for t and ~t to see exactly how accurate your formula is at each point.
Finally, issue the Print command (in the File menu) to print your open windows. Congratulations! We hope you've found this exercise to be intellectually stimulating. Enjoy!
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