A Lesson with string data on the Excel

Gooyeon Kim and Eduarda Moura

The following data corresponds to the measurements from the bridge of a guitar to each of the frets:

 Fret # Length to the fret Fret # Length to the fret 0 64.5 11 34.2 1 60.7 12 32.3 2 57.4 13 30.4 3 54.1 14 28.7 4 51.1 15 27.1 5 48.3 16 25.6 6 45.5 17 24.2 7 43 18 22.8 8 40.6 19 21.6 9 38.4 20 20.4 10 36.3

That graphs into:

Students might conjecture that the graph corresponds to a branch of a parabola or that corresponds to a exponential function. Depending on the conjecture the teacher might suggest the students to compose a table of second differences for the parabola or ratios for the exponential.

Computing the ratios of the values corresponding to 2 successive frets we get the following sequence:

 Fret# Length to Fret Ratio Fret# Length to Fret Ratio 0 64.5 0.941085271 11 34.2 0.944444444 1 60.7 0.945634267 12 32.3 0.941176471 2 57.4 0.942508711 13 30.4 0.944078947 3 54.1 0.944547135 14 28.7 0.944250871 4 51.1 0.945205479 15 27.1 0.944649446 5 48.3 0.942028986 16 25.6 0.9453125 6 45.5 0.945054945 17 24.2 0.94214876 7 43 0.944186047 18 22.8 0.947368421 8 40.6 0.945812808 19 21.6 0.944444444 9 38.4 0.9453125 20 20.4 10 36.3 0.94214876

The ratio seems to be around the value 0.944. In fact, if we average the values in the third column we get the value 0.94406996. The function seems to be exponential with first value 64.5. If we produce the values of this function in excel we have:

 x x 0 64.5 11 34.217766 1 60.888 12 32.3015711 2 57.478272 13 30.4926831 3 54.2594888 14 28.7850929 4 51.2209574 15 27.1731277 5 48.3525838 16 25.6514325 6 45.6448391 17 24.2149523 7 43.0887281 18 22.858915 8 40.6757593 19 21.5788157 9 38.3979168 20 10 36.2476335

And if we graph simultaneously the original data and the values given by this new table we can see that the values of the function:

approximates quite well the string data:

What if we change the length of the open string to 50 or 10 or 1?

 x 0 50 1 10 1 47.2 0.944 9.44 2 44.5568 0.891136 8.91136 3 42.0616192 0.84123238 8.41232384 4 39.7061685 0.79412337 7.9412337 5 37.4826231 0.74965246 7.49652462 6 35.3835962 0.70767192 7.07671924 7 33.4021148 0.6680423 6.68042296 8 31.5315964 0.63063193 6.30631928 9 29.765827 0.59531654 5.9531654 10 28.0989407 0.56197881 5.61978813 11 26.5254 0.530508 5.30508 12 25.0399776 0.50079955 5.00799552 13 23.6377388 0.47275478 4.72754777 14 22.3140255 0.44628051 4.46280509 15 21.06444 0.4212888 4.21288801 16 19.8848314 0.39769663 3.97696628 17 18.7712808 0.37542562 3.75425617 18 17.7200891 0.35440178 3.54401782 19 16.7277641 0.33455528 3.34555283

Then the graphs are:

The students might want to look at better graphs of the functions and graph them in another application like the the Graphical Calculator where the functions can be seen from different zooms.

If the students decide to explore the possibility of the data fitting a parabola we can ask them to make second differences, discuss what do the second differences look like if the curve is a parabola.

The table is:

 fret data 1st Differences 2nd Differences 0 64.5 3.8 0.5 1 60.7 3.3 7.10543E-15 2 57.4 3.3 0.3 3 54.1 3 0.2 4 51.1 2.8 7.10543E-15 5 48.3 2.8 0.3 6 45.5 2.5 0.1 7 43 2.4 0.2 8 40.6 2.2 0.1 9 38.4 2.1 7.10543E-15 10 36.3 2.1 0.2 11 34.2 1.9 7.10543E-15 12 32.3 1.9 0.2 13 30.4 1.7 0.1 14 28.7 1.6 0.1 15 27.1 1.5 0.1 16 25.6 1.4 3.55271E-15 17 24.2 1.4 0.2 18 22.8 1.2 -3.55271E-15 19 21.6 1.2 1.2 20 20.4

The teacher can also plot the points one by one in the graphical calculator and ask students to construct the parabola that best fits the data. Then, to draw the function:

and to discuss again the fitness of the graphs.

The parabola that we found to best fit the data was:

The following is a graph of both functions: