This lesson focuses on the investigation of geometric sequences by considering the placement of frets on a stringed instrument. A major component of this lesson is the measurement of distances between frets, and without correct measurements, the mathematics of the lesson will be lost. It is imperative to stress the importance of accurate measurements for this lesson, so you may wish to begin the lesson with a discussion of precision and accuracy.
As a first step, display the Pre Activity Questions from the first page of the Overheads. Questions 2 and 3 are warm-up questions that allow students to review some of the skills that will be needed for the lesson. Question 1, however, will lead into a nice discussion about precision of measuring devices.
During the activities, students will be required to measure lengths on fretted musical instruments and calculate the ratios of consecutive lengths. Although there are advantages to students making their own decisions about which measurement system to use, a discussion about precision and accuracy might be useful in helping students choose a system so that their measurements are as accurate as possible and their calculations are less tedious. In general, the metric system is less tedious, less likely to result in measurement and calculation errors, and is the more precise than most U.S. measuring tapes. (A U.S. ruler with 1/16-inch precision, converted to metric, has precision of 1.6 mm. Comparatively, a metric ruler typically has 1-mm precision.)
Display the second page of the Overheads (which shows three rulers with various levels of precision) and discuss the difference between accuracy and precision. Students should understand that precision refers to the smallest unit that a tool will measure, while accuracy refers to the range of possible values that might be described by a particular measurement.
Precision is the smallest unit of a particular measuring tool. The actual length could be as small as the measured length minus one-half the precision of the measuring tape, or as long as the measured length plus one-half the precision of the tape. For instance, a correctly measured length of 4½" inches, measured with a measuring tape with ½-inch precision, could have an actual length of 4½ ± ¼ , or anywhere between 4¼" and 4¾". If the same length were measured as 115 mm, with a metric measuring tape of precision 1 mm (and accuracy of 0.5 mm), the actual measurement could be anywhere between 339.5 and 340.5 mm. The first two rulers below demonstrate precision and accuracy within the U.S. system; the first has ¼‑inch precision, whereas the second has 1/16‑inch precision. The third ruler is a metric ruler with 1‑mm precision.
Students may convert between metric and U.S. measurements to determine which has greater precision; however, they might also "eyeball" the markings to determine which has greater precision. If the smallest marking on a U.S. tape is 1/8 inch, it is easy to see that 1 mm is the smaller measure, so the metric tape has greater precision. If the smallest marking is 1/16 inch, students may have to convert between metric and U.S. measurements to make the comparison. (However, they can probably still see the difference—a millimeter is smaller than 1/16 inch.)
Divide students into groups of four for the first activity, and distribute the To Fret activity sheet.
In this activity, students measure the distance from the bridge to the nut and to each fret on an instrument. (See figure below.) Two students can measure while the other students record information or do calculations. Any instruments with frets will work, such as guitars, ukuleles, and mandolins. Students should discover that the ratio of consecutive lengths will be constant, regardless of the measuring system or musical instrument. The ratio is approximately 0.94, meaning that the distance from the (n + 1)st fret to the bridge is about 0.94 times the distance from the nth fret to the bridge. Although measurements may differ from group to group, the ratio should be the same for all groups. Slight variations will occur due to measurement or calculation errors; other variations may occur because of errors in the instrument itself, such as incorrect bridge or fret placement.
After students complete the To Fret activity sheet, review the results as a class. Allow students to point out similarities and differences in what they found.
Rearrange students into new groups, and then distribute the Exact Ratio activity sheet. In the previous activity, students should have realized that the ratio between lengths is approximately 0.94. For this activity, students use the exact ratio of 2-1/12 to generate a geometric sequence that gives the exact location of each fret. The points of this geometric sequence can be approximated by an exponential equation, and the y-intercept of the graph represents the scale length of the instrument.
The last row of the table in Question 5 on the Exact Ratio activity sheet reveals an interesting occurrence. Using the exact ratio, the value of the calculated length from the 12th fret to the bridge is ar12 = a(2-1/12)12 = ½a, meaning that the 12th fret occurs exactly halfway between the nut and the bridge. A string played "open" (that is, played without pressing any fret on the instrument) is exactly one octave lower than the same string played while pressing just behind the 12th fret. A string played open is twice as long as a string played when the 12th fret is pressed, so the open string vibrates at half the speed. (It is this very occurrence that explains why the exact ratio is 2-1/12.)
In the next lesson of this unit, Fretting, students will use what they’ve learned about the geometric sequence that occurs with frets to place frets on an instrument.
