Rearrange the groups from the previous day.
Say to students, "In the previous lesson, you determined the
placement of frets on a stringed instrument. Today, you will decide
where to put frets on an instrument." Many stringed instruments, such
as violins, do not contain frets. Musicians play these instruments by
sliding their fingers up and down the strings to the appropriate spots,
but there are no frets to guide them. For this activity, explain to
students that they will be determining where the frets would be (i.e.,
where a musician should place his or her fingers).
If available, have each group of students work with a fretless
instrument. (You might wish to borrow instruments from the music
department for this activity.) Alternatively, you can hand out the Not To Fret
activity sheet, which shows the neck of a fretless stringed instrument.
The sheet shows the location of the nut and the 12th fret, and students
are to determine the placement of the 1st through 11th frets for this
Students should remember that the distance from the nut to the
12th fret is half the distance from the nut to the bridge. On the
activity sheet, the distance from the nut to 12th fret is about 19 cm,
so the distance from the nut to the bridge is 38 cm. That means that
the 1st fret will occur at 38 × 2-1/12 ≈ 38 × 0.9439 = 35.9 cm
from the bridge, or 38 − 35.87 = 2.13 cm from the nut. The same process
can be used to find the 2nd fret (that is, multiply 35.87 × 0.9439, and
then subtract the result from 35.87), but a more insightful solution is
this: students should realize that the distances between frets form a
geometric sequence, too. Therefore, if the distance from the nut to the
1st fret is 2.13 cm, then the distance from the 1st fret to the
2nd fret is approximately 2.13 × 0.9439 = 2.01 cm. By continually
multiplying by 0.9439, the successive distances between frets can be
found. On a calculator, this can be accomplished easily by multiplying
the previous answer by 0.9439 and then repeatedly hitting the Enter key.
If students are using the activity sheet, they should indicate
the location of the frets by drawing them. If using an actual
instrument, students should use chalk or masking tape to indicate the
location of the frets, to prevent damage to the instrument.
The following form can be used to determine the fret placement
on various instruments. It is currently set with a scale length of 38
to match the instrument depicted on the Not To Fret activity sheet, but its value may be changed. Additionally, the number of frets can be changed from 12, but the maximum is 30.
Distribute the Placing Frets
activity sheet, which allows students to consider the two types of
curves—discrete and continuous—that surfaced during this lesson.
As a class, discuss which letters in the equations y = arn and y = arx
are variables and which or constants. This is an opportunity to deepen
student understanding of variables beyond being just a letter that
represents a number. In this case, y is the length of string
from a fret or finger position to the bridge; as such, it varies as the
finger position changes. Students should clearly see that r is a
constant since that ratio is the same (approximately) regardless of the
instrument or from which adjacent lengths it is based. Students may
have a hard time recognizing that a is a constant because a changes from instrument to instrument; however, for a particular instrument, the value of a does not change as the finger position changes, and it represents the length from nut to bridge. The value of n or x relates to a particular fret or finger position, so they are variables.
To conclude the lesson, display the last page of the Overheads,
which contains a set of summary questions. Allow students to answer
these questions as well as to ask any questions that they may have.
Observe student conversation about continuous versus discrete graphs.
Are they connecting the "slide effect" possible on a violin with
continuous changes in x in the equation y = arx and the numbered frets as related to the values for n (n = 0, 1, 2, 3, …) in the equation y = arn?
Questions for Students
1. Which type of stringed instrument, fretted or fretless, gives a musician more flexibility in playing accurate pitches, if the instrument itself is out of tune?
[With a fretless instrument, a musician slides his or her fingers to move from note to note. Because the finger position is decided by distance from other notes and is not dictated by fret position, a musician may be more likely to play correct pitches on a fretless instrument.]
- Did students understand the differences between continuous and discrete
functions? If not, what could be done to make the differences more