## Fretting

In the second lesson of this unit, students will use their discoveries from the first lesson to place frets on a fretless instrument. They will then compare geometric sequences with exponential functions.

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 instrument.

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.

Fret Placement | |

Scale Length: | |

Number of Frets: |

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* = *ar ^{n}* and

*y*=

*ar*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,

^{x}*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.

- Fretless instruments (such as violins, basses, etc.)
- Metric and U.S. rulers
- Not To Fret Activity Sheet
- Placing Frets Activity Sheet

**Assessment Option**

Observe student conversation about continuous versus discrete graphs. Are they connecting the "slide effect" possible on a violin with continuous changes in

xin the equationy=arand the numbered frets as related to the values for^{x}n(n= 0, 1, 2, 3, …) in the equationy=ar?^{n}

**Question for Students**

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.]

**Teacher Reflection**

- Did students understand the differences between continuous and discrete functions? If not, what could be done to make the differences more apparent?

### Exploring Measurement, Sequences, and Curves with Stringed Instruments

### Learning Objectives

Students will:

- Distinguish between discrete and continuous curves.
- Compare exponential growth curves to geometric sequences.

### NCTM Standards and Expectations

- Use geometric models to gain insights into, and answer questions in, other areas of mathematics.

- Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP8

Look for and express regularity in repeated reasoning.