What is algebraic thinking?

NCTM’s definition:

Algebraic thinking includes recognizing and analyzing patterns, studying and representing relationships, making generalizations, and analyzing how things change. Of course, facility in using algebraic symbols is an integral part of becoming proficient in applying algebra to solve problems.

Music provides a forest so that kids can study their trees.

Every culture’s music follows the same basic laws or principles.

Exponential Growth

Patterns of music grow exponentially. If there is one, then there will be two. If there are two, then there will be four, and so on.

Cadence

Musical patterns drive to a cadence, a place of rest or repetition.

Form

An entire composition, whether 2:00 minutes or 20:00 minutes will have one of the following structures: AABA, AAAB, ABACAD... In longer compositions, music takes on a fractal structure.

*Note: In the 20th century, composers attempted to move away from these three principles with avant-garde music. The general public did not embrace these experimental compositions, which is one reason symphony orchestras do not perform avante-garde music regularly.

Measuring Music

Historically, music patterns were notated in a codified system that helped transmit compositions from one musician to another, and we continue to use this system of notation today. Choirs, bands, and orchestras play/sing music written in measures. Although measures are essential for visual notation, time measurements are part of math and science. For this reason, music puzzles are assembled based on timing (mm:ss:hnd) and the exact placement of sound files within the composition’s overall structure.

Music patterns as a single variable.

Let x represent a composition’s total time.

  • composition = x
  • composition = 2(.5x)
  • composition = 4(.25x) … n.

Music patterns as multiple variables.

Let x represent one fraction of the total composition.

Let y represent a different fraction of the total composition.

  • composition = (?)x + (?)y

 

Challenges:

  1. If z represents a composition’s total time, then z = 2x + 4y, what fraction is x, and what fraction is y?
  2. Create a composition that uses three different fractions (three variables).