NOTE BY NOTE;

MEASURE BY MEASURE

Mike ReinersBob Horton

FRACTIONS: A MAJOR STUMBLING BLOCK

• Algebra: a Major Gatekeeper

• Problems: Often Not the Algebra

• Areas that Hinder: Lack of Understanding and Skills with Fractions (and integers)

• Often Taught Too Abstractly

• Music Can Help!

THEORETICAL FRAMEWORK:4E X 2 INSTRUCTIONAL MODEL

SIMPLE MEASURES IN 4|4 TIME

CONNECTING MUSIC TO MATH

• Let’s “Sing” Their Durations

• How Might We Record the Durations of Each Note?

• Can We Use Addition for Each Measure?

• Can We Use a Single Fraction for Each Measure?

• How Can We Show Their Equivalence?

LOOKING FOR EQUIVALENTS

• How many beats does a half note receive?

• How many quarter notes does it take to occupy that same number of beats?

• How many eighth notes does it take?

• How many sixteenth notes does it take?

• Express these equivalent relationships in mathematical symbols. Confirm your results.

LOOKING FOR EQUIVALENTS (cont’d.)

• How many beats does a quarter note receive?

• How many eighth notes does it take to occupy that same number of beats?

• How many sixteenth notes?

• Express these relationships in mathematical symbols. Confirm your results.

BUILDING TOWARD THE COMPLEX

• What fraction of a measure does a dotted quarter note occupy?

• What makes this type of fraction more “complex” than other fractions?

• How might we represent this using mathematical symbols?

BUILDING TOWARD THE COMPLEX (cont’d.)

• How many eighth notes occupy the same number of beats as a dotted quarter note?

• How many sixteenth notes occupy the same number of beats as a dotted quarter note?

• How can you express these equivalent relationships mathematically? Confirm your results.

FINDING SUMSFind as many sets of notes as you can that are musically equivalent to 3 quarter notes. You can combine different types of notes in doing so. Use musical and mathematical notation to demonstrate your sets. Then, confirm your results.

RHYTHMS-FRACTIONS-MELODIES The following represents the rhythm of a phrase from a well-known song. Write the duration of each note in each measure in fractional form; then add them together to determine the time signature. Can you guess the song?

TRY ANOTHER

AND ONE MORE

EQUAL OR EQUIVALENT?

Name another fraction that is equivalent to . Why is some music written in 3|4 time while other music is written in a time signature that is different, but equivalent?

MUSIC: THE INFINITE MEASURE

• Our standard notes are the whole note, half note, quarter note, eighth note, and so on.

• Starting with the half note, write the first five note values in descending order using words, fractional notation, and musical notation.

• What is the ratio of the duration of consecutive notes?

MUSIC: THE INFINITE MEASURE

• Write a partial measure in 4|4 time with:–A half and a quarter–A half, a quarter, and an eighth–A half, a quarter, an eighth, and a sixteenth–A half, quarter, eighth, sixteenth, and thirty-second

• Determine how many total beats are accounted for in each partial measure. Also indicate what fraction of the measure the notes occupy in total.

• What would it take for a full measure?

CLOSING THOUGHTS• Mathematics is everywhere!

• Context should be used to motivate students and help them learn. Music is one possibility.

• Start with the familiar and simple.

• Concepts BEFORE skills

• Explore BEFORE Explain (and let students explain)

REFERENCE and CONTACT INFORMATION

Fostering Mathematical Thinking through Music; Reiners & Horton; Casio Education.

Mike Reiners: mreiners@chof.net

Bob Horton: bhorton@clemson.edu