NOTE BY NOTE; MEASURE BY MEASURE Mike Reiners Bob 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 SUMS Find 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?

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: Bob Horton: