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Fractions, Multiples, Beats, and Measures

Module by: Catherine Schmidt-Jones

Summary: Interdisciplinary lesson plan for music-oriented activities that help students understand the basic math concepts underlying multiples and fractions.

Students are more likely to grasp key math concepts if several concrete examples of different types are given. Music is one area that can provide interesting, explorable, concrete examples of mathematical ideas. There are many strong relationships between math and music. One is meter, which is all about is all about organizing time into fractions and multiples. (See Duration and Time Signature, and Meter for more on this.)
    Goals and Assessment
  • Goals - students will learn about key musical concepts (beats, measures, meter, and time signature) necessary to discuss, analyze, read, or perform music. The student will use these basic musical concepts to explore the mathematical concepts of multiplication of whole numbers, and of the use of fractions to represent equal portions (in this case, equal portions of time in music).
  • Grade Level - grade 3, or any students focusing on basic multiplication and fractions concepts, or on time signature, meter, and reading rhythms correctly. This lesson plan is appropriate for either a math or a music class.
  • Student Prerequisites - Students should have studied or should be studying the basic math concepts underlying fractions, multiplication, and/or division.
  • Teacher Expertise - Teacher training in music education is not necessary to present this activity. The teacher should be familiar and comfortable with the terms and concepts regarding beats and measures, and must be able to identify and count beats and measures when listening to music. Teachers who wish to present the extension activity do not have to be adept at reading written rhythms, but should understand the concepts behind rhythm notation.
  • Time Requirements - One (approximately 45-minute) class period to do both the regular and extension activity, or 10-30 minutes (depending on the number of musical examples used) to do the basic activity only.
  • Objectives - While listening to music, the students will clap on the beat. As a group, the students will identify the strong and weak beats and use a different kind of clap for each. For a given musical example, students will count the number of beats and the number of strong beats. The students will then construct true multiplication, division, and/or fraction statements using three numbers: the total number of beats, the total number of strong beats (measures), and the number of beats per measure. In the extension activity, different groups of students will simultaneously clap different types of note values, and students will demonstrate understanding of fractions by making true math statements about the note length values.
  • Music Standards Addressed - National Standards for Music Education standards 8 (understanding relationships between music, the other arts, and disciplines outside the arts), 6 (listening to, analyzing, and describing music), and 5 (reading and notating music).
  • Other Subjects Addressed - The activity also addresses national standards in math in the focal points of multiplication and fractions for grade 3.
  • Evaluation - Assess students on ability to write or say correct math statements based on the musical examples.
  • Extensions - Advanced or gifted students, or students who have learned or are learning to read music, can do the extension activity.
    Materials and Preparation
  • Decide on at least two songs to use. Songs with a strong, easy-to-feel beat and with beats that are obviously stronger or weaker are best. If the class regularly sings songs together, you may use those. (But note that "Happy Birthday to You" and other songs that begin in the middle of a measure may be confusing.) Recordings of songs you think they'll know or like are also good. Try to pick songs that have a different rhythmic feel; for example one is fast and has a 1-2-1-2 rhythm, and the other slow with a 1-2-3-1-2-3 rhythm. Keep your selections short; one verse or one refrain is plenty.
  • Read through the procedure, following. Go through your selections as the children will, counting beats and measures, so that you know the correct numbers.
    Procedure
  1. Sing a verse or refrain of the song together, or play a recording of it. Have the students clap along on the beats.
  2. Repeat the song. This time ask them to clap louder on the stronger beats, and more softly on other beats. They should be able to reach a consensus in which they are steadily clapping loud-soft-loud-soft, or loud-soft-soft-loud-soft-soft, or possibly loud-soft-soft-soft.
  3. Explain that they are now clapping measures or bars as well as beats. Each loud clap starts a new measure.
  4. Ask for a volunteer to count beats (all claps) and another to count measures (only loud claps). Counters don't have to sing or clap, just count. Have everyone else sing and clap measures again for the counters.
  5. Let the counters report their numbers. If they are incorrect, repeat the previous step, while helping them count. If they are correct, have the students write them down for all to see.
  6. Ask the students how many beats (claps) there are in every measure.
  7. Ask the students if they see how the three numbers are related. Give hints or show them the answers if necessary. Depending on the concepts you are studying, you can just point out the basic multiplication and division statements involved, or you can treat each measure as a whole divided into fractions. For example, "Frere Jaque" has eight measures of two beats, for sixteen beats in all. That's 8x2=16, 16/8=2, etc.
  8. Repeat the above steps with the other songs for different multiplication and division statements.
    Extension Activity
  1. This works best with music that is not too fast. Students who already know something about music may catch on to this faster, but musical knowledge is not necessary. Ask the students if they've ever heard of a "whole note" or a "quarter note". Tell the students that most music has a time signature that looks and acts a lot like a fraction, and that each measure gets filled up with fraction notes. For example, if a measure is 4/4, it can be filled up with four quarter notes, or one whole note, or any other combination that equals "four quarters" or "one whole". Demonstrate by playing music that has a clear "4" beat. (Almost any pop music will do.) Have the students clap loud-soft-soft-soft, along with the song. Tell them they are clapping quarter notes, because there are four equal notes (of one beat) in each measure; each note is one quarter of a measure long. Play the same song and have the students clap only at the beginning of each measure (clap-wait-wait-wait). Now they are clapping whole notes; each clap represents a whole measure.
  2. Now ask the class if they can clap half notes (two claps to a measure; one clap every other beat) or eighth notes (eight per measure, two claps every beat).
  3. To demonstrate the relationships between various fractions, you can divide the class into two groups and have each group clap a different type of note For example, have one group clap half notes and another clap quarter notes. Then ask the students to make a math statement (for example "a half equals two quarters" or "a quarter is half of a half") that is demonstrated by their clapping.
  4. If the students are learning to read music, you may follow or precede this exercise with a discussion of rhythm notation. (See time signature, Duration: Note Length, Duration: Rest Lengths, and Simple Rhythm Activities for ideas and suggestions.)

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