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        Math + Arts | Fractions in Music

        In this lesson, students explore the relationship between fractions and musical pitch.

        Lesson Summary

        There are three options for this lesson, depending on class needs and time available:

        1. Frame, Focus, and Reflection (view and discuss): students watch a video about fractions and discuss.

        2. Short hands-on activity: students play a fraction game.

        3. Project: students explore the relationship between fractions and musical pitch using an online interactive. An extension provides a hands-on project building instruments (water bottles) that demonstrate pitch and fractional relationships.

        Time Allotment

        1. Frame, Focus, and Reflection (view and discuss): 1/2 class period

        2. Short hands-on activity: 1/2 class period

        3. Project: 1-2 class periods

        Learning Objectives


        I can compare fractions with different numerators and denominators.

        Arts and Humanities

        I can begin to recognize and identify pitch and melody.

        I can listen to and explore how changing elements results in different musical effects.

        I can compose a melody on the computer.

        I can compose a group melody on water bottles.

        Prep for Teachers

        Watch "What Is Music?" from the Music Toolkit.

        (The entire clip is very informative and worthwhile. For this lesson, pay close attention to the discussion of pitch, from 0:21 to 1:22.)




        Computer with internet access

        MathMastery Teacher’s Guide (page 1)

        "I Have, Who Has?" game (cut before the lesson)


        For Making Instruments

        Five identical bottles


        Metal spoons

        Media Resources

        Cyberchase: The Puzzle of the Amulet of Amagansett

        Music Toolkit: What Is Music?

        Philtuga website: Musical Fraction Bars

        Philtuga website: Water Bottle Xylophone

        Introductory Activity

        Students should have a beginning understanding of fractional parts of a whole.

        Learning Activities

        Frame, Focus, and Reflection

        Frame and Focus:

        Explain that fractions are special numbers. When are they useful? You’ve probably noticed how a fraction consists of a number on top, the numerator, and a number on the bottom, the denominator.

        Write the number 3/4 on the board. Show students the numerator and denominator.

        Ask: "What does each of these numbers mean?" "Can you give examples of some fractions that you know?"

        Show Cyberchase: "The Puzzle of the Amulet of Amagansett."

        As students watch this segment, pause the video to ask questions.

        "What does the denominator tell us?"

        "What does the numerator tell us?"


        Ask students to turn to their partners and discuss what they saw in the video.

        Ask the pairs of students to think of everyday things that can be divided into fractions. (Pizza, cake, candy, etc.)

        Allow students to share and list ideas on the board.

        Hand out MathMastery Teacher’s Guide (page 1) to each student. Allow time for students to finish before moving on to the hands-on activity.

        Short Activity

        Tell students they will now be playing a game using fractions.

        Draw and shade in the following for the fraction 11/16.

        Ask students what the denominator should be (16)

        What should the numerator be? (11)

        What is our fraction? (11/16)

        Give one of the “I Have...Who Has...” cards to each student (mixed up)

        The teacher should go first.

        Call out what the fraction is on your card and ask who has the next fraction.

        The student with that fraction will then come up and stand beside the previous person.

        Continue until every student has a turn to participate.



        Have music playing as students enter the classroom. It can be any music.

        Ask, “Did you know that music is full of fractions?” (You may get some good answers from students, especially if they have had private lessons or have discussed notation and pitch in music classes.)

        Say, “We’re going to work with music and fractions.”

        Show the “What Is Music” video segment for pitch. Lead students in discussing how melodies are composed by combining various pitches.

        Pitch Fractions

        Say, “Now we’re going to explore fractions in musical pitches.” Show the “What Is Music” video segment for pitch.

        Explain that, scientifically, each pitch in music has a number, based on how many sound vibrations it has per second. The relationships between different pitches are very mathematical relationships.

        Ask if any of your students have heard of the term “A-440.” Some may have—celebrate their knowledge! When musicians use electronic tuners, many times they will see the term A-440 on the tuner. Members of an orchestra listen to a lead musician play A-440 and then tune their own instruments based on that pitch.

        Explain that in music, a scale is made up of eight pitches in sequence. Tell students that in their computer activity today, they’ll be working with a C-major scale. (Some of the students may know what you’re talking about and others won’t; It’s OK... just put the terminology out there anyway.) Each pitch in the scale is expressed as a fraction.

        The designer of this activity came up with the fractions for the pitches in a pretty cool way: he measured the fraction of a string that vibrates to create each pitch. If the entire string vibrates to create the pitch “C,” then creating the pitch “D” requires 8/9 of the string to vibrate. The pitch “E” requires 4/5 of the string to vibrate. He had to use a LOT of equivalent fractions to figure that out!

        The main idea is that shorter bars, or smaller fractions, make higher pitches; while longer bars, or larger fractions, make lower pitches.

        Take a look at the Philtulga website’s “Musical Fraction Bars” interactive activity page with your students, and go over the information and instructions with them.

        Before allowing students to begin creating their own fraction-based songs, select one of the examples (“Twinkle, Twinkle, Little Star” is a great one!) from the “Songs” tab at the bottom of the page. Before you play it, ask students to look at the bars and predict which will play the lowest pitches and which will play the highest ones. Press the “play” button and listen to see if the predictions were right.

        Play another example or two, and then allow students to work on creating their own tunes. Some students may need more encouragement than others in this activity. Working in pairs might be a good way to get started. You can have students move into working individually as they gain confidence.

        Encourage students to share their original and/or collaborative compositions with the class. Allow some positive discussion about how each piece sounded, what choices the composers made, and why.


        Make Our Own Instrument

        Say, “Remember playing the Musical Fraction Bars on the computer? Well, now we’re going to research and discover how to create a scale using glass water bottles in our classroom.”

        Bring up the “Water Bottle Xylophone” page and scroll down to the Virtual Water Bottle Xylophone (you can also click on the bookmark link at the top to get there).

        With the students, read the section that begins, “As you will hear...”

        Ask, “Do you remember that, in the Musical Fraction Bars activity, the longer bars played lower pitches? Which bottles will play lower pitches: the fuller ones or the emptier ones?” (the fuller ones).

        Explain that the reason for that is, the more area there is to vibrate, the lower the pitch will sound. The less area that vibrates, the higher the sound.

        Play a song or two on the Philtulga website’s Virtual Water Bottle Xylophone, and allow students to try it also.

        Divide students into groups of 3 or 4 (or the number you need based on space and availability of glass bottles).

        Give each group a copy of the pitch and measurement table on the “Water Bottle Xylophone” activity page.

        Each group will also need five of the same type of glass bottle (the groups may have different bottles, but each group needs a matched set of five), a measuring cup that has markings for ounces (oz.) and/or milliliters (mL), a metal spoon or other utensil, and access to water.

        Before you allow students to begin experimenting, you may want to make a statement about appropriate use of water in this exercise, and let the students know what your expectations are about careful pouring of water and handling of the bottles.

        Instruct students to carefully measure the water into their bottles, following the instructions on the measuring table. When they have done that, they should test their success by trying to play a song on their five-note scale. They can try one of the songs notated on the activity page, or they can notate their own—or both!

        If there is time, allow each group to share a tune on their Water Bottle Xylophone. Compare how each group’s xylophone sounds in relation to the others’.

        Show the “What Is Music” video segment for pitch.

        Formative Assessment

        What are the indicators of student progress toward or achievement of each learning target?

        Math Assessment Problems

        Observation/review of fraction sequences created in Rhythm Fractions

        Differentiated levels of questioning

        Arts and Humanities

        Pitch Fractions—Teacher Observation—Did students successfully create a tune?

        Exit slip—what is the relationship between the length of a vibrating string or bar and the pitch it creates?

        Program Review

        Where does this fit in? How should you document it?

        This activity contributes to your school’s overall efforts in art programming in several areas, depending on whether you implement just the Frame, Focus, and Reflection portion or you implement the entire project.

        Curriculum and Instruction: Aligned and Rigorous Curriculum

        a) To what extent does the school ensure that the arts curriculum encompasses creating, performing, and responding and is fully aligned with the Kentucky Core Academic Standards?

        b) To what extent does the school ensure that the arts curriculum provides for the development of arts literacy in all four arts discipline and also utilizes the Common Core Standards for English/Language Arts?

        c) To what extent does the school ensure that the school’s curriculum provides opportunities for integration as natural cross-curricular connections are made between the arts and other content areas?

        Curriculum and Instruction: Instructional Strategies

        a) To what extent do teachers systematically incorporate all three components of arts study: creating, performing, and responding into the arts?

        b) To what extent do teachers provide models of exemplary artistic performances and products to enhance students’ understanding of an arts discipline and to develop their performance/production skills?

        c) To what extent do arts teachers provide for the development of artistic theory, skills, and techniques through the development of student performances or products that are relevant and developmentally ap propriate for students?

        Curriculum and Instruction: Student Performance

        a) To what extent are students actively engaged in creating, performing, and responding to the arts?

        b) To what extent do students identify a purpose and generate original and varied art works or performances that are highly expressive with teacher guidance?


        Lesson Creators: Emily Jackson, Dean Cornett, Dawn Hibbard, and Judy Sizemore


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