NCDPI – AIG Instructional Resource: Background Information
Resource Title: Musical Fractions
Subject Area/Grade Level (s): Math/4
Time Frame: One-two days
Common Core Standards Addressed:
Number and Operation - Fractions
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an
equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 2 1/8 = 1 + 1 + 1/8 =
8/8 + 8/8 + 1/8.
c. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g by using
visual fraction models and equations to represent the problem.
Number and Operation - Fractions
Use equivalent fractions as a strategy to add and subtract fractions.
5.NF.1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in
such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In
general, a/b + c/d = (ad + bc)/bd.)
5.NF.2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators,
e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate
mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
Mathematical Practice:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with Mathematics.
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project
Additional Standards Addressed:
North Carolina Essential Standards for Music:
4.ML.2 Interpret the sound and symbol systems of music.
 4.ML.2.1 Interpret rhythm patterns, including whole, half, dotted half, quarter, and eighth notes and rests in 2/4, 3/4, and 4/4 meter
signatures.
 4.ML.2.4 Use standard symbols to notate rhythm, meter, and dynamics in simple patterns.
4.ML.2 Create music using a variety of sound and notational sources.
 4.ML.3.2 Create compositions and arrangements using a variety of traditional and non-traditional sound sources.
 4.ML.3.3 Create rhythmic compositions which include the use of whole, dotted half, half and quarter notes; whole, half and quarter rests; and
beamed eighth notes in duple and triple time and which are arranged using a variety of sound sources.
Brief Description of Lesson/Task/Activity: This task uses music notes to develop a deeper understanding of composing and decomposing
fractions. Students will discover that music notes actually represent fractional parts of measures of music. Students will apply this
understanding to create their own music arrangements, using mathematics to support the accuracy of their creations.
Type of Differentiation for AIGs (include all that apply):
Adaptations for AIGs:
x Content
x Process
Enrichment
x Extension
Acceleration
x Product
Explanation of How Resource is Appropriate for AIGs: This task requires students to transfer knowledge of composing and decomposing
fractions to music concepts. Students must apply understanding of the fourth grade concept of decomposing fractions as well as fifth grade
standards that require students to work with unlike denominators. Students must use this knowledge and individual creativity to produce a
musical arrangement, which directly relates with Bloom’s highest level of thinking, creating. The open-endedness and opportunity to use a
variety of solution strategies also make this an excellent task for AIG students.
Needed Resources/Materials
•
Sample piece of music
•
Blank staff paper
•
Music Notes Chart
•
Composer Warm Up sheet
•
Computer, internet access, and headphones for each student
Sources: NA
TEACHER NOTES: NA
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project
NCDPI AIG Curriculum Resource Outline
STAGE ONE: ENGAGE
Play a short piece of simple music (e.g., Minuet in G by Bach). Explain that whenever we listen to music, we are actually hearing a series of fractions.
Display the Music Notes Chart. Identify and describe the following items:
 Measure: a measure is a unit of time in music.
 Time signature: a fraction that tells what the total value of the notes in each measure. This time signature (which is read as “four-four”)
states that notes in a whole measure should be worth 4/4 (or one whole). Today, we will only work with the 4/4 time signature.
 Whole note: a whole note is held for the whole measure.
 Half note: a half note is held for 1/2 of the whole measure.
 Quarter note: a quarter note is held for 1/4 of the whole measure.
 Eighth note: the eighth note is held for 1/8 of the whole measure.
Ask: How many half notes must be played to equal one whole measure? How many quarter notes must be played to equal one whole measure?
How many whole notes must be played to equal one whole measure? How many eighth notes must be played to equal one whole measure?
How can use mathematics to prove your answers? (e.g., ½ + ½ = 1)
Show the Composer Warm Up and have students justify whether or not the notes in each measure equal one whole.
STAGE TWO: ELABORATE
Explain: Today you will compose your own music. You can use as many different types of notes in a single measure as you want. Just be sure that the
notes in each measure equal one whole measure. As you noticed on the Composer Warm Up, notes can be placed on any line or space between two
lines. The higher a note, the higher its pitch. Once you have composed your piece of music, check your accuracy using mathematics; record your
work. Then, play your composition at http://www.classicsforkids.com/games/compose/compose.html. Here, you can check to make sure that the
notes in each of your measures equal one whole measure. In addition, you can listen to your music and change the pitch of the notes to suit you.
Once all students have completed pieces, have the whole group listen to each piece. Then, engage in a whole group conversation. Ask students who
decided to put more than one type of note in a single measure to share strategies for proving that the notes were equal to one whole measure. Have
students justify their solutions using mathematics (including equation, words, and models). Ask others to add on to the strategies, determine if they
make sense, and critique each other’s reasoning.
STAGE THREE: EVALUATE
Students should be assessed based on their hand-written composition. Using the written composition, students should be able to articulate why each
measure is correct and explain strategies for placing notes in each measure (e.g., explain why placing four eighth notes and two quarter notes in a single
measure would be correct). Additionally, students should be assessed as they justify thinking and critique other’s reasoning in the whole group conversation.
TEACHER NOTES: If students do not opt to use a variety of notes in a single measure, encourage them to do so in at least one-two measures of their
arrangement. It may also help to require students to include a given number of measures to ensure an adequate and appropriate challenge for them.
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project