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 Instructional Focus DocumentMathematical Models with Applications
 TITLE : Unit 05: Applications of Algebra and Geometry in Music SUGGESTED DURATION : 10 days

Unit Overview

This unit bundles student expectations that address modeling music with period functions and analyzing patterns and structure in music using transformations and proportions. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this unit, in Grade 6, students used scale factors involving ratios and rates to solve problems. In Grades 7 and 8, students studied proportionality and proportional change. In Algebra I, students analyzed the characteristics and transformations of the parent functions. In Geometry, students studied trigonometric ratios.

During this unit, students investigate music using trigonometric functions (sine function) to model periodic behavior. Students connect periodic motion to sound, especially musical notes, using technology. Periodic models are analyzed to determine volume (amplitude) modeled by a vertical dilation and pitch (frequency) by a horizontal dilation. Students examine a periodic sound wave of a note to determine the period of the note and calculate the reciprocal of the period to determine the frequency. Students identify the period and frequency of a note as an inverse proportional relationship. Students collect real data using technology to model periodic motion of music and analyze the characteristics of the graph in terms of music. Students examine representative pieces of music to determine the relationship between rhythm, beats per note, and the number of beats per measure using the time signature. Students transpose (translated by proportions) frequency up or down for a given interval (e.g., a third, a fifth, an octave, etc.).

After this unit, in Algebra II and Precalculus, students will use patterns in transformations of functions to solve problems. Throughout Math Models with Applications, students will be required to take given information or collected data and determine tools and methods needed to solve the problem situation. The concepts in this unit will be applied in subsequent mathematics courses.

This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning B1; III. Geometric Reasoning A3, B1, B2, B3; IV. Measurement Reasoning C3; VII. Functions B1, C1; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the Connections Standard for Grades 9-12 from the National Council of Teachers of Mathematics (NCTM), “Instructional programs from pre-kindergarten through grade 12 should enable students to:

• recognize and use connections among mathematical ideas;
• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• recognize and apply mathematics in contexts outside of mathematics.

When students can see the connections across different mathematical content areas, they develop a view of mathematics as an integrated whole. As they build on their previous mathematical understandings while learning new concepts, students become increasingly aware of the connections among various mathematical topics. As students' knowledge of mathematics, their ability to use a wide range of mathematical representations, and their access to sophisticated technology and software increase, the connections they make with other academic disciplines, especially the sciences and social sciences, give them greater mathematical power” (NCTM, 2000, p. 354).

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics: Connections standard for grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.

OVERARCHING UNDERSTANDINGS and QUESTIONS

Relationships can be used to describe real-world patterns.

• Why is it important to describe the relationships found in numeric patterns?
• What relationships can be found in patterns?

Functions can be classified into different families with each function family having its own unique graphs, attributes, and relationships.

• Why are functions classified into families of functions?
• How are functions classified as a family of functions?
• What graphs, key attributes, and characteristics are unique to each family of functions?
• What patterns of covariation are associated with the different families of functions?
• How are the parent functions and their families used to model real-world situations?

Proportional reasoning can be used to describe and solve problems in everyday life.

• Why can proportional reasoning be used to make predictions and comparisons in problem situations?
• How are ratios used in a proportional relationship?

Transformation(s) of a parent function create a new function within that family of functions.

• Why are transformations of parent functions necessary?
• How do transformations affect a function?
• How can transformations be interpreted from various representations?

Function models for problem situations can be determined by collecting and analyzing data using a variety of representations and applied to make predictions and critical judgments in terms of the problem situation.

• Why is it important to determine and apply function models for problem situations?
• What representations can be used to analyze collected data and how are the representations interrelated?
• Why is it important to analyze various representations of data when determining appropriate function models for problem situations?
• How do the key attributes and characteristics of the function differ from the key attributes and characteristics of the function model for the problem situation?
• How does technology aid in the analysis and application of modeling and solving problem situations?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Patterns/Rules
• Ratios/Rates

Function

• Non-Linear Functions
• Proportional Relationships

Geometric Reasoning

• Coordinate Planes
• Geometric Attributes/Properties
• Scale Factors
• Transformations

Measurement Reasoning

• Length

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Graphs of trigonometric functions model qualities of music such as pitch (frequency) and volume (amplitude).

• What are trigonometric functions and how are they used to model periodic motion in fields such as music?
• What are the characteristics of trigonometric functions and their graphs?
• How are trigonometric functions used to find the pitch of a musical note?
• What characteristic of the function’s graph determines pitch?
• How are trigonometric functions used to find the volume of a musical note?
• What characteristic of the function’s graph determines volume?

Transformations are used to demonstrate changes in periodic functions.

• What kind of transformation is demonstrated by a change in the pitch of the sound?
• What kind of transformation in the periodic function is demonstrated by a change in the volume of the sound?

The period and frequency of sound waves are inversely proportional.

• How can the graph of a sound wave be used to determine the period?
• How can the graph of a sound wave be used to determine the frequency?
• How are the period and frequency of a sound wave related?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Patterns/Rules
• Ratios/Rates

Function

• Non-Linear Functions
• Proportional Relationships

Geometric Reasoning

• Geometric Attributes/Properties
• Scale Factors
• Transformations

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

The rhythm in a piece of music is expressed through a time signature which defines the number of beats per measure and the type of note that gets one full beat.

• What are the relationships between the different types of notes such as a whole note, half note, quarter note, and eighth note?
• How is the time signature represented on a piece of music?
• How does the time signature distinguish which type of note gets assigned one full beat?
• How does the time signature determine the number of beats per measure?
• How does knowing the type of note that gets one beat determine the number of beats for the remaining type of notes?

A proportional relationship exists between the different frequencies of musical notes.

• What relationship exists between notes with different frequencies?
• How can the interval between two different frequencies be defined?
• How is a frequency transposed (translated by proportions) up or down for a given interval (e.g., a third, a fifth, an octave, etc.)?

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• When studying a graph of sound or music, students may relate the amplitude to the pitch instead of to the volume, and students may relate the frequency to the volume instead of to the pitch.
• Students may confuse period and frequency when analyzing periodic graphs of sound and have difficulty describing the sound that created the graphical representation.

Unit Vocabulary

• Amplitude – refers to half the difference between the minimum and maximum y-values on the graph of the trigonometric function. A change in amplitude is a vertical dilation.
• Dilation – similarity transformation in which a figure is enlarged or reduced using a scale factor and a center of dilation
• Frequency of a sound wave – the number of cycles (completed patterns) per second with seconds (time) measured along the x-axis
• Horizontal (phase) shift –  translation of the graph of a periodic function along the x-axis
• Measure of music – a division of sets of notes in music defined by the time signature
• Period – the distance along the x-axis over which a periodic function completes one cycle of its pattern
• Sound waves – disturbance pattern caused by the movement of energy through air, liquid, or solid
• Time signature – numerical symbol seen at the beginning of a piece of music
• Trigonometric functions – a function of an angle expressed as the ratio of two of the sides of a right triangle that contain that angle

Related Vocabulary:

 Bilateral symmetry Chords Cycle Harmony Horizontal dilation Periodic Pitch Proportional change   Ratio Rhythm Scale factor Sine Tone Vertical dilation Volume
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Mathematical Models with Applications Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
M.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
M.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
M.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
M.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
M.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
M.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION
Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.7 Mathematical modeling in fine arts. The student uses mathematical processes with algebra and geometry to study patterns and analyze data as it applies to fine arts. The student is expected to:
M.7A

Use trigonometric ratios and functions available through technology to model periodic behavior in art and music.

Use

TRIGONOMETRIC FUNCTIONS AVAILABLE THROUGH TECHNOLOGY TO MODEL PERIODIC BEHAVIOR IN MUSIC

Including, but not limited to:

• Trigonometric functions – a function of an angle expressed as the ratio of two of the sides of a right triangle that contain that angle
• Symmetry in trigonometric functions
• Amplitude – refers to half the difference between the minimum and maximum y-values on the graph of the trigonometric function. A change in amplitude is a vertical dilation.
• Vertical stretch or compression of the trigonometric function
• Music amplitude determines volume. The larger the amplitude of the graph, the louder the sound.
• Horizontal (phase) shift – translation of the graph of a periodic function along the x-axis
• Period – the distance along the x-axis over which a periodic function completes one cycle of its pattern
• A change in the period of a periodic funciton is a horizontal dilation of the graph of the function.
• The period is expressed as x units per cycle.
• Frequency of a sound wave is the number of cycles (completed patterns) per second with seconds (time) measured along the x-axis.
• Music frequencies are determined by the reciprocal of the period.

Note(s):

• Geometry studied trigonometric ratios.
• Precalculus will study trigonometric ratios and functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions.
• III. Geometric reasoning
• A3 – Recognize and apply right triangle relationships including basic trigonometry.
• B1 – Identify and apply transformations to figures.
• B2 – Identify the symmetries of a plane figure.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
M.7C Use geometric transformations, proportions, and periodic motion to describe mathematical patterns and structure in music.

Use

GEOMETRIC TRANSFORMATIONS, PROPORTIONS, AND PERIODIC MOTION

Including, but not limited to:

• Translations, dilations, and reflections used in repeated patterns of notes to create melodies
• Periodic motion related to sound waves
• Ratios and proportions within frequencies of tones
• Ratios and proportions related to rhythm, time duration for different types of notes in music

To Describe

MATHEMATICAL PATTERNS AND STRUCTURE IN MUSIC

Including, but not limited to:

• Rhythm
• Time signature – numerical symbol seen at the beginning of a piece of music
• Time signature is written like a fraction with one number over another where the upper number represents the number of beats in a measure and the lower number represents the type of note that receives one beat.
• Measure of music – a division of sets of notes in music defined by the time signature
• Measure marks the divisions between sets of notes that complete the number of beats specified in the time signature.
• Measures are separated on the music by vertical lines through the staff from top to bottom, called bar lines.
• Ratios and proportions of note duration values
• Relationships between the different notated notes with different time durations as specified by the time signature.
• Beginning with the whole note, each different subsequent note has a duration that is half the one before it.
• A quarter note has a duration  that of a half note and a duration of  that of the whole note. This pattern continues for all the different types of notes.
• Chords and harmonies
• Frequencies
• Intervals
• Ratios and proportions
• Sound waves – disturbance pattern caused by the movement of energy through air, liquid, or solid
• Sound can be modeled by a trigonometric (circular, periodic) function, most commonly f(x)= sin(x).
• Sound model graphs demonstrate symmetry and transform geometrically through translation, reflection, and/or dilation as the attributes of the sound changes.

Note(s):

• Grades 6 and 7 studied proportions.
• Precalculus will address periodic behavior trigonometric functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions.
• III. Geometric reasoning
• B1 – Identify and apply transformations to figures.
• B2 – Identify the symmetries of a plane figure.
• B3 – Use congruence transformations and dilations to investigate congruence, similarity, and symmetries of plane figures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections