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Introduction
Project
Description:
Students will spend the length of this project
exploring music in ways they never have before. Students will learn how
to read music traditionally as well as with mathematics. After learning
the mathematical concepts of permutations, translations and symmetry,
students will apply this knowledge to develop methods to notate the
patterns they find in the music. They will then analyze these findings.
Through this analysis, the mathematics that lives beneath the surface
of music will be discovered.
Driving
Question:
Is there really math in music?
Goals :
- Students will be able to apply their understand
of permutation, translations and symmetry to music
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- Students will discover the mathematics that
drives music
- Engage students in mathematics
Project
Objectives:
- define permutation·
- define translation·
- define symmetry
- define set
- apply concept of permutation to set theory
- apply concept of translation to set theory
- apply concept of symmetry to set theory
- develop a method of notation
- analyze various sets for patterns
Rationale:
Take a quick poll of the people walking down any
street in America and you will find that a disheartening proportion of
our citizens view mathematics as impossibly difficult, dry, and the
source of anxiety in their educational experiences. Those who have
pursued their mathematical studies past algebra, trigonometry, and
calculus know that it is in fact a highly creative and graceful art!
While future engineers may thrive in the algorithmic setting of a
standard high school mathematics course, most others will begin to
develop a life-long aversion to the subject.
This problem is more significant than it first may
seem. Studies have shown that mathematical aptitude often coincides
with a creative, artistic aptitude. Unfortunately, high school
mathematics course content and project topics typically are geared
towards computation and scientific applications. While many students
enjoy such activities, a significant student population – including
many of those who may excel in higher-level mathematics – can fail to
be engaged by such activities.
In this four-week project, designed for use in a
Mathematical Models with Applications course, students will not only be
exposed to elegant mathematics typically reserved for upper-division
and graduate level college students, but will also be given the
opportunity to explore intrinsic connections between mathematics and
music composition. It is reasonable to propose that most high school
students have an interest, at any level, in music. This interest will
be utilized as students seek out examples of set theory, combinatorics,
probability, and symmetry in the compositions of Steve Reich and begin
to see that mathematics can be found in unexpected places.
Background:
1 -2 pages of background info (content specific)
Standards
addressed:
TEKS addressed:
Geometry
(a) Basic understandings.
(3) Geometric figures and their properties. Geometry consists of the
study
of geometric figures of zero, one, two, and three dimensions and the
relationships among them. Students study properties and relationships
having to do with size, shape, location, direction, and orientation of
these figures.
(4)
The relationship between geometry, other mathematics, and other
disciplines. Geometry can be used to model and represent many
mathematical and real-world situations. Students perceive the
connection between geometry and the real and mathematical worlds
and use geometric ideas, relationships, and properties to solve
problems.
(5) Tools for geometric thinking. Techniques for working with spatial
figures and their properties are essential in understanding underlying
relationships. Students use a variety of representations (concrete,
pictorial, algebraic, and coordinate), tools, and technology,
including,
but not limited to, powerful and accessible hand-held calculators and
computers with graphing capabilities to solve meaningful problems by
representing figures, transforming figures, analyzing relationships,
and
proving things about them.
(b) Geometric structure: knowledge and skills and performance
descriptions.
(2) The student analyzes geometric relationships in order to make and
verify conjectures. Following are performance descriptions.
(A) The student uses constructions to explore attributes of geometric
figures and to make conjectures about geometric relationships.
(c) Geometric
patterns: knowledge and skills and performance descriptions.
The student identifies, analyzes, and describes patterns that emerge
from two- and three-dimensional geometric figures. Following are
performance descriptions.
(2) The student uses properties of transformations
and their compositions to make connections between mathematics and the
real world in applications such as tessellations or fractals.
Algebra:
(b) Foundations
for functions: knowledge and skills and performance
descriptions.
(1) The student understands that a function represents a dependence of
one quantity on another and can be described in a variety of ways.
Following are performance descriptions.
(B) The student gathers and records data, or uses data sets, to
determine functional (systematic) relationships between quantities.
NCTM Standards:
Apply transformations and use symmetry to analyze mathematical
situations
- understand and represent translations,
reflections, rotations, and dilations of objects in the plane by using
sketches, coordinates, vectors, function notation, and matrices;
- use various representations to help understand
the effects of simple transformations and their compositions.
Assessment
Students will be assessed throughout the course
through question and answer sessions. Class discussions will be held to
address any concerns the students may have. During lab days the teacher
will walk around to students and address their progress. The students
will be given quizzes over topics they have learned at the beginning of
the project. The intermediate assessment will allow students to
critique each other. This assessment will allow the teacher to make
sure the student is on the right track and redirect them if necessary.
The main assessment will come from their final presentation at the
conclusion of the project. They will present their findings at this
time and the teacher will be able to assess the students understanding
as well as their ability to stay within the parameters of the project.
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