Name(s): Emily Buschang
Title of Lesson: Symmetry, Rotation, Permutations, Translations with Sets in Music
Source
of Lesson:
http://www.utc.edu/Faculty/Christopher-Mawata/transformations/translations/
Length of Lesson: Two 55 minutes (2 class periods)
Description of the Class: High School Math Students – After-school Program
TEKS
Address
Algebra
I:
(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.
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.
(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.
Students will use their understanding of symmetry, rotation, permutations, and translations in geometry and apply it to sets. Students will then compare sets and music.
The student will be able to:
¨ Identify symmetry in sets.
¨ Identify the translation function needed to get from one set to another.
¨ Identify the various permutations possible given a set.
¨ Look at music as sets as label it as such.
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Engage: Time:
10 minutes Go to website and play with applets. http://www.utc.edu/Faculty/Christopher-Mawata/transformations/translations/
Ask the students the following:
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Explore: Time: 30 minutes Tell students that they are to come up with a (nontrivial) set of size five that has rotational symmetry. Students will already be familiar with sets, permutations and translations from lessons earlier in the week. Instruct students to trade their set with the set their neighbor came up with. Draw a triangle of the board. Label vertices A, B and C. Do examples of permutations of vertices so students are able to visualize that it. Instruct
students to find all permutations of size n=3 of the set. Have students trade their set with a different neighbor. Ask students to check the permutations that were found and correct any errors they found. They are then to come up with a translation of the set. (Do not write the relationship on the paper) Ask students to trade back with the last person they traded with. Allow time for students to discuss any disagreements they have about the permutations. Have students figure out what the relationship was for the translation. After a few minutes, have them check back with their neighbor to see if they are correct, or if they cannot figure out the relationship have the students explain it to each other.
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Students listen to instructions.
Students trade paper. Students find permutations of size n=3. Students trade papers. Students check permutations. Students come up with a translation. Students trade papers and discuss possible disagreements about permutations.
Students figure out what the translation relationship is. They check their solution with their neighbor. |
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Explain: Time: 15 minutes Discuss any common problems with exploration. |
Students discuss drawings with the class. |
End Day One of Two Day Lesson
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Engage: Time:
5 minutes Review definitions of Symmetry, Permutation and Translations. Address and student questions/concerns. |
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Extend/Elaborate: Time:
30 minutes Music is made up of notes. The notes have a finite high and low
value. Write two measures on the
board.
i) A B D G and D G A B ii) A B D G and G D B A iii)
A B D G and C D F B Ask Students: What do you notice about i, ii, and iii? How
can you make music look like a set in the sense that we normally think about
it? How do you decide what number or letter to label the note? So what does it mean if music is really just a set? |
¨ (i) has rotated by two notes. (ii) if backwards, or reflected. (iii) has been translated up two notes. ¨ Label each note with a letter or a number. ¨ It doesn’t really matter as long as you’re consistent. ¨ We can look at music and figure out if it has any kind of symmetry or if it is a translation or a permutation of itself.
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Evaluate: 20
minutes Have students come up with pieces of music
that are ten to twenty measures long.
It should contain instances of rotations, symmetry and
translations. Have them justify the
method they are using to label the notes in the music.
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Compose short pieces of music. |