6. Kaleidoscopes, Hubcaps, and Mirrors - CMS Goals

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Understanding important properties of symmetry.

Recognize and describe symmetries of figures.

Use tools to examine symmetries and transformations.

Create figures with specified symmetries

Identify basic design elements that can be used to replicate a given design.

Perform symmetry transformations of figures, including reflections, translations, and rotations.

Give precise mathematical directions for performing reflections, rotations, and translations.

Write coordinate rules for specifying the image of a general point (x,y) under particular transformations.

Combine transformations and find a single transformation that will produce the same result.

Find the symmetries of geometric figures and make tables showing the results of combining symmetry transformations.

Learn to appreciate the power of transformational geometry to describe motions, patterns and designs in the real world.

 

1.     3 Types of Sym

    1.1   Reflection Symmetry

   1.2   Rotational Symmetry

   1.3   Sym in Kaleidoscope Des

   1.4   Translational Symmetry

 

To explore reflectional, rotational, and translational symmetry informally.

To explore the use of tools, such as tracing paper, to analyze designs to determine their symmetries.

To design shapes that have specified symmetries.

To identify basic design elements that can be used to replicate a design

 

2. Symmetry Transformations

    2.1   Describing Line Reflections

   2.2  Describing Translations

   2.3  Describing Rotations

   2.4  Combining Translations

        

 

To examine reflections, translations, and rotations to determine how to specify such transformational precisely.

To use the properties of reflections, translations, and rotations to perform transformations.

To find lines of reflections, magnitudes and directions of translations and centers and angles of rotation.

To examine the results of combining reflections over two intersecting lines or two parallel lines; two translations; or two rotations to find single a transformation that with produce the same result

3.    Transforming Coordinates 

    3.1   Writing Rules for Reflections

   3.2  Writing Rules for Translations

   3.3  Writing Rules for Rotations

   3.4  Relating Symmetry to Congruence

 

To write directions for drawing figures composed of line segments.

To analyze the vertices of a figure under a transformation and to specify translations with coordinate rules.

To recognize that a transformation of the form (x,y) Ð> (x+a, y+b) is a translation of point (x,y) a units in the x directions and b units in the y directions.

To specify rotations of 90¡, 180¡, 270¡, & 360¡ with coordinate rules.

To specify reflections over the x-axis, the y-axis and the line y=x

To combine transformations to find single, equivalent transformations.

To understand the relationship between symmetry transformations and congruence.

 

4. Symmetry & Algebra 

   4.1   Modeling Real-Life Events

   4.2  Simulating Cookies

   4.3  Exploring Graphs

            

 

To determine the possible symmetry transformations for a given polygon.

To construct a table showing all possible results of combining two symmetry transformations of a given polygon.

To analyze such a table to determine whether (1) there is an identity element for the "and then" operations, (2) each element has an inverse for the "and then" operation, and (3) the "and then" operation is commutative.