**Title:** Describing Rotations

**Description:**
In this lesson, one of a multi-part unit from Illuminations, students learn how to describe rotations of a figure. They then use an interactive Java applet to investigate the effect of rotations through different angles and on different shapes.
**Standard(s): **

[MA2015] GEO (9-12) 5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. [G-CO5]

**Subject:**Mathematics

**Title:**Describing Rotations

**Description:**In this lesson, one of a multi-part unit from Illuminations, students learn how to describe rotations of a figure. They then use an interactive Java applet to investigate the effect of rotations through different angles and on different shapes.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Symmetries II: Conclusions

**Description:**
In this lesson, one of a multi-part unit from Illuminations, students reflect on what they learned in the three previous lessons. Eight thought questions (with a link to the answers) are provided.
**Standard(s): **

[MA2015] MI1 (9-12) 9: Analyze works of visual art and architecture for mathematical relationships. (Alabama)

**Subject:**Mathematics

**Title:**Symmetries II: Conclusions

**Description:**In this lesson, one of a multi-part unit from Illuminations, students reflect on what they learned in the three previous lessons. Eight thought questions (with a link to the answers) are provided.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Covering the Plane with Rep-Tiles

**Description:**
In this lesson, from Illuminations, students discover and explore a special kind of tiling of the plane. Rep-tiles are geometric figures such that n copies can fit together to form a larger, similar figure. Students experiment with various shapes and values of n. Spatial sense is encouraged by the need to visualize and perform transformations with the shapes involved.
**Standard(s): **

[MA2015] GEO (9-12) 16: Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. [G-SRT3]

**Subject:**Mathematics

**Title:**Covering the Plane with Rep-Tiles

**Description:**In this lesson, from Illuminations, students discover and explore a special kind of tiling of the plane. Rep-tiles are geometric figures such that n copies can fit together to form a larger, similar figure. Students experiment with various shapes and values of n. Spatial sense is encouraged by the need to visualize and perform transformations with the shapes involved.

**Thinkfinity Partner:**Illuminations

**Grade Span:**3,4,5,6,7,8

**Title:** Describing Reflections

**Description:**
In this lesson, one of a multi-part unit from Illuminations, students learn how reflections work and what happens when two or more reflections are applied one after the other. They use interactive Java applets to examine the reflection of a point and how to describe reflections.
**Standard(s): **

[MA2015] GEO (9-12) 6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. [G-CO6]

**Subject:**Mathematics

**Title:**Describing Reflections

**Description:**In this lesson, one of a multi-part unit from Illuminations, students learn how reflections work and what happens when two or more reflections are applied one after the other. They use interactive Java applets to examine the reflection of a point and how to describe reflections.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Polygons

**Description:**
This reproducible activity sheet, from an Illuminations lesson, features line drawings of several polygons including squares, triangles, rectangles, parallelograms, and trapezoids.
**Standard(s): **

[MA2015] GEO (9-12) 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [G-CO3]

**Subject:**Mathematics

**Title:**Polygons

**Description:**This reproducible activity sheet, from an Illuminations lesson, features line drawings of several polygons including squares, triangles, rectangles, parallelograms, and trapezoids.

**Thinkfinity Partner:**Illuminations

**Grade Span:**6,7,8

**Title:** Symmetries I

**Description:**
In this unit of four lessons, from Illuminations, investigate rotational symmetry. They learn about the mathematical properties of rotations and have an opportunity to make their own designs.
**Standard(s): **

[MA2015] GEO (9-12) 14: Verify experimentally the properties of dilations given by a center and a scale factor. [G-SRT1]

**Subject:**Mathematics

**Title:**Symmetries I

**Description:**In this unit of four lessons, from Illuminations, investigate rotational symmetry. They learn about the mathematical properties of rotations and have an opportunity to make their own designs.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Symmetries II

**Description:**
In this unit of four lessons, from Illuminations, students use Java applets to investigate reflection, mirror, or bilateral symmetry. They learn about the mathematical properties of mirror symmetry and have a chance to create designs with mirror symmetry.
**Standard(s): **

[MA2015] GEO (9-12) 16: Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. [G-SRT3]

**Subject:**Mathematics

**Title:**Symmetries II

**Description:**In this unit of four lessons, from Illuminations, students use Java applets to investigate reflection, mirror, or bilateral symmetry. They learn about the mathematical properties of mirror symmetry and have a chance to create designs with mirror symmetry.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Symmetries III

**Description:**
This lesson, from Illuminations, helps students to understand how translations work and what happens when two or more translations are applied one after the other. Students discover that all band ornaments have translational symmetry and all wallpaper patterns have translational symmetry in at least two directions.
**Standard(s): **

[MA2015] MI1 (9-12) 9: Analyze works of visual art and architecture for mathematical relationships. (Alabama)

**Subject:**Mathematics

**Title:**Symmetries III

**Description:**This lesson, from Illuminations, helps students to understand how translations work and what happens when two or more translations are applied one after the other. Students discover that all band ornaments have translational symmetry and all wallpaper patterns have translational symmetry in at least two directions.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Symmetries IV

**Description:**
This lesson, from Illuminations, helps students to understand and identify glide reflections. With the help of a Java applet, students construct glide reflections using a translation and a reflection. Students then identify glide reflections from groups of band ornaments and wallpaper patterns.
**Standard(s): **

[MA2015] GEO (9-12) 12: Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. *Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.* [G-CO12]

**Subject:**Mathematics

**Title:**Symmetries IV

**Description:**This lesson, from Illuminations, helps students to understand and identify glide reflections. With the help of a Java applet, students construct glide reflections using a translation and a reflection. Students then identify glide reflections from groups of band ornaments and wallpaper patterns.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Recognizing Transformations

**Description:**
This lesson introduces students to the world of symmetry and rotation in figures and patterns. Students learn how to recognize and classify symmetry in decorative figures and frieze patterns, and get the chance to create and classify their own figures and patterns using JavaSketchpad applets.
**Standard(s): **

[MA2015] GEO (9-12) 5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. [G-CO5]

**Subject:**Arts,Mathematics

**Title:**Recognizing Transformations

**Description:**This lesson introduces students to the world of symmetry and rotation in figures and patterns. Students learn how to recognize and classify symmetry in decorative figures and frieze patterns, and get the chance to create and classify their own figures and patterns using JavaSketchpad applets.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Classifying Transformations

**Description:**
Students will identify and classify reflections and symmetries in figures and patterns. They will also have the opportunity to create frieze patterns from each of the seven classes using the supplemental on-line activities.
**Standard(s): **

[MA2015] ALC (9-12) 9: Analyze aesthetics of physical models for line symmetry, rotational symmetry, or the golden ratio. (Alabama)

**Subject:**Arts,Mathematics

**Title:**Classifying Transformations

**Description:**Students will identify and classify reflections and symmetries in figures and patterns. They will also have the opportunity to create frieze patterns from each of the seven classes using the supplemental on-line activities.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Pinwheel

**Description:**
In this Illuminations lesson, students create parallelograms from square sheets of paper and connect them to form an octagon. During the construction, students consider angle measures, segment lengths, and areas in terms of the original square. At the end of the lesson, the octagon is transformed into a pinwheel, and students discover a surprising result. There is also a link to a related activity sheet.
**Standard(s): **

[MA2015] GEO (9-12) 34: Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics. (Alabama)

**Subject:**Mathematics

**Title:**Pinwheel

**Description:**In this Illuminations lesson, students create parallelograms from square sheets of paper and connect them to form an octagon. During the construction, students consider angle measures, segment lengths, and areas in terms of the original square. At the end of the lesson, the octagon is transformed into a pinwheel, and students discover a surprising result. There is also a link to a related activity sheet.

**Thinkfinity Partner:**Illuminations

**Grade Span:**6,7,8,9,10,11,12

**Title:** Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Visualizing Transformations

**Description:**
The interactive figures in this four-part example from Illuminations allow a user to manipulate a shape and observe its behavior under a particular transformation or composition of transformations. e-Math Investigations are selected e-examples from the electronic version of the Principles and Standards of School Mathematics (PSSM). The e-examples are part of the electronic version of the PSSM document. Given their interactive nature and focused discussion tied to the PSSM document, the e-examples are natural companions to the i-Math investigations.
**Standard(s): **

[MA2015] GEO (9-12) 12: Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. *Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.* [G-CO12]

**Subject:**Mathematics

**Title:**Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Visualizing Transformations

**Description:**The interactive figures in this four-part example from Illuminations allow a user to manipulate a shape and observe its behavior under a particular transformation or composition of transformations. e-Math Investigations are selected e-examples from the electronic version of the Principles and Standards of School Mathematics (PSSM). The e-examples are part of the electronic version of the PSSM document. Given their interactive nature and focused discussion tied to the PSSM document, the e-examples are natural companions to the i-Math investigations.

**Thinkfinity Partner:**Illuminations

**Grade Span:**6,7,8

**Title:** Shape Cutter

**Description:**
This student interactive, from Illuminations, allows students to explore how they can decompose shapes and recompose them to make other shapes. Students draw and cut shapes, and also use slides, turns, and flips to move the cut pieces around.
**Standard(s): **

[MA2015] GEO (9-12) 5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. [G-CO5]

**Subject:**Mathematics

**Title:**Shape Cutter

**Description:**This student interactive, from Illuminations, allows students to explore how they can decompose shapes and recompose them to make other shapes. Students draw and cut shapes, and also use slides, turns, and flips to move the cut pieces around.

**Thinkfinity Partner:**Illuminations

**Grade Span:**K,PreK,1,2,3,4,5,6,7,8,9,10,11,12

**Title:** Cyclic Figures

**Description:**
Students will recognize rotation symmetry in figures and examining various rotation symmetries.
**Standard(s): **

[MA2015] GEO (9-12) 6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. [G-CO6]

**Subject:**Mathematics

**Title:**Cyclic Figures

**Description:**Students will recognize rotation symmetry in figures and examining various rotation symmetries.

**Thinkfinity Partner:**Illuminations

**Grade Span:**6,7,8

**Title:** Dihedral Figures

**Description:**
Students will recognize dihedral symmetry and reflections in figures and examining various symmetries.
**Standard(s): **

[MA2015] GEO (9-12) 15: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. [G-SRT2]

**Subject:**Mathematics

**Title:**Dihedral Figures

**Description:**Students will recognize dihedral symmetry and reflections in figures and examining various symmetries.

**Thinkfinity Partner:**Illuminations

**Grade Span:**6,7,8,9,10,11,12

**Title:** Square Graphs

**Description:**
This student interactive, from an Illuminations lesson, allows students to view six different graphs. Each graph plots one characteristic of the square along the horizontal axis and another along the vertical axis. Students then determine which two characteristics are used to make each graph.
**Standard(s): **

[MA2015] GEO (9-12) 14: Verify experimentally the properties of dilations given by a center and a scale factor. [G-SRT1]

**Subject:**Mathematics

**Title:**Square Graphs

**Description:**This student interactive, from an Illuminations lesson, allows students to view six different graphs. Each graph plots one characteristic of the square along the horizontal axis and another along the vertical axis. Students then determine which two characteristics are used to make each graph.

**Thinkfinity Partner:**Illuminations

**Grade Span:**9,10,11,12

**Title:** Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Identifying Unknown Transformations

**Description:**
**Standard(s): **

[MA2015] GEO (9-12) 12: Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. *Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.* [G-CO12]

**Subject:**Mathematics

**Title:**Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Identifying Unknown Transformations

**Description:**

**Thinkfinity Partner:**Illuminations

**Grade Span:**6,7,8

**Title:** Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Composing Reflections

**Description:**
This is part three of a four-part e-example from Illuminations that features interactive figures that allow a user to manipulate a shape and observe its behavior under a particular transformation or composition of transformations. In this part, Composing Reflections, users can examine the result of reflecting a shape successively through two different lines. e-Math Investigations are selected e-examples from the electronic version of the Principles and Standards for School Mathematics (PSSM). Given their interactive nature and focused discussion tied to the PSSM document, the e-examples are natural companions to the i-Math Investigations.
**Standard(s): **

[MA2015] GEO (9-12) 12: Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software.

**Subject:**Mathematics

**Title:**Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Composing Reflections

**Description:**This is part three of a four-part e-example from Illuminations that features interactive figures that allow a user to manipulate a shape and observe its behavior under a particular transformation or composition of transformations. In this part, Composing Reflections, users can examine the result of reflecting a shape successively through two different lines. e-Math Investigations are selected e-examples from the electronic version of the Principles and Standards for School Mathematics (PSSM). Given their interactive nature and focused discussion tied to the PSSM document, the e-examples are natural companions to the i-Math Investigations.

**Thinkfinity Partner:**Illuminations

**Grade Span:**6,7,8

**Title:** Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Composing Transformations

**Description:**
This is part four of a four-part e-example from Illuminations that features interactive figures that allow a user to manipulate a shape and observe its behavior under a particular transformation or composition of transformations. In this part, Composing Transformations, the users are challenged to compose equivalent transformations in two different ways. e-Math Investigations are selected e-examples from the electronic version of the Principles and Standards for School Mathematics (PSSM). Given their interactive nature and focused discussion tied to the PSSM document, the e-examples are natural companions to the i-Math Investigations.
**Standard(s): **

[MA2015] GEO (9-12) 12: Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software.

**Subject:**Mathematics

**Title:**Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Composing Transformations

**Description:**This is part four of a four-part e-example from Illuminations that features interactive figures that allow a user to manipulate a shape and observe its behavior under a particular transformation or composition of transformations. In this part, Composing Transformations, the users are challenged to compose equivalent transformations in two different ways. e-Math Investigations are selected e-examples from the electronic version of the Principles and Standards for School Mathematics (PSSM). Given their interactive nature and focused discussion tied to the PSSM document, the e-examples are natural companions to the i-Math Investigations.

**Thinkfinity Partner:**Illuminations

**Grade Span:**6,7,8