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Title: Using Exploration and Discovery To Determine the Vertices of Transformed Images
Description:
This handson lesson uses exploration and discovery to help students determine the coordinates of the vertices of various figures when translated, rotated, or reflected in a coordinate plane. The exploration phase will access students' prior knowledge and prepare them for the assigned task of using and applying their understanding of transformed images to new situations. The final project will be a cooperative activity, requiring collaboration in the application phase to discover the patterns and create graphs following directions unique to each group.
Standard(s): [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3]
Subject: Mathematics (8)
Title: Using Exploration and Discovery To Determine the Vertices of Transformed Images
Description: This handson lesson uses exploration and discovery to help students determine the coordinates of the vertices of various figures when translated, rotated, or reflected in a coordinate plane. The exploration phase will access students' prior knowledge and prepare them for the assigned task of using and applying their understanding of transformed images to new situations. The final project will be a cooperative activity, requiring collaboration in the application phase to discover the patterns and create graphs following directions unique to each group.
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Title: Geometric Quilts
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Students will use a square to investigate other polygonal shapes that can be formed. Students will work with these polygons to make designs that will later constitute cooperatively designed quilts. The quilts will be made by translating, rotating, or reflecting a thirtysix patch design. The completed quilts will then be used for a discussion of area and perimeter. Students will blog about what they have learned while doing the lesson and will include pictures of their finished product in our class blog.
Standard(s): [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3] [MA2013] GEO (912) 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. [GCO5] [MA2013] GEO (912) 34: Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics. (Alabama)
Subject: Mathematics (8  12)
Title: Geometric Quilts
Description: Students will use a square to investigate other polygonal shapes that can be formed. Students will work with these polygons to make designs that will later constitute cooperatively designed quilts. The quilts will be made by translating, rotating, or reflecting a thirtysix patch design. The completed quilts will then be used for a discussion of area and perimeter. Students will blog about what they have learned while doing the lesson and will include pictures of their finished product in our class blog.
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Title: Using a Graphing Calculator to Determine Transformation(s)
Description:
Students will use graphing calculators(TI83 Plus) to find transformations of polygons.This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation.
Standard(s): [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3] [MA2013] (8) 19: Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them. [8G4]
Subject: Mathematics (8)
Title: Using a Graphing Calculator to Determine Transformation(s)
Description: Students will use graphing calculators(TI83 Plus) to find transformations of polygons.This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation.
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Title: Geometry City
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In this lesson, students will review coordinate plane concepts, practice map skills, review shapes, transform geometric figures, and calculate the area and perimeter of figures.This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation.
Standard(s): [TC2] (68) 9: Practice responsible and legal use of technology systems and digital content. [TC2] (68) 11: Use digital tools and strategies to locate, collect, organize, evaluate, and synthesize
information. [MA2013] (6) 21: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems. [6G1] [MA2013] (6) 23: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems. [6G3] [MA2013] (7) 16: Solve realworld and mathematical problems involving area, volume, and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. [7G6] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3] [MA2013] (8) 19: Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them. [8G4]
Subject: Mathematics (6  8), or Technology Education (6  8)
Title: Geometry City
Description: In this lesson, students will review coordinate plane concepts, practice map skills, review shapes, transform geometric figures, and calculate the area and perimeter of figures.This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation.
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Title: Creating a Reflection
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In this lesson, the students will create geometric reflections. This is a hands on activity where the student will be able to conceptualize how geometric shapes are reflected across the x and y axis of a coordinate graph. This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation.
Standard(s): [MA2013] (5) 3: Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. [5OA3] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3] [MA2013] (5) 23: Use a pair of perpendicular number lines, called axes, to define a coordinate system with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., xaxis and xcoordinate, yaxis and ycoordinate). [5G1] [MA2013] (5) 24: Represent realworld and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. [5G2] [MA2013] (5) 25: Understand that attributes belonging to a category of twodimensional figures also belong to all subcategories of that category. [5G3] [MA2013] (5) 26: Classify twodimensional figures in a hierarchy based on properties. [5G4]
Subject: Mathematics (5  8)
Title: Creating a Reflection
Description: In this lesson, the students will create geometric reflections. This is a hands on activity where the student will be able to conceptualize how geometric shapes are reflected across the x and y axis of a coordinate graph. This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation.
Thinkfinity Lesson Plans
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Title: Relationships Between Reflections and Symmetry
Description:
In this lesson, one of a multipart unit from Illuminations, students learn the properties of designs that have bilateral symmetry. They also make their own designs and investigate dihedral symmetry.
Standard(s): [MA2013] (8) 16: Verify experimentally the properties of rotations, reflections, and translations: [8G1] [MA2013] (8) 17: Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8G2] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3] [MA2013] (8) 19: Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them. [8G4]
Subject: Mathematics Title: Relationships Between Reflections and Symmetry
Description: In this lesson, one of a multipart unit from Illuminations, students learn the properties of designs that have bilateral symmetry. They also make their own designs and investigate dihedral symmetry. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Symmetries I
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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): [MA2013] (8) 16: Verify experimentally the properties of rotations, reflections, and translations: [8G1] [MA2013] (8) 17: Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8G2] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3] [MA2013] (8) 19: Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them. [8G4] [MA2013] GEO (912) 2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). [GCO2] [MA2013] GEO (912) 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [GCO3] [MA2013] GEO (912) 4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [GCO4] [MA2013] GEO (912) 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. [GCO5] [MA2013] GEO (912) 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. [GCO6] [MA2013] GEO (912) 7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. [GCO7] [MA2013] GEO (912) 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. [GCO12] [MA2013] GEO (912) 14: Verify experimentally the properties of dilations given by a center and a scale factor. [GSRT1]
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
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Title: Symmetries II
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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): [MA2013] (8) 16: Verify experimentally the properties of rotations, reflections, and translations: [8G1] [MA2013] (8) 17: Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8G2] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3] [MA2013] (8) 19: Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them. [8G4] [MA2013] GEO (912) 2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). [GCO2] [MA2013] GEO (912) 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [GCO3] [MA2013] GEO (912) 4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [GCO4] [MA2013] GEO (912) 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. [GCO5] [MA2013] GEO (912) 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. [GCO6] [MA2013] GEO (912) 7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. [GCO7] [MA2013] GEO (912) 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. [GCO12] [MA2013] GEO (912) 14: Verify experimentally the properties of dilations given by a center and a scale factor. [GSRT1] [MA2013] GEO (912) 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. [GSRT2] [MA2013] GEO (912) 16: Use the properties of similarity transformations to establish the angleangle (AA) criterion for two triangles to be similar. [GSRT3]
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
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Title: Go the Distance
Description:
This lesson is the fourth part of a fourpart Illuminations unit titled '' Analyzing Numeric and Geometric Patterns of Paper Pool.'' The interactive paper pool game in this unit provides an opportunity for students to further develop their understanding of ratio, proportion, and least common multiple. At the end of the lesson, students write reports on their findings. This resource is referenced in the Illuminations unit titled '' Analyzing Numeric and Geometric Patterns of Paper Pool'' and is related to the Illuminations lessons titled '' Paper Pool Game,'' '' Explore More Tables,'' '' Look for Patterns.''
Standard(s): [MA2013] (6) 21: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems. [6G1] [MA2013] (6) 24: Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems. [6G4] [MA2013] (8) 16: Verify experimentally the properties of rotations, reflections, and translations: [8G1] [MA2013] (8) 17: Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8G2] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3]
Subject: Language Arts,Mathematics Title: Go the Distance
Description: This lesson is the fourth part of a fourpart Illuminations unit titled '' Analyzing Numeric and Geometric Patterns of Paper Pool.'' The interactive paper pool game in this unit provides an opportunity for students to further develop their understanding of ratio, proportion, and least common multiple. At the end of the lesson, students write reports on their findings. This resource is referenced in the Illuminations unit titled '' Analyzing Numeric and Geometric Patterns of Paper Pool'' and is related to the Illuminations lessons titled '' Paper Pool Game,'' '' Explore More Tables,'' '' Look for Patterns.'' Thinkfinity Partner: Illuminations Grade Span: 6,7,8
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Title: Algebraic Transformations
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In this unit of two lessons, from Illuminations, students learn about the commutative and associative properties using geometric shapes. They investigate the concepts first using a rectangle then using a plus sign shape and a triangle.
Standard(s): [MA2013] (8) 16: Verify experimentally the properties of rotations, reflections, and translations: [8G1] [MA2013] (8) 17: Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8G2] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3]
Subject: Mathematics Title: Algebraic Transformations
Description: In this unit of two lessons, from Illuminations, students learn about the commutative and associative properties using geometric shapes. They investigate the concepts first using a rectangle then using a plus sign shape and a triangle. Thinkfinity Partner: Illuminations Grade Span: 6,7,8
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Title: Commutative Rectangle
Description:
In this lesson, one of a multipart unit from Illuminations, students investigate the concepts of identity, inverse, commutativity, and associativity using a geometric model. Moves are performed with a rectangle, and the results of an operation that combines two moves are analyzed. Students determine if the operation is commutative or associative, if an identity element exists, or if there are inverses for any of the moves.
Standard(s): [MA2013] (8) 16: Verify experimentally the properties of rotations, reflections, and translations: [8G1] [MA2013] (8) 17: Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8G2] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3]
Subject: Mathematics Title: Commutative Rectangle
Description: In this lesson, one of a multipart unit from Illuminations, students investigate the concepts of identity, inverse, commutativity, and associativity using a geometric model. Moves are performed with a rectangle, and the results of an operation that combines two moves are analyzed. Students determine if the operation is commutative or associative, if an identity element exists, or if there are inverses for any of the moves. Thinkfinity Partner: Illuminations Grade Span: 6,7,8
ALEX Podcasts
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Title: Geometric Quilts
Overview:
This podcast is intended to accompany the lesson "Geometric Quilts." It is intended to be shown to the class between the design and creation segments of the project. Students will get tips to putting their quilt together properly and will get to see other students' completed quilts. Students will also see some quilts that were not properly constructed in hopes that they will not make the same mistakes. Standard(s):
[MA2013] (8) 16: Verify experimentally the properties of rotations, reflections, and translations: [8G1] [MA2013] (8) 17: Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8G2] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3] [MA2013] GEO (912) 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [GCO3] [MA2013] GEO (912) 34: Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics. (Alabama)
Geometric Quilts Overview: This podcast is intended to accompany the lesson "Geometric Quilts." It is intended to be shown to the class between the design and creation segments of the project. Students will get tips to putting their quilt together properly and will get to see other students' completed quilts. Students will also see some quilts that were not properly constructed in hopes that they will not make the same mistakes.
Thinkfinity Interactive Games
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Title: Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Visualizing Transformations
Description:
The interactive figures in this fourpart example from Illuminations allow a user to manipulate a shape and observe its behavior under a particular transformation or composition of transformations. eMath Investigations are selected eexamples from the electronic version of the Principles and Standards of School Mathematics (PSSM). The eexamples are part of the electronic version of the PSSM document. Given their interactive nature and focused discussion tied to the PSSM document, the eexamples are natural companions to the iMath investigations.
Standard(s): [MA2013] (8) 16: Verify experimentally the properties of rotations, reflections, and translations: [8G1] [MA2013] (8) 17: Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8G2] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3] [MA2013] GEO (912) 2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). [GCO2] [MA2013] GEO (912) 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [GCO3] [MA2013] GEO (912) 4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [GCO4] [MA2013] GEO (912) 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. [GCO5] [MA2013] GEO (912) 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. [GCO6] [MA2013] GEO (912) 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. [GCO12]
Subject: Mathematics Title: Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Visualizing Transformations
Description: The interactive figures in this fourpart example from Illuminations allow a user to manipulate a shape and observe its behavior under a particular transformation or composition of transformations. eMath Investigations are selected eexamples from the electronic version of the Principles and Standards of School Mathematics (PSSM). The eexamples are part of the electronic version of the PSSM document. Given their interactive nature and focused discussion tied to the PSSM document, the eexamples are natural companions to the iMath investigations. Thinkfinity Partner: Illuminations Grade Span: 6,7,8
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Title: Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Identifying Unknown Transformations
Description:
Standard(s): [MA2013] (8) 17: Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8G2] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3] [MA2013] GEO (912) 2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). [GCO2] [MA2013] GEO (912) 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [GCO3] [MA2013] GEO (912) 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. [GCO5] [MA2013] GEO (912) 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. [GCO6] [MA2013] GEO (912) 7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. [GCO7] [MA2013] GEO (912) 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. [GCO12]
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
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Title: Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Composing Reflections
Description:
This is part three of a fourpart eexample 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. eMath Investigations are selected eexamples 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 eexamples are natural companions to the iMath Investigations.
Standard(s): [MA2013] (8) 16: Verify experimentally the properties of rotations, reflections, and translations: [8G1] [MA2013] (8) 17: Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8G2] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3] [MA2013] (8) 19: Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them. [8G4] [MA2013] GEO (912) 2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). [GCO2] [MA2013] GEO (912) 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [GCO3] [MA2013] GEO (912) 4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [GCO4] [MA2013] GEO (912) 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. [GCO5] [MA2013] GEO (912) 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. [GCO12]
Subject: Mathematics Title: Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Composing Reflections
Description: This is part three of a fourpart eexample 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. eMath Investigations are selected eexamples 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 eexamples are natural companions to the iMath Investigations. Thinkfinity Partner: Illuminations Grade Span: 6,7,8
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Title: Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Composing Transformations
Description:
This is part four of a fourpart eexample 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. eMath Investigations are selected eexamples 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 eexamples are natural companions to the iMath Investigations.
Standard(s): [MA2013] (8) 16: Verify experimentally the properties of rotations, reflections, and translations: [8G1] [MA2013] (8) 17: Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8G2] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3] [MA2013] (8) 19: Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them. [8G4] [MA2013] GEO (912) 2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). [GCO2] [MA2013] GEO (912) 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [GCO3] [MA2013] GEO (912) 4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [GCO4] [MA2013] GEO (912) 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. [GCO5] [MA2013] GEO (912) 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. [GCO6] [MA2013] GEO (912) 7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. [GCO7] [MA2013] GEO (912) 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. [GCO12]
Subject: Mathematics Title: Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Composing Transformations
Description: This is part four of a fourpart eexample 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. eMath Investigations are selected eexamples 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 eexamples are natural companions to the iMath Investigations. Thinkfinity Partner: Illuminations Grade Span: 6,7,8
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Title: Calculation Nation
Description:
Become a citizen of Calculation Nation! Play online math strategy games to learn about fractions, factors, multiples, symmetry and more, as well as practice important skills like basic multiplication and calculating area! Calculation Nation uses the power of the Web to let students challenge themselves and opponents from anywhere in the world. The element of competition adds an extra layer of excitement.
Standard(s): [MA2013] (3) 1: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. [3OA1] [MA2013] (3) 2: Interpret wholenumber quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. [3OA2] [MA2013] (3) 13: Understand a fraction ^{1}/_{b} as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction ^{a}/_{b} as the quantity formed by a parts and size ^{1}/_{b}. [3NF1] [MA2013] (3) 15: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. [3NF3] [MA2013] (4) 12: Explain why a fraction ^{a}/_{b} is equivalent to a fraction ^{nxa}/_{nxb} by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. [4NF1] [MA2013] (4) 13: Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators or by comparing to a benchmark fraction such as ^{1}/_{2}. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. [4NF2] [MA2013] (4) 28: Recognize a line of symmetry for a twodimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify linesymmetric figures and draw lines of symmetry. [4G3] [MA2013] (6) 2: Understand the concept of a unit rate ^{a}/_{b} associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. [6RP2] [MA2013] (6) 4: Interpret and compute quotients of fractions, and solve word problems involving division of fractions, e.g., by using visual fraction models and equations to represent the problem. [6NS1] [MA2013] (6) 21: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems. [6G1] [MA2013] (7) 1: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. [7RP1] [MA2013] (7) 4: Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. [7NS1] [MA2013] (7) 5: Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. [7NS2] [MA2013] (7) 6: Solve realworld and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.) [7NS3] [MA2013] (7) 10: Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. [7EE4] [MA2013] (8) 17: Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8G2] [MA2013] (8) 18: Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. [8G3] [MA2013] DM1 (912) 1: Analyze topics from elementary number theory, including perfect numbers and prime numbers, to determine properties of integers. (Alabama)
Subject: Mathematics Title: Calculation Nation
Description: Become a citizen of Calculation Nation! Play online math strategy games to learn about fractions, factors, multiples, symmetry and more, as well as practice important skills like basic multiplication and calculating area! Calculation Nation uses the power of the Web to let students challenge themselves and opponents from anywhere in the world. The element of competition adds an extra layer of excitement. Thinkfinity Partner: Illuminations Grade Span: 3,4,5,6,7,8,9

