Geometric Transformations - Lesson Plan

Objectives

• Given a figure (pre-image) and a transformation specify the figure resulting from the transformation (image).
• Given a transformed figure (image) and a transformation, specify the original figure (pre-image).
• Given a figure (pre-image) and a transformation draw the result of the transformation.
• Given two figures, and a transformation recognize whether or not the second figure is the result of applying the transformation to the first figure.
• Given a figure (pre-image) and another figure that is a transformation of the first (image), describe the transformation that generates the second figure.
• Given a figure, recognize whether or not it has line and point symmetry and specify points and lines of symmetry.
• Recognize and produce instances of the concepts: pre-image, image,line of reflection, center of rotation, rotation angle,translation vector, center of dilation, dilation factor.

Types of transformations addressed are: translations, rotations, reflections, and dilations.Figures are referred to using both vertex names and cartesian coordinates.

Teaching Plan (1 week)

Mappings (10 minutes)

The objectives of this lesson are to help students identify images and preimages of a mapping and recognize an isometry.

This lesson could also be taught as a presentation with the whole class working together.

Introduction

 Using a computer projector, demonstrate how to go to the eNLVM website, go to your school and class, and login. (5 minutes) (2 activities) Direct students to Mappings: Introduction. Use the first two pages to demonstrate to students how to enter their answer and move between activites. Present some background on Esher and tesselations. Ask students to enter their observations and continue through the lesson on their own. (4 activities) Students are presented with the definition of preimage and image. They are also introduced to the 4 transformations taught in the eModule and are asked to identify characteristics of each. (1 activity) Students are presented with the definition of a mapping and a transformation. (1 activity) Students identify different transformations. (2 activities) Students are introduced to mapping notation. (2 activities) Students are presented with a definition of congruence, isometry, and similarity. They are asked to identify if a translation is congruent or similar and justify their answer. (1 activity) Students are presented with how to prove that a mapping is an isometry (1 activity) Questions check for the students' understanding of concepts and terms. When students have completed the Introduction, bring the class together and discuss the concepts they learned. Asess their understanding and clarify questions before moving onto the practice.

Practice

 (5 activities) Direct students to the Practice section. Students will name and identify images and preimages. (1 activity) Students create figures by rotating, translating, and flipping (reflecting) blocks, then identify the number of flips needed. (1 activity) Students identify isometries. (1 activity) Students identify a rotation and create a tessalation. (1 activity) Students describe a scene with transformations. (1 activity) Students pick a transformation and create a scene.

Reflections (40 - 50 minutes)

The objective of the second lesson is to name a reflection image, to recognize line and point symmetry, and to draw a reflection image.

Introduction

 Direct students to comple the section on reflections. Observe the students working and provide help and clarifcation when needed. (3 activities) Students are presented with examples and definitions of reflection, line of reflection, and point of reflection. (3 activities) Students are presented with the following properties of reflections: reflections preserve collinearity, betweeness, angle and distance measures. (1 activity) Students are shown how to construct a reflection image over a line. (1 activity) Students are presented with the definition of line of symmetry. (2 activities) Students are taught how to identify a line of symmetry. (2 activities) Students practice identifying lines of symmetry. (2 activities) Students are presented with the definitintion of point of symmetry. (3 activities) Students are presented with examples of points of symmetry. (1 activities) Students are given questions that check their understanding of concepts. Bring the class together if you want to provide any clarification.

Practice

 (1 activity) Students name reflections. (2 activities) Students draw the reflection image of figures across a line. (2 activities) Students identify lines of symmetry. (1 activity) Students draw lines of symmetry. (1 activity) Students identify lines and points of symmetry. (2 activities) Students draw reflection images through a point. (2 activities) Students describe how to check for a line of symmetry using a reflection and a point symmetry using rotation. (2 activities) Students create a figure with point symmetry or line symmetry. (17 activities) If you identify a need or if a student has extra time, a section of Extra Practice is available.

Translations (50 minutes)

The objective of this lesson is to name a translation image with respect to parallel lines and to draw a such a translation image.

Introduction

 Review the concepts of the previous day and address any problems you observed. Direct students to complete the Translations section. Continue to observe their work and correct and clarify as needed. (1 activitiy) Students are presented with an example and a definition of a translation. (2 activities) Students are shown how to construct a translation as a reflection over two parallel lines. (1 activity) Students are presented with properties preserved by reflections. Students are asked to recall the difference between an isometry and a similarity transform. (1 activity) Students are shown how to build 3D images with translations. (1 activity) Students work with an example of how translations can be used in tessalation wallpaper.

Practice

 (2 activities) Students identify translations. (2 activities) Students draw the translation images of figures across parallel lines. (2 activities) Students identify translation images. (1 activity) Students name reflection and translation images. (2 activities) Students draw translation images with respect to two parallel lines. (1 activity) Students plan a proof that translations preserve certain qualities. (1 activitiy) Students create 3D images with translations. (1 activity) Students create a translation. (10 activities) If you identify a need or if a student has extra time, a section of Extra Practice is available.

Rotations (50 minutes)

The objective of this lesson is to help students name a rotation image with respect to intersecting lines and to draw a such a rotation image.

Introduction

 Review the concepts of the previous day and address any problems you observed. Direct students to complete the Rotations section. Continue to observe their work and correct and clarify as needed. (2 activities) Students are presented with the definitions of rotation and angle of rotation. (2 activities) Students identify angle of rotation and construct rotation images. (1 activity) Students identify the relationship between angle of rotation and an angle between intersecting lines. (1 activity) Students identify angles of rotation. (1 activity) Students are presented questions that check their understanting of the definitions and concepts. Bring the class together if you wish to provide any clarification.

Practice

 (1 activity) Students name reflection and rotation images. (2 activities) Students draw the rotation images of figures across intersecting lines. (2 activities) Students identify rotations and angles of rotation. (2 activities) Students determine angle of rotation. (1 activity) Students identify which properties are preserved by rotation. (1 activity) Students identify rotations and reflections. (1 activitiy) Students create 3D images with translations. (7activities) If you identify a need, or if a student has extra time, a section of Extra Practice is available.

Dilations (50 minutes)

The objective of this lesson is to help students understand the meaning of scale factors, identify the center and scale factor of a dilation, and draw a dilation given a center and scale factors.

Introduction

 Review the concepts of the previous day and address any problems you observed. Direct students to complete the Dilations section. Continue to observe their work and correct and clarify as needed. (2 activities) Students are presented with the definition of a dilation or similarity transform and the scale factor of a dilation. (3 activities) Students explore the effect of the scale factor on a dilation. (1 activity) Students are shown how to construct a dilation image on a geoboard. (2 activities) Students are shown how to identify the center of dilation and the scale factor. (1 activity) Students practice finding a center and scale factors

Practice

 (2 activities) Students find scale factors given a dilation. (2 activities) Given a dilation, students find the center of dilation and scale factor. (4 activities) Students construct a dilation image of a figure given the center and scale factor. (1 activity) Students identify which transformations will move the ball to win a game of golf..

Review

Review the content periodically as misunderstandings become apparent. Spend time after all the lessons are completed to review properties of each transformation and how each transformation is constructed before having students complete the quiz.

Assessment

Ask students to complete the online Quiz. If they have time, ask them to complete the Student Feedback Questionaire as well.

Credits

 Lesson Design Joel Duffin, Jean Culbertson Web Development Joel Duffin, Liz Hart Mathlets Argyll Home Education Centre, MyMaths, NLVM, Shodor Foundation Inc., Stephen Webber Images Denise Chandler, US Library of Congress

Correlation to Standards

Correlation to NCTM Standards

Geometry - Standard 3. Apply transformations and use symmetry to analyze mathematical situations.

• Recognize and apply slides, flips, and turns.
• Recognize and create shapes that have symmetry.
• Predict and describe the results of sliding, flipping, and turning two-dimensional shapes.
• Describe a motion or a series of motions that will show that two shapes are congruent.
• Identify and describe line and rotational symmetry in two- and three-dimensional shapes and designs.
• Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.
• Examine the congruence, similarity, and line or rotational symmetry of objects using transformations.
• 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.

Correlation to Utah Standards

• Math 3 - 3.3 Visualize and identify geometric shapes after applying transformations.
• Demonstrate the effect of a slide (translation) or flip (reflection) on a figure, using manipulatives.
• Determine whether two polygons are congruent by sliding, flipping, or turning to physically fit one object on top of the other.
• Math 4 - 3.3 Visualize and identify geometric shapes after applying transformations.
• Identify a slide (translation) or a flip (reflection) of a geometric shape using manipulatives.
• Math 5 - 3.3 Visualize and identify geometric shapes after applying transformations.
• Identify a slide (translation) or a flip (reflection) of a shape across a line.
• Demonstrate the effect of a turn (rotation) on a figure using manipulatives.
• Math 6 - 3.3 Visualize and identify geometric shapes after applying transformations.
• Turn (rotate) a shape around a point and identify the location of the new vertices.
• Slide (translate) a polygon either horizontally or vertically on a coordinate grid and identify the location of the new vertices.
• Flip (reflect) a shape across either the x- or y-axis and identify the location of the new vertices.
• Math 7 - 3.3 Visualize and identify geometric shapes after applying transformations, and identify lines of symmetry.
• Identify line(s) of symmetry in plane figures.
• Transform geometric shapes using translations (slides), rotations (turns), and reflections (flips).
• Geometry - 3.2.4. Perform and analyze transformations (translations, rotations, reflections, and dilations) using coordinate geometry.

Correlation to Arizona Standards

Strand 4. Concept 2. Apply spatial reasoning to create transformations and use symmetry to analyze mathematical situations.

• Demonstrate translation using geometric figures.
• Demonstrate reflections using geometric figures.
• Describe the transformations that created a tessellation.