Geometry Transformational Geometry
Course: Geometry Unit #5: Transformational Geometry Overarching Question:
What impact does each type of transformation (reflection, rotation, translation, and dilation) have on the location, size, and orientation of geometric objects? Previous Unit:
This Unit:
Quadrilaterals and Other Polygons
Questions to Focus Assessment and Instruction: 1. What impact does changing the center of dilation have on the location of the image? 2. Which isometries preserve distance? Shape? Orientation? 3. How does the value of the scale factor in a dilation influence the size of the image? 4. How might geometric transformations be represented algebraically? 5. How might you investigate the composition of two or more isometries to determine which combinations are commutative? Key Concepts: Preimage Image Isometry Dilation Symmetry
Next Unit:
Transformational Geometry
Scale factor or magnitude Center of dilation Reflection Translation Glide reflection Congruence
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Right Triangle Trigonometry
Intellectual Processes(Standards for Mathematical Practice): Model with mathematics: Use transformations to model and interpret physical, and mathematical phenomena Look for and make sense of structure: Understand how algebraic and geometric ideas of transformations interconnect and build on one another to produce a coherent whole Use appropriate tools strategically: Apply and adapt a variety of appropriate strategies using transformations to solve problems Rotation Angle of rotation Center of Rotation Transformation Composition of two or more transformations
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Geometry Transformational Geometry
Unit Abstract The study of transformations in geometry gives students a visual perspective of the outcome of using reflections, rotations, translations, glide reflections, and dilations. Including the task of using coordinates to look for functions rules for each of these transformations helps connect the geometric and algebraic views. When transforming function rules in later algebra courses, the visual perspective and understanding of the effect the transformation has on the coordinates will assist in connecting the geometric and algebraic representations. Throughout the study of geometry, the use of a computer software program or graphing calculator with geometric drawing capabilities that allow students to draw and manipulate figures to make conjectures and reach conclusions should be encouraged. In particular, using available software to perform transformations allows students to make conjectures about algebraic or coordinate rules that model these transformations. Common Core State Standards Geometry-Congruence (G-CO)____________________________________________________ Experiment with transformations in the plane 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 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). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 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. Understand congruence in terms of rigid motions 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. Make geometric constructions 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing This document is the property of MAISA.
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Geometry Transformational Geometry
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. Geometry-Similarity, Right Triangles, and Trigonometry (G-SRT)________________________ Understand similarity in terms of similarity transformations 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. 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. Instructional Resources NCTM Illuminations (http://illuminations.nctm.org) Symmetries I: Students will learn about the mathematical properties of rotation. http://illuminations.nctm.org/LessonDetail.aspx?ID=U138 Symmetries II: Students investigate reflection. http://illuminations.nctm.org/LessonDetail.aspx?ID=U139 Symmetries III: Students will learn how translations work and what happens when two or more translations are applied one after another http://illuminations.nctm.org/LessonDetail.aspx?id=L474 Symmetries IV: Students will learn about glide reflection. http://illuminations.nctm.org/LessonDetail.aspx?id=L475 Transformations and Frieze Patterns: http://illuminations.nctm.org/LessonDetail.aspx?ID=U181 Texas Instruments (http://education.ti.com) Connecting Translations, Reflections, and Rotations(TI-84): In this activity, students will investigate the relationship among the three types of rigid transformations - translations, rotations, and reflections. http://education.ti.com/xchange/US/Math/Geometry/4046/CabriJr_Interactive_Act%2 009.pdf Scale Factor (TI-84): Students dilate polygons and find the perimeter and area of both the pre-image and image. http://education.ti.com/calculators/timath/US/Activities/Detail?sa=5024&id=10233 Also for (TI-Nspire): In this activity, students will dilate polygons and find the perimeter and area of both the pre-image and image. Then they find the ratios of the perimeters and areas. When they change the scale factor, all of these values automatically update and students can see that the ratio of the perimeters equals the scale factor and the ratio of the areas equals the square of the scale factor. Students will see that this is true for both enlargements and reductions. This document is the property of MAISA.
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Geometry Transformational Geometry
http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=8299&MICROS ITE=ACTIVITYEXCHANGE Interactive Geometry Activities and Investigations: http://education.ti.com/educationportal/sites/US/nonProductSingle/activitybook_cabrij r_interactive.html Cabri™ Jr.: Interactive Geometry Activities and Investigations: Explore geometry through the 29 acivities in this book. The focus here is on concepts rather than keystrokes and allows students to explore fice areas of geometry in depth. http://education.ti.com/educationportal/sites/US/nonProductSingle/activitybook_cabrijr_intera ctive.html NCTM Reasoning and Sense Making Task Bank: (www.nctm.org/hsfocus) Taking a Spin: Although students are often asked to find the angles of rotational symmetry for given regular polygons, in this task they are asked to find the regular polygons for a given angle of rotational symmetry, a reversal that yields some surprising results. This task would be most appropriate with students who have at least some experience in exploring rotational symmetry. http://www.nctm.org/uploadedFiles/Journals_and_Books/Books/FHSM/RSM-Task/RSM_TaskTaking_a_Spin.pdf NCTM Navigations Series Day, R., Kelley, P., Krussel, L., Lott,J. W., & Hirstein, J. (2002). Navigating through Geometry in Grades 9–12 (with CD-ROM). Reston,VA: NCTM. Approaching geometry through a transformational lens, this book concentrates on topics such as the use of transformations, coordinates and matrices, and congruence and similarity. A Useful Activity Students are able to practice or demonstrate understanding of transformations by moving shapes on a coordinate grid to new locations following specific requirements. http://www.teachnet-uk.org.uk/2006%20Projects/MathsTransformations/Tristan%20Jones/Home.htm Geometer’s Sketchpad Investigate reflections, translations, dilations, glide reflections, rotations of 90º, 180º, 270º, rotations of any degree, and composition of reflections with this menu driven exploration. http://mathbits.com/MathBits/GSP/Transformations.htm Investigate the result of two different geometric transformations when combined. Determine whether (or under what conditions) the order of transformations may be reversed, and which compositions may be written as a single transformation. http://www.uni.illinois.edu//~hcrussel/CompositionOfTransformations.htm This unit is intended to guide students through the study of transformational geometry of isometries using Geometer's SketchPad as the major tool for investigation and production. Students should work through each day's assignment, printing their results for teacher approval. After the investigations and productions have been performed, students will be asked to take a final assessment that will require them to use all that they have learned This document is the property of MAISA.
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about transformations of isometries. Homework assignments each day will be answered in a journal that students must keep to write about the processes they have experienced and the discoveries they have made. http://jwilson.coe.uga.edu/emt668/EMAT6680.Folders/Maddox/Transformational.Geo.html Other Computer Applets A series of 23 interactive online math lessons such as Basic Transformation Concepts, Reflections, Rotations, Translations, and Dilations http://enlvm.usu.edu/ma/nav/toc.jsp?sid=__shared&cid=emready@transformations&cf=activ ity Use the virtual pantograph in this applet to create similar figures. http://www.ies.co.jp/math/products/geo1/applets/panta/panta.html A free interactive math textbook on the web, defining and giving examples for transformational geometry http://www.mathopenref.com/translate.html A variety of interactive geometry activities, with some especially good activities on 3-D and 2-D views of objects
http://www.internet4classrooms.com/eoc_geometry.htm CPMP-Tools Course 1 – Tilings with Triangles or Quadrilaterals Tilings with Penrose Tiles Course 2 - Animate Shuttle Roll Over Geogebra Pre-image and image points and coordinates are shown. Student works to determine what type of transformation was done. http://www.geogebra.org/en/upload/files/knwilson/Transformations.html The scale factor is dynamic and students are asked to observe how the scale factor effects the lengths of the pre-image and image. http://www.geogebra.org/en/wiki/index.php/Dilations Manipulatives Mira Geoboard Linkage strips Patty paper Assessments This assessment is pre-built in Geometer’s Sketchpad. It has a number of figures and a design. The students must use transformations to move the figures onto the design with as few transformations as possible. http://jwilson.coe.uga.edu/emt668/EMAT6680.Folders/Maddox/student%20directions/directio ns.html This document is the property of MAISA.
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Geometry Transformational Geometry
The quiz provided here has the students do a variety of activities on the computer. Some are manipulating figures to show transformations, others are answering questions regarding properties and results of doing transformations. http://enlvm.usu.edu/ma/nav/activity.jsp?sid=__shared&cid=emready@transformations&lid=1
This resource could serve as a formative assessment task on transformations and does not require the use of any technology.
http://www.lewiscentral.org/pages/uploaded_files/Geometry%201%20Practice%20Qu iz%20%237.pdf Professional Resources NCTM (www.nctm.org) Focus in High School Mathematics: Reasoning and Sense Making: This publication elevates reasoning and sense making to a primary focus of secondary mathematics teaching. It shifts the teachers’ role from acting as the main source of information to fostering students’ reasoning to make sense of the mathematics. http://www.nctm.org/catalog/product.aspx?ID=13494 Focus in High School Mathematics: Reasoning and Sense Making In Geometry: Classically, geometry has been the subject in which students encounter mathematical proof based on formal deduction. Attention to proof in the geometry curriculum is strengthened by a focus on reasoning and sense making. The authors examine the four key elements (conjecturing about geometric objects, construction and evaluation of geometric arguments, multiple geometric approaches, and geometric connections and modeling) identified in Focus in High School Mathematics: Reasoning and Sense Making in more detail and elaborates on the associated reasoning habits. http://www.nctm.org/catalog/product.aspx?ID=13525 Articles from National Council of Teachers of Mathematics (www.nctm.org) Articles available as free downloads to NCTM members, or for a fee to non-members Sheats Harkness, S. (2005). Geometry of Transformations: Teacher and Unit Under Construction. Mathematics Teacher, 99 (2), 88-92. Retrieved December 6, 2010 from www.nctm.org/eresources/article_summary.asp?URI=MT2005-09-88a&from=B Devaney, R. L. (2004). Fractal Patterns and Chaos Games. Mathematics Teacher, 98 (4), 228-233. Retrieved December 6, 2010 from www.nctm.org/eresources/article_summary.asp?URI=MT2004-11-228a&from=B Siegrist , R. (2009). Activities for Students: Inquiry into Fractals. Mathematics Teacher, 103 (3), 206-211. Retrieved December 6, 2010 from www.nctm.org/eresources/article_summary.asp?URI=MT2009-10-206a&from=B Edwards, M.T. (2005). Activities for Students: Using Overhead Projectors to Explore Size Change Transformations. Mathematics Teacher, 98 (7), 498-505. Retrieved December 6, 2010 from www.nctm.org/eresources/article_summary.asp?URI=MT2005-03-498a&from=B This document is the property of MAISA.
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Cooper, B. D. and Barger, R. (2009). Listening to Geometry. Mathematics Teacher, 103 (2), 108-115. Retrieved December 6, 2010 from www.nctm.org/eresources/article_summary.asp?URI=MT2009-09-108a&from=B
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