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The Concept of Transformations in a High School Geometry Course

The Concept of Transformations in a High School Geometry Course. A workshop prepared for the Rhode Island Department of Education by Monique Rousselle Maynard momaynard86@gmail.com. Transformations. “ Means to an end .”  Hung-Hsi Wu. Goals.

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The Concept of Transformations in a High School Geometry Course

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  1. The Concept of Transformations in a High School Geometry Course A workshop prepared for the Rhode Island Department of Education by Monique Rousselle Maynard momaynard86@gmail.com

  2. Transformations “Means to an end.” Hung-Hsi Wu

  3. Goals Experience the transition from a hands-on and concrete experience with transformations to a more formalized and precise experience with transformations in a high school Geometry course.  Write precise definitions for Rotation, Reflection, and Translation.  Distinguish the Properties of the Rigid Transformations.

  4. Transformation Progression Focus is on translation, reflection, rotation, and dilation. Middle School High School Informal  Formal Hands-on  Definitions Descriptive  Functions on the Plane*  Transformations of a function from plane to plane.

  5. Rotation: in the Real World

  6. Preliminary Notion of Rotation https://tube.geogebra.org/student/m50296

  7. How do I rotate a figure around a point? http://www.youtube.com/watch?v=U4Hv494HwrQ Debrief. Q: What information does one require in order to perform a rotation? A: pre-image, center of rotation, degree (angle) of rotation, direction of rotation (in some cases) Q: What tools are required? A: paper, protractor, straightedge, pencil, colored pencils

  8. Q: If, on an assessment, you were asked to rotate a geometric shape, what evidence should you provide to support the location of the image? A: the angle of rotation, direction of rotation, labels on points of the pre-image and image, and compass work

  9. Rotation Sketch the image of the figure L after it is rotated 60 counterclockwise around point O (optionally denoted RoO, 60). Resources • Handout with figure and point O on it • Compass • Protractor • Pencil

  10. Develop a Precise Definition of Rotation In developing a precise definition of rotation, we need to consider the effects of various centers of rotation, various degrees of rotation, and different directions of rotation. Thus, we will conduct a guided investigation to inform our definition. Rotation Guided Investigation Definition Jigsaw (Poster Paper)

  11. Precise Definition: Rotation The rotation Ro of t degrees(180t 180) around a given point O, called the center of the rotation, is a transformation of the plane. Given a point P, the point Ro(P) is defined according to the following conditions. The rotation is counterclockwise or clockwise depending on whether the degree is positive or negative, respectively. For definiteness, we first deal with the case where 0  t 180. • If P = O, then by definition, Ro(O) = O. • If P is distinct from O, then by definition, Ro(P) is the point Q on the circle with center O and radius |OP| such that |mQOP|= tand such that Q is in the counterclockwise direction of the point P. We claim that this assignment is unambiguous (i.e., there cannot be more than one such Q). • If t= 180, then Q is the point on the circle so that is a diameter of the circle. • If t= 0, then Q = P; and Ro is the identity transformation I of the plane. Hence, if 0 < t < 180, then all the Q's in the counterclockwise direction of the point P with the property 0 < |mQOP| < 180lie in the fixed half-plane of that contains Q Thus Ro is well-defined, in the sense that the rule of assignment is unambiguous. Now, if t < 0, then by definition, we rotate the given point P clockwise on the circle that is centered at O with radius |OP|. Everything remains the same except that the point Q is now the point on the circle so that |mQOP| = |t| and Q is in the clockwise direction of P. Thus, we define Ro(P) = Q.

  12. Reflection: in the Real World

  13. Preliminary Notion of Reflection http://www.harpercollege.edu/~skoswatt/RigidMotions/reflection.html

  14. Optional Additional Support: Performing a Reflection http://www.youtube.com/watch?v=nAt212f6Uds

  15. Develop a Precise Definition of Reflection In developing a precise definition of reflection, we need to distinguish between the reflections of points that do and do not lie on the line of reflection. Use Reflection Definition-Writing Activity Materials: Patty Paper, Straightedge, Pencil

  16. Precise Definition: Reflection The reflection R across a given line l, where l is called the line of reflection, assigns • to each point on line l, the point itself, and • to any point P not on line l, the point R(P) that is symmetric to it with respect to line l, in the sense that line l is the perpendicular bisector of the segment joining P to R(P).

  17. Translation: in the Real World

  18. Preliminary Notion of Translation http://tube.geogebra.org/material/show/id/18530

  19. Performing a Translation Along a Vector Translation with Patty Paper: http://www.youtube.com/watch?v=aPo0X6u_W-I&list=PLl4sjkH9L9JBu1dG4UkAwdogsxMDB2VFe

  20. Develop a Precise Definition of Translation Along a Vector In developing a precise definition of Translation we need to distinguish between a vector with and without length. Translation Along a Vector Activity

  21. Precise Definition: Translation The translation T along a given vector assigns the point D to a given point C. Let the starting point and endpoint of be A and B, respectively. Assume C does not lie on . Draw the line l parallel to passing through C.* The line passing through B and parallel to then intersects line l at a point D; we call the line.** By definition, T assigns the point D to C; that is, T(C) = D. If C lies on , then the image D is by definition the point on to such that the direction from C to D is the same direction as from A to B such that |CD| = |AB|. If the vector is , the zero vector (i.e., the vector with zero length), then the translation along is the identify transformation I.

  22. Rigid vs. Non-RigidTransformations: What is the Difference?

  23. Properties of Isometries Rotations, Reflections, and Translations: • Map lines to lines, rays to rays, and segments to segments. • Are distance-preserving. • Are degree-preserving.

  24. Connecting Today’s Workto the CCSS During the course of today’s session, our activities have connected with several CCSS for Geometry and Mathematical Practices. Can you identify the standards and briefly explain the connection.

  25. Teacher Resources

  26. Illustrative Math Activity:Defining Rotations (G-CO.A.4) *Alternative: Provide students with these definitions and ask them to critique their accuracy. MP3 http://www.illustrativemathematics.org/standards/hs

  27. Illustrative Math Activity:Defining Reflections (G-CO.A.4) http://www.illustrativemathematics.org/standards/hs

  28. Properties of Rotations, Reflections, and Translations Activities can be similarly developed that will lead students to visualize or develop the properties of the individual rigid transformations.

  29. Properties of Rotations The distance of a point on the pre-image from the center of rotation is equal to the distance of its corresponding point on the image from the center. ** Although demonstrated to be the most difficult transformation for students, it has been observed that spatial imagery cognitive style can significantly improve performance in rotation tasks (Xenia & Demetra, 2009).

  30. Activity. An opportunityto demonstrate your understandingof the properties of rotation. Analytic Activity. Find the coordinates of the image of the triangle after a 90 clockwise rotation about the point (3, 5).

  31. Properties of Reflections A reflection is a transformation of a plane having the following properties: • The line joining the pre-image and corresponding image is perpendicular to the line of reflection (which is a perpendicular bisector of the line joining any two corresponding points). • Any point on the reflected pre-image is the same distance as its corresponding image point from the line of reflection. • All points on the line of reflection are unchanged or are not affected by the reflection. • The pre-image and the image are oppositely congruent to each other.

  32. Activity. An opportunityto demonstrate one’s understandingof the properties of reflection. Graphical Activity. Draw the image of the triangle, given as follows, under a reflection about the line y = 4. y 4 x

  33. Properties of Translations • http://www.ixl.com/math/geometry • http://nrich.maths.org/5457 • http://nrich.maths.org/public/leg.php?code=130

  34. Activity. An opportunityto demonstrate one’s understandingof the properties of translation. Algebraic Activity. The vertices of a triangle are A(4, 1), B(2, 1), and C(4, 5). If ABC is translated by vector , find the coordinates of the vertices of its image. **In a study carried out by Xenia & Demetra (2009) it emerged that students perform better in translation tasks than the other types.

  35. Representative CCSS Vocabulary for HS Geometry • Algebraic • Alternate interior angles • Arc • Area • Base angles • Central angle • Chords • Circle • Circumference • Circumscribed angle • Collinear • Compass • Complete the square • Cone • Congruent • Constructions • Coordinate geometry • Coordinate plane • Coplanar • Corresponding sides • Corresponding angles • Cross-section • Cylinder • Derive • Diagonal • Dilation • Directrix • Distance formula • Distinct • Endpoint • Equidistant • Equilateral triangle • Experiment • Focus • Geometric • Inscribed angles • Interior angle • Interpret • Isometry • Isosceles triangle • Line • Line segment • Locus • Median • Midpoint • Parallel • Parallelogram • Perimeter • Perpendicular • Perpendicular bisector • Plane • Point • Preserve angle • Preserve distance • Proof • Proportion • Pythagorean Theorem • Radian • Radii • Ratio • Rectangle • Reflection • Regular hexagon • Regular polygon • Rigid motion • Rotation • Scale factor • Sector • Sequence • Skew • Slope • Solution • Square • Sphere • Straightedge • Symmetry • Tangent • Theorem • Three dimensional • Transformation • Translation • Transversal • Trapezoid • Triangle • Triangle congruence • Trig ratios • Two dimensional • Undefined • Vector • Vertical angles • Volume • xy-Coordinate axis

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