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 Why focus on transformations? Because the CCSS are written in such a way that give the concept of Transformations a new life in a high school Geometry course. Rather than an isolated concept restricted to one unit of the course, the concept of transformations will permeate the year through. As Dr. Wu believes, transformations are a means to the end.

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. Our goals for today are to make clear for participants the transition expected of students as they advance from middle school mathematics to a high school Geometry course, specifically as it pertains to transformations. We will apply a familiarity with hands-on work in completing activities that will inform our goals of writing precise definitions of the three rigid transformations: rotation, reflection, and translations. If time allows, we will also distinguish properties of the rigid transformations from non-rigid transformations. **Disclaimer. We will be using YouTube videos as part of our instruction today. We do so at the risk of introducing language and/or notation that is not precise or otherwise inaccurate. Thus, let’s be mindful, in our role as teacher, of the responsibility to make those errors as well as the “fixes” transparent for our students.

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. According to the Common Core State Standards, students encounter transformations in Grade 8, and they do so in a very concrete hands-on manner. Eighth graders confirm the properties of the rigid transformations, describe sequences of transformations that bring one geometric figure to coincide with another given congruent geometric figure, describe the effects of transformations on a geometric figure on the xy-coordinate plane, and understand that transformations can be used to support the similarity of two-dimensional figures. In high school, students formalize their discussion of transformations; write precise definitions; treat transformations as functions on the plane; and use transformations to prove congruence, similarity, and geometric theorems.

Let’s enter the classroom and set about our goal to, first, write precise definitions of the three rigid transformations. Our protocol will be to: #1. make fresh in our minds real-world representations of the mathematical concept; #2. recollect images and understandings from our concrete 8th grade encounter with the transformation; and #3. perform the transformation and analyze the results.

Rotation: in the Real World We will start with ROTATION. Please connect the mathematical concept of rotation to the physical world. Think “rotation.” What thoughts come to mind? Tire on a bike or car Carousel Stopwatch

Preliminary Notion of Rotation https://tube.geogebra.org/student/m50296 Let us build upon those real-world images by recalling your 8th grade experience with rotations. Here we view three controlled animations that show rotations in motion. Can someone express what they see? <Take Away The center of rotation appears to be the origin (0,0). The distance of each identified point from the center of rotation remains constant. The images created by the rotation appear to be congruent to the pre-image. *The farther a point is away from the center of rotation, the longer the distance it is rotated (in terms of linear distance; connects to the concept of arc length and radians).

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 We begin to formalize our prior experience. Let us first watch a video on YouTube that will either refresh our memories or inform us of how to rotate a geometric figure around a point. View video. [4 minutes] In case some of us have doubts about our personal abilities to re-perform a rotation after watching that video, can anyone summarize for us all how to perform a rotation? And, within that explanation, refer to the information and tools required to perform a rotation. Perhaps your words will resonate with the image of the video and make the whole concept clearer for us all. <Take Away> Need to know. Center, Angle, and Direction of Rotation Tools. Compass, Straightedge, Protractor, Patty Paper (optional) Steps. Sketch a segment from center (say, O) to the pre-image point (say, P). Use protractor to measure/draw angle of rotation (make the terminal side of the angle at least as long as 𝑂𝑃 ). With its width adjusted to equal the length of 𝑂𝑃 , draw an arc from the initial side of the angle to the terminal side of the angle. Label the point where the arc intersects the terminal side of the angle as point P’. Follow-up questions …

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 Just so that the expectations for your work are clear, in your rotation work, what pieces of evidence should you provide? <Take Away> Segments joining the center to the points of the pre-image The angle of rotation from each point of the pre-image Arcs from each point of the pre-image The image with appropriately labeled points.

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 Let us practice. Within the folder labeled “Rotations” that is located at your table, please remove a copy of the “Exercise A. You Try It.” activity for everyone at your table and rotate the figure L as directed. Troubleshoot student difficulties. Encourage productive collaboration. Upon completion: Reflect upon your work for a moment. Do we see the pieces of evidence within your work that we established just a few moments ago? [center of rotation, direction of rotation, angle of rotation, corresponding points on pre-image and image, compass work]

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) Since we have demonstrated that everyone is familiar with how to rotate a figure a certain distance around a point, we are going to move into an investigation. The goal of this guided investigation is to write a precise definition of rotation. You and your table mates will work cooperatively to meet this goal. Again, from the “Rotation” folder, please ensure that everyone at your table gets one copy of both Parts 1 and 2. Your work will begin with Part 1. Please take a moment to read the Protocol. <Pause.> Q: Any questions? Is everyone clear on the expectations of their charge? When you are done with the work in Part 1, compare the results of your rotations before completing Part 2. So, individually work on the rotation exercises; collaborate on the definition. Any questions? Please begin. Report out on poster paper. Assign different portions (7) of the definition to different groups.

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 𝑃𝑄 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 𝑂𝑃 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. RE: Rotation. Concepts of direction, that is, clockwise and counterclockwise, on a circle are accepted as self-evident. This is the complete definition as you all just completed it. Any comments?

Reflection: in the Real World Just as we did with rotation, please connect the mathematical concept of reflection to the physical world. Think “Reflection.” What thoughts come to mind? One’s reflection in the bathroom mirror every morning …

Preliminary Notion of Reflection http://www.harpercollege.edu/~skoswatt/RigidMotions/reflection.html Let us build upon those real-world images by again recalling your 8th grade experience with reflections. Here we view two animations that show reflection in motion. Can someone express what they see? <Take Away> R moves every point in the plane not lying on line <script lc ell> to the “opposite side” of line <script lc ell>.

Optional Additional Support: Performing a Reflection http://www.youtube.com/watch?v=nAt212f6Uds Before we move onto reflections in depth, is there anyone who feels they need help in how to make a basic reflection of a point or object across a line of reflection? It is important to have a level of confidence with this before we move on, so it is certainly okay to spend some time revisiting the process first. <Use video link.> [2 minutes] If anyone answers in the affirmative, facilitate one paper-folding activity. First: Fold to establish a line of reflection and label it. Second: Within the smaller region, use a straightedge and your pencil to sketch a geometric figure (a scalene triangle will suffice) and label all vertices. Third: Fold the paper backward along the line of reflection. Fourth: Trace the pre-image, and label its vertices appropriately. <What do I mean by appropriately?>

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 For this activity, you will work in groups of four students. On your table, if you look within the folder labeled “Reflections,” you will find copies of the Reflection Definition-Writing Activity. Please note that there is a sticker on the top of the first page of each packet. Why the sticker? Right, for this activity you will work with those students who have the same sticker on their packet. Also in the folder are a set of folded colored papers labeled Hint Card #1, Hint Card #2, and Hint Card #3. Let us read through the protocol together and address any questions that might arise before forming our work groups and getting to work.

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). This is the complete definition as you all just completed it. Any comments?

Translation: in the Real World Just as we did with reflection, please connect the mathematical concept of translation to the physical world. Think “translation.” What thoughts come to mind? The foot press machine at the gym “Pass the potatoes please.” Rearrange furniture

Preliminary Notion of Translation http://tube.geogebra.org/material/show/id/18530 Let us build upon those real-world images by again recalling your 8th grade experience with translations. Here we view one animation that shows translation in motion. Can someone express what they see? <Take Away> A vector that specifies the direction and distance along which the pre-image will be moved. [A vector is a segment together with the designation of one of its two endpoints as a starting point; the other endpoint will be referred to simply as the endpoint of the vector and distinguished by an arrowhead. Familiar notation: 𝑣 <-3, 5> or −3 5 .] Under translation, a geometric figure is moved in the same direction and same distance as the given vector.  

Performing a Translation Along a Vector Translation with Patty Paper: http://www.youtube.com/watch?v=aPo0X6u_W-I&list=PLl4sjkH9L9JBu1dG4UkAwdogsxMDB2VFe Before we attempt to perform translations, let us view two videos, compliments of YouTube. The first demonstrates how to translate a triangle along an arc using patty paper. The second demonstrates how to translate a point along an arc using construction tools. The third link, which we will not watch together today, demonstrates how to translate a triangle along an arc using construction tools. Optional. Translate a point by construction: http://www.youtube.com/watch?v=i17f24w7zAM Translate a triangle by construction: http://www.youtube.com/watch?v=HXMbkvIzjB4 (for those with construction experience)

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 For this activity, you will work in groups of four students. As you walked in the room today, you were given a folder in which to organize your work and handouts. Within that folder, there is a sticker. You will work with those students with matching stickers. On your table, if you look within the folder labeled, “Translations,” one item you will find copies of the Translation Definition-Writing Activity. Vector without length? How can that be? The zero vector has zero magnitude. Its physical representation is subjective. As we did for our previous activity, let us read through the protocol together, make clear the expectations for your work, form groups, and set about writing a precise definition of “translation.” Protocol *There are six translations in total. Within your four-student group, two students perform translations #13 and two students perform translations #46. *Upon completion, compare your translations with other members of your group. *Scaffolds and supports were included in the last two definition-writing activities to help you build momentum and confidence. They have been removed from this activity. Part 3 asks that you brainstorm individually for a few minutes, record your ideas, and then collaborate on writing a precise definition of “Translation.” *When you settle upon the final version of your definition, please write it on chart paper and be prepared to defend it.

Precise Definition: Translation The translation T along a given vector 𝒗 assigns the point D to a given point C. Let the starting point and endpoint of 𝑣 be A and B, respectively. Assume C does not lie on 𝐴𝐵 . Draw the line l parallel to 𝐴𝐵 passing through C.* The line passing through B and parallel to 𝐴𝐶 then intersects line l at a point D; we call the line 𝐵𝐷 .** By definition, T assigns the point D to C; that is, T(C) = D. If C lies on 𝐴𝐵 , then the image D is by definition the point on to 𝐴𝐵 such that the direction from C to D is the same direction as from A to B such that |CD| = |AB|. If the vector 𝑣 is 0 , the zero vector (i.e., the vector with zero length), then the translation along 0 is the identify transformation I. This is the target definition. How does your definition compare with it? Any comments? If concerns arise about the notion of a zero vector, advise students that its physical representation is subjective. *Guaranteed to exist by the Parallel Postulate. **Lines 𝐵𝐷 and <script lc ell> must intersect because the Parallel Postulate says that … “Given any straight line and a point not on it, there "exists one and only one straight line which passes" through that point and never intersects the first line, no matter how far they are extended. This statement is equivalent to the fifth of Euclid's postulates, which Euclid himself avoided using until proposition 29 in the Elements.” http://mathworld.wolfram.com/ParallelPostulate.html

Rigid vs. Non-Rigid Transformations: What is the Difference? When you entered the room today, you were asked to complete an activity prior to the start of class. Please find that work so that we may discuss your findings. [Note that this activity was designed to give students a better appreciation of the basic rigid motions and a sense for their distance- and angle-preserving properties.] Although it might be worthwhile conversation to allow students to talk about any of the transformations and things they might find peculiar or interesting about them, time will limit today’s conversation to the properties and justification for them. The focus is on distinguishing Rigid and Non-Rigid Transformations. What is another term we use to refer to the class of Rigid Transformations? “Isometries” So, what are your conclusions about the properties of isometries and the evidence that support them? <Take Away> See next slide.

Properties of Isometries Rotations, Reflections, and Translations: Map lines to lines, rays to rays, and segments to segments. Are distance-preserving. Are degree-preserving. Students are very likely to contribute the second and third bullets however may not think to suggest the first. To be consistent with the language “-preserving” one might accept “shape-preserving”. Try to elicit the explicit language of “line to line, etc.” from students.

Connecting Today’s Work to 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. Because our collaboration here today is all due to the implementation of the CCSS, please take out your copy of the standards so that we can connect today’s work with the appropriate standards. <Take Away> In order: G-CO.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. … G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G-CO. 2 Represent transformation in the plane … Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). MP3 Construct viable arguments and critique the reasoning of others. MP5 Use appropriate tools strategically. MP6 Attend to precision. And perhaps upon your review of the standards you saw several other standards that can be supported by the work of this session.

Teacher Resources The following slides provide some additional resources that deal with the topic of Transformations. Included are some activities, web sites, and a relevant vocabulary list.

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

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

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.

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).

Activity. An opportunity to demonstrate your understanding of 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).

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.

Activity. An opportunity to demonstrate one’s understanding of 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

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

Activity. An opportunity to demonstrate one’s understanding of 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 6 2 , 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.

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