Geometric Transformations Sequence

Transcription

1 Geometric Transformations Sequence Geometric transformations are implicitly present in mathematics teaching from the very early stages of grade school all the way up to advanced topics in secondary school and the university. They provide an important unifying concept from early intuitive observations to more abstract algebraic notions. They can be used to unify understanding of simple geometric ideas, the concepts of advanced transformations in geometry and later the conceptualization of functions and their graphs. Geometric transformations also provide connections to complex numbers, matrices, vector algebra, in general, and to advanced mathematical topics. In this brief, geometric transformations (including rigid transformations, referred to as isometries, and those that preserve shape but not necessarily size, referred to as dilations, or similarities where dilations are composed with isometries) are presented as a sequence of ideas that evolve and develop, independently of curricula in different countries, to different stages at different age of students. 1. In the early grades of schooling, the very first approach should be intuitive. a) Students should begin by observing simple symmetric shapes. i. As they observe examples of shapes that are symmetric over a line or about a point, students should be challenged to describe the shapes and what makes the shapes symmetric. In further examples students can be challenged to fill out incomplete shapes to make them symmetric. ii. Students observe congruent and similar figures, starting with simple shapes that are non mathematical. Then the shapes are switched to triangles, rectangles and other mathematical geometric figures. b) Constructions naturally evolve from observations of properties to having those properties explained by transformations, albeit informally. i. Through observations and discussions, students determine that congruent figures can be moved to exactly cover each other, and similar figures can be moved and enlarged or shrunk to coincide with each other. Through observations, students also determine that in symmetric and congruent figures, corresponding distances are preserved and that in similar figures, all the corresponding distances enlarge or shrink proportionally while lines remain parallel. They also note that measures of corresponding angles remain unchanged in all of the transformations. ii. Movements of congruent figures are described and specified in terms of the isometries: translations, rotations, reflections and glide reflections. With these movements, corresponding distances and angle measures remain unchanged. Changing the size but not the shape of figures is described in terms of dilations or similarities. With the latter transformations, all corresponding distances change by the same factor and all corresponding 1

2 iii. iv. angle measures remain unchanged. Students should be challenged by combining two or more transformations consecutively. Students become familiar with congruent figures obtained by isometries and similarities. Students practice geometric transformations more formally using a compass and a ruler to draw congruent and similar figures, or using dynamic geometry software, as required by specific tasks. Students become acquainted with problems where proofs are required and where proportional reasoning is used in practical geometric problems. (See examples 1 and 2 later in the brief.) Exploring isometries in relation to geometry of solids can aid in developing students' spatial visualization. 2. An analytic approach to geometric transformations in a coordinate system is introduced after students are familiar with coordinates. a) Geometric transformations studied without coordinates are reviewed and interpreted in a coordinate system. All previously introduced properties are interpreted within the coordinate system. Now distances can be understood and calculated with knowledge of coordinates (using, for example, the Pythagorean theorem). Students consider reflections over coordinate lines and rotations about the origin. b) Algebraic descriptions of transformations are introduced. First examples include a rotation of 180º about the origin (sometimes called a reflection over the origin) addressed first with concrete points, for example (1, 1) >( 1, 1) before proceeding to the general (x, y) >( x, y). Similarly, reflections over the x and y axes should first be considered using concrete points and then in general (x, y) >(x, y) for the x axis and (x, y) >( x, y) for the y axis. General notation for the three mentioned transformations follow: R O, 180º: (x, y) >( x, y), r x: (x, y) >(x, y) and r y: (x, y) >( x, y) should also be introduced. Translations should start with simple notation, for example, the translation move right 2 units shifts the point (3, 4) to the point (5, 4). Moving the point (1, 1) left 2 and down 3 would give the point ( 1, 2). Moving the general point (x, y) by (1, 2) (right 1 and up 2) gives the point (x + 1, y + 2). Then after even more generalization moving (x, y) to (x + a, y + b), the notation of the translation in the form of T a,b:(x, y) >(x + a, y + b) is finally introduced. Transformations with regard to the origin and the coordinate axes should be considered with special care; reflections with regard to general lines could be considered using the coordinates. Other transformations could also be introduced step by step. Rotations for 90 o, 270 o and 0 o, or 360 o, can be described for all students. For example for the rotation for 90 o with the origin as center in the form R O, 90 o:(x, y) >( y, x). If possible, where students have a background in trigonometry, rotations should be described in the general form as R O,φ:(x, y) >(xcos(φ) ysin(φ), xsin(φ) + ycos(φ)). 2

3 c) Special attention should be given to the fact that the formal notation of geometric transformations might be quite abstract for students. This problem should be addressed by enough practical exercises that develop students' understanding of new notations step by step. Further understanding of the notation can be achieved by introducing a wide variety of different transformations within concrete problems and proofs, by discussing issues like inverse transformation and finally by composing two or more transformations by formal composition (for example, T a,b R O, 90 o). d) Translating and dilating are reviewed with simple figures in the coordinate system. Analysis is shifted to the question, what would happen to a shape like a parabola when translations are applied? Students notice themselves or are guided to the realization that these are just the shifts and dilations of the graphs they know analytically. Students visualize that, for example, a graph of a quadratic function y = (x p) 2 + q corresponds to the translation of the graph of a function y = x 2 p units right and q units up. Students should be exposed to several similar situations. Finally formal notation for transformations should be introduced in the following form. The graph of a quadratic function y = x 2 consists of all the points (x, x 2 ) that are transformed as T (p,q): (x, x 2 ) > (x + p, x 2 + q). Probably the hardest step for students to comprehend is that points (x + p, x 2 + q) fit the equation x 2 + q = (x + p p) 2 + q. Or taking (X, Y)=(x + p, x 2 + q), one has Y = (X p) 2 + q. This idea can be used to explore the fact that different transformations transform parabolas to parabolas. Also for example, a dilation D O,k: (x, y) > (kx, ky) transforms the graph of a quadratic function y = ax 2 + bx + c to the graph of a new quadratic function with related coefficients as follows:. 3. Making connections to other math topics a) If and when students are familiar with vectors, transformations (especially translations) can be represented by the use of vectors. The connection deepens the comprehension of both topics. For example, a translation right 2 units and up 3 units can be represented as simply adding the vector (2, 3). Therefore previous notation T a,b:(x, y) >(x + a, y + b) can be described simply in terms of :, where, and,. See example 4 as an exercise. b) If and when students are familiar with complex numbers, as with vectors, translations can simply be presented as addition of complex numbers. The above translation would be described as :. A dilation can be represented as simple multiplication of complex numbers by real numbers. That is a dilation with center at the origin D O,k (x, y) (kx, ky) can be represented in the form D O,k: x + iy kx + iky. A reflection over the x axis is given by conjugation. Complex numbers provide also a powerful tool to 3

4 represent rotations. For example, a rotation by 90 o about the origin is simply multiplication by the imaginary unit i. If complex numbers are known to students in more advanced levels using polar form, the rotation of an object through angle φ around the origin can be represented by multiplication by the complex number cos(φ) + i sin(φ) or, if the notation is known to students, by the number e iφ. c) If and when students are familiar with matrices, it is easy to show that a dilation with center at the origin can be presented in matrix form as 0 0 where the dilation previously noted as D O,k is now presented as an operator 0. Further, a rotation about the origin through the angle φ is given by the 0 matrix cosφ sinφ sinφ cosφ. Composition of all mentioned transformations, except translations, can now be efficiently represented by a product of matrices. Translations cannot be represented by multiplication of 2 by 2 matrices, as matrices represent linear transformations that always preserve the origin, while translations do not. However, translations can be represented as addition of 2 x 1 matrices in the form 4. (Additionally, 3 x 3 matrices can be used to depict all isometries as seen in step 4 below.) d) Examples 5 and 6 help students acquire deeper understanding of geometric transformations. e) Example 7 shows how an algebraic problem can be solved by the means of geometric transformations. f) Example 8 is geometric and is usually solved by students with the use of trigonometric equations that follow from geometric properties. However, transformations offer an elegant solution. 4. Extended perspective of transformations using 3 x 3 matrices Geometric transformations in 2D can also be considered using special cases of 3 by 3 matrices. For example, the matrix below represents a rotation of φ through the origin.

5 cos() sin() 0 sin() cos() Using homogeneous coordinates, first discussed by August Ferdinand Möbius in 1827, a point with coordinates (x, y) may be identified as the triple (x, y, 1) so that the matrix dimensions work and the matrix multiplication can be accomplished as follows: cos() sin() 0 sin() cos() x y 1 x cos() ysin() xsin() ycos() 1 A primary reason for using 3 x 3 matrices with geometric transformations is that translations do not have to be treated separately with a different form of matrix and can be composed with any other transformations simply by multiplying matrices. For example using 3 x 3 matrices, the translation T A, B: (x, y) >(x + a, y + b) is seen below where a general point in the plane has coordinates (x, y, 1): 1 0 a 0 1 = Transformations may also be considered within their algebraic group structures with the operation of composition. For example, isometries all have inverses, have an identity and the property of associativity, providing the structure of a group. Further, considering all the rotations about the origin and all the reflections over a line through the origin, the subgroup of all orthogonal transformations is obtained. In the notion of matrices, this group is defined by all orthogonal matrices, that is to say that each row and each column in the matrix considered as a vector has length 1 and is orthogonal (perpendicular) to the other one. All these matrices have determinant 1 or 1. Composition of transformations is not commutative in general as students can easily see by construction within the geometric interpretation or can be seen by considering the matrix product. Examples: 1. Find the area of a square with diagonal 6 cm in Figure 1. 5

6 Figure 1: Square with diagonal 6 cm While the problem can easily be solved by the means of the Pythagorean theorem, the problem can also be solved by splitting the square into four isosceles right triangles, each with legs 3 cm, and therefore the area is 18. An elegant solution can also be obtained by the use of geometric transformations. As Figure 2 below indicates, the rotation of the triangle (half of the square) produces a bigger square with the side 6 cm, and half of this square's area is exactly the area of the original square. Therefore the area is 18. Figure 2: Transformation solution to the area of a square whose diagonal is 6 units 2. Inscribe a square in any given triangle, so that one side of the square lies on one of the sides of the triangle. The problem can easily be solved by dilation. Draw 'a small square' as seen in Figure 3 below. By a dilation (line through upper right corner) enlarge the square as seen in the figure. The proof of the construction follows from the fact that corresponding sides are proportional. 6

7 Figure 3: Triangle containing a square with a side on the triangle 3. An equilateral triangle sits on a square (all the sides are equal in length, say equal to a) as seen in Figure 4. Find the radius of the circle circumscribing the triangle ABC. Figure 4: Square with equilateral triangle atop The problem is not trivial and (advanced) students would usually find a solution using trigonometric equations obtained from the geometric properties of the figure. Geometric transformations offer an elegant and simple solution. By the use of a translation the triangle is moved down as seen in Figure 5. Figure 5: Square with triangle atop with a translation 7

8 It is obvious that the 'move' CD of the length a where also AD = BD = DC = a. Therefore the radius of the circumscribed circle is a. 4. Consider the composition of two 180º rotations in different centers, A and B. Let the two points A and B in plane be given as in Figure 6. If R A, 180º and R B, 180º are the rotations in points A and B respectively, then what is R B, 180 R A, 180? By the use of vectors, R A, 180 (X) 2A X using the equivalence X X O. So, R B, 180 R A, 180 (X) R B, 180 (2A X) 2B (2A X) X 2(B A) Therefore, the composition of two 180º rotations in points A and B is simply a translation of 2. Figure 6: Composition of 180º rotations in two points A and B 5. Show that any rigid transformation can be obtained by composition of at most three reflections over lines. The sketch of the proof of the statement is presented in the following sequence of arguments. a) Note, that any rigid transformation is given by image of three noncollinear points. Therefore, assume a rigid motion transforms triangle ABC to congruent triangle A'B'C' as seen in Figure 7(a) below. b By reflection over the bisector of CC' (assume ), triangle ABC is transformed onto triangle. Note that triangles A'B'C' and are congruent, as in Figure 7(b). c Similarly by reflection over the bisector of, triangle is transformed onto triangle. Note that triangles A'B'C' and are congruent. d By reflection over the bisector of, triangle is transformed onto triangle A'B'C'. 8

9 e Therefore, by the composition of the three reflections the triangle ABC is transformed onto triangle A'B'C', which proves that any rigid transformation can be obtained by the composition of at most three reflections. (a) (b) Figure 7: Reflections in lines 6. Use the notions in Example 5 to show that any isometry is either a reflection, translation, rotation or glide reflection. The proof here uses Example 5. With Example 5, we know that any isometry can be written as the composition of no more than 3 reflections in lines. If there is only one line needed, then the isometry is clearly a reflection. If two lines of reflection are used, then either the two lines are parallel in which case it is a translation, or the lines intersect in which case it is a rotation about the point of intersection. In the case that three lines are required, it may take some time to show this with cases, but it can be shown that it is a glide reflection. 7. Find the number of solutions of the system of equations for specific values of a. 1 The geometric representations of the two equations produce Figure 8 below. The circle with center (a, 0) and radius 1 corresponding to the second equation can be visualized as translated from right to left, and the number of its intersections with the two lines corresponding to the first equation will give the number of solutions of the system of equations. It can be deduced that there are two solutions when 2, that there are three solutions when 1, four when 2 and 1 and none when 2. 9

10 Figure 8: Circle with two lines Contributing authors Bjorkqvist, Ole; Abo Akademi University,Vasa, Finland Castellón, Libni Berenice; Universidad Pedagógica Nacional Francisco,Tegucigalpa, Honduras Govender, Vasuthavan (Nico); University of Fort Hare, Alice,Port Elizabeth, South Africa Hernendez, Maria; North Carolina School of Science and Mathematics, Durham, North Carolina, USA Kobal, Damjan; University of Ljubljana, Ljubljana, Slovenia Kejzar, Bogdan; Gimnazija Kranj, Kranj, Slovenia Lott, Johnny; Retired, University of Montana, Oxford, Mississippi, USA Li, Shiqi; East China Normal University, Shanghai, China Osterberg, Leif; Korsholms Gymnasium, Kvevlax, Finland Sambo, Jobe Nhlanhla; Dlumana High School, Hluvukani, South Africa Soto, Luis; Colonia Miraflores Sur, Tegucigalpa, Honduras Tao, Yexin (Madeline); Weiyu High School, Shanghai, China 10