Transformational Geometry in the Coordinate Plane Rotations and Reflections

Transcription

1 Concepts Patterns Coordinate geometry Rotation and reflection transformations of the plane Materials Student activity sheet Transformational Geometry in the Coordinate Plane Rotations and Reflections Construction sheet Directions for Constructing a Circle, Drawing Radii, and Creating a Rotation of a Point on the Circle Voyage 200/TI-92 Plus Introduction Two transformations of the plane that have many applications in algebra as well as in other disciplines are rotations and reflections. These transformations are known as rigid motion transformations because the angles and lengths of segments of the original object remain the same in the transformed figures. In this activity you will investigate changes in the coordinates of points when they undergo reflective and rotational transformations and observe the resulting patterns. There are three investigations in this activity that require you to complete drawings using the geometry program on the Voyage 200/TI-92 Plus. If you are unfamiliar with the geometry program on the Voyage 200/TI-92 Plus, work through the special section of the activity, Directions for Constructing a Circle, Drawing Radii, and Creating a Rotation of a Point on the Circle, first.. In the first investigation you will rotate a single point and in the third investigation you will rotate an object. The second investigation involves reflecting a point across the x- and y-axes and comparing the result to a rotation. PTE: Dynamic Geometry Page 1

2 Student Activity Sheet Investigation #1: Rotation Follow the instructions #1-5 on the Directions for Constructing a Circle, Drawing Radii, and Creating a Rotation of a Point on the Circle sheet to complete the drawing in Figure 1. a. Open a new TI-92 Geometry figure. b. Display a rectangular coordinate axes and define a grid that has 1 as the unit of the grid. c. Draw and label a point P in the first quadrant at a grid point where the x-coordinate is not equal to the y-coordinate. Label point P with its coordinates. d. Construct a circle centered at the origin. e. Construct a radius of the circle to point P. f. Rotate point P about the origin through a 90 angle. Label the rotated point as P. Label point P with its coordinates. g. Construct a radius of the circle to point P. h. Mark the angle created by point P, the origin, and point P. Figure 1 After you have completed the drawing, answer the following questions. 1. Compare the coordinates of point P with the coordinates of point P. Drag point P around the screen and check your conjecture. Give two conjectures. 2. Fill in Table 1 with the absolute values of the x- and y-coordinates of two other points P and P after dragging point P around. How do the table values relate to your conjectures? Table 1: Sum of Absolute Values of X- and Y-Coordinates of Points P and P Point P Point P x- y- Sum of Coordinates x- y- Sum of Coordinates coordinate coordinate coordinate coordinate PTE: Dynamic Geometry Page 2

3 3. Keep point P at the same location and change the angle of rotation. Use Numerical Edit (under the F7: DISPLAY toolbar menu) to change the angle of rotation of point P by 10 at a time. Rotate point P through angles of 180, 270, and 360. Record your results in Table 2. Table 2: Coordinates of Rotation of P (2,1) Through Given Angles of Rotation Angle of Coordinates of P Angle of Coordinates of P Rotation Rotation 90 (-1, 2) (-1.33, 1.8) What do you notice about the coordinates of P compared to P for these angles of rotation? For which angle of rotation do the coordinates of P and P have the same value but opposite signs? PTE: Dynamic Geometry Page 3

4 4. Drag point P to different locations such as to the points (4,2), (3, -2), and (-3, -1). Repeat the directions for #3. Record your results in Table 3. Table 3: Coordinates of Rotation of Point P by Multiples of 90 Angle of P (x, y) for P (4, 2) P (x, y) for P(3, -2) P (x, y) for P -3, -1) Rotation What do you notice about the coordinates of P compared to P at these angles of rotation? What conclusion can you draw from the table entries? PTE: Dynamic Geometry Page 4

5 Investigation #2: Reflection Delete all objects in the window except for point P. 1. Place a point P on a grid point in the first quadrant where the coordinates are not the same. Use the Reflection tool under the Transformations toolbar menu (F5) to a. reflect point P across the y-axis to point R b. reflect point R across the x-axis to point R c. reflect point R across the y-axis to point R Show the coordinates of the points on the screen. Move the coordinates out of the way of the points, if necessary. See Figure 2. Figure 2 2. Explain where the reflection points R, R, and R are located in relationship to point P and to each other and why. 3. Drag point P around the screen and make a conjecture about the location of the reflection points. Based on the results, fill-in Table 4 Table 4: Coordinates of Reflected Points Given Point P Trial# P(x,y) R(x,y) R (x,y) R (x,y,) How do the coordinates of the reflection points R, R, and R compare to the coordinates of point P? 4. Rotate point P through a 90 angle centered at the origin. Compare the location of the rotated point P to the location of the reflection points R, R, and R. HINT: Change the angle of rotation to locate point P. 5. Make some conjectures about any relationships you observe between the reflected points (R, R, and R ), the rotated point (P ), and the point P. Explain why they might be true. Explain any special cases and why they are true. PTE: Dynamic Geometry Page 5

6 Investigation #3: Rotation Hide all coordinates for the points P, R, R, and R. Move point P to the first quadrant. 1. Construct a triangle in the first quadrant with point P as one vertex and the other two vertices on grid points (Figure 3). Figure 3 2. Use the Reflection tool under the Transformations toolbar menu (F5) to a. reflect triangle P across the y-axis to triangle R b. reflect triangle R across the x-axis to triangle R c. reflect triangle R across the y-axis to triangle R See Figure 4. Figure 4 3. Rotate triangle P around the origin using the angle measure of 90. Label the triangle P. Change the angle measure using Numerical Edit from the DISPLAY toolbar menu (F7) for rotations of 180, 270, and 360. For what angle measure does the rotated triangle P match one of the reflected triangles? Which triangle? 4. How do these results of rotating triangle P compare to the results of rotating point P in Activity #2? 5. Free each of the vertices of triangle P from the grid. Move triangle P around the screen. Explain whether or not the results from question #3 change. Reflective Analysis: Using the results of the three investigations you completed, summarize your conclusions about reflections and rotations. PTE: Dynamic Geometry Page 6

7 Directions for Constructing a Circle, Drawing Radii, and Creating a Rotation of a Point on the Circle 1. Open a new TI-92 Plus Geometry figure in the Geometry application found under the APPS menu. a. Select New and press ENTER (Figure 1). b. Press ENTER, cursor down and type reflec1 in the Variable line, and press ENTER (Figure 2). At this time you should have a blank working space on the screen. Figure 1 Figure 2 2. Display a rectangular coordinate axes and define a grid. a. Select the FILE toolbar menu (F8) from the menu at the top of the screen, select 9: Format (Figure 3), and press ENTER (Figure 4). b. Move the right directional key to open the menu for Coordinate Axes, then use the down directional key to select 2:Rectangular, and press ENTER (Figure 5). Repeat the steps to turn on the Grid (Figure 5). Press ENTER to get back to the main screen. Figure 3 Figure 4 Figure 5 3. Drag a positive x-axis unit toward the origin to change the scale of the screen such that 1 is the unit of the grid on the screen. a. Move the cursor until it reads THIS UNIT and press ENTER (Figure 6). b. Hold down the Hand key and move the left directional key until the labels on both axes change to 1. Release the Hand key and the left directional key (Figures 7 and 8). Figure 6 Figure 7 Figure 8 4. Draw and label point P in the first quadrant at a grid point where the x-coordinate is not equal to the y-coordinate. Show the coordinates of point P. a. Select the POINTS AND LINES toolbar menu (F2). Select 1: Point and then press ENTER (Figure 9). b. Move the cursor to a point in the first quadrant using the right-left-up-down directional keys and then press ENTER (Figure 10). PTE: Dynamic Geometry Page 7

8 c. Select the MEASUREMENT toolbar menu (F6). Select 5: Equation & Coordinates and press ENTER (Figure 11). d. Move the cursor to the point. When the screen says, COORDINATES OF THIS POINT (Figure 12), press ENTER to have the coordinates appear on the screen. e. Select the DISPLAY toolbar menu (F7), select 4: Label, and press ENTER (Figure 13). Press ENTER. Move the cursor until the message THIS POINT is on the screen and press ENTER to get the text box. Type a capital P by holding down the up arrow key (found on the QWERTY keyboard above the ON key) and typing a P (Figure 14). Point P can be moved by selecting the point, grabbing it with the Hand, and using the directional keys to move it to a new location. Figure 9 Figure 10 Figure Construct a circle centered at the origin and attached to point P. Construct a radius of the circle to point P. a. Select the CURVES AND POLYGONS toolbar menu (F3), select 1: Circles, and press ENTER (Figure 15). Move the cursor to the origin. Stop when the message THIS CENTER POINT is on the screen; press ENTER (Figure 16). Move the cursor to point P to get the message THIS RADIUS POINT ; press ENTER (Figure 17). Figure 12 Figure 13 Figure 14 b. Select the POINTS AND LINES toolbar menu (F2), select 5: Segment, and press ENTER (Figure 18). Move the cursor to point P to read the message THIS POINT and press ENTER (Figure 19). Move the cursor to the origin to read the message THIS POINT and press ENTER (Figure 20). Figure 15 Figure 16 Figure 17 PTE: Dynamic Geometry Page 8

9 Figure 18 Figure 19 Figure Rotate the image of point, P, 90 around the origin. a. Select the DISPLAY toolbar menu (F7), select 6: Numerical Edit, and press ENTER (Figure 21). Move the cursor to any position in the third quadrant, press ENTER, type 90 in the numerical edit box, and press ENTER (Figure 22). b. Select the TRANSFORMATION toolbar (F5), select 2: Rotation, and press ENTER (Figure 23). Figure 21 Figure 22 Figure 23 c. The Rotation tool takes three arguments: the point P, the point of rotation, and the angle of rotation. Move the cursor to point P to read the message ROTATE THIS POINT and press ENTER (Figure 24). Move the cursor to the origin to read the message AROUND THIS POINT and press ENTER (Figure 25). Move the cursor to the numeral 90 to read the message USING THIS ANGLE and press ENTER. A point appears on the circle in the second quadrant. Label this point P following the directions in #4e above (Figure 26). The prime mark is found using the 2 nd key with the B. Drag the labels of P and P outside of the circle for easier reading. Figure 24 Figure 25 Figure 26 d. Draw another radii of the circle to point P following the directions in #5c above (Figure 27). e. Select the DISPLAY toolbar menu (F7), select 7: Mark Angle, and press ENTER (Figure 28). Move the cursor to point P, the origin, and point P ; press ENTER after each one (Figure 29). Figure 27 Figure 28 Figure 29 PTE: Dynamic Geometry Page 9

10 f. Follow the directions in #4c to label P with its coordinates (Figure 30). Figure 30 Delete all objects in a current window by selecting the object and pressing DEL. 7. To hide an object, select FILE toolbar menu (F8), select 1: Hide/Show, press ENTER, and then select object (Figure 31). Figure Construct a triangle by using the CURVES AND POLYGONS toolbar menu (F3), select 3: Triangle, and press ENTER. 9. Redefine the points of a triangle to be unattached to a grid a. Turn off grid by following directions in #2b. b. Select CONSTRUCTION toolbar menu (F4), select B: Redefine, and press ENTER (Figure 32). c. Select a point to be changed, select 1:Point, and press ENTER (Figure 33). d. Select the POINTER toolbar menu (F1), select 1: Pointer, select the point, grab the point with the Hand and drag it to a new location (Figure 34). Figure 32 Figure 33 Figure 34 PTE: Dynamic Geometry Page 10

11 Teacher Notes Introduction Two transformations of the plane that have many applications in algebra as well as in other disciplines are rotations and reflections. These transformations are known as rigid motion transformations because the angles and lengths of segments of the original object remain the same in the transformed figures. In this activity you will investigate changes in the coordinates of points when they undergo reflective and rotational transformations and observe the resulting patterns. There are three investigations in this activity that require you to complete drawings using the geometry program on the Voyage 200/TI-92 Plus. If students are unfamiliar with the geometry program on the Voyage 200/TI-92 Plus, have them work through the special section of the activity Directions for Constructing a Circle, Drawing Radii, and Creating a Rotation of a Point on the Circle first. Additional information is available from the Voyage 200/TI-92 Plus manual. In the first investigation students rotate a single point and in the third investigation they rotate an object. The second investigation involves reflecting a point across the x- and y-axes and comparing the result to a rotation. Instructions The default screen scale for Cabri is adequate for this activity. The default screen of the Voyage 200/TI-92 Cabri Geometry is a bit small for integer coordinate points. By dragging an x-axis unit toward the origin, the scale of both the x- and y-axes are changed equally. This keeps the screen in a square coordinate system with and aspect ratio equal to 1. However, when the default screen of either program is changed, the coordinate system no longer matches the measurement system. The measurement system is always the same no matter how the coordinate system s scale is set. This is the same reason that the scale factor is computed when graphing functions interactively. Investigation #1: Rotation Starting this activity at an integer point in the first quadrant increases the chances for students to make connections easily. Starting at a point with the x-coordinate not equal to the y-coordinate avoids the special cases like a rotation of 90 being equal to a reflection over the y-axis. Constructing the circle and the two radii to points P and P help students visualize what a rotation means. Marking the angle also helps them focus on the angle of rotation and how it changes. The Mark Angle tool under the DISPLAY toolbar menu has a bad habit of jumping to the opposite side of an angle, especially when the angle becomes reflex. To change this, simply drag the mark back through the vertex of the angle. 1. In general, if the coordinates of point P are (x, y), then the coordinates of P rotated around the origin by 90 are (-y, x) (Figure 1). Encourage students to drag point P to different locations to check their conjectures for this and other rotations. Figure 1 2. A conjecture implied by the table values is that the sum of the coordinates for P equals the sum of the coordinates for its rotated point P. Values in Table 1 will vary depending on where point P is dragged to the different location than indicated. PTE: Dynamic Geometry Page 11

12 Table 1: Sum of Absolute Values of X- and Y-Coordinates of Points P and P Point P Point P x- y- Sum of Coordinates x- y- Sum of Coordinates coordinate coordinate coordinate coordinate Changing the angle of rotation by 10 at a step will make the exploration go smoother, especially on the Voyage 200/TI-92 Plus. In general, rotating the point P (x, y) through a 180 angle around the origin yields the point P (-x, y). This transformation is sometimes called a half-turn or reflection through a point. The Symmetry tool under the TRANSFORMATIONS toolbar menu (F5) executes this transformation by reflecting an object through a specified point. This fact may become more evident after students do Investigation #2. A rotation of point P (x, y) through a 270 angle around the origin yields the point P (y, -x). An interesting way to look at this rotation is as a 180 rotation from the point of a 90 rotation. A rotation of 180 reverses the sign of the coordinates, but not the position of the coordinates in each pair. If a 90 rotation gives the point (-y, x), then a 270 rotation would be the point (y, -x), which is the case (Figure 1). A rotation of 360 is the point itself. Table 2: Coordinates of P From Rotating P (2,1) Through Given Angles of Rotation Angle of Coordinates of P Angle of Coordinates of P Rotation Rotation 90 (-1, 2) 230 (-0.52, -2.17) 100 (-1.33, 1.8) 240 (-0.15, -2.23) 110 (-1.62, 1.54) 250 (0.26, -2.22) 120 (-1.87, 1.23) 260 (0.64, -2.14) 130 (-2.05, 0.89) 270 (1, -2) 140 (-2.17, 0.52) 280 (1.33, -1.8) 150 (-2.23, 0.13) 290 (1.62, -1.54) 160 (-2.22, -0.26) 300 (1.87, -1.23) 170 (-2.14, -0.64) 310 (2.05, -0.89) 180 (-2, -1) 320 (2,.17, -0.52) 190 (-1.8, -1.33) 330 (2.23, -0.13) 200 (-1.54, -1.62) 340 (2.22, 0.26) 210 (-1.23, -1.87) 350 (2.14, 0.64) 220 (-0.89, -2.05) 360 (2, 1) PTE: Dynamic Geometry Page 12

13 4. Dragging the point P to different locations on the screen produces the same results even if point P is not in the first quadrant. An interesting extension is to have students continue to rotate the point in a positive direction or to rotate it in a negative direction and extend their generalizations using the cyclic nature of this activity. Table 3: Coordinates of Rotation of Point P by Multiples of 90 Angle of P (x, y) for P (4, 2) P (x, y) for P (3, -2) P (x, y) for P (-3, -1) Rotation 90 (-2, 4) (2, 3) (1, -3) 180 (-4, -2) (-3, 2) (3, 1) 270 (2, -4) (-2, -3) (-1, 3) 360 (4, 2) (3, -2) (-3, -1) Investigation #2 1. Starting this investigation on a point with the x-coordinate not equal to the y-coordinate avoids the problem of special cases. These special cases will be noted, but not as the first instance of the relationship. 2. In general, reflecting point P (x, y) across the y-axis yields the point R (-x, y) for any location of point P. All reflections across the y-axis reverse the sign of the x-coordinate and leave the sign of the y-coordinate the same. If students have done any work with transformations of the graphs of functions, this relationship is the same as the comparison between f(x) and f(-x), a flip over the y- axis. 3. Answers may vary depending on the location point P is dragged to on the screen. Table 4: Coordinates of Reflected Points Given Point P Trial# P(x, y) R(x, y) R (x, y) R (x, y) 1 (4, 2) (-4, 2) (-4, -2) (4, -2) 2 (1, 3) (-1. 3) (-1, -3) (1. -3) 3 (-3, 4) (3, 4) (3, -4) (-3. -4) 4 (-2, -1) (2, -1) (2, 1) (-2. 1) 4. Point R and point P do not coincide showing that in general, a flip over the y-axis is not equivalent to a rotation of 90. However, if point P is located on the line y = x, then P and R are the same point since the angle between the positive x-axis and the radius to point P is Reflecting the point R (-x, y) across the x-axis yield the point R (-x, -y). Any reflection across the x-axis leaves the x-coordinate the same and reverses the sign of the y-coordinate. In this case, since R is the product of two reflections across the y- and x-axis, the signs of both coordinates are reversed. It may become apparent to students that this double reflection produces the same result as a rotation of 180, especially if the radii of the circle to the reflection points are drawn (Figure 2). This is the same relationship that is called a reflection through the origin as with the graph of an odd function like f(x) = x 3 or f(x) = sin (x). The same transformation can be done using the Symmetry tool from the TRANFORMATIONS (F5) toolbar menu. Figure 2 PTE: Dynamic Geometry Page 13

14 In general, rotations of point P by 90 or 270 do not coincide with reflections of P across the x- and/or y-axes. Rotation of point P 180 around the origin yields the same point as two reflections across the x- and y-axes in either order. Figure 3 shows that points R and P coincide. Drag point P anywhere on the screen and these relationships remain true. Rotations of 90 or 270 correspond to reflections over the axes only when P is on the line y = x or y = -x. Figure 3 Investigation # This investigation displays an important property of reflections not obvious with individual points. The orientation of a reflection figure is reversed compared to the original. The orientation of a figure after two reflections (or any even number of reflections) is the same as the original. The orientation of a rotated figure is the same as the original no matter what the size of the angle of rotation. 3. A rotated figure can only coincide with a figure reflected an even number of times, such as over the x- and y-axes. Figure 4 shows a triangle reflected over the axes and rotated 90, 180, and 270 around the origin. Only the 180 rotation coincides with two reflections over the axes. There are no special cases for the triangle like there are for a single point because of the orientation change for reflections. 4. A triangle, or any geometric shape or graph, can be thought of as a collection of individual points. Geometric figures can be defined by the coordinates of their vertices. In any case, since we can represent figures as points, each point of the figure reacts like any individual point. Rotations of the triangle through 90 and 270 around the origin do not coincide with reflections of the triangle over the axes (Figure 4). Figure 4 A rotation of 180 does correspond to two reflections over the x- and y-axes. 5. The results are the same as those in question #3. 6. Any vertex of a geometric figure has the same properties as a single point. In Figure 4 the vertex of the triangle at the point (2,1) is reflected and rotated showing these properties. A circle centered at the origin and passing through the point (2,1) also passes through all of the reflected and rotated points. PTE: Dynamic Geometry Page 14