Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Rotation formalisms in three dimensions wikipedia , lookup
Plane of rotation wikipedia , lookup
Duality (projective geometry) wikipedia , lookup
Perspective (graphical) wikipedia , lookup
Derivations of the Lorentz transformations wikipedia , lookup
Rational trigonometry wikipedia , lookup
Cartesian coordinate system wikipedia , lookup
Rotation group SO(3) wikipedia , lookup
Euclidean geometry wikipedia , lookup
! ! Pomona Unified Math News Domain: 8 th Grade Geometry (G) 8.G.1: Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. Suggested Standards for Mathematical Practice (MP): MP 1: Make sense of problems and persevere in solving them. MP 5: Use appropriate tools strategically: students use tools such as Mira’s, patty paper, geometric software, rulers, compasses and graph paper. MP 6: Attend to precision: When graphing points or figures, students must be accurate with measurements, vocabulary, and calculations to ensure that the transformations are completed correctly. MP 7: Look for and make use of structure: Student’s develop/find patterns between a preimage and image of a figure when using geometric transformations. Vocabulary: (Note: vocabulary will be taught in the context of the lesson, not before or separate from the lesson.) Reflection: A transformation in which a geometric figure is reflected across a line, creating a mirror image. That line is called the axis or line of reflection. A reflection is often referred to as “flipping” rotation to a point and its image form an angle called the angle of rotation. (Notation Rdegrees ) Translation: A transformation in which every point of the pre-image is moved the same distance and in the same direction to form the image. 10 A' 8 6 B' A C' 4 B 2 C 5 10 In the translation above, ∆ABC is moved 3 units to the right and 4 units up to form the image ∆A’B’C’. In this translation the points (x, y) Line segment: The part of a line that connects two points. Parallel lines: Two lines in a plane that never intersect. They are always the same distance apart. Below, line AB is parallel to line CD, however line FE intersects or crosses through the two lines and is not parallel to either. Triangle ABC is reflected over the y-axis. The resulting image is triangle A’B’C’ We use the term “prime” when referring to the ‘, so the image is A prime B prime C prime. Rotation: A transformation that turns a figure around a fixed point known as the center of rotation. The figure may be rotated clockwise or counterclockwise. Rays drawn from the center of Angle measure: A numerical measure of an angle. In 8th grade, this measure is given in degrees. ! ! Pomona Unified Math News Domain: 8 th Grade Geometry (G) 8.G.1: Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. 2. Construct a segment from the axis of symmetry congruent to the segments from each vertex and perpendicular to the axis. A: $-5.00, 2.00+ B: $-2.00, 5.00+ C: $-3.00, -1.00+ B 4 A -10 Rigid transformation: a transformation that does not change the shape or size of the preimage. Reflections, rotations and translations are rigid transformations. Rigid transformations are also called isometries. -2 3. Label the endpoints of each segment with the letter that represents the vertex from which we started – but add a prime symbol to the letter. A: %-5.00, 2.00, B: %-2.00, 5.00, C: %-3.00, -1.00, A': %5.00, 2.00, B': %2.00, 5.00, C': %3.00, -1.00, B 6 B' 4 A A' 2 C' -5 C 5 -2 4. Connect the new vertices to form ∆ A’B’C’, the reflection of ∆ ABC over the y-axis. Reflections: How to reflect a triangle over the y-axis: A: $-5.00, 2.00+ B: $-2.00, 5.00+ C: $-3.00, -1.00+ B 6 4 A A: %-5.00, 2.00, B: %-2.00, 5.00, C: %-3.00, -1.00, A': %5.00, 2.00, B': %2.00, 5.00, C': %3.00, -1.00, -10 B 6 B' 4 A A' 2 2 -10 -5 C B 6 4 A 2 -5 5 C 5 -2 th -2 1. From each vertex, drop a perpendicular line to the y-axis (line of symmetry) A: $-5.00, 2.00+ B: $-2.00, 5.00+ C: $-3.00, -1.00+ C' -5 5 C -10 5 C -10 Examples and Explanations: 2 -5 Pre-image: the original point or figure that will undergo a transformation. Image: the resulting point or figure that is obtained after a transformation 6 -2 Reflections in 8 grade will also include those with the line of symmetry being the x-axis, the line y = x and the line y = -x. Questions to Ponder: • What happens to the coordinates of each vertex as it is reflected across the y-axis? • What would happen to the coordinates if you reflected the triangle across the x-axis? • What happens to the coordinates of each vertex as it is reflected across the line y = x? • What happens to the coordinates of each ! ! Pomona Unified Math News Domain: 8 th Grade Geometry (G) 8.G.1: Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. vertex as it is reflected across the line y = -x? Possible Responses: • Each coordinate (a, b) becomes (-a, b) when reflected over the y-axis. • Each coordinate (a, b) becomes (a, -b) when reflected over the x-axis. • Each coordinate (a, b) becomes (b, a) when reflected over the line y = x. • Each coordinate (a, b) becomes (-b, -a) when reflected over the line y = -x. Properties of Reflections over a line: 1. Distance - lengths of segments are not changed under a reflection: Eg. 𝐴𝐵 ≅ 𝐴′𝐵′ 2. Angle measures are not changed under a reflection: Eg.∠𝐴 ≅ ∠𝐴′ 3. Parallelism (parallel lines remain parallel): 4. Colinearity – points stay on the same line: Eg. Points A and B were on line AB and are now on line A’B” 5. Midpoint (midpoints remain the same in each figure) 6. Orientation -lettering order is NOT preserved. Order is reversed. Eg. ∆ ABC’s orientation changed in our example to ∆A’C’B’ when the angles are read in a counterclockwise order. Rotations: Below is a diagram of a series of rotations, beginning with ∆ABC. 8 Center: (1.00, 2.00. B' C' 6 B 4 A' C'' -5 2 A C A'' Center A''' B'' 5 -2 C''' B''' In the diagram above, begin with ∆ABC. The center of rotation is the point (1, 2). • First rotation: ∆ABC is rotated 90° (R90°) in a counterclockwise direction around the center to result in ∆ A’B’C’. • Second rotation: ∆ A’B’C’ is rotated 90° (R90°) in a counterclockwise direction around the center to result in ∆ A’’B’’C’’. • Third rotation: ∆ A’’B’’C’’ is rotated 90° (R90°) in a counterclockwise direction around the center to result in ∆ A’’’B’’’C’’’. Properties of Rotations around a point: 1. Distance - lengths of segments are not changed under a reflection: Eg. 𝐴𝐵 ≅ 𝐴′𝐵′ 2. Angle measures are not changed under a reflection: Eg.∠𝐴 ≅ ∠𝐴′ 3. Parallelism (parallel lines remain parallel): 4. Colinearity – points stay on the same line: Eg. Points A and B were on line AB and are now on line A’B” 5. Midpoint (midpoints remain the same in each figure) 6. Orientation -lettering order is preserved. Eg. ∆ ABC’s orientation under the first rotation remained in the same orientation ∆A’B’C’. When the angles are read in a counterclockwise order. How to rotate about the Origin: In the example below, we will rotate point C of ∆ABC about the origin, 90°, 180°, and 270°. Note: when we rotate a positive number of degrees, this means that the rotation is counterclockwise. ! ! Pomona Unified Math News Domain: 8 th Grade Geometry (G) 8.G.1: Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. B 6 B 6 4 4 2 A -5 C 2 A C O 5 -5 5 -2 -2 D -4 -4 -6 D: $-5.00, -2.00+ • First I construct a circle with center at the origin and point C on the circle. 4 2 • Point D (-5, -2) would be the coordinate of C’ if we rotated ∆ABC 180° about the origin. • Next we create a perpendicular to line DC through the Origin. B 6 A -5 C -6 • 5 B 6 -2 4 -4 2 -6 • A C O -5 Next I construct a line through point C and the center of the circle, point O. -2 D B 6 5 -4 4 2 D: $-5.00, -2.00+ A C O -5 5 -2 • -6 The points where this perpendicular line intersects the circle give us the coordinates of the image of point C if we rotate it 90° and 270° about the origin. -4 B 6 E: $-2.00, 5.00+ -6 • E The intersection of this line with the circle will be the image of point C under a rotation of 180°. 4 2 A C O -5 D 5 -2 -4 D: $-5.00, -2.00+ -6 F F: $2.00, -5.00+ ! ! Pomona Unified Math News Domain: 8 th Grade Geometry (G) 8.G.1: Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. Translation: • • • Point E (-2, 5) is the image of point C when ∆ABC is rotated 90° about the origin. Point F (2, -5) is the image of point C when ∆ABC is rotated 270° about the origin. What, if anything do you notice about these points? o C (5, 2) R90° becomes (-2, 5) o C (5, 2) R180° becomes (-5, -2) o C (5, 2) R270° becomes (2, -5) o What would C become under R360°? What we have done above is an example of determining a pattern. (MP 7) Example: Determine the coordinates of the image of points A and B be under the rotations listed below. Explain your reasoning. 1. 90° 2. 180° 3. 270° 4. 360° How to translate a figure: 4 B 2 C -5 5 A -2 -4 -6 To translate the triangle above a units to the right and b units up, we move each vertex a units to the right and b units up: (x, y) -> (x + a, y + b). For example, to translate this triangle 3 units right and 5 units down: A (-4, -1) -> (-4 + 3, -1 – 5) -> A’ (-1, -6) B (-2, -3) -> (-2 + 3, -3 – 5) ->B’ (1, -8) C (5, 2) -> (5 + 3, 2 – 5) -> C’ (8, -3) Possible Response 4 B 2 1. Under a rotation of 90° the image points would be as follows: • A (2, 2) -> A’ (-2, 2) • B (5, 6) -> B’ (-6, 5) 2. Under a rotation of 180° the image points would be as follows: • A (2, 2) -> A’ (-2, -2) • B (5, 6) -> B’ (-5, -6) 3. Under a rotation of 270° the image points would be as follows: • A (2, 2) -> A’ (2, -2) • B (5, 6) -> B’ (6, -5) 4. Under a rotation of 360° the image points would be as follows: • A (2, 2) -> A’ (2, 2) • B (5, 6) -> B’ (5, 6) C -5 5 A -2 10 B' C' -4 A' -6 Properties of Translations: 1. Distance - lengths of segments are not changed under a translation: Eg. 𝐴𝐵 ≅ 𝐴′𝐵′ 2. Angle measures are not changed under a translation: Eg.∠𝐴 ≅ ∠𝐴′ 3. Parallelism (parallel lines remain parallel): 4. Colinearity – points stay on the same line: Eg. Points A and B were on line AB and are now on line A’B’. 5. Midpoint (midpoints remain the same in each figure) continued on next page ! ! Pomona Unified Math News Domain: 8 th Grade Geometry (G) 8.G.1: Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. 6. Orientation -lettering order is preserved. Eg. ∆ ABC’s orientation under our transformation remained in the same orientation ∆A’B’C’, when the angles are read in a counterclockwise order. • • More Explanations and Examples: Example 1: Triangle ABC was reflected over the x-axis. Determine whether the Image is a true reflection of the pre-image. Support your reasoning. • • 6 B (2,4) 4 2 • C (4,2) A (-2,1) -5 A' (-2,-1) 5 -2 -4 C (4,-2) • B' (2,-4) -6 Possible Solution: I believe that triangle ABC was correctly reflected over the x-axis to create triangle A’B’C’. When reflecting over the x-axis the y-values of each point switch signs, while the x-values remain the same. Also, after being reflected, each point in the image is the same distance from the line of symmetry (line of reflection) that it is from the original image. Important Points: • Students use patty paper, compasses, protractors and rulers or technology to explore figures created from translations, reflections and rotations. (MP 5) • Characteristics of figures, such as lengths of line segments, angle measures and parallel lines are explored before the transformation (pre-image) and after the transformation (image). • • • Students understand that rigid transformations produce images of exactly the same size and shape as the pre-image and are known as isometries. (MP 1) Students need multiple opportunities to explore the transformation of figures to appreciate that points stay the same distance apart and lines stay at the same angle after they have been rotated, reflected, and/or translated. (MP 1) Students are not expected to work formally with properties of dilations until high school. Students appropriately label figures, angles, lines, line segments, congruent parts, and images (primes or double primes). (MP 6) Students use logical thinking, expressed in words using correct terminology. They are NOT expected to use theorems, axioms, postulates or a formal format of proof as in two-column proofs. (MP 1, 6) The concepts behind each type of transformation and the effects that each transformation has on an object should be stressed before working within the coordinate system. (MP 2) Discussions include the description of the relationship between the original figure and its image(s) in regards to preserving size and the description of movement, including the attributes of transformations (line of symmetry, distance to be moved, center of rotation, angle of rotation and the amount of dilation). (MP 7) Students observe and discuss such questions as “What happens to the polygon?” and “How does the transformation affect the vertices?” Understandings should include generalizations about the changes that maintain size or maintain shape, as well as the changes that create distortions of the polygon (dilations). Students describe the transformations required to go from an original figure to a ! ! Pomona Unified Math News Domain: 8 th Grade Geometry (G) 8.G.1: Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. • • • transformed figure (image). Provide opportunities for students to discuss the procedure used, whether different procedures can obtain the same results, and if there is a more efficient procedure to obtain the same results. (MP 3, 7, 8) Students learn to describe transformations with both words and numbers. (MP 6) Through understanding symmetry and congruence, conclusions can be made about the relationships of line segments and angles within figures. Students relate rigid motions to the concept of symmetry and to use this to prove congruence or similarity of two figures. (MP 2) Provide opportunities for students to physically manipulate figures to discover properties of similar and congruent figures, for example, the corresponding angles of similar figures are equal. Additionally use drawings of parallel lines cut by a transversal to investigate the relationship among the angles. (MP 3) Common Misconceptions: • • Students confuse the rules for transforming two-dimensional figures because they rely too heavily on rules as opposed to understanding what happens to figures as they translate, rotate, reflect, and dilate. It is important to have students describe the effects of each of the transformations on twodimensional figures through the coordinates but also through the visual transformations that result. By definition, congruent figures are also similar. It is incorrect to say that similar figures are the same shape, such as a triangle, just a different size. This thinking leads students to misconceptions such as that all triangles are similar. It is important to add to that definition, the property of proportionality among similar figures. • • • • • • Student errors with the Pythagorean theorem may involve mistaking one of the legs as the hypotenuse, multiplying the legs and the hypotenuse by 2 as opposed to squaring them, or using the theorem to find missing sides for a triangle that is not right. Students often confuse situations that require adding with multiplicative situations in regard to scale factor. Providing experiences with geometric figures and coordinate grids may help students visualize the different. Students have difficulty differentiating between congruency and similarity. They assume any combination of three angles will form a congruence condition. Students also have problems recognizing congruent figures because of different orientations. Students confuse terms such as clockwise and counter-clockwise. Students often believe that the line of reflection must be vertical or horizontal (e.g. across the y- axis or x-axis. Reflections can occur over any line. Students do not realize that rotations can be about any point. Web Help Links (Use a QR scanner to take you directly to the website) Vocabulary Cards http://www.virtualnerd.com/middlemath/integers-coordinate-plane/transformations/ ! ! Pomona Unified Math News Domain: 8 th Grade Geometry (G) 8.G.1: Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. Pearson Lessons: Translations http://www.phschool.com/atschool/academy123/ english/academy123_content/wl-book-demo/ph092s.html http://www.phschool.com/atschool/academy123/ english/academy123_content/wl-book-demo/ph438s.html Reflections http://www.phschool.com/atschool/academy123/ english/academy123_content/wl-book-demo/ph889s.html Rotations http://www.phschool.com/atschool/academy123/ english/academy123_content/wl-book-demo/ph440s.html