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Unit 12 Transformations • This unit addresses transformations in the Coordinate Plane. • It includes transformations, translations, dilations, reflections, rotations, symmetry (including line/plane and rotational), compositions of reflections in parallel lines, glide reflections, and the fundamental theorem of Isometries. • It also includes tessellations, symmetry in tessellations, and proportions of dilations.
• • • • • • • Standards SPI’s taught in this unit: SPI 3108. 1. 1 Give precise mathematical descriptions or definitions of geometric shapes in the plane and space. SPI 3108. 1. 3 Use geometric understanding and spatial visualization of geometric solids to solve problems and/or create drawings. SPI 3108. 2. 2 Perform operations on vectors in various representations. SPI 3108. 3. 1 Use algebra and coordinate geometry to analyze and solve problems about geometric figures (including circles). SPI 3108. 3. 2 Use coordinate geometry to prove characteristics of polygonal figures. SPI 3108. 3. 3 Describe algebraically the effect of a single transformation (reflections in the x- or y-axis, rotations, translations, and dilations) on twodimensional geometric shapes in the coordinate plane. SPI 3108. 4. 10 Identify, describe, and/or apply transformations on two and three dimensional geometric shapes. CLE (Course Level Expectations) found in Unit 12: CLE 3108. 1. 1 Use mathematical language, symbols, definitions, proofs and counterexamples correctly and precisely in mathematical reasoning. CLE 3108. 1. 4 Move flexibly between multiple representations (contextual, physical written, verbal, iconic/pictorial, graphical, tabular, and symbolic), to solve problems, to model mathematical ideas, and to communicate solution strategies. CLE 3108. 1. 5 Recognize and use mathematical ideas and processes that arise in different settings, with an emphasis on formulating a problem in mathematical terms, interpreting the solutions, mathematical ideas, and communication of solution strategies. CLE 3108. 2. 2 Explore vectors as a numeric system, focusing on graphic representations and the properties of the operation. CLE 3108. 4. 7 Apply the major concepts of transformation geometry to analyzing geometric objects and symmetry. CFU (Checks for Understanding) applied to Unit 12: 3108. 1. 5 Use technology, hands-on activities, and manipulatives to develop the language and the concepts of geometry, including specialized vocabulary (e. g. graphing calculators, interactive geometry software such as Geometer’s Sketchpad and Cabri, algebra tiles, pattern blocks, tessellation tiles, MIRAs, mirrors, spinners, geoboards, conic section models, volume demonstration kits, Polyhedrons, measurement tools, compasses, Penta. Blocks, pentominoes, cubes, tangrams). 3108. 1. 10 Use visualization, spatial reasoning, and geometric modeling to solve problems. 3108. 1. 11 Identify and sketch solids formed by revolving two-dimensional figures around lines. 3108. 2. 4 Add vectors graphically and algebraically. 3108. 2. 5 Multiply a vector by a scalar graphically and algebraically. 3108. 3. 5 Use mapping notation to identify the image of a transformation given the coordinates of the pre-image. 3108. 3. 6 Identify a transformation given its mapping notation. 3108. 4. 13 Locate, describe, and draw a locus in a plane or space (e. g. , fixed distance from a point on a plane, fixed distance from a point in space, fixed distance from a line, equidistant from two points, equidistant from two parallel lines, and equidistant from two intersecting lines). 3108. 4. 29 Extend the effect of a scale factor k in similar objects to include the impact on volume calculations and transformations 3108. 4. 31 Use properties of single transformations and compositions of transformations to determine their effect on geometric figures (e. g. reflections across lines of symmetry, rotations, translations, glide reflections, and dilations). 3108. 4. 32 Recognize, identify and apply types of symmetries (point, line, rotational) of two- and three- dimensional figures. 3108. 4. 33 Use transformations to create and analyze tessellations and investigate the use of tessellations in architecture, mosaics, and artwork. 3108. 4. 34 Create and analyze geometric designs using rigid motions (compositions of reflections, translations, and rotations).
Definitions • Transformations -A change in a geometric figure’s position, shape, or size. • Preimage -This is the original figure before it is transformed • Image -This is the resulting image after it is transformed • Isometry -A transformation in which the preimage and the image are congruent. Examples of transformations which would result in isometry would include: flips, slides, and turns
Examples • Flip • Slide • Turn -
Transformation Maps • A transformation maps a figure onto its image and may be described with an arrow notation: • Prime notation: (‘) is sometimes used to identify image points. In this diagram, K’ is the image of K (K K’) K’ K J Q Q’ J’
Reflections • A reflection -or flip- is an isometry in which a figure (preimage) and its image have opposite orientations. Thus, a reflected image in a mirror appears backwards. • Here, triangle ABC is reflected along a line to produce triangle A’B’C’. Since the reflection is isometry, triangle ABC is congruent to triangle A’B’C’ A A’ B C C’ B’
Properties of Reflections • Below is a reflection in line r, where the following properties are true: Line of reflection B r A=A’ B’ • If a point A is on line r, then the image of A (remember, the “after transformation” creates the image) is A itself that is, A = A’ In other words, the reflection of a point is the same point • If a point B is not on line r, then line r is the perpendicular bisector of the line segment B/B’
Quiz (50 Points) 1. 2. 3. 4. 5. 6. 7. 8. 9. What is a Transformation? What is a Pre-Image? What is an Image? What is Isometry? Give an example of Isometry What is the name for this symbology? __’__ What is a reflection? What is one property concerning reflections and points? What is the second property concerning reflections and points? 10. Are a Flip and a Slide the same type of Transformation?
In Class Assignment • Text, page 636 -637 1 -11 • You may hand draw # 10 and #11, as long as the images are correct, and all points are labeled properly • Workbook, page 144 # 2 -8 • 20 Points
Translation • A translation (or slide) is an isometry that maps all points of a figure the same distance in the same direction. Thus you can use a vector to describe a translation. • Mr Bass’ definition of a vector (for translations): an ordered pair (x, y) that tells me where to go from where I am
Translation Example • What if you had a point A, at coordinate (-3, -1), and you were told to use the vector (6, 6) to find the image of A (we’ll call it A’). What is the coordinate of A’? • Use the vector in the same manner you use slope. So you would go right 6, and up 6 (rise over run). • The new coordinates would thus be (3, 5) A’ (3, 5) A(-3, -1)
Another way to work with Translations • You may see a pre-image, and an image, and be asked to identify the vector. Again, this is no different that determining slope. • From A (-3, 3) to A’ (3, 5) you must go right 6, and up 2, so the vector would be (6, 2) A’ A (3, 5) (-3, 3)
Next Assignment • Page 643, 1 -21
Rotations • A rotation occurs around a point -like the center of a circle. These hold true: – You need to know the center of the rotation -we’ll call this a point such as r. After the rotation (the image after the pre-image) this point gets renamed -such as r’ – You need to know the angle of rotation -this will be a positive number of degrees– You need to know whether the rotation is clockwise or counterclockwise – Unless stated otherwise, rotations in the book are counterclockwise
Rotations Continued • Having this information a rotation is conducted by: • Rotating x degrees about a point (such as r) where a transformation occurs such that: • The image of point r is itself (that is, r = r’) • For any point (such as V), r. V’ = r. V (the distance of each segment) and the measure of Vr. V’ = x degrees V’ r and r’ X degrees V
Drawing a Rotation • Example: Draw the image (after picture) of triangle LOB for a 100 degree rotation about C. • Mark a point (i. e. ) C. Draw a line from C to a point on triangle LOB (i. e. O). • Measure 100 degrees on the protractor from that line, and draw another line
Drawing a Rotation Continued • Use a compass to measure the distance from C to the point on the line. • Use the compass to mark the same distance on the second line • Do the same for each succeeding corner (mark a 100 degree angle and mark the distance) • It looks like this….
Draw a Rotation : ) C 100 degrees O B’ O’ L B L’
Assignment • Page 650, 10 - 16 • Page 651 35, 37, 38
Compositions of Reflections • Theorem 12. 1 - A translation or rotation is a composition of two reflections • Theorem 12. 2 - A composition of reflections in two parallel lines is a translation • Theorem 12. 3 - A composition of reflections in two intersecting lines is a rotation
Composition of Reflections in Parallel Lines • This image is reflected on line l. Then it is reflected on line m. The result is a a translation (slide). l m
Composition of Reflections in Intersecting Lines • This image is reflected in line A, then line B. The result is a rotation. a b c
Fundamental Theorem of Isometries • • • Theorem 12. 4 - In a plane, one of two congruent figures can be mapped onto the other by a composition of at most three reflections. Isometry Classification Theorem: Theorem 12. 5 - There are only four Isometries. They are: 1. 2. 3. 4. Reflection Translation (slide) Rotation Glide Reflection
Example of a Glide Reflection • Remember -First you glide, then reflect – Find the image of a triangle TEX for a glide reflection where the glide vector is (0, -5) and the reflection line is x = 0. T X E First, use the vector, and move (glide) down -5. Then, create a reflection based on the x = 0 line. (Numbers have been omitted for clarity of drawing). T’ X’ E’
Homework • Page 657 1 -9 ( on 4 -9, just draw the images, don’t describe them).
Symmetry • A figure has symmetry (is symmetrical) if there is an isometry that maps the figure onto itself. • If the isometry is the reflection of a plane figure, the figure has reflectional symmetry or line symmetry. • This is something most people already know (the idea of symmetry), they just haven’t applied it to geometry. – For example, a face has symmetry, if you draw a line of reflection down the middle -through the nose, chin and so on
Symmetry • A figure can have more than one line of symmetry. • Consider a hexagon: How many lines of symmetry are there in a hexagon? It looks like a lot, but there actually 6
Symmetry • How many lines of symmetry are there in this rectangle? There are only 2. Why aren’t there any for the corners?
Rotational Symmetry • Rotational Symmetry: Where a figure is it’s own image (after transformation) for some rotation of 180 degrees or less. For example, this equilateral triangle. The angle of rotation is 120 degrees. -if you rotate the triangle 120 degrees, you get the exact same image (isometry). 120 degrees
Rotational Symmetry • Would a square have rotational symmetry? If so, what would the angle of rotation be? • Would a rectangle have rotational symmetry? If so, what is the angle of rotation? • Would a hexagon have rotational symmetry? What would the angle of rotation be? • Does this figure have rotational symmetry? If so, what is the angle of rotation?
Point Symmetry • A figure that has point symmetry has 180 degree rotational symmetry. • Thinking back to the examples given (Square, Rectangle, Equilateral Triangle, unusual polygon), which had Point symmetry? • NOTE: 3 dimensional objects can have various types of symmetry, including rotational symmetry about a line, and reflectional symmetry in a plane. • A pencil could have rotational symmetry about a line. • A house reflected in a lake of water could have reflectional symmetry in a plane.
3 Dimensional Examples • Imagine a ping pong paddle. – Does it have rotational symmetry? – Does it have reflectional symmetry? • How about a coffee mug? • How about an umbrella?
Classwork • Page 664 1 - 19
Tessellations • Tessellation (or Tiling) occurs when you repeat a pattern of figures that completely covers a plane -without gaps or overlaps. • These tessellations can be created with translations, rotations, or reflections. • You see it in art, nature (honeycombs), and everyday life (tiled floors).
Example • Identify the transformation, and the repeating figure • It is a translation (slide) • The figure is a hexagon
Determining Figures that Tessellate • Because the figures in a tessellation do not overlap or leave gaps, the sum of the measures of the angles around any vertex must be 360 degrees. If the angles around a vertex are all congruent (they are) then the measure of each angle must be a factor of 360. • To determine whether a regular n-gon will tessellate, you must calculate one angle around it’s vertex. • Then you determine whether the angle is a factor of 360.
Determine the measure of one angle to determine Tessellation • A = (180(n-2))/n • For example, determine one angle for an 18 gon • A = (180(18 -2))/18 • A = 160 • Is 160 a factor of 360 degrees? No. Therefore there is no pattern possible where you can tessellate an 18 -gon, and have no overlaps or gaps
Examples • What about an equilateral triangle? • A = (180(3 -2))/3 ---or 60. Thus, one interior angle of an equilateral triangle is 60 degrees. • 60 is a factor of 360, so equilateral triangles do in fact tessellate.
Theorems and Ideas • A figure does not have to be a regular polygon to tessellate. It should be noted, however, that the method we used to determine tessellation only works on regular polygons • Theorem 12. 6 - Every Triangle (whether it is equilateral or not) tessellates • Theorem 12. 7 - Every quadrilateral tessellates (whether they are regular or not -it just has to be a 4 sided figure)
Symmetries in Tessellations • You will often find symmetries in Tessellations. • These would include – Line Symmetry – Rotational Symmetry – Glide Reflectional Symmetry – Translational Symmetry
Classwork • Page 670 1 -16 • Page 672 40 -44
Dilations • A dilation is a transformation whose preimage and image are similar. • This is the geometric definition of similar – Angles must have the same measure – Sides must be proportional – Therefore a dilation is a similarity transformation, but not, in general, an isometry
Properties of Dilations • Every dilation has a center, and a scale factor n, where n > 0. This scale factor describes the size change from the original figure to the image. • Generally, a dilation with a center (C for example), and a scale factor n is a transformation where the following hold true:
Properties of Dilations • The image (after picture) of C (the center) is itself. That is C = C’ • For any point R, R’ is on ray CR and CR’ = n x CR C = C’ R CR’ = 25 R’ • In this example, if n = 2. 5, and CR = 10, what is CR’? • The dilation is an enlargement if the scale factor is greater than 1. It is a reduction if the scale factor is between 0 and 1. • Is this example an enlargement or reduction?
Finding a Scale Factor • Here, the larger triangle is a dilation of the smaller one • The center is X. Because the image is larger than the pre-image, it is an enlargement T’ • X’T’/XT = (4+8)/4 = 3 8 T R’ R 4 • Therefore, the dilation has center X and a scale factor of 3. X=X’
Classwork • Page 676 1 -14