August 3 - August 31

Transcription

1 Mathematics Georgia Standards of Excellence Geometry Parent Guide Unit 1 A All About Our Unit of Study Transformations in the Coordinate Plane August 3 - August 31 In this unit students will perform transformations in the coordinate plane, describe a sequence of transformations that will map one figure onto another, and describe transformations that will map a figure onto itself. Students will compare transformations that preserve distance and angle to those that do not. The Big Ideas of this Unit Students should walk away from this unit understanding that The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles (and therefore shapes in general). Reflections and rotations each explain a particular type of symmetry, and the symmetries of an object offer insight into its attributes. Engage with Standards for Mathematical Practice (SMPs) The Standards for Mathematical Practice (SMP) are practices that we want to develop in students as they engage in mathematics. SMP1: Make sense of problems and persevere in solving them. While doing tasks in this unit, students will need to make sense of the geometric transformations of shapes by representing those transformations algebraically. Students must understand and apply the various types of transformations and will need to persevere through the problem solving process in order to arrive at a solution. SMP2: Reason abstractly and quantitatively. Students connect graphical representations with the function (symbolic) representation of translations. SMP3: Construct viable arguments and critique the reasoning of others. Students should be able to justify why the image represents a particular transformation. Students should be able to critique the reasoning of others by making comparisons between the visual representations of a transformation to their verbal/ algebraic descriptions of the transformations. SMP4: Model with mathematics. In this unit, it will be critical for students to make sense of geometric concepts by modeling them with algebraic tools. By applying coordinates to a geometric transformation, students will be able to generalize what is happening to the shapes. Students use computer software to demonstrate transformations and represent transformations graphically. SMP5: Use appropriate tools strategically. Students can use a variety of tools to help them perform and understand transformations. Students will need to select appropriate tools (graph paper, Desmos, Geogebra, sketchpad, geoboard, mira, and transparencies) to model transformations.

2 SMP7: Look for and make use of structure. Students can use software (geogebra, desmos, TI graphing calculator, sketchpad) to perform transformations, and then try to generalize and understand the transformation that has taken place. Students generalize what they see with reflections/rotations/dilations etc. into a formal definition of this transformation. SMP8: Look for and express regularity in repeated reasoning. Students look for patterns in their results on various transformations to develop a general rule for the symbolic representation of the transformation. Ask your child some of the following questions: Can you explain how the geometrical shape has been transformed? Is the shape preserved? Is the size preserved? What effects do transformations have on geometric figures? Can transformation change an object s position, orientation and/ or size? How do transformations of geometric figures and functions compare? Explain the various types of transformations and their properties. How did you know a particular transformation has taken place? Why did you decide to use a specific transformation for a given problem? Is there another transformation you could have used? Does this method always work? How do you know? Where can you apply transformations in a real-world situation? How do you know that your image under a transformation is reasonable? How can you be sure? Are there other problems that are similar to this one? If I told you that your image under a transformation was not accurate, how would you convince me that I was wrong? Key Online Resources for Mathematics Learning Math.com provided practice problems, calculators & tools, and games : Lessons and games aligned to Common Core: Video Lessons: Online Geometry textbook: Using digital tools to learn transformations: Unpacking standards:

3 Angle: A figure created by two distinct rays that share a common endpoint (also known as a vertex). ABC or B or CBA indicate the same angle with vertex B. Angle of Rotation: The amount of rotation in degrees) of a figure about a fixed point such as the origin. Axis of Reflection. The "mirror line" of a reflection. That is, the line across which a reflection takes place. Bisector: A point, line or line segment that divides a segment or angle into two equal parts. Circle: The set of all points equidistant from a point in a plane. Congruent: Having the same size, shape and measure. A _ B indicates that angle A is congruent to angle B. Corresponding angles: Angles that have the same relative position in geometric figures. Corresponding sides: Sides that have the same relative position in geometric figures. Endpoint: The point at each end of a line segment or at the beginning of a ray. Image: The result of a transformation. Intersection: The point at which two or more lines intersect or cross. Isometry: a distance preserving map of a geometric figure to another location using a reflection, rotation or translation indicates an isometry of the figure M to a new location M. M and M remain congruent. Line: One of the undefined terms of geometry that represents an infinite set of points with no thickness and its length continues in two opposite directions indefinitely. indicates a line that passes through points A and B. Line segment: A part of a line between two points on the line. indicates the line segment between points A and B. Line symmetry: If you can reflect (or flip) a figure over a line and the figure appears unchanged, then the figure has reflection symmetry or line symmetry Line of Symmetry: The line that you reflect over is called the line of symmetry. A line of symmetry divides a figure into two mirror-image halves. Parallel lines: Two lines are parallel if they lie in the same plane and do not intersect. Key Terms to Know Perpendicular lines: Two lines are perpendicular if they intersect to form right angles. indicates that line AB is perpendicular to line CD. Point: One of the basic undefined terms of geometry that represents a location. A dot is used to symbolize it and it is thought of as having no length, width or thickness. Pre image: A figure before a transformation has taken place. Ray: A part of a line that begins at a point and continues forever in one direction. indicates a ray that begins at point A and continues in the direction of point B indefinitely. Rectangle: a parallelogram whose angles are all right angles Regular Polygon: a polygon having all sides (and hence all angles) equal Reflection: A transformation of a figure that creates a mirror image, flips, over a line. Reflection Line (or line of reflection): A line that acts as a mirror so that corresponding points are the same distance from the mirror. Rhombus: A quadrilateral with all sides equal (Every rhombus is a parallelogram) Rotation: A transformation that turns a figure about a fixed point through a given angle and a given direction, such as 90 clockwise. Rotational Symmetry: An object has rotational symmetry if there is a center point around which the object is turned (rotated) a certain number of degrees and the object looks the same. The number of positions in which the object looks exactly the same is called the order of the symmetry. Symmetry: the quality of being made up of exactly similar parts facing each other or around an axis Transformation: The mapping, or movement, of all points of a figure in a plane according to a common operation, such as translation, reflection or rotation. Translation: A transformation that slides each point of a figure the same distance in the same direction. Trapezoid: a quadrilateral with one pair of opposite sides parallel. Vertex: The location at which two lines, line segments or rays intersect.

4 indicates that line AB is parallel to line CD. Parallelogram: a quadrilateral with opposite sides parallel (and therefore opposite angles equal). Polygon: a two-dimensional closed figure made up of line segments Support Learning at Home 1. Students have learned about transformations in middle school and can relate to demonstrate transformations using concrete objects. Encourage your child to use appropriate math vocabulary to define their transformations. 2. Ask your child to show/prove to you that the resultant image of a translation followed by a reflection is same as the resultant image of the same object under the same reflection followed by the same translation. Ask them to verify if this is true for any two transformations. 3. Research says that students understand and retain mathematical concepts by teaching others. Ask your child to teach you the functional (algebraic) representations of transformations and how images can be drawn on a coordinate plane. 4. Cut out a square, a rectangle, and a regular hexagon and a regular octagon(stop sign). Ask your child to explain rotational symmetry using the cut figures. Sample Problems 1. Which of the following would be considered a true line, in geometric terms? (A) You, a few friends, and thousands of others standing outside a stadium waiting for tickets to the hottest concert of the year (B) The direction along which an astronomer is looking through a telescope for a new planet thought to be farther from Earth than Neptune (C) The equator (D) A row of cars, stuck in a traffic jam so long that you can't see the end neither in front, nor behind you Correct Answer: The direction along which an astronomer is looking through a telescope for a new planet thought to be farther from Earth than Neptune Answer Explanation: Remember that a line is infinite in length. Though long, the lines of (A), (C), and (D) are, in fact, finite. However, the line of sight along which the scientist in (B) is stretching to find this new alleged farthest planet is unknown. Even if that planet were in a galaxy far, far away, the line that the astronomer uses to find it is infinite. 2. Reflect the triangle with vertices (-4, 1), (-4, 4) and (-1, 4) across the y-axis on a geo board and write your conclusion based on your observations. The point with coordinates (-4,1) is mapped to (4, 1) under the reflection across y-axis

5 The point with coordinates (-4,4) is mapped to (4, 4) under the reflection across y-axis The point with coordinates (-1,4) is mapped to (1, 4) under the reflection across y-axis Conclusion: The point (x, y) is mapped to (-x, y) under the reflection across y-axis. 3. What is the image of the triangle ABC with coordinates at (2, 4), (4, -3) and (5, 1) under the transformation f(x, y) = (-x, -y). Describe the transformation. Solution: The image of (2, 4) under the given transformation is (-2, -4) The image of (4, -3) under the given transformation is (-4, 3) The image of (5, 1) under the given transformation is (-5, -1) Plot the points of the pre-image and the image and draw the two triangles. Use a patty paper and draw the pre-image. Secure the center of rotation by placing a pencil on top of the patty paper at the origin. Now rotate the patty paper; check to see if it coincides with the image any time. If it coincides, the transformation is a rotation. Find the angle between OA and OA to find the angle of rotation in the transformation. In this case, the transformation is a rotation through an angle of in the counterclockwise direction. 4. Below are four road signs. Which has the greatest order of rotational symmetry? A. B. C. D. Correct Answer: Answer Explanation: This is partly defined by its line segments, its angles, and the circular arcs on the face of the sign. Just imagine rotating each of the signs a full 360 and imagine how many times the image repeats itself. The three arrows in (A) indicate an order of 3, while (B) has an order of 2. Although (C) is symmetrical, it has no rotational symmetry, which gives it an order of 1. While the shape of the stop sign has an order of 8, the word "STOP" gives the sign an order of 1 as well. Our highest is (A).