# COORDINATE GEOMETRY AND TRANSFORMATIONS

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1 COORDINATE GEOMETRY AND TRANSFORMATIONS i 2 t 2 Final Project 5-Day Unit Plan 8 th Grade Math Lab Helen Roseler December 1,

2 Preface Math Lab is an additional math class designed to deliver Academic Intervention Services to eighth grade students Instruction is supportive or additional to the initial instruction provided by the regular eighth grade math teacher The eighth grade math teacher gives homework assignments I do not have access to any computers as yet and the only graphing calculator I have is the one given to me this summer Resources Mathematics Teaching in the Middle School, November 2003, volume 9, number 3 Turning Traditional Textbook Problems into Open-Ended Problems, Mary S Kabiri and Nancy L Smith, (pgs ) Mathematics Curriculum Benchmarks, 8 Buffalo Public Schools Mathematics Committee, 2001 Mathematics Resource Guide with Core Curriculum State Education Department: The University of the State of New York, 1999 Principles & Standards for School Mathematics Reston, VA: The National Council of Teachers of Mathematics, Inc, 2000 Woodward, Ernest and Hamel, Thomas Geometric Constructions and Investigations with a Mira Maine: J Weston Walch, 1992 Usiskin, Z, Feldman, CH, Davis, S, Mallo, S, Sanders, G, Witonski, D, Flanders, D, Polonsky, L, Porter, S, Viktora, S S Transition Mathematics Illinois: Scott Foresman and Company,

3 Equipment, Materials and Manipulatives Equipment: Overhead Projector Materials: White paper Blacklines Graph paper Copies Geoboard paper Transparencies Teaching Aids Textbook Manipulatives: Geoboards/geobands Miras Overhead geoboard Overhead tangram pieces Tangram pieces Rulers Colored pencils Pencils 3

4 OVERALL OBJECTIVES Objectives; Lesson 1 Use the coordinate grid to explore geometric ideas Plot and name points on a coordinate grid Create and plot geometric shapes on the coordinate grid Identify vertices using ordered pairs Describes properties of the shapes Objectives; Lesson 2 Create congruent and similar shapes Graph and identify congruent shapes, their pre-images, and images Use coordinate graphing to perform 2-dimensional translations Read and use vocabulary of translations to interpret translations on a coordinate Graph Learn that a translation is a type of transformation Objectives; Lesson 3 Learn the language of reflections Draw a reflection image of a figure over a line of reflection Perform 2-dimensional reflections using a Mira Recognize when a figure is not symmetric, any reflection will look reversed from the original Objectives; Lesson 4 Identify a line of reflection as a line of symmetry Identify reflections as congruent figures Draw a reflection using a perpendicular bisector Objectives; Lesson 5 Know that a reflection is a transformation Construct the reflection image of a figure on a coordinate graph Apply the relationships between figures and their reflection images Interpret reflections on a coordinate grid 4

5 Overview Lesson 1: Coordinate Graphing 1) Students review fundamentals of coordinate graphing using geoboards 2) Students create and draw a figure using ordered pairs as vertices Lesson 2: Translations on a Coordinate Grid 1) Students use tangram pieces to demonstrate their understanding of congruent and similar 2) Teacher introduces vocabulary associated with transformations and translations 3) Students explore translations of a triangle 4) Students explain the changes in coordinates as a result of a translation Lesson 3: Reflections with Miras 1) Students learn how to use Miras 2) Students use Miras to explore and understand reflections of symmetric and non-symmetric figures 3) Students discover lines of reflection Lesson 4: More Reflections 1) Students use a fold line and a perpendicular bisector for reflection 2) Students identify reflection lines as lines of symmetry 3) Students learn reflections result in congruent figures that are usually flipped and equal distant from a line of symmetry Lesson 5: Reflections on a Coordinate Grid 1) Students learn reflections and changes in pre-image, thus reflections are transformations 2) Students learn to construct a reflection on a coordinate grid 3) Students interpret the reflection using the coordinates of the pre-image and image 5

6 STANDARDS New York State Standards for Math Grades 7 and 8 / BPS Benchmarks 1A: Students apply a variety of reasoning strategies 1B: Make and evaluate conjectures and arguments, using appropriate language 2A: Understand, represent, and use numbers in a variety of equivalent forms (integer, fraction, decimal, percent, exponential, expanded, and scientific notation) 3A: Add, subtract, multiply, and divide fractions, decimals, and integers 4A: Identify and construct two-dimensional and three-dimensional shapes 4C: Use the coordinate plane to explore geometric ideas 4H: Investigate two-dimensional transformations 4I: Use appropriate tools to construct and verify geometric relationships 7A: Recognize, describe, and generalize a wide variety of patterns and functions 7H: Explore relationships involving points, lines, angles, and planes NCTM Principles and Standards for School Mathematics Grades 6-8 Understand numbers, ways of representing numbers, relationships among numbers, and number systems Use mathematical models to represent and understand quantitative relationships Analyze characteristics and properties of two-dimensional geometric shapes and develop Mathematical arguments about geometric relationships Specify locations and describe spatial relationships using coordinate geometry and other representational systems Apply transformations and use symmetry to analyze mathematical situations Use visualization, spatial reasoning, and geometric modeling to solve problems Make and investigate mathematical thinking through communication Communicate their mathematical thinking coherently and clearly to peers, teachers, and others 6

8 8 Worksheet 1 Name Use the geoboard template and a ruler to draw and label the following: x-axis y-axis origin quadrants I, II, III, IV Record the triangle you made on the geoboard, placing one vertex in any three of the four quadrants Label its vertices using ordered pairs

9 9 Sample Name Use the geoboard template and a ruler to draw and label the following: x-axis, y-axis, origin, quadrants I, II, III, IV Record the triangle you made on the geoboard, placing one vertex in any three of the four quadrants Label its vertices using ordered pairs y-axis II 2,5 I x-axis -6,2 6,-3 III IV

11 Worksheet 2 Name Ticket out of class: Use the graph paper to graph the quadrilateral; S (-5, -2), T (-4, 1), O (-1, -2), P (-3, -5) Write an explanation of what happens to any pre-image if you add 4 to the first coordinate of every point and you add 2 to every second coordinate? What will be the new coordinates for S 1, T 1, O 1 and P 1 Pre-Image x y S -5-2 T -4 1 O -1-2 P -3-5 Image x S T O P y 11

12 SAMPLE Name Ticket out of class: Use the graph paper to graph the quadrilateral; S (-5, -2), T (-4, 1), O (-1, -2), P (-3, -5) Write an explanation of what happens to any pre-image if you add 4 to the first coordinate of every point and you add 2 to every second coordinate What will be the new coordinates for S 1, T 1, O 1 and P 1? Pre-Image x y S -5-2 T -4 1 O -1-2 P -3-5 Image x y S T O P

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17 LESSON 4 Objectives: Identify a line of reflection as a line of symmetry Identify reflections as congruent figures Draw a reflection using a perpendicular bisector Materials: Miras, Rulers, transparency of Teaching Aid 91, and copies of Teaching Aids 91 and 92, work from closing activity in previous lesson Opening Activity: Let s look at your notebooks and your responses to questions Which letters resulted from a reflection that appears reversed? (T, O, M, A, H, Y, C, E, K, and B) Yes, the color-coding helps you see the reversals Which letters resulted from a reflection that does not appear reversed? (W, and I) Have you been able to figure out why? (If you look at the line of reflection on W and I their pre-images are symmetric and the color-coding helps you see it) Can you thing of another name for a line of reflection? (A line of symmetry) You re right Do you recall the meaning of congruent? (Yes, figures that are the same size and same shape) Would you say that both sides of these letters are congruent figures? (Yes, both sides of the letters are the same size and the same shape Most of them are just flipped or reversed) When we were reflecting the boy onto the swing did it result in a congruent figure? (Yes, and the hats were congruent figures too) Great! Now we are going to explore reflection in a different way Developmental Activity: Teacher places Transparency of Teaching Aid 91 on the overhead projector She passes out copies of Teaching Aid 91 to the students and also gives them rulers We are going to reflect point P over line m Any ideas about how we can do it? (Just guess We can fold the paper back on line m and then hold it up to the light or on the window and fill in point P on the opposite side of the line) A good idea What does the fold line become? (The line of symmetry, or the line of reflection) What do we call the new point P? (P prime) You re on the ball, everybody reflect point P over line m using a fold if you haven t already done so Now let s see if we can reflect an image another way When you are finished with point P 1 look up here at triangle ABC I am going to use my ruler to determine the distance of point A from line m I can t do it just any way, I have to use a line that is perpendicular to line m How do I determine if my ruler is perpendicular to line m? (Perpendicular lines form right angles) Yeah for you Camilo So, I am going to line up a line on my ruler with line m and that should give me right angles The teacher illustrates by using a transparent ruler with the transparency and measures 2 centimeters from point A to line m Next I have to measure 2 centimeters from line m to the opposite side, still using the perpendicular line, also called the perpendicular bisector, and mark the point A 1 The students watch as the teacher uses the same procedure to find B 1 and C 1 Now I want you to reflect triangle ABC over line m The teacher monitors as the students work and she helps as needed She goes on to the reflection of point E, on the reflecting line She tells the students: when a point is on the reflecting line, then it is its own image We say it coincides with (takes the same position as) its image The teacher again models the reflection of lines l and n over line m (see Teaching Aid 91) the students complete the reflections on their own copies Closing Activity: Students complete Teaching Aid 92, items 9-11 on their own 17

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21 Name Worksheet 3 - Use the graph paper to create a coordinate grid - Label the origin, the x-axis and the y-axis - Plot the following points and connect them in the order in which they are given: - S (-2,1); H (-3,4); I (-4,1); N (-2,3); E (-4,3); back to S ( 2,1) - Reflect this figure over the y- axis Pre-image Image Write an explanation as to why translations are sometimes called slides, and reflections are sometimes called flips Give examples to support your reasoning 21

22 SAMPLE Name - Use the graph paper to create a coordinate grid - Label the origin, the x-axis and the y-axis - Plot the following points and connect them in the order in which they are given: - S (-2,1); H (-3,4); I (-4,1); N (-2,3); E (-4,3); back to S ( 2,1) - Reflect this figure over the y- axis Pre-image Image x y x y S -2 1 S H -3 4 H I -4 1 I N -2 3 N E -4 3 E S -2 1 S Write an explanation as to why translations are sometimes called slides, and reflections are sometimes called flips Give examples to support your reasoning Translations are sometimes called slides because a figure remains congruent but moves according to the change in the first or second coordinate (ie T(2,3); A (3,5); N (5,4) If I add 3 to each second coordinate the figure will move 3 units up on the grid Reflections are sometimes called flips because a figure remains congruent but is flipped over a line of some kind (ie T(2,3); A(3,5); N(5,4) If I flip it over the line of the x-axis the second coordinate remains the same but changes to a negative coordinate 22

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