NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 8 2. Grade 8 Module 2,The Concept of Congruence, In this module students learn about translations reflections and rotations in the plane and more. importantly how to use them to precisely define the concept of congruence Up to this point congruence. has been taken to mean intuitively same size and same shape Because this module begins a serious study. of geometry this intuitive definition must be replaced by a precise definition This module is a first step its. goal is to provide the needed intuitive background for the precise definitions that are introduced in this. module for the first time, Translations reflections and rotations are examples of rigid motions which are intuitively rules of moving. points in the plane in such a way that preserves distance For the sake of brevity these three rigid motions. will be referred to exclusively as the basic rigid motions Initially the exploration of these basic rigid motions. is done via hands on activities using an overhead projector transparency but with the availability of geometry. software the use of technology in this learning environment is inevitable and some general guidelines for. this usage will be laid out at the end of Lesson 2 What needs to be emphasized is that the importance of. these basic rigid motions lies not in the fun activities they bring but in the mathematical purpose they serve in. clarifying the meaning of congruence, Throughout Topic A on the definitions and properties of the basic rigid motions students verify. experimentally their basic properties and when feasible deepen their understanding of these properties. using reasoning In particular what students learned in Grade 4 about angles and angle measurement. 4 MD 5 will be put to good use here they learn that the basic rigid motions preserve angle measurements. as well as segment lengths, Topic B is a critical foundation to the understanding of congruence All the lessons of Topic B demonstrate to. students the ability to sequence various combinations of rigid motions while maintaining the basic properties. of individual rigid motions Lesson 7 begins this work with a sequence of translations Students verify. experimentally that a sequence of translations have the same properties as a single translation Lessons 8. and 9 demonstrate sequences of reflections and translations and sequences of rotations The concept of. sequencing a combination of all three rigid motions is introduced in Lesson 10 this paves the way for the. study of congruence in the next topic, In Topic C on the definition and properties of congruence students learn that congruence is just a sequence. of basic rigid motions The fundamental properties shared by all the basic rigid motions are then inherited by. congruence congruence moves lines to lines and angles to angles and it is both distance and degree. preserving Lesson 11 In Grade 7 students used facts about supplementary complementary vertical and. adjacent angles to find the measures of unknown angles 7 G 5 This module extends that knowledge to. angle relationships that are formed when two parallel lines are cut by a transversal In Topic C on angle. relationships related to parallel lines students learn that pairs of angles are congruent because they are. angles that have been translated along a transversal rotated around a point or reflected across a line. Module 2 The Concept of Congruence,Date 9 19 13 2,This work is licensed under a. 2013 Common Core Inc Some rights reserved commoncore org. Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 8 2. Students use this knowledge of angle relationships in Lessons 13 and 14 to show why a triangle has a sum of. interior angles equal to 180 and why the exterior angles of a triangle is the sum of the two remote interior. angles of the triangle, Optional Topic D begins the learning of Pythagorean Theorem Students are shown the square within a. square proof of the Pythagorean Theorem The proof uses concepts learned in previous topics of the. module i e the concept of congruence and concepts related to degrees of angles Students begin the work. of finding the length of a leg or hypotenuse of a right triangle using 2 2 2 Note that this topic will. not be assessed until Module 7,Focus Standards, Understand congruence and similarity using physical models transparencies or geometry. 8 G A 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. 8 G A 2 Understand that a two dimensional figure is congruent to another if the second can be. obtained from the first by a sequence of rotations reflections and translations given two. congruent figures describe a sequence that exhibits the congruence between them. 8 G A 5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles. about the angles created when parallel lines are cut by a transversal and the angle angle. criterion for similarity of triangles For example arrange three copies of the same triangle so. that the sum of the three angles appears to form a line and give an argument in terms of. transversals why this is so,Understand and apply the Pythagorean Theorem. 8 G B 6 Explain a proof of the Pythagorean Theorem and its converse. 8 G B 7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real. world and mathematical problems in two and three dimensions. Module 2 The Concept of Congruence,Date 9 19 13 3,This work is licensed under a. 2013 Common Core Inc Some rights reserved commoncore org. Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 8 2. Foundational Standards, Geometric measurement understand concepts of angle and measure angles. 4 MD C 5 Recognize angles as geometric shapes that are formed wherever two rays share a common. endpoint and understand concepts of angle measurement. a An angle is measured with reference to a circle with its center at the common endpoint of. the rays by considering the fraction of the circular arc between the points where the two. rays intersect the circle An angle that turns through 1 360 of a circle is called a one. degree angle and can be used to measure angles, b An angle that turns through one degree angles is said to have an angle measure of. Draw and identify lines and angles and classify shapes by properties of their lines and. 4 G A 1 Draw points lines line segments rays angles and perpendicular and parallel lines Identify. these in two dimensional figures, 4 G A 2 Classify two dimensional figures based on the presence or absence of parallel or. perpendicular lines or the presence or absence of angles of a specified size Recognize right. triangles as a category and identify right triangles. 4 G A 3 Recognize a line of symmetry for a two dimensional figure as a line across the figure such that. the figure can be folded along the line into matching parts Identify line symmetric figures. and draw lines of symmetry, Solve real life and mathematical problems involving angle measure area surface area and. 7 G B 5 Use facts about supplementary complementary vertical and adjacent angles in a multi step. problem to write and solve simple equations for an unknown angle in a figure. Focus Standards for Mathematical Practice, MP 2 Reason abstractly and quantitatively This module is rich with notation that requires. students to decontextualize and contextualize throughout Students work with figures and. their transformed images using symbolic representations and need to attend to the meaning. of the symbolic notation to contextualize problems Students use facts learned about rigid. motions in order to make sense of problems involving congruence. MP 3 Construct viable arguments and critique the reasoning of others Throughout this module. students construct arguments around the properties of rigid motions Students make. assumptions about parallel and perpendicular lines and use properties of rigid motions to. directly or indirectly prove their assumptions Students use definitions to describe a sequence. Module 2 The Concept of Congruence,Date 9 19 13 4,This work is licensed under a. 2013 Common Core Inc Some rights reserved commoncore org. Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 8 2. of rigid motions to prove or disprove congruence Students build a logical progression of. statements to show relationships between angles of parallel lines cut by a transversal the. angle sum of triangles and properties of polygons like rectangles and parallelograms. MP 5 Use appropriate tools strategically This module relies on students fundamental. understanding of rigid motions As a means to this end students use a variety of tools but. none as important as an overhead transparency Students verify experimentally the. properties of rigid motions using physical models and transparencies Students use. transparencies when learning about translation rotation reflection and congruence in. general Students determine when they need to use the transparency as a tool to justify. conjectures or when critiquing the reasoning of others. MP 6 Attend to precision This module begins with precise definitions related to transformations. and statements about transformations being distance and angle preserving Students are. expected to attend to the precision of these definitions and statements consistently and. appropriately as they communicate with others Students describe sequences of motions. precisely and carefully label diagrams so that there is clarity about figures and their. transformed images Students attend to precision in their verbal and written descriptions of. rays segments points angles and transformations in general. Terminology,New or Recently Introduced Terms, Transformation A rule to be denoted by that assigns each point of the plane a unique point. which is denoted by, Basic Rigid Motion A basic rigid motion is a rotation reflection or translation of the plane. Basic rigid motions are examples of transformations Given a transformation the image of a. point is the point the transformation maps the point to in the plane. Translation A basic rigid motion that moves a figure along a given vector. Rotation A basic rigid motion that moves a figure around a point degrees. Reflection A basic rigid motion that moves a figure across a line. Image of a point image of a figure Image refers to the location of a point or figure after it has. been transformed, Sequence Composition of Transformations More than one transformation Given. transformations and is called the composition of and. Vector A Euclidean vector or directed segment is the line segment together with a. direction given by connecting an initial point to a terminal point. Congruence A congruence is a sequence of basic rigid motions rotations reflections translations. of the plane, Transversal Given a pair of lines and in a plane a third line is a transversal if it intersects at. a single point and intersects at a single but different point. Module 2 The Concept of Congruence,Date 9 19 13 5,This work is licensed under a. 2013 Common Core Inc Some rights reserved commoncore org. Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 8 2. Familiar Terms and Symbols 2,Ray line line segment angle. Parallel and perpendicular lines, Supplementary complementary vertical and adjacent angles. Triangle quadrilateral,Area and perimeter,Suggested Tools and Representations. Transparency or patty paper, Wet or dry erase markers for use with transparency. Optional geometry software, Composition of Rigid Motions http youtu be O2XPy3ZLU7Y. ASA http www youtube com watch v yIZdenw5U4,Assessment Summary. Assessment Type Administered Format Standards Addressed. Mid Module, After Topic B Constructed response with rubric 8 G A 1. Assessment Task,End of Module, After Topic C Constructed response with rubric 8 G A 2 8 G A 5. Assessment Task, These are terms and symbols students have seen previously. Module 2 The Concept of Congruence,Date 9 19 13 6,This work is licensed under a. 2013 Common Core Inc Some rights reserved commoncore org. Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. New York State Common Core,Mathematics Curriculum,GRADE 8 MODULE 2. Definitions and Properties of the Basic Rigid, Focus Standard 8 G A 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. b Angles are taken to angles of the same measure,c Parallel lines are taken to parallel lines. Instructional Days 6,Lesson 1 Why Move Things Around E. Lesson 2 Definition of Translation and Three Basic Properties P. Lesson 3 Translating Lines S, Lesson 4 Definition of Reflection and Basic Properties P. Lesson 5 Definition of Rotation and Basic Properties S. Lesson 6 Rotations of 180 Degrees P,In Topic A students learn about. NYS COMMON CORE MATHEMATICS CURRICULUM 8 of geometry this intuitive definition must be replaced by a precise definition This module is a first step its goal is to provide the needed intuitive background for the precise definitions that are introduced in this module for the first time Translations reflections and rotations are examples of rigid motions which are intuitively rules

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