Focus of this Unit: Connections to Subsequent Learning: Approximate Time Frame: 4-6 weeks Connections to Previous Learning:

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1 Approximate Time Frame: 4-6 weeks Connections to Previous Learning: In Grade 8, students are introduced to the concepts of congruence and similarity through the use of physical models and dynamic geometry software. They are introduced to four types of transformations: rotations, reflections, translations, and dilations. These transformations are then used to establish whether or not two-dimensional figures are congruent to one another and whether or not two-dimensional figures are similar to one another. Students are also introduced to the notion of proof through informal arguments that establish theorems such as the sum of the angles of a triangle is 180 and the Pythagorean Theorem. Focus of this Unit: Since concepts such as rotation, reflection, and translation were treated in the Grade 8 standards mostly in the context of hands-on activities, and with an emphasis on geometric intuition, one focus of this unit is to develop more precise definitions for these transformations. Precision and proof are critical elements of this unit. Students' understanding of what constitutes proof will be developed as students are introduced to the logical structure behind hypotheses and conclusions. Geometric objects will be more precisely defined using the basic building blocks of geometry, point, line, and distance. One very specific outcome of the unit is to develop a precise definition of congruence in terms of rigid motions, to use this definition to develop criteria for determining when two triangles are congruent, and finally, to develop triangle congruence theorems which follow from the definition of congruence in terms of rigid motions. In addition, attributes of geometric objects observed at an informal level in earlier grades will now be looked at more precisely through proof, and in many instances, these proofs will make use of triangle congruence and its consequences. Along with physical models, dynamic geometry environments will provide students with tools for investigating, experimenting with, conjecturing about, and modeling geometric phenomena. Connections to Subsequent Learning: Rigid motions will be combined with the non-rigid motion of dilation to develop a definition of similarity in terms of these motions. The definitions and properties of geometric objects considered in this unit will reappear in future units that deal with other geometric concepts from an analytic perspective (coordinate geometry) and with modeling. Understanding of the criteria for triangle congruence will help students, not just during their investigations of the properties of lines, triangles, and quadrilaterals in this unit, but also in their future investigations of circles and trigonometric ratios and in many geometric modeling tasks. Proof concepts will be spiraled into subsequent units and courses as it will be used later on in both geometric and non-geometric settings. The problem solving techniques that will be used to solve problems from this unit will help students deal with similar types of problems that they will see in later units, particularly those units which involve modeling. 10/9/ :09:22 AM Adapted from UbD framework Page 1

2 Desired Outcomes Standard(s): Understand congruence in terms of rigid motions. G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effects of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G.CO.7 Use definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motion. Prove geometric theorems. G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Experiment with transformations in the plane. G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.CO.3 Given a rectangle parallelogram trapezoid, or regular polygon, describe the rotations and reflections that carry onto itself. G.CO.4 Develop definitions of rotations, reflections, and translations, in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g. graph paper, tracing paper, or geometry software. Specify a sequence of transformation that will carry a given figure onto another. Transfers: Students will be able use their geometry skills of rigid motion to create proofs of theorems. Ex. Using rigid motion to take the vertices of a triangle and transform them in to a straight angle, to prove the sum of the angles of a triangle is 180 o. 10/9/ :09:22 AM Adapted from UbD framework Page 2

3 WIDA Standard: (English Language Learners) English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. English language learners benefit from: explicit vocabulary instruction with regard to geometric tools and terminology. explicit instruction with regard to theorems and their meanings. Understandings: Students will understand Two geometric figures are congruent if there is a sequence of rigid motions (rotations, reflections, or translations) that carries one onto the other. A proof consists of a hypothesis and conclusion connected with a series of logical steps. The basic building blocks of geometric objects are formed from the undefined notions of point, line, distance along a line, and distance around a circular arc. Two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles of the triangles are congruent. It is possible to prove two triangles congruent without proving corresponding pairs of sides and corresponding pairs of angles of the triangle are congruent if certain subsets of these 6 congruence relationships are known to be true (e.g. SSS, SAS, ASA, but not SSA). Different observed relationships between lines, between angles, between triangles, and between parallelograms are provable using basic geometric building blocks and previously proven relationships between these building blocks and between other geometric objects. The geometric relationships that come from proving triangles congruent may be used to prove relationships between geometric objects. Essential Questions: In terms of rigid motions, when are two geometric figures congruent? What are the undefined building blocks of geometry and how are they used? What are possible conditions that are necessary to prove two triangles congruent? What are the roles of hypothesis and conclusion in a proof? What criteria are necessary in proving a theorem? 10/9/ :09:22 AM Adapted from UbD framework Page 3

4 Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) * 1. Make sense of problems and persevere in solving them. Students recognize the hypothesis and conclusion in a proof statement and generate the requisite proof using givens, along with known facts, definitions, postulates, and theorems. * 2. Reason abstractly and quantitatively. Students use figures and information, including auxiliary lines, pertaining to a specific geometric object as an aid in reasoning about that geometric object in general. * 3. Construct viable arguments and critique the reasoning of others. Students create and present proofs, and critique the proofs and deductive reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. Students use physical models, drawings, and dynamic geometry software to form conjectures about geometric objects and to reason from information about the geometric object provided by these tools. * 6. Attend to precision. Students recognize that incorrect initial attempts at definitions, conjectures, and theorems may be corrected through a process of refinement. * 7. Look for and make use of structure. Students use the structure of geometric objects to gain insights into, make conjectures about, and create proofs pertaining to these objects. 8. Look for and express regularity in repeated reasoning. Prerequisite Skills/Concepts: Students should already be able to: Understand what constitutes a rigid motion. Have an informal understanding of what it means for two geometric figures to be congruent. Given an object, perform specific reflections, rotations, and translations to that object. Understand geometric objects such a point, line, angle, triangle, parallelogram, circle, and so on. Use informal methods of proof. Advanced Skills/Concepts: Some students may be ready to: Recognize how counterexamples can be used to refute conjectures. Know the definitions of, make conjectures about, and create proofs pertaining to quadrilaterals that are not parallelograms, such as kites or cyclic quadrilaterals. 10/9/ :09:22 AM Adapted from UbD framework Page 4

5 Knowledge: Students will know Precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Skills: Students will be able to Develop definitions of rotations, reflections, and translations, in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Represent transformations in the plane. Predict the effects of a given rigid motion on a given figure. Use the definition of congruence in terms of rigid motions to decide if 2 given figures are congruent. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle and those that do not. Given two geometric objects, recognize if and describe how one can be transformed to the other through a sequence of rigid motions. Describe the necessary transformations to carry a figure onto itself. Draw a figure that represents specific given information. Construct a hypothesis and proof from a given conjecture. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove theorems about lines, angles, triangles and parallelograms. 10/9/ :09:22 AM Adapted from UbD framework Page 5

6 Academic Vocabulary: Critical Terms: Transformation Rigid motion Congruence Hypothesis Conclusion Proof Postulate Theorem Corresponding Supplemental Terms: Rotation Reflection Translation Dilation Point Line Angle Assessment Pre-Assessments Formative Assessments Summative Assessments Self-Assessments Pre-Assessment Four Corners Unit 2 Pre-Assessment Vocabulary Activity Vocabulary Crossword Puzzle G.CO.7-8 Formative Assessment G.CO.11 Spot the Parallelogram Vocabulary Worksheet G.CO.11 Prove It! G.CO.9 Build Your Own City Quilting with Transformations (use with dynamic geometry software) 1. Transformations G.CO.1-7 (Look at as a reference) 2. Theorems about Lines and Angles G.CO.9 3. Congruence and Theorems about Triangles G.CO.10, G.CO.8 4. Theorems about Parallelograms G.CO.11 (Model Lesson) Sample Lesson Sequence Transformations Summative Assessment G.CO.1-7 Summative Assessment G.CO.9-11 Post-Assessment Four Corners Vocabulary Graphic Organizer 10/9/ :09:22 AM Adapted from UbD framework Page 6