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Geometry Mathematics Curriculum Guide 2015 – 2016 Unit 4: Triangle Properties and Congruence Time Frame: 16 days Primary Focus There are three primary parts of this unit: A) Building off the student understanding of transformations that was developed in Unit 2, students will use rigid motion to develop proofs for triangle congruence. B) Students will continue to work with triangles proving theorems about triangles. C) Develop theorems of parallelograms using theorems of lines and angles. Common Core State Standards for Mathematical Practice Standards for Mathematical Practice MP1 - Make sense of problems and persevere in solving them. MP2 - Reason abstractly and quantitatively. MP3 - Construct viable arguments and critique the reasoning of others. MP4 - Model with mathematics. MP5 - Use appropriate tools strategically. MP6 - Attend to precision. MP7 - Look for and make use of structure. MP8 - Look for and express regularity in repeated reasoning. Unit 4 How It Applies to this Topic… Analyze given information to develop possible strategies for solving the problem. Make connections between the abstract theorems and their real-world applications. Justify (orally and in written form) the argument by deductive reasoning, including how it fits in the context from which the problem arose. Use a variety of methods to model, represent, and solve real-world problems. Select and use appropriate tools to best model/solve problems. Transform figures efficiently and accurately and label them appropriately. Use patterns or structure to make sense of mathematics and connect prior knowledge to similar situations and extend to novel situations. Generalize the process to create a shortcut which may lead to developing rules or creating a formula. Clover Park School District 2015-2016 Page 1 Geometry Mathematics Curriculum Guide 2015 – 2016 Stage 1 Desired Results Transfer Goals Students will be able to independently use their learning to… Determine if 2 figures are congruent by comparing corresponding parts Prove triangles are congruent without having to compare all corresponding parts Show the angles and sides of Isosceles and Equilateral triangles have special relationships. Meaning Goals UNDERSTANDINGS ESSENTIAL QUESTIONS Students will understand that… How can the properties of rigid motion be used to prove that two triangles are • The properties of transformations that are rigid motion can be used to congruent (ASA, SAS, SSS)? identify and prove congruence of figures in a plane. • Constructing a viable argument using the precise vocabulary of transformations and congruence to prove geometric theorems in a variety of formats is important to Geometry proof. • Triangles have special properties that allow you to use shortcuts for proving triangles congruent. Acquisition Goals Students will know and will be skilled at… • Define congruent polygons as a one-to-one relationship between the corresponding congruent parts. • Demonstrate that two polygons are congruent by mapping one onto another using rigid transformation. • Determine the composition of rigid transformations that map two congruent polygons onto each other. • Determine the minimum number of transformations that are needed to map one congruent polygon onto another. • Identify the minimum number of parts of a triangle and their relationships to each other (SAS, ASA, SSS) that can be transformed to result in congruent triangles. • Understand that AAA and SSA criteria do not necessarily create congruent triangles; use compass and straightedge constructions to create two non-congruent triangles that satisfy AAA and SSA. Stage 1 Established Goals: Common Core State Standards for Mathematics Cluster: 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 effect of a given rigid motion on a given figure; given two figures, use the Unit 4 Clover Park School District 2015-2016 Page 2 Geometry Mathematics Curriculum Guide 2015 – 2016 definition of congruence in terms of rigid motions to decide if they are congruent. G-CO.7 Use the 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 of triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. 2008 Geometry Standard connection: G.3.B Explanations, Examples, and Comments What students should know prior to this unit and may need to be reviewed Describe and apply the different types of transformations and be able to differentiate between them Perform translations, reflections, and rotations with and without the use of technology; including reflecting over parallel lines and reflecting over intersecting lines Describe and perform the composition of transformations that will map a given object onto a congruent object in the plane with or without technology Perform all the basic constructions using a compass and straightedge Explanations: A rigid motion is a transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions are assumed to preserve distances and angle measures. Students may use geometric software to explore the effects of rigid motion on a figure(s) and theorems about lines and angles. Congruence of triangles Two triangles are said to be congruent if one can be exactly superimposed on the other by a rigid motion, and the congruence theorems specify the conditions under which this can occur.Develop the relationship between transformations and congruency. Allow adequate time and provide hands-on activities for students to visually and physically explore rigid motions and congruence. Use graph paper, tracing paper or dynamic geometry software to obtain images of a given figure under specified rigid motions. Note that size and shape are preserved. Use rigid motions (translations, reflections and rotations) to determine if two figures are congruent. Compare a given triangle and its image to verify that corresponding sides and corresponding angles are congruent. Work backwards – given two figures that have the same size and shape, find a sequence of rigid motions that will map one onto the other. Unit 4 Clover Park School District 2015-2016 Stage 3 MATERIALS BY STANDARD(S): Teacher should use assessment data to determine which of the materials below best meet student instructional needs. All materials listed may not be needed. Holt Geometry Lesson 4-3 Congruent Triangles Holt Geometry Lesson 4-4 Triangle Congruence SSS and SAS Holt Geometry Lesson 4-5 Triangle Congruence ASA, AAS, and HL Holt Geometry Lesson 4-6 Triangle Congruence CPCTC Supplemental Materials Discovering Geometry 4.4 Are There Congruence Shortcuts? Discovering Geometry 4.5 Are There Other Congruence Shortcuts? Discovering Geometry 4.6 Corresponding Parts of Congruent Triangles Discovering Geometry 4.7 Flowchart Thinking Discovering Geometry 13.3 Triangle Proofs Performance Tasks MAP: Analyzing Congruence Proofs MVP: Congruent Triangles MVP: Congruent Triangles to the Rescue Georgia CCGPS: Proving Two Triangles are Congruent Page 3 Geometry Mathematics Curriculum Guide 2015 – 2016 Build on previous learning of transformations and congruency to develop a formal criterion for proving the congruency of triangles. Construct pairs of triangles that satisfy the ASA, SAS or SSS congruence criteria, and use rigid motions to verify that they satisfy the definition of congruent figures. Investigate rigid motions and congruence both algebraically (using coordinates) and logically (using proofs). Cluster: Standard(s) Prove geometric theorems 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 2008 Geometry Standard connection: G.3.A Explanations, Examples, and Comments What students should know prior to this unit and may need to be reviewed Understand and apply the definitions of: interior angles, isosceles triangles, parts of isosceles triangles, mid-segment of a triangle, medians of a triangle. Theorems Included in these standards: Triangle sum theorem Isosceles triangle theorem Mid-segment of a triangle is parallel to a side and half the length Medians of a triangle are concurrent Stage 3 MATERIALS BY STANDARD(S): Teacher should use assessment data to determine which of the materials below best meet student instructional needs. All materials listed may not be needed. Holt Geometry Lesson 4-2 Angle Relationships in Triangles Holt Geometry Lesson 5-2 Triangle Bisectors Holt Geometry Lesson 5-3 Medians &Altitudes of Triangles Holt Geometry Lesson 5-4 Triangle Mid-segment Holt Geometry Lesson 4-8 Isosceles and Equilateral Triangles Supplemental Material Discovering Geometry 3.4 Constructing Angle Bisectors Discovering Geometry 3.7 Constructing Points of Concurrency Discovering Geometry 3.8 The Centroid Performance Tasks Performance Tasks MVP: It’s All In Your Head MVP: Conjectures and Proofs MVP: Centers of Triangles Georgia CCGPS: Triangle Proofs Prove geometric theorems G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram Unit 4 Clover Park School District 2015-2016 Page 4 Geometry Mathematics Curriculum Guide 2015 – 2016 bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. 2008 Standards Connection: G.3.F, G.3.G, G.4.C The focus of this standard is to use lines and angles to prove theorems about parallelograms. What students should know prior to this unit and may need to be reviewed Understand and apply the definitions of: parallelograms, diagonals, opposite sides, opposite angles, consecutive angles, rectangles Theorems Included in this standards: Opposite sides of a parallelogram are congruent Opposite angles of a parallelogram are congruent Diagonals of a parallelogram bisect each other Rectangles are parallelograms with congruent diagonals Georgia Analytical Geometry EOCT Study Guide; pgs 42-46 provides an overview of content. MATERIALS BY STANDARD(S): Teacher should use assessment data to determine which of the materials below best meet student instructional needs. All materials listed may not be needed. Holt Geometry Lesson 6-1 Properties and Attributes of Polygons Holt Geometry Lesson 6-2 Properties of Parallelograms Holt Geometry Lesson 6-3 Conditions of Parallelograms Holt Geometry Lesson 6-4 Properties of Special Parallelograms Holt Geometry Lesson 6-5 Conditions for Special Parallelograms Holt Geometry Lesson 6-6 Kites and Trapezoids Supplemental Materials Parallelograms and other Quadrilaterals Discovering Geometry 1.6 Special Quadrilaterals Discovering Geometry 5.3 Kites and Trapezoid Properties Discovering Geometry 5.5 Properties of Parallelograms Discovering Geometry 5.6 Properties of Special Parallelograms Discovering Geometry 5.7 Proving Quadrilateral Properties Discovering Geometry 13.4 Quadrilateral Proofs Other Materials EngageNY Geometry Module 1 Lesson 22-27 MAP: Square Georgia CCGPS: Constructing Diagonals Georgia CCGPS: Proving Quadrilaterals in the Coordinate Plane MVP: Parallelogram Conjectures and Proof Unit 4 Clover Park School District 2015-2016 Page 5 Geometry Mathematics Curriculum Guide Evaluative Criteria/Assessment Level Descriptors (ALDs): Claim 1 Clusters: 2015 – 2016 Stage 2 - Evidence Sample Assessment Evidence Concepts and Procedures NONE Claim 2 Clusters: Problem Solving NONE Claim 3 Clusters: Prove geometric theorems Understand congruence in terms of rigid motions Go here for Sample SBAC items Communicating Reasoning Level 3 students should be able to use stated assumptions, definitions, and previously established results and examples to test and support their reasoning or to identify, explain, and repair the flaw in an argument. Students should be able to break an argument into cases to determine when the argument does or does not hold. Level 4 students should be able to use stated assumptions, definitions, and previously established results to support their reasoning or repair and explain the flaw in an argument. They should be able to construct a chain of logic to justify or refute a proposition or conjecture and to determine the conditions under which an argument does or does not apply. Go here for more information about the Achievement Level Descriptors for Mathematics: Other Assessment – at or near the completion of this unit (approximately school day 82) give the Geometry 1st Semester Summative. This assessment will be sent to building administrators. Unit 4 Clover Park School District 2015-2016 Page 6 Geometry Mathematics Curriculum Guide 2015 – 2016 Stage 3 – Learning Plan: Sample Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning LEARNING ACTIVITIES: Suggested Sections of the unit NOTES: Triangle centers can be included in this unit or in the circles unit because they can relate to standards in either unit. Parallelograms Holt Geometry Chapter 6 Lesson 2 & 3 Triangle Properties: Holt Geometry Chapter 4 Lesson 3 Holt Geometry Chapter 4 Lesson 8 Holt Geometry Chapter 5 Lesson 3 The sections of this unit can be rearranged to building philosophy. Triangle Congruence EngageNY Geometry Module 1, Lesson 22, 24, 29, 30 Holt Geometry Chapter 4 Lesson 4-6 Daily Lesson Components Learning Target Warm-up Activities • Whole Group: • Small Group/Guided/Collaborative/Independent: • Whole Group: Checking for Understanding (before, during and after): Assessments Unit 4 Clover Park School District 2015-2016 Page 7