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Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles Unit 3: Lines & Angles 2016 – 2017 Time Frame: 8 Days Primary Focus There are 2 main parts to this unit: a) Students will build on their work in the previous unit on coordinate geometry and further develop their understanding of slope b) Students will use slope to develop theorems about parallel perpendicular lines 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. Unit 3 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 observations and prior knowledge (stated assumptions, definitions, and previous established results) to make conjectures and construct arguments. Clover Park School District 2016-2017 Page 1 Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles 2016 – 2017 Stage 1 Desired Results Transfer Goals Students will be able to independently use their learning to… Prove properties of two- and three-dimensional geometric figures when solving increasingly complex problems involving geometric figures and their measurements. UNDERSTANDINGS Students will understand that… Meaning Goals ESSENTIAL QUESTIONS The basic properties of parallel and perpendicular lines, their respective slopes, and the properties of the angles formed when parallel lines are intersected by a transversal are important when proving related theorems and solving both mathematical and practical problems. • • • How do you summarize and explain basic theorems? How do you know that two lines are parallel? How do you know that two lines perpendicular to each other? Acquisition Goals Students will know and will be skilled at… Determining the coordinates of a point described geometrically. Explaining and performing basic compass and straightedge constructions related to parallel and perpendicular lines. Proving that if two parallel lines are cut by a transversal, and then alternate-interior angles are equal. Determining the equation of a line in the coordinate plane described geometrically from: (two points, parallel line through a given point, and perpendicular line through a given point). Proving and applying theorems about parallel and perpendicular lines. Proving and applying theorems about angles, including angles that arise from parallel lines intersected by a transversal. Summarizing and explaining basic theorems. Unit 3 Clover Park School District 2016-2017 Page 2 Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles 2016 – 2017 Stage 1 Established Goals: Common Core State Standards for Mathematics Note on Proofs for this unit: Students may use geometric simulations (computer software or graphing calculator) to explore theorems about lines and angles. Use inductive and deductive reasoning, students will solve problems, proofs, and real world situations involving parallel lines, perpendicular lines, and the angles relationships formed by those lines Proofs in high school geometry should not be restricted to the two-column format. Most proofs at the college level are done in paragraph form, with the writer explaining and defending a conjecture. In many cases, the two-column format can hinder the student from making sense of the geometry by paying too much attention to format rather than mathematical reasoning. That being said, as it relates to high school math, formal proofs have been deemphasized in the common core geometry progression. Providing references for students will help them utilize and apply precise language more efficiently while also keeping proofs from disproportionately taking up too much class time. Vocabulary: hypothesis, transitive property, symmetric property, reflexive property, law of syllogism, law of detachment, counterexample, converse Cluster: Standard(s) Use coordinates to prove simple geometric theorems algebraically G.GPE.5 – Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Explanations, Examples, and Comments Lines can be horizontal, vertical, or neither. Stage 3 MATERIALS BY STANDARD(S): Students may use a variety of different methods to construct a parallel or perpendicular line to a given line and calculate the slopes to compare the relationships. Teacher should use assessment data to determine which of the materials below best meet student instructional needs. All materials listed may not be needed. Relate work on parallel lines in G.GPE.5 to work on A.REI.5 in High School Algebra I involving systems of equations having no solution or infinitely many solutions Review the concept of slope as the rate of change of the y-coordinate with respect to the x-coordinate for a point moving along a line, and derive the slope formula. Use similar triangles to show that every non-vertical line has a constant slope. Review the point-slope, slope-intercept and standard forms for equations of lines. Investigate pairs of lines that are known to be parallel or perpendicular to each other and discover that their slopes are either equal or have a product of –1, respectively. Unit 3 Clover Park School District 2016-2017 Holt Geometry Lesson 3-6 Lines in the Coordinate Plane Supplemental Materials Discovering Geometry Using your Algebra Skills 3: p167 Slopes of Parallel and Perpendicular Lines Page 3 Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles Pay special attention to the slope of a line and its applications in analyzing properties of lines. Allow adequate time for students to become familiar with slopes and equations of lines and methods of computing them. Connect to G.GPE.4,6 and 7 Have student given two points, find the following: slope, distance, and midpoint 2016 – 2017 Performance Tasks: MVP Task: Slippery Slopes Georgia CCGPS: Discovery Task: Slopes of Special Pairs of Lines Georgia Coordinate Algebra EOCT Study Guide Unit 6; pgs 166 – 179 provides an overview of content. Cluster: Standard(s) Make geometric constructions G.CO.12 -- Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Explanations, Examples, and Comments Stage 3 MATERIALS BY STANDARD(S): Teacher should use assessment data to determine which Focus on: constructing perpendicular lines, including the perpendicular bisector of a line segment; and of the materials below best meet student instructional constructing a line parallel to a given line through a point not on the line. needs. All materials listed may not be needed. Additional Resources for Constructions Construction Resource Constructions Parallel Lines: Holt Geometry Lesson 3-3 & 3-3 Lab Perpendicular Lines: Holt Geometry Lesson 3-4 & 34 Lab Supplemental Material: Discovering Geometry 3.2 Constructing Perpendicular Bisectors Discovering Geometry 3.3 Constructing Perpendiculars to a Line Discovering Geometry 3.5 Constructing Parallel Lines Georgia Analytical Geometry EOCT Study Guide; pgs 47-59 provides additional resources for Constructions Unit 3 Clover Park School District 2016-2017 Page 4 Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles 2016 – 2017 Cluster: Standard(s) 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. Explanations, Examples, and Comments What students should know prior to this unit and may need to be reviewed Understand and apply the definitions of: linear pair of angles, parallel lines and angles formed by transversals, corresponding angles, corresponding sides, perpendicular lines, perpendicular bisector, and equidistance Stage 3 MATERIALS BY STANDARD(S): Theorems included in this standard: Vertical angle theorem Alternate interior angle theorem Corresponding angle theorem Perpendicular bisector theorem Holt Geometry Lesson 3-1 Lines and Angles Holt Geometry Lesson 3-2 Angles Formed by Parallel Line and Transversals Holt Geometry Lesson 3-3 Proving Lines Parallel Holt Geometry Lesson 3-4 Perpendicular Lines Georgia Analytical Geometry EOCT Study Guide; pgs 42-46 provides an overview of content. Supplemental Materials Discovering Geometry 2.5 Angle Relationships Discovering Geometry 2.6 Special Angles on Parallel Lines Discovering Geometry 3.5 Constructing Parallel Lines Theorems Vertical angles are congruent -Geometry Holt Unit 2: Lesson 7 Alternate interior angles -Geometry Holt Unit 3: Lesson 2 Points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.-Geometry Holt Unit 5: Lesson 1 Unit 3 Clover Park School District 2016-2017 Teacher should use assessment data to determine which of the materials below best meet student instructional needs. All materials listed may not be needed. Other Resources GSE Coordinate Algebra: Unit 6: Connecting Algebra and Geometry Through Coordinates: Slopes of Special Pairs of Lines MVP: Parallelism Preserved and Protected EngageNY Geometry Module 1: Lessons 6-11 Page 5 Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles 2016 – 2017 Stage 2 - Evidence Evaluative Criteria/Assessment Level Descriptors (ALDs): Claim 1 Clusters: Sample Assessment Evidence Concepts and Procedures NONE Claim 2 Clusters: Problem Solving NONE Claim 3 Clusters: Prove geometric theorems 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 Sample SBAC items Go here for more information about the Achievement Level Descriptors for Mathematics Common Assessment See Sample Assessments for Geometry Units. Unit 3 Clover Park School District 2016-2017 Page 6 Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles 2016 – 2017 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 Sequence of the Unit (See NOTES section for more detail for Extended Geometry.) Day 1: Holt Geometry Chapter 3 Lesson 6 - Exit Task, see pg 201 or 205 Day 2: Georgia Coordinate Algebra: Unit 6: Connecting Algebra and Geometry Through Coordinates: Slopes of Special Pairs of Lines (Discovery Task) Days 3-4: Holt Geometry Chapter 2 Lessons 1,5,6,7 - Provide EngageNY Geo Module 1 Teacher Guide pgs 58-60 to students for their reference. The emphasis is not on memorization, but on logical use of given theorems. - Holt 2-2 was removed because it does not align with Common Core. Only Converse is needed, and it was taught in 8th Grade. You may review the term converse. - Exit Tasks, see pgs103,127,132,133 Days 5-7: EngageNY Geometry Module 1, Lesson 6,7, and 10 (use to develop Theorems) Teacher - Student - See note above about providing reference pages to students. - Exit Tickets provided. Common Assessment (Includes use of reference pages noted above.) Daily Lesson Components Learning Target Opening Activities Activities Whole Group: Small Group/Guided/Collaborative/Independent: Whole Group: Checking for Understanding (before, during and after): Assessments Unit 3 Clover Park School District 2016-2017 NOTES: Scaffolding that may be needed: Slope and Linear Function Review from Alg. 1 Review from 8th grade identifying angle pairs formed by parallel lines cut by a transversal Students should get supportive scaffolds with simple proofs, both two-column and paragraph. Proofs have been de-emphasized in progression documents. Extended Geometry classes should select two or three of these Discovering Geometry lessons to support student success in the core curriculum. Discovering Geometry 2.5 Angle Relationships Discovering Geometry 2.6 Special Angles on Parallel Lines Discovering Geometry 3.5 Constructing Parallel Lines Portions of Holt Geometry Lessons 3-1 through 3-4 may also help students. Other Resources MVP: Parallelism Preserved and Protected Page 7