Line and Angle Proofs

Transcription

1 Line and Angle Proofs Overview Number of instruction days: 8-10 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Develop and prove conjectures through logical thinking. Prove theorems about lines and angles that show that vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. 2 Reason abstractly and quantitatively. Make sense of relationships among lines and angles in order to write informal and formal proofs. 3 Construct viable arguments and critique the reasoning of others. Use relationships among lines and angles, justify conclusions, communicate to others, and respond to the arguments of others. Critique the arguments of others by comparing the effectiveness of two plausible arguments, distinguishing correct logic or reasoning from that which is flawed, and, if there is a flaw in an argument, explain what it is. 6 Attend to precision. Utilize precise definitions in construction of arguments and in discussions with others for a common understanding. 7 Look for and make use of structure. Identify and use relationships in order to construct viable arguments or proofs. Providence Public Schools D-35

2 Line and Angle Proofs (8-10 days) Essential Questions What are the different kinds of proofs that can be used to show logical reasoning? How do you prove or disprove conjectures using logical thinking? Why is it important to use reasoning to prove geometric concepts such as the relationship among line segments and angles? How is this study of Geometry similar to and different from the study of Geometry in previous grades? When do you use formal proofs to justify your thinking? Standards Common Core State Standards for Mathematical Content Congruence G-CO Prove geometric theorems [Focus on validity of underlying reasoning while using variety of ways of writing proofs] 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. Common Core State Standards for Mathematical Practice 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the D-36 Providence Public Schools

3 Line and Angle Proofs (8-10 days) Geometry, Quarter 2, Unit 2.1 units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Providence Public Schools D-37

4 Line and Angle Proofs (8-10 days) Clarifying the Standards Prior Learning In previous grades, students informally used reasoning to draw conclusions and to make and justify conjectures and arguments in both written and oral form. In Grade 8, students used ideas about distance and angles and how they behave under translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students showed that the sum of the angles in a triangle is the angle formed by a straight line and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines. Students also applied theorems, such as the Pythagorean Theorem, to solve problems. Current Learning In this unit, students develop informal and formal proofs of previously learned concepts and apply them as they did in earlier grades. Students also use measurement of line segments as a tool in developing the idea of proof. For example, students prove theorems relating to congruence of vertical angles, transversals and parallel lines, and points on a perpendicular bisector of a line segment. Geometry students learn to write proofs in multiple ways, including narrative paragraphs, flow diagrams, the twocolumn format, and diagrams without words. According to Appendix A, Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Proving geometric theorems is major content as defined by the PARCC Model Frameworks for Mathematics. Future Learning Students will continue to need the ability to reason and justify through proof, both formally and informally, in this course. Later in Algebra II and Precalculus, as well as in real-world situations, students will continue to use logical reasoning in all problem situations. Additional Findings Using logical thinking to prove theorems is challenging for students because it requires them to think in a new and precise way about situations presented to them. Many students have difficulty with formulating justifications for their conclusions, especially in mathematics. Proof is a form of justification, but not all justifications are proofs. (Adding It Up, p. 132) Proofs are challenging for both students and teachers, since [c]onjectures are not the same as proofs. Finding precise descriptions of conditions for the first step is important. Students are not accustomed to justifying their reasoning using formalized methods of proof. (Principles and Standards for School Mathematics, p. 311) D-38 Providence Public Schools

5 Line and Angle Proofs (8-10 days) Geometry, Quarter 2, Unit 2.1 Another challenge is that as students progress through levels of learning and understanding in mathematics, [e]ach level has its own language and way of thinking; teachers unaware of this hierarchy of language and concepts can easily misinterpret students understanding of geometric ideas. At Level 4, students can establish theorems within an axiomatic system. A misconception to be overcome is that students have difficulty understanding the difference between an explanation and a justification. (A Research Companion to Principles and Standards for School Mathematics, p. 152) Assessment When constructing an end-of-unit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Demonstrate logical thinking by developing and proving conjectures. Justify validity of the underlying geometry reasoning using a variety of methods. Formally and informally prove theorems about lines and angles, including the following: Vertical angles are congruent. Angle pairs resulting from when a transversal crosses parallel lines. Points on a perpendicular bisector of a line segment. Providence Public Schools D-39

6 Line and Angle Proofs (8-10 days) Instruction Learning Objectives Students will be able to Identify and use basic postulates involving points, lines, and planes to write paragraph proofs. Use algebra to write two-column proofs. Use the properties of equality to write geometric proofs. Write proofs involving segment addition and segment congruence. Prove theorems involving parallel lines and transversals. Apply an understanding of formal proof involving vertical, alternate interior and corresponding angles. Apply an understanding of formal proof involving points on a perpendicular bisector of a segment. Demonstrate and review knowledge of informal and formal proofs as a process of logical thought in order to develop definitions or rules. Resources Geometry, Glencoe McGraw-Hill, 2010, Student/Teacher Editions Section (pp ) Study Notebook 2-5 Postulates and Paragraph Proofs (pp ) 2-6 Algebraic Proof (pp ) 2-8 Proving Angle Relationships (pp ) 5-1 Bisectors of Triangles (pp ) Chapter 2 Resource Masters Chapter 5 Resource Masters Glencoe McGraw-Hill Online D-40 Providence Public Schools

7 Line and Angle Proofs (8-10 days) Geometry, Quarter 2, Unit 2.1 Teacher Works CD-ROM Interactive Classroom CD (PowerPoint Presentations) Exam View Assessment Suite Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations. Materials White paper, graphic organizer (optional) Instructional Considerations Key Vocabulary axiom converse counter example postulate proof theorem Planning for Effective Instructional Design and Delivery Reinforced vocabulary from previous grades or units: definition, conclusion, and hypothesis. Begin with postulates and paragraph proofs. If then statements may be reviewed, but they are addressed in earlier grade levels. Expand the thought process to include segment and angle addition, which is an example of inductive reasoning. Follow with the idea of step-by-step justification. This skill is used in both algebraic and geometric proof. Students have had experience using this skill when they have solved an equation. Students see the logical process, extend the properties of equality to the properties of congruence, and use these in proving theorems involving angle relationships, using the paragraph and two-column proof format. As students investigate the concept of geometric proof, use a cues, questions, and advance organizers strategy by asking appropriate questions at appropriate times. For example, How did you arrive at your conclusion? [Students should reply with mathematical language other than I did this..., I did that... ] What is another way to develop the same result? If changed, what effect would it have on your conclusion? Providence Public Schools D-41

8 Line and Angle Proofs (8-10 days) These questions can help students access prior knowledge, as they have had many opportunities to prove their conclusions or results. Students have done this by supporting the result with evidence of the algorithm information or by convincing a peer of the correctness of their result through conversation or written form. In this unit, students formalize their conversation and written format of that same correctness or proof of their conclusions. The foldable on page 170 is an excellent organizational tool, as it requires students to analyze the subject, determine what is essential for understanding, and write the results of their analysis in their own words. This initial work with proofs may be differentiated to include additional theorems and postulates. The following additional opportunity for differentiation is provided on page 139 in the teacher s edition. Provide students with algebraic and geometric proofs that are missing the justifications for each step. At least one proof should contain errors. Have students fill in the justifications and explain the errors. Using the 5-minute check transparencies is an excellent way of assessing student knowledge using the cues, questions, and advance organizers strategy. The questions help students review prior knowledge necessary for the lesson, and they should be used at the beginning of a new section. D-42 Providence Public Schools

9 Line and Angle Proofs (8-10 days) Geometry, Quarter 2, Unit 2.1 Notes Providence Public Schools D-43