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Subject Geometry Honors Grade 10 Unit Name Overview Angles, Lines, Proof & Constructions Montclair Public Schools CCSS Geometry Honors Unit: Marshall A.b Unit # 1 Pacing 10 weeks Students will begin by understanding the building blocks of geometry. Basic terms, theorems, and proofs will be introduced. Students will explore these ideas by learning how to create proofs (formal and informal) and by generating constructions (using compass and straightedge, or geometric software and other methods). Standard # Standard MC, SC, or AC SLO # Student Learning Objectives Depth of Knowledge G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. SC 1 Define precisely the key vocabulary – angle, circle, perpendicular line, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 1 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 MC 2 Calculate the measures of angles or lengths of segments using the angle addition or segment addition postulates. 2 3 Formally prove vertical angles congruence theorem. 3 4 Formally prove the congruent supplement / congruent complement theorem. 3 5 Formally prove all right angles are congruent. 3 6 Apply properties of angles formed when a transversal intersects two parallel lines (e.g. AIA, SSIA, AEA, CA) 3 7 Prove that lines are parallel given special angle 4 20132014 relationships. 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, find the distance between a line and a point not on the line, and the distance between parallel lines). Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper G.CO.12 20132014 MC SC 8 Create an informal proof for properties of transversals and parallel lines (e.g. alternate interior angles are congruent, corresponding angles are congruent, sameside interior angles are supplementary) 3 9 Create an informal proof for the theorem that points on a perpendicular bisector of a line segment are exactly those equidistant from the segments endpoints.) Write an equation of a line in point-slope form and slope-intercept form when given the coordinates of two points, when given the line on a coordinate, or when given the coordinates of one point and the slope of the line. 3 11 Write the equation of a parallel or perpendicular line when given the same criteria. 2 12 Use the distance formula for a directed line segment (x1, y1) (x2, y2) 2 13 Calculate the distance between a line and a point not on the line given the equation of the line and the coordinates of the point. 3 14 Calculate the distance between two parallel lines. 3 15 Simplify complicated radical expressions without the use of calculators Use a compass and a straightedge to construct a) congruent segments b) perpendicular bisectors 2 10 16 2 2 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 G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Mathematical Practice # c) congruent angles d) angle bisectors e) parallel lines, given a line and a point not on the line MC 17 Calculate the midpoint of a line segment, given two points or from a coordinate. 2 18 Find the common midpoint of diagonals in a parallelogram drawn in a coordinate plane. 2 19 Extend the midpoint concept to calculate the coordinates of a point that separates a line with endpoints P1 & P2 into the ratio is a/b. Calculate the coordinates to be ((x1+(a/b)(x2-x1)), (y1+(a/b)(y2-y1)) 3 Selected Opportunities for Connections to Mathematical Practices Big Ideas Undefined terms (point, line, plane) are the building blocks of geometry. Inductive and deductive reasoning are used to prove valid geometric statement true. Geometric constructions help students discover and explore geometric concepts and interpret geometric concepts. Parallel line properties (CA, AIA, SSIA, AEA and their converses) can be used to find missing angles and in proofs. Equations can also be used to determine parallel and perpendicular lines Essential Questions • Explain the significance of undefined terms (point, line, plane) to the study of geometry. 20132014 • Why is it important to provide every logical step in a proof? • How do you determine the measures of all the angles created by two parallel lines cut by a transversal, given only one angle’s measure? How can you use equations of lines to determine if they are parallel or perpendicular or neither? • What are the basic tools for a geometric construction and how does a construction differ from a measurement? Assessments Key Vocabulary Construction Coordinate Postulate Theorem Transversal Suggested Resources (list specific chapters and or page numbers from existing text that correspond to the SLOs and Standards) Prentice Hall Geometry Textbook Chapter 1 Sections: 1-1 (conjecture and counterexamples only), 1-2, 1-3, 1-4, 1-5, 1-6. (Focus on problems at the end of each set of exercises.) Chapter 2 Sections: 2-4, 2-5 Chapter 3 Sections: 3-1, 3-2, 3-5, 3-6, 3-7. 20132014