Rochester City School District



Geometry-R Module 4Topic BPerpendicular and Parallel Lines 6 Days Topic CPerimeters and Areas 3 Days Topic DPartitioning and Extending Segments 7 Days Review /Test 2 DaysIn Lesson 5, students explain the connection between the Pythagorean theorem and the criterion for perpendicularity (G-GPE.B.4). Lesson 6 extends that study by generalizing the criterion for perpendicularity to any two segments and applying this criterion to determine if segments are perpendicular. In Lesson 7, students recognize that when a line and a normal segment intersect at the origin, the segment from (0,0) to (a1,a2) is the normal segment, with a slope of a2a1, and the equation of the line is a1x+a2y=c with a slope of –a1a2. Lesson 8 concludes Topic B when students recognize parallel and perpendicular lines from their slopes and create equations for parallel and perpendicular lines. The criterion for parallel and perpendicular lines and the work from this topic with the distance formula is extended in the last two topics of this module as students use these foundations to determine perimeter and area of polygonal regions in the coordinate plane defined by systems of inequalities. Additionally, students study the proportionality of segments formed by diagonals of polygons.Lesson 9 begins Topic C with students finding the perimeter of triangular regions using the distance formula and deriving the formula for the area of a triangle with vertices 0, 0,x1,y1, (x2,y2) as A= 12x1y2-x2y1 (G-GPE.B.7). Students compare the traditional formula for area and area by decomposition of figures and see that the “shoelace” formula is much more efficient in some cases. Lesson 11 concludes this work as the regions are described by a system of inequalities. Students sketch the regions, determine points of intersection (vertices), and use the distance formula to calculate perimeter and the “shoelace” formula to determine area of these regions. Students return to the real-world application of programming a robot and extend this work to robots not just confined to straight line motion but also motion bound by regions described by inequalities and defined areas Topic D concludes the work of Module 4. In Lesson 12, students find midpoints of segments and points that divide segments into 3 or more equal and proportional parts. Students also find locations on a directed line segment between two given points that partition the segment in given ratios (G-GPE.B.6). Lesson 13 requires students to show that if B' and C' cut AB and AC proportionately, then the intersection of BC' and B'C lies on the median of △ABC from vertex A and connects this work to proving classical results in geometry (G-GPE.B.4). For instance, the diagonals of a parallelogram bisect one another, and the medians of a triangle meet at the point 23 of the way from the vertex for each. Suggested Lessons Big IdeaEmphasize Sample Regents Module 4 June 2015 # 09, 27, 31, 36August 2015 # 10, 15,22, 31, 33January 2016 # #2,18, 15,27 TOPIC BImportant Concepts to Focus On:Focus on the distance formula, the midpoint and partition of a segment. The focus should also be on the coordinate geometry and proving quadrilaterals Lesson 1 Teacher created materials Lesson 1: Graphing Linear equations with two variable and systems of equations Objective: I can graph linear equations and systems of linear equations and I can identify solution(s) to systems of equations Suggested Day(s): ( 1) Use coordinates to prove simple geometric theorems algebraically.1FG-GPE.B.4Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1,3) lies on the circle centered at the origin and containing the point (0, 2).G-GPE.B.5Prove2F 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).G-GPE.B.6Find the point on a directed line segment between two given points that partitions the segment in a given ratio.G-GPE.B.7Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★Foundational StandardsSolve systems of equations.A-REI.C.6Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.Lesson 2Teacher created Materials Lesson 2: Writing the equation of a line in slope intercept form and point – slope form Objective: Students will write the equations of lines in slope- intercept and slope-point form when two points are given, a graph is given, and the slope and one point is given Suggested Day(s): ( 1) Lesson 5 in EngageNYLesson 3: Criterion for PerpendicularityObjective: Students explain the connection between the Pythagorean theorem and the criterion for perpendicularitySuggested day(s) : 1 Lesson 7 in Engage Ny Lesson 4 : Equations for Lines Using Normal Segments Objective: Students recognize a segment perpendicular to a line, with one of its endpoints on the line as a normal segment.Students recognize that when a line and a normal segment intersect at the origin, the segment from (0,0) to (a1,a2) is the normal segment, with a slope of a2a1, and the equation of the line is a1x+a2y=c with a slope of –a1a2.Suggested Day(s): ( 1) Lesson 8 in EngageNYLesson 5 Parallel and perpendicular Lines Objective: Students recognize parallel and perpendicular lines from slope. Students create equations for lines satisfying criteria of the kind: “Contains a given point and is parallel/perpendicular to a given line.”Suggested Day(s): ( 2) Topic C Important concepts to focus on The focus will be using the distance formula to calculate perimeter and determine area of these regions.Lesson 9 in EngageNy Lesson 6: Perimeter and Area of Triangles in the Cartesian planeObjective: Students find the perimeter of a triangle in the coordinate plane using the distance formula.Students state and apply the formula for area of a triangle with vertices (0,0),(x1, y1), and (x2 , y2).Suggested Day(s): ( 1)Lesson 10 In Enagageny Lesson 7: Perimeter and Area of Polygonal Regions in the PlaneObjective: I can find the perimeter of a quadrilateral in the coordinate plane given its vertices and edges.I can find the area of a quadrilateral in the coordinate plane given its vertices and edges Suggested Day(s): ( 1)Lesson 11-optional In Engageny Lesson 8: Triangle inequality and types of triangles Objective: Students find the perimeter of a triangle or quadrilateral in the coordinate plane given a description by inequalities.Students find the area of a triangle or quadrilateral in the coordinate plane given a description by inequalities by employing Green’s theorem.Suggested Day(s): ( 1 days )Topic D Important concepts to focus on The focus will be the midpoints of segments and points that divide segments into 3 or more equal and proportional parts and extend this concept prove classical results in geometry. Focus will also be on profs using coordinate geometry Lesson 12 In Enagageny Lesson 9: Midpoint and Partition Objective: Students find midpoints of segments and points that divide segments into 3, 4, or more proportional, equal parts Suggested Day(s): ( 2)Lesson 13 In Enagageny Lesson 10: Analytic Proofs of Theorems Previously Proved by Synthetic MeansObjective: Using coordinates, students prove that the intersection of the medians of a triangle meet at a point that is two-thirds of the way along each median from the intersected vertex.Using coordinates, students prove the diagonals of a parallelogram bisect one another and meet at the intersection of the segments joining the midpoints of opposite sides.Suggested Day(s): ( 3)Teacher created assessment Review of unit assessment Suggested Day(s): ( 2) ................
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