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Sand Creek Zone Curriculum Map Subject: Geometry Strand/Concept Q1 Unit 1 Engage NY Mod 1:A,B Grade: Student Expectation Student Friendly Learning Objective (SLO) Quarter: Level of thinking G-CO.A.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment I can precisely define angle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, and distance along a line. Knowledge Angle Distance End point Line Line segment Parallel lines Point Precision Perpendicular lines Ray Sides of an angle Undefined notion Vertex of an angle G-CO.D.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). I can make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geographic software Application Compass Formal geometric Construction Straightedge CG-CO.D.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. I can construct an equilateral triangle Application Construction Inscribe Strand: Congruence Concept: Experiment with transformations in the plane Strand: Congruence Concept: Make geometric constructions Strand: Congruence Concept: Make geometric constructions Vocabulary 1 Sand Creek Zone Curriculum Map Subject: Geometry Strand/Concept Strand: Congruence Grade: Student Expectation Student Friendly Learning Objective (SLO) Quarter: Level of thinking G-CO.C.9: Prove theorems about lines and angles. I can prove theorems about lines and angles Analysis Interior angles Line Prove Theorem G-CO.A.2: Represent transformations in the plane I can represent transformations in the planning using, e.g., transparencies and geometry software Comprehension Compare Describe Distance Function Horizontal stretch Input Output Plane Point Represent Transformation Translation Concept: Prove geometric theorems Q1/2 Unit 2 Engage NY Mod 1:C Strand: Congruence I can describe transformations as functions that take points in the plane as inputs and give other points as outputs Concept: Experiment with transformations in the plane Strand: Congruence Concept: Experiment with transformations in the plane Strand: Congruence Concept: Experiment with transformations in the plane Vocabulary G-CO.A.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Given a rectangle, parallelogram, trapezoids, or regular polygon, I can describe the rotations and reflections that carry it onto itself Application Describe Polygon Rectangle Reflection Rotation Trapezoid G-CO.A.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. I can develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Application Circle Describe Line Segment Parallel lines Reflection Rotation Translation 2 Sand Creek Zone Curriculum Map Subject: Geometry Strand/Concept Strand: Congruence Concept: Experiment with transformations in the plane Grade: Student Expectation G-CO.A.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure Student Friendly Learning Objective (SLO) Given a geometric figure and a rotation, reflection, or translation, I can draw the transformed figure, e.g., graph paper, tracing paper, or geometry software Quarter: Level of thinking Comprehension Application Geometric figure Reflection Rotation Sequence of transformations Specify Transform Application Comprehension Congruent Decide Geometric description Rigid motion Transform Analysis Congruent Corresponding angles Corresponding sides Rigid motion Triangle I can specify a sequence of transformations that will carry a given figure onto another Strand: Congruence Concept: Understand congruence in terms of rigid motions G-CO.B.6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure I can use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure Vocabulary Given two figures, I can use the definition of congruence in terms of rigid motions to decide if they are congruent Q2 Unit 3 Engage NY Mod 1:D Strand: Congruence G-CO.B.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. I can 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 angle are congruent Concept: Understand congruence in terms of rigid motions 3 Sand Creek Zone Curriculum Map Subject: Geometry Strand/Concept Strand: Congruence Concept: Understand congruence in terms of rigid motions Grade: Student Expectation G-CO.B.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Student Friendly Learning Objective (SLO) I can explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid notions Quarter: Level of thinking Analysis Vocabulary ASA (angle-side-angle postulate) Congruence Criteria for triangle congruence Explain Rigid motion SAS (side-angle-side postulate) SSS(Side-side-side postulate) Interior angles Strand: Congruence Concept: Prove geometric theorems Strand: Congruence Concept: Prove geometric theorems G-CO.C.9: Prove1 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. I can prove theorems about lines and angles Analysis Line Prove Theorem G-CO.C.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. I can prove theorems about triangles Analysis Interior angle Prove Theorem Triangle 4 Sand Creek Zone Curriculum Map Subject: Geometry Strand/Concept Strand: Congruence Concept: Prove geometric theorems Q3 Unit 4 Engage NY Mod 2:E Strand: Similarity, Right Triangles, and Trigonometry Grade: Student Expectation Student Friendly Learning Objective (SLO) Quarter: Level of thinking Vocabulary G-CO.C.11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. I can prove theorems about parallelograms Analysis Prove Theorem G-SRT.C.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. I can state that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles Application Property Right triangle Side ratio Similarity Triangle Trigonometric ratio Understand G-SRT.C.7: Explain and use the relationship between the sine and cosine of complementary angles. I can explain and use the relationship between the sine and cosine of complementary angles Application Explain Complementary angle Cosine Relationship Sine Concept: Prove theorems involving similarity Strand: Similarity, Right Triangles, and Trigonometry Concept: Define trigonometric ratios and solve problems involving right triangles 5 Sand Creek Zone Curriculum Map Subject: Geometry Strand/Concept Strand: Similarity, Right Triangles, and Trigonometry Grade: Student Expectation Student Friendly Learning Objective (SLO) Quarter: Level of thinking G-SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. I can use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applies problems Application Applied problem Pythagorean theorem Right triangle Solve Trigonometric ration G-SRT.A.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, I can use the definition of similarity in terms of similarity transformations to decide if they are similar Application I can explain using similarity transformations, the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides Comprehension Corresponding angles Corresponding sides Decide Equality Explain Proportion Similar Similarity transformation Triangle G-SRT.A.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. I can use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Analysis Concept: Define trigonometric ratios and solve problems involving right triangles Strand: Similarity, Right Triangles, and Trigonometry Concept: understand similarity in terms of similarity transformations Strand: Similarity, Right Triangles, and Trigonometry Concept: understand similarity in terms of similarity transformations Vocabulary AA criterion (angle-angle) Properties of similarity transformations Similar Triangle 6 Sand Creek Zone Curriculum Map Subject: Geometry Strand/Concept Strand: Similarity, Right Triangles, and Trigonometry Grade: Student Expectation Student Friendly Learning Objective (SLO) Quarter: Level of thinking G-SRT.B.5: Use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures. I can use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures. Analysis Prove Solve Geometric figure Relationship Similarity criteria Triangle G-MG.A.1: Using geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder) I can use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder) Application Cylinder Describe Geometric Measure Model Property Shape G-GMD.A.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. I can five and informal argument for the formulas for the circumference of a circle Analysis Argument Circle Circumference Formula Informal Concept: Prove theorems involving similarity Strand: Modeling with Geometry Concept: Apply geometric concepts in modeling situations Q3 Unit 5 Engage NY Mod 3:A,B Strand: Geometric Measurement and Dimension Vocabulary Context: Explain volume formulas and use them to solve problems 7 Sand Creek Zone Curriculum Map Subject: Geometry Strand/Concept Strand: Geometric Measurement and Dimension Grade: Student Expectation Student Friendly Learning Objective (SLO) Quarter: Level of thinking G-GMD.A.3: Use volume formulas for cylinders, pyramids, cones and spheres to solve problems. I can use volume formulas for cylinders, pyramids, cones and spheres to solve problems. Application Cone Cylinder Solve Pyramid Sphere Volume formula G-GMD.B.4: Identify the shapes of twodimensional cross-sections of threedimensional objects, and identify threedimensional objects generated by rotations of I can identify the shapes of two-dimensional cross-sections of three-dimensional objects, Application Cross section Generate Identify Rotation Three-dimensional Two-dimensional Shape G-GMD.A.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. I can five and informal argument for the formulas for the circumference of a circle Analysis Context: Explain volume formulas and use them to solve problems Strand: Geometric Measurement and Dimension Concept: Visualize relationships between two-dimensional and three-dimensional objects Strand: Geometric Measurement and Dimension Concept: Visualize relationships between two-dimensional and three-dimensional objects Vocabulary 8 Sand Creek Zone Curriculum Map Subject: Geometry Strand/Concept Strand: Geometric Measurement and Dimension Grade: Student Expectation Student Friendly Learning Objective (SLO) Quarter: Level of thinking G-GMD.A.3: Use volume formulas for cylinders, pyramids, cones and spheres to solve problems. I can use volume formulas for cylinders, pyramids, cones and spheres to solve problems. Application Cylinder Describe Geometric Measure Model Property Shape G-GMD.B.4: Identify the shapes of twodimensional cross-sections of threedimensional objects, and identify threedimensional objects generated by rotations of two- dimensional objects. I can identify the shapes of two-dimensional cross-sections of three-dimensional objects, Application Cross section Generate Identify Rotation Three-dimensional Two-dimensional Shape G-GPE.B.4: Use 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). I can use coordinates to prove simple geometric theorems algebraically Concept: Visualize relationships between two-dimensional and three-dimensional objects Strand: Modeling with Geometry Concept: Apply geometric concepts in modeling situations Q4 Unit 7 Engage NY Mod 4 Strand: Expressing Geometric Properties and Equations Vocabulary I can identify three-dimensional objects generated by rotations of two- dimensional objects Concept: Use coordinates to prove simple geometric theorems algebraically 9 Sand Creek Zone Curriculum Map Subject: Geometry Strand/Concept Strand: Expressing Geometric Properties and Equations Concept: Use coordinates to prove simple geometric theorems algebraically Strand: Expressing Geometric Properties and Equations Grade: Student Expectation Student Friendly Learning Objective (SLO) Quarter: Level of thinking Vocabulary G-GPE.B.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). I can 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). Analysis G-GPE.B.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio. I can find the point on a directed line segment between two given points that partitions the segment in a given ratio. Application Directed line segments Partition Point Ratio G-GPE.B.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula I can use coordinates to compute perimeters of polygons e.g., using the distance formula Application I can compute areas of triangles and rectangles e.g., using the distance formula Application Compute Coordinate Perimeter Polygon Rectangle Triangle Application Equation Geometric problem Line Parallel lines Point Prove Slope criteria Solve Concept: Use coordinates to prove simple geometric theorems algebraically Strand: Expressing Geometric Properties and Equations Concept: Use coordinates to prove simple geometric theorems algebraically 10 Sand Creek Zone Curriculum Map Subject: Geometry Strand/Concept Q4 Unit 6 Engage NY Mod 5:all Strand: Circles Concept: Understand and apply theorems to circles Strand: Circles Concept: Understand and apply theorems to circles Strand: Circles Concept :Find arc lengths and areas of sectors of circles Grade: Student Expectation Student Friendly Learning Objective (SLO) Quarter: Level of thinking G-C.A.2: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. I can identify and describe relationships among inscribed angles, radii, and chords. Comprehension I can describe relationships among inscribed angles, radii, and chords. Application G-C.A.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.. I can construct the inscribed ad circumscribed circles of a triangle Application I can prove properties of angles for a quadrilateral inscribed in a circle Analysis G-C.B.5: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. I can derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius Analysis I can define the radian measure of the angle as the constant of proportionality Comprehension I can derive the formula for the area of sector Analysis Vocabulary Chord Describe Diameter Identify Inscribe Inscribed angle Radius Relationship Circle Circumscribed circle of a triangle Construct Inscribe Inscribed circle of a triangle Prove Quadrilateral Arc Constant of proportionality Define Derive Formula Intercept Proportion Radian Radius Sector Similarity 11 Sand Creek Zone Curriculum Map Subject: Geometry Strand/Concept Grade: Student Expectation Student Friendly Learning Objective (SLO) Quarter: Level of thinking Vocabulary G-C.A.1: Prove that all circles are similar 12 COMMON CORE AND COLORADO ACADEMIC STANDARDS 13