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Curriculum Guide 2014-2015 High School Geometry Poudre School District Geometry Standards Aligned With the Geometry PARCC Assessment Performance Based Assessment (PBA/MYA) and End Of Year (EOY) Cluster Standard PBA/MYA Unit 1: Congruence, Proof, and Constructions HS.G.CO.A HS.G.CO.A.1 X Experiment with transformations in the plane. HS.G.CO.A.2 X HS.G.CO.B Understand congruence in terms of rigid motions. HS.G.CO.C Prove Geometric theorems. HS.G.CO.D Make geometric constructions. HS.G.CO.A.3 X HS.G.CO.B.6 X HS.G.CO.A.4 HS.G.CO.A.5 HS.G.CO.B.7 HS.G.CO.B.8 HS.G.CO.C.9 HS.G.CO.C.10 HS.G.CO.C.11 HS.G.CO.D.12 HS.G.CO.D.13 Unit 2: Similarity, Proof, and Trigonometry HS.G.SRT.A Understand similarity in terms of similarity transformations. HS.G.SRT.B Prove theorems involving similarity. HS.G.SRT.C Define trigonometric ratios and solve problems involving right triangles. HS.G.SRT.D Apply trigonometry to general triangles. X X X X HS.G.SRT.C.7 HS.G.SRT.C.8 HS.G.SRT.D.9 HS.G.SRT.D.10 X X X HS.G.SRT.C.6 X X X HS.G.SRT.B.4 HS.G.SRT.B.5 X X X HS.G.SRT.A.3 X X HS.G.SRT.A.1 HS.G.SRT.A.2 EOY X X X X X X X X X X X X X HS.G.SRT.D.11 Page | 1 Curriculum Guide 2014-2015 High School Geometry Poudre School District Geometry Standards Aligned With the Geometry PARCC Assessment Performance Based Assessment (PBA/MYA) and End Of Year (EOY) Cluster Standard PBA/MYA Unit 3: Connecting Algebra and Geometry through Coordinates HS.G.GPE.B HS.G.GPE.B.4 X Use coordinates to prove simple geometric theorems. HS.G.GPE.B.5 X HS.G.GPE.B.6 HS.G.GPE.B.7 HS.G.GPE.A Translate between the geometric description and the HS.G.GPE.A.2 equation for a conic section. Unit 4: Circles With and Without Coordinates HS.G.MG.A HS.G.MG.A.1 Apply geometric concepts in modeling situations. HS.G.C.A HS.G.C.A.1 Understand and apply theorems about circles. HS.G.C.A.2 HS.G.C.A.3 HS.G.C.A.4 HS.G.C.B HS.G.C.B.5 Find arc lengths and areas of sectors of circles. HS.G.GPE.A Translate between the geometric description and the HS.G.GPE.A.1 equation for a conic section. HS.G.GPE.B Use coordinates to prove simple geometric theorems HS.G.GPE.B.4 algebraically. Unit 5: Extending to Three Dimensions HS.G.MG.A HS.G.MG.A.1 Apply geometric concepts in modeling situations. HS.G.MG.A.2 HS.G.GMD.A Explain volume formulas and use them to solve problems. HS.G.GMD.B Visualize the relation between two-dimensional and three-dimensional objects. HS.G.MG.A.3 HS.G.GMD.A.1 HS.G.GMD.A.3 HS.G.GMD.B.4 X X EOY X X X X X X X X X X X X X X X Page | 2 Curriculum Guide 2014-2015 High School Geometry Poudre School District Geometry Standards Aligned With the Geometry PARCC Assessment Performance Based Assessment (PBA/MYA) and End Of Year (EOY) Cluster Standard Unit 6: Applications of Probability HS.S.CP.A HS.S.CP.A.1 Understand independence and conditional probability HS.S.CP.A.2 and use them to interpret data. HS.S.CP.A.3 HS.S.CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability. HS.S.MD.B.6 Use probability to evaluate outcomes of decisions. PBA/MYA EOY HS.S.CP.A.4 HS.S.CP.A.5 HS.S.CP.B.6 HS.S.CP.B.7 HS.S.CP.B.8 HS.S.CP.B.9 HS.S.MD.B.6 HS.S.MD.B.7 Page | 3 Curriculum Guide 2014-2015 High School Geometry Course Overview Poudre School District The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences from the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments. Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high school CCSS. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas, organized into six units are as follows. Unit 1: Congruence, Proof, and Constructions In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motion: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorem – using a variety of formats – and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work. Students will be able to… • Experiment with transformations in the plane. HS.G.CO.A.1, HS.G.CO.A.2, HS.G.CO.A.3, HS.G.CO.A.4, HS.G.CO.A.5 • Understand congruence in terms of rigid motions. HS.G.CO.B.6, HS.G.CO.B.7, HS.G.CO.B.8 • Prove geometric theorems. HS.G.CO.C.9, HS.G.CO.C.10, HS.G.CO.C.11 • Make geometric constructions. HS.G.CO.D.12, HS.G.CO.D.13 Page | 4 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 2: Similarity, Proof, and Trigonometry Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem. Students develop the Laws of Sines and Cosines in order to find missing measures of general (not necessarily right) triangles. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles. Students will be able to… • Understand similarity in terms of similarity transformations. HS.G.SRT.A.1, HS.G.SRT.A.2, HS.G.SRT.A.3 • Prove theorems involving similarity. HS.G.SRT.B.4, HS.G.SRT.B.5 • Define trigonometric ratios and solve problems involving right triangles. HS.G.SRT.C.6, HS.G.SRT.C7, HS.G.SRT.C.8 • Apply trigonometry to general triangles. HS.G.SRT.D.9, HS.G.SRT.D.10, HS.G.SRT.D.11 Unit 3: Connecting Algebra and Geometry through Coordinates Building on their work with the Pythagorean Theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola. Students will be able to… • Use coordinates to prove simple geometric theorems. HS.G.GPE.B.4, HS.G.GPE.B.5, HS.G.GPE.B.6, HS.G.GPE.B.7 • Translate between the geometric description and the equation for a conic section. HS.G.GPE.A.2 Page | 5 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 4: Circles With and Without Coordinates In this unit, students prove basic theorems about circles, with particular attention to perpendicular and inscribed angles, in order to see symmetry in circles and as an application of triangle congruence criteria. They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations to determine intersections between lines and circles or parabolas and between two circles. Students will be able to… • Apply geometric concepts in modeling situations. HS.G.MG.A.1 • Understand and apply theorems about circles. HS.G.C.A.1, HS.G.C.A.2, HS.G.C.A.3, HS.G.C.A.4 • Find arc lengths and areas of sectors of circles. HS.G.C.B.5 • Translate between the geometric description and the equation for a conic section. HS.G.GPE.A.1 • Use coordinates to prove simple geometric theorems algebraically. HS.G.GPE.B.4 Unit 5: Extending to Three Dimensions Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line. Students will be able to… • Apply geometric concepts in modeling situations. HS.G.MG.A.1, HS.G.MG.A.2, HS.G.MG.A.3 • Explain volume formulas and use them to solve problems. HS.G.GMD.A.1, HS.G.GMD.A.3 • Visualize the relation between two-dimensional and three-dimensional objects. HS.G.GMD.B.4 Page | 6 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 6: Applications of Probability Building on probability concepts that began in the middle grades, students use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions. Students will be able to… • Understand independence and conditional probability and use them to interpret data. HS.S.CP.A.1, HS.S.CP.A.2, HS.S.CP.A.3, HS.S.CP.A.4, HS.S.CP.A.5 • Use the rules of probability to compute probabilities of compound events in a uniform probability. HS.S.CP.B.6, HS.S.CP.B.7, HS.S.CP.B.8, HS.S.CP.B.9 • Use probability to evaluate outcomes of decisions. HS.S.MD.B.6, HS.S.MD.B.7 Page | 7 Curriculum Guide 2014-2015 High School Geometry Unit 1: Congruence, Proof, and Constructions Poudre School District In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motion: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorem – using a variety of formats – and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work. Sub-Unit A: Transformations HS.G.CO.A.1 HS.G.CO.D.12 HS.G.CO.D.13 HS.G.CO.A.4 HS.G.CO.A.2 HS.G.CO.B.6 HS.G.CO.A.5 HS.G.CO.A.3 Sub-Unit B: Congruence HS.G.CO.B.7 HS.G.CO.C.9 HS.G.CO.B.8 HS.G.CO.C.11 HS.G.CO.C.10 Page | 8 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 1: Congruence, Proof, and Constructions Sub-Unit A: Transformations Start of Quarter 1 – October 3, 2014 Common Core State Standards Explanations/Examples Resources HS.G.CO.A: Experiment with transformations in the plane. Build on student experience with rigid motion from earlier grades. Point out the basis of rigid motions in geometric concepts, e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle. HS.G.CO.A.1 HS.G.CO.A.1 Lessons Know precise definitions of angle, circle, HS.G.CO.A.1 perpendicular line, parallel line, and line Prentice Hall segment, based on the undefined notions of • PH G point, line, distance along a line, and distance o Ch 1.3-1.6 around a circular arc. o Ch 3.3 o Ch 10.6 HS.G.CO.A.2 HS.G.CO.A.2 Represent transformations in the plane using, Students may use geometry software and/or HS.G.CO.A.2/HS.G.CO.A.4 e.g., transparencies and geometry software; manipulatives to model and compare transformations. Prentice Hall describe transformations as functions that take • PH G points in the plane as inputs and give other o Ch 9.1-9.3 points as outputs. Compare transformations that o p. 477 preserve distance and angle to those that do not o p. 483 (e.g., translation versus horizontal stretch). o p. 490 o Ch 9.5-9.6 HS.G.CO.A.3 HS.G.CO.A.3 Given a rectangle, parallelogram, trapezoid, or Students may use geometry software and/or HS.G.CO.A.2 regular polygon, describe the rotations and manipulatives to model transformations. Howard County reflections that carry it onto itself. • Experiment with Transformations • Similarity and Congruence HS.G.CO.A.4 HS.G.CO.A.4 Develop definitions of rotations, reflections, and Students may use geometry software and/or HS.G.CO.A.3 translations in terms of angles, circles, manipulatives to model transformations. Students may Prentice Hall perpendicular lines, parallel lines, and line observe patterns and develop definitions of rotations, • PH G segments. reflections, and translations. o Ch 9.4 Page | 9 Curriculum Guide 2014-2015 Common Core State Standards HS.G.CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. High School Geometry Unit 1: Congruence, Proof, and Constructions Sub-Unit A: Transformations Start of Quarter 1 – October 3, 2014 Explanations/Examples HS.G.CO.A.5 Students may use geometry software and/or manipulatives to model transformations and demonstrate a sequence of transformations that will carry a given figure onto another. Poudre School District Resources Lessons (continued) HS.G.CO.A.3 (continued) Howard County • Carrying a Figure onto Itself HS.G.CO.A.4 Howard County • Defining Transformations that Create Congruent Figures HS.G.CO.A.5 Prentice Hall • PH G o Ch 9.6 o p. 514 Tasks HS.G.CO.A.1/HS.G.CO.4 Inside Mathematics • Once Upon a Time HS.G.CO.A.2/HS.G.CO.A.3/HS.G.CO.A.4 Inside Mathematics • Between the Lines HS.G.CO.A.4/HS.G.CO.A.5 Inside Mathematics • The Shape of Things Page | 10 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 1: Congruence, Proof, and Constructions Sub-Unit A: Transformations Start of Quarter 1 – October 3, 2014 Explanations/Examples Poudre School District Resources Tasks (continued) HS.G.CO.A.5 Howard County • Transformation Sensation Illustrative Mathematics • Reflected Triangles • Regular Tessellations of the Plane Activities HS.G.CO.A.1 NCTM Illuminations • Shape Cutter HS.G.CO.A.2 Howard County • Patty Paper Tutorial Transformations that Preserve Distance NCTM Illuminations • Mirror Tool HS.G.CO.A.5 NCTM Illuminations • Algebraic Transformations Page | 11 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 1: Congruence, Proof, and Constructions Sub-Unit A: Transformations Start of Quarter 1 – October 3, 2014 Explanations/Examples Poudre School District Resources Practice HS.G.CO.A.1 Prentice Hall • PH G o p. 21 #55-60, 65-67 (C) o p. 571 #59 (A) Assessments HS.G.CO.D: Make geometric constructions. Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. HS.G.CO.D.12 HS.G.CO.D.12 Lessons Make formal geometric constructions with a Students may use geometric software to make HS.G.CO.D.12 variety of tools and methods (compass and geometric constructions. Prentice Hall straightedge, string, reflective devices, paper • PH G • Construct a triangle given the lengths of two sides folding, dynamic geometric software, etc.). o Ch 1.7 and the measure of the angle between the two Copying a segment; copying an angle; bisecting a o p. 51 sides. segment; bisecting an angle; constructing o p. 303 Construct the circumcenter of a given triangle. perpendicular lines, including the perpendicular o Ch 3.8 bisector of a line segment; and constructing a line o p. 319-320 parallel to a given line through a point not on the o p. 328 line. o Ch 6.3-6.4 HS.G.CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. HS.G.CO.D.13 Students may use geometric software to make geometric constructions. Page | 12 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 1: Congruence, Proof, and Constructions Sub-Unit A: Transformations Start of Quarter 1 – October 3, 2014 Explanations/Examples Poudre School District Resources Lessons (continued) HS.G.CO.D.12/HS.G.CO.D.13 Howard County • Investigating Properties of Parallelograms MARS Shell Center • Inscribing and Circumscribing Right Triangles HS.G.CO.D.13 Prentice Hall • PH G o Ch 6.2 Tasks HS.G.CO.D.12 Howard County • First to Finish • Safe Crossings Illustrative Mathematics • Angle Bisection and Midpoints of Line Segments • Bisecting an Angle • Construction of Perpendicular Bisector • Locating Warehouse • Reflected Triangles Page | 13 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 1: Congruence, Proof, and Constructions Sub-Unit A: Transformations Start of Quarter 1 – October 3, 2014 Explanations/Examples Poudre School District Resources Tasks (continued) HS.G.CO.D.12 (continued) Inside Mathematics • Once Upon a Time • Polly Gone • The Shape of Things • What's Your Angle? HS.G.CO.D.12/HS.G.CO.D.13 Illustrative Mathematics • Placing a Fire Hydrant Inside Mathematics • Circular Reasoning Activities HS.G.CO.D.12 Howard County • Compass Tutorial Copying a Segment or Angle • Compass Tutorial Constructing Parallel and Perpendicular Lines • Patty Paper Construction of Parallel and Perpendicular Lines Tutorial • Patty Paper Tutorial Copying a Segment or Angle • Sketchpad Constructions of Parallel and Perpendicular Lines Tutorial Page | 14 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 1: Congruence, Proof, and Constructions Sub-Unit A: Transformations Start of Quarter 1 – October 3, 2014 Explanations/Examples Poudre School District Resources Activities (continued) HS.G.CO.D.12/HS.G.CO.D.13 Howard County • Patty Paper Tutorial Angles in Parallelogram • Patty Paper Tutorial Diagonals of Rectangles and Parallelograms • Sketchpad Tutorial Parallelogram Theorems Practice Assessments HS.G.CO.B: Understand congruence in terms of rigid motions. Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can be used to prove other theorems. HS.G.CO.B.6 HS.G.CO.B.6 Lessons Use geometric descriptions of rigid motions to A rigid motion is a transformation of points in space HS.G.CO.B.6 transform figures and to predict the effect of a consisting of a sequence of one or more translations, Prentice Hall given rigid motion on a given figure; given two reflections, and/or rotations. Rigid motions are • PH G figures, use the definition of congruence in terms assumed to preserve distances and angle measures. o p. 504 of rigid motions to decide if they are congruent. o p. 513-514 Students may use geometric software to explore the o p. 517 effects of rigid motion on a figure(s). o p. 522 Howard County • Congruent Figures Page | 15 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 1: Congruence, Proof, and Constructions Sub-Unit A: Transformations Start of Quarter 1 – October 3, 2014 Explanations/Examples Poudre School District Resources Tasks HS.G.CO.B.6 Illustrative Mathematics • Building a Tile Pattern by Reflecting Hexagons • Building a Tile Pattern by Reflecting Octagons • Reflections and Equilateral Triangles I • Reflections and Equilateral Triangles II • Symmetries of a Circle Activities Practice Assessments Page | 16 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 1: Congruence, Proof, and Constructions Sub-Unit B: Congruence October 6, 2014 – December 6, 2014 Common Core State Standards Explanations/Examples Resources HS.G.CO.B: Understand congruence in terms of rigid motions. Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. HS.G.CO.B.7 HS.G.CO.B.7 Lessons Use the definition of congruence in terms of rigid A rigid motion is a transformation of points in space HS.G.CO.B.7 motions to show that two triangles are congruent consisting of a sequence of one or more translations, MARS Shell Center if and only if corresponding pairs of sides and reflections, and/or rotations. Rigid motions are • Analyzing Congruence Proofs corresponding pairs of angles are congruent. assumed to preserve distances and angle measures. Tasks Congruence of triangles HS.G.CO.B.7 Two triangles are said to be congruent if one can be Illustrative Mathematics exactly superimposed on the other by a rigid motion, • Are the Triangles Congruent? and the congruence theorems specify the conditions under which this can occur. Inside Mathematics HS.G.CO.B.8 HS.G.CO.B.8 • Between the Lines Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of HS.G.CO.B.8 congruence in terms of rigid motions. Illustrative Mathematics • Reflections and Isosceles Triangles • When Does SSA Work to Determine Triangle Congruence? • Why Does ASA Work? • Why Does SAS Work? • Why Does SSS Work? Inside Mathematics • The Shape of Things Page | 17 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 1: Congruence, Proof, and Constructions Sub-Unit B: Congruence October 6, 2014 – December 6, 2014 Explanations/Examples Poudre School District Resources Activities HS.G.CO.B.7 NCTM Illuminations • Congruence Theorems Practice Assessments HS.G.CO.C: Prove geometric theorems. Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of F.CO.10 may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for G.C.3 in Unit 4. HS.G.CO.C.9 HS.G.CO.C.9 Lessons Prove theorems about lines and angles. Theorems Students may use geometric simulations (computer HS.G.CO.C.9/HS.G.CO.C.10 include: vertical angles are congruent; when a software or graphing calculator) to explore theorems Prentice Hall transversal crosses parallel lines, alternate about lines and angles. • PH G interior angles are congruent and corresponding o Ch 3.1-3.4 angles are congruent; points on a perpendicular o Ch 5.1-5.3 bisector of a line segment are exactly those equidistant from the segment’s endpoints. HS.G.CO.C.9 Prentice Hall HS.G.CO.C.10 HS.G.CO.C.10 • PH G Prove theorems about triangles. Theorems Students may use geometric simulations (computer o Ch 2.5 software or graphing calculator) to explore theorems include: measures of interior angles of a triangle o p. 126 about triangles. sum to 180°; base angles of isosceles triangles are o p. 145 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. Page | 18 Curriculum Guide 2014-2015 Common Core State Standards HS.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. High School Geometry Unit 1: Congruence, Proof, and Constructions Sub-Unit B: Congruence October 6, 2014 – December 6, 2014 Explanations/Examples HS.G.CO.C.11 Students may use geometric simulations (computer software or graphing calculator) to explore theorems about parallelograms. Poudre School District Resources Lessons (continued) HS.G.CO.C.10 Prentice Hall • PH G o Ch 4.5 o p. 397 o Ch 7.5 HS.G.CO.C.11 Prentice Hall • PH G o p. 271 o Ch 6.2 o Ch 6.4 Tasks HS.G.CO.C.9/HS.G.CO.C.10 Inside Mathematics • Between the Lines HS.G.CO.C.9 Illustrative Mathematics • Points Equidistance from Two Points in a Plane • Tangent Lines and the Radius of a Circle Page | 19 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 1: Congruence, Proof, and Constructions Sub-Unit B: Congruence October 6, 2014 – December 6, 2014 Explanations/Examples Poudre School District Resources Tasks (continued) HS.G.CO.C.10/HS.G.CO.C.11 Inside Mathematics • The Shape of Things HS.G.CO.C.10 Illustrative Mathematics • Finding the Area of an Equilateral Triangle • Seven Circles I HS.G.CO.C.11 Illustrative Mathematics • Congruence of Parallelograms • Is This a Parallelogram? • Midpoints of the Sides of a Parallelogram • Quadrilaterals Activities HS.G.CO.C.9 Howard County • Patty Paper Tutorial AlA/Corresponding Angles • Patty Paper Tutorial Vertical Angles Proof • Proof Block Tutorial Vertical Angles • Sketchpad Tutorial AIA/Corresponding Angles • Sketchpad Tutorial Vertical Angles Proof Page | 20 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 1: Congruence, Proof, and Constructions Sub-Unit B: Congruence October 6, 2014 – December 6, 2014 Explanations/Examples Poudre School District Resources Activities (continued) HS.G.CO.C.10 Prentice Hall • PH G o p. 259 o p. 272 HS.G.CO.C.11 NCTM Illuminations • Parallelogram Exploration Tool Practice Assessments HS.G.CO.C.9/HS.G.CO.C.10 Inside Mathematics • Circles in Triangles HS.G.CO.C.9 Inside Mathematics • Quadrilaterals Page | 21 Curriculum Guide 2014-2015 High School Geometry Unit 2: Similarity, Proof, and Trigonometry Poudre School District Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem. Students develop the Laws of Sines and Cosines in order to find missing measures of general (not necessarily right) triangles. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles. Sub-Unit A: Similarity HS.G.SRT.A.1 HS.G.SRT.A.2 HS.G.SRT.A.3 HS.G.SRT.B.4 HS.G.SRT.B.5 Sub-Unit B: Trigonometry HS.G.SRT.C.6 HS.G.SRT.C.8 HS.G.SRT.D.10 HS.G.SRT.D.11 HS.G.SRT.C.7 HS.F.SRT.D.9 Page | 22 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 2: Similarity, Proof, and Trigonometry Sub-Unit A: Similarity January 6, 2015 – January 16, 2015 Common Core State Standards Explanations/Examples Resources HS.G.SRT.A: Understand similarity in terms of similarity transformations. HS.G.SRT.A.1 HS.G.SRT.A.1 Lessons Verify experimentally the properties of dilations A dilation is a transformation that moves each point HS.G.SRT.A.1/HS.G.SRT.A.2 given by a center and a scale factor: along the ray through the point emanating from a fixed Prentice Hall a. A dilation takes a line not passing through center, and multiplies distances from the center by a • PH G the center of the dilation to a parallel line, common scale factor. o Ch 9.5 and leaves a line passing through the center Students may use geometric simulation software to unchanged. HS.G.SRT.A.1 b. The dilation of a line segment is longer or model transformations. Students may observe patterns Howard County shorter in the ratio given by the scale factor. and verify experimentally the properties of dilations. • Verifying Dilation HS.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. HS.G.SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. HS.G.SRT.A.2 A similarity transformation is a rigid motion followed by a dilation. Students may use geometric simulation software to model transformations and demonstrate a sequence of transformations to show congruence or similarity of figures. HS.G.SRT.A.3 HS.G.SRT.A.2/HS.G.SRT.A.3 Prentice Hall • PH G o Ch 7.3 MARS Shell Center • Analyzing Congruence Proofs HS.G.SRT.A.2 Howard County • Similar Figures MARS Shell Center • Geometry Problems: Circles and Triangles Page | 23 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 2: Similarity, Proof, and Trigonometry Sub-Unit A: Similarity January 6, 2015 – January 16, 2015 Explanations/Examples Poudre School District Resources Tasks HS.G.SRT.A.1 Illustrative Mathematics • Dilating a Line Inside Mathematics • Surrounded and Covered HS.G.SRT.A.2 Illustrative Mathematics • Are They Similar? HS.G.SRT.A.3 Illustrative Mathematics • Similar triangles Activities Practice Assessments HS.G.SRT.A.2 Inside Mathematics • Hopewell Geometry • Rhombuses Page | 24 Curriculum Guide 2014-2015 Common Core State Standards HS.G.SRT.B.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. HS.G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. High School Geometry Unit 2: Similarity, Proof, and Trigonometry Sub-Unit A: Similarity January 6, 2015 – January 16, 2015 Explanations/Examples HS.G.SRT.B: Prove theorem involving similarity. HS.G.SRT.B.4 Students may use geometric simulation software to model transformations and demonstrate a sequence of transformations to show congruence or similarity of figures. HS.G.SRT.B.5 Similarity postulates include SSS, SAS, and AA. Congruence postulates include SSS, SAS, ASA, AAS, and H-L. Students may use geometric simulation software to model transformations and demonstrate a sequence of transformations to show congruence or similarity of figures. Poudre School District Resources Lessons HS.G.SRT.B.4/HS.G.SRT.B.5 Prentice Hall • PH G o Ch 4 o Ch 7.3-7.5 HS.G.SRT.B.4 Prentice Hall • PH G o p. 226-227 o p. 258 o Ch 8.1 MARS Shell Center • Modeling: Rolling Cups • Proofs of the Pythagorean Theorem • Solving Geometry Problems: Floodlights Tasks HS.G.SRT.B.4 Illustrative Mathematics • Joining Two Midpoints of Sides of a Triangle Page | 25 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 2: Similarity, Proof, and Trigonometry Sub-Unit A: Similarity January 6, 2015 – January 16, 2015 Explanations/Examples Poudre School District Resources Tasks (continued) HS.G.SRT.B.5 Illustrative Mathematics • Bank Shot • Congruence of Parallelograms • Extensions, Bisections and Dissections in a Rectangle • Is This a Rectangle? • Points from Directions • Unit Squares and Triangles Activities Practice Assessments HS.G.SRT.B.5 Inside Mathematics • Hopewell Geometry • Rhombuses Page | 26 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 2: Similarity, Proof, and Trigonometry Sub-Unit B: Trigonometry January 20, 2015 – February 13, 2015 Common Core State Standards Explanations/Examples Resources HS.G.SRT.C: Define trigonometric ratios and solve problems involving right triangles. HS.G.SRT.C.6 HS.G.SRT.C.6 Lessons Understand that by similarity, side Students may use applets to explore the range of values of the HS.G.SRT.C ratios in right triangles are properties trigonometric ratios as θ ranges from 0 to 90 degrees. Prentice Hall of the angles in the triangle, leading to • PH G definitions of trigonometric ratios for o Ch 8 hypotenuse acute angles. opposite of θ HS.G.SRT.C.6 θ Prentice Hall Adjacent to • PH G θ o p. 431-432 hypotenuse opposite o p. 438 cosecant of θ = csc θ = sine of θ = sin θ = opposite hypotenuse adjacent hypotenuse opposite tangent of θ = tan θ = adjacent cosine of θ = cos θ = HS.G.SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. hypotenuse adjacent adjacent cotangent of θ = cot θ = opposite secant of θ = sec θ = HS.G.SRT.C.7 Geometric simulation software, applets, and graphing calculators can be used to explore the relationship between sine and cosine. MARS Shell Center • Calculating Volumes of Compound Objects • Geometry Problems: Circles and Triangles • Inscribing and Circumscribing Right Triangles • Solving Quadratic Equations: Cutting Corners HS.G.SRT.C.8 Prentice Hall • PH G o p. 444 o p. 451 o Ch 10.5 Page | 27 Curriculum Guide 2014-2015 Common Core State Standards HS.G.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★ High School Geometry Unit 2: Similarity, Proof, and Trigonometry Sub-Unit B: Trigonometry January 20, 2015 – February 13, 2015 Explanations/Examples HS.G.SRT.C.8 Students may use graphing calculators or programs, tables, spreadsheets, or computer algebra systems to solve right triangle problems. • Find the height of a tree to the nearest tenth if the angle of elevation of the sun is 28° and the shadow of the tree is 50 ft. Poudre School District Resources Tasks HS.G.SRT.C.6 Illustrative Mathematics • Finding the Area of an Equilateral Triangle • Mt. Whitney to Death Valley • Seven Circles I HS.G.SRT.C.8 Illustrative Mathematics • Coins in a Circular Pattern • Neglecting the Curvature of the Earth • Setting Up Sprinklers • Seven Circles III • Shortest Line Segment from a Point P to a Line L Activities Practice Assessments HS.G.SRT.C.6 Inside Mathematics • Hopewell Geometry Page | 28 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 2: Similarity, Proof, and Trigonometry Sub-Unit B: Trigonometry January 20, 2015 – February 13, 2015 Common Core State Standards Explanations/Examples Resources HS.G.SRT.D: Apply trigonometry to general triangles. With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles. HS.G.SRT.D.9 HS.G.SRT.D.9 Lessons 1 HS.G.SRT.D.9 Derive the formula 𝐴𝐴 = 2 𝑎𝑎𝑎𝑎 sin(𝐶𝐶) Prentice Hall for the area of a triangle by drawing • PH G an auxiliary line from a vertex o Ch 10.5 perpendicular to the opposite side. • Example 2 HS.G.SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems. HS.G.SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). HS.G.SRT.D.10 HS.G.SRT.D.11 Tara wants to fix the location of a mountain by taking measurements from two positions 3 miles apart. From the first position, the angle between the mountain and the second position is 78o. From the second position, the angle between the mountain and the first position is 53o. How can Tara determine the distance of the mountain from each position, and what is the distance from each position? • Tasks HS.G.SRT.D.10/HS.G.SRT.D.11 Illustrative Mathematics • Seven Circles III HS.G.SRT.D.11 Inside Mathematics • Piece it Together • The Shape of Things • What's Your Angle? Activities HS.G.SRT.D.10/HS.G.SRT.D.11 Prentice Hall • PH G o p. 565 Practice Assessments Page | 29 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 3: Connecting Algebra and Geometry through Coordinates Building on their work with the Pythagorean Theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola. HS.G.CPE.B.6 HS.G.GPE.B.4 HS.G.GPE.B.5 HS.G.GPE.B.7 HS.G.GPE.A.2 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 3: Connecting Algebra and Geometry through Coordinates February 17, 2015 – February 27, 2015 Common Core State Standards Explanations/Examples Resources HS.G.GPE.B: Use coordinates to prove simple geometric theorem algebraically. This unit has a close connection with the next unit. For example, a curriculum might merge G.GPE.1 and the Unit 4 treatment of G.GPE.4 with the standards in the unit. Reasoning with triangles in this unit is limited to right triangles; e.g., derive the equation for a line through two points using similar right triangles. Relate wok on parallel lines in G.GPE.5 to work on A.REI.5 in High School Algebra 1 involving systems of equations having no solution or infinitely many solutions. G.GPE.7 provides practice with the distance formula and its connection with the Pythagorean Theorem. HS.G.GPE.B.4 HS.G.GPE.B.4 Lessons Use coordinates to prove simple geometric Students may use geometric simulation software to HS.G.GPE.B.4 theorems algebraically. For example, prove or model figures and prove simple geometric theorems. Prentice Hall disprove that a figure defined by four given points Use slope and distance formula to verify the polygon • PH G formed by connecting the points in the coordinate plane is a rectangle; prove or o Ch 1.8 (-3, -2), (5, 3), (9, 9), (1, 4) is a parallelogram. o p. 60 disprove that the point �1, √3� lies on the circle o Ch 6.6-6.7 centered at the origin and containing the point (0, 2). HS.G.GPE.B.5 HS.G.GPE.B.5 HS.G.GPE.B.5 Prentice Hall Prove the slope criteria for parallel and Lines can be horizontal, vertical, or neither. • PH G perpendicular lines and use them to solve o Ch 3.7 geometric problems (e.g., find the equation of a Students may use a variety of different methods to line parallel or perpendicular to a given line that construct a parallel or perpendicular line to a given MARS Shell Center passes through a given point). line and calculate the slopes to compare the • Finding Equations of Parallel and relationships. Perpendicular Lines HS.G.GPE.B.6 HE.G.GPE.B.6 Find the point on a directed line segment Students may use geometric simulation software to HS.G.GPE.B.6 between two given points that partitions the model figures or line segments. Howard County • Given A(3, 2) and B(6, 11), segment in a given ratio. • Partitioning a Segment o Find the point that divides the line segment AB two-thirds of the way from A to B. The point two-thirds of the way from A to B has Tasks HS.G.GPE.B.4/HS.G.GPE.B.5 x-coordinate two-thirds of the way from 3 to 6 Illustrative Mathematics and y coordinate two-thirds of the way from 2 • Unit Squares and Triangles to 11. Page | 31 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 3: Connecting Algebra and Geometry through Coordinates February 17, 2015 – February 27, 2015 Common Core State Standards Explanations/Examples Resources HS.G.GPE.B.6 (continued) HS.G.GPE.B.6 (continued) Tasks (continued) So, (5, 8) is the point that is two-thirds from HS.G.GPE.B.4 point A to point B. Howard County • Pooltastic o Find the midpoint of line segment AB. Illustrative Mathematics HS.G.GPE.B.7 HS.G.GPE.B.7 • Is this a Rectangle Use coordinates to compute perimeters of Students may use geometric simulation software to polygons and areas of triangles and rectangles, model figures. HS.G.GPE.B.5 Illustrative Mathematics e.g., using the distance formula. ★ • A Midpoint Miracle • Triangles Inscribed in a Circle Activities Practice HS.G.GPE.B.4/HS.G.GPE.B.7 Shmoop • Proving Geometric Shapes via Coordinate Grid Worksheet HS.G.GPE.B.4 Prentice Hall • PH G o p. 326 #22-24 Page | 32 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 3: Connecting Algebra and Geometry through Coordinates February 17, 2015 – February 27, 2015 Common Core State Standards Explanations/Examples Resources Practice (continued) HS.G.GPE.B.7 Prentice Hall • PH G o Ch 10.1-10.3 • p. 538 #29-33 • p. 453 #25-27 • p. 550 #37 Assessments HS.G.GPE.A: Translate between the geometric description and the equation for a conic section. The directrix should be parallel to a coordinate axis. HS.G.GPE.A.2 HS.G.GPE.A.2 Lessons Derive the equation of a parabola given a focus Students may use geometric simulation software to Tasks and directrix. explore parabolas. Write and graph an equation for a parabola with focus (2, 3) and directrix y = 1. Activities Practice Assessments Page | 33 Curriculum Guide 2014-2015 High School Geometry Unit 4: Circles With and Without Coordinates Poudre School District In this unit, students prove basic theorems about circles, with particular attention to perpendicular and inscribed angles, in order to see symmetry in circles and as an application of triangle congruence criteria. They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations to determine intersections between lines and circles or parabolas and between two circles. HS.G.MG.A.1 HS.G.C.A.1 HS.G.GPE.A.1 HS.G.C.B.5 HS.G.GPE.B.4 HS.G.C.A.3 HS.G.C.A.2 HS.G.C.A.4 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 4: Circles With and Without Coordinates March 2, 2015 – March 27, 2015 Common Core State Standards Explanations/Examples Resources HS.G.MG.A: Apply geometric concepts in modeling situations. Focus on situations in which the analysis of circles is required. HS.G.MG.A.1 HS.G.MG.A.1 Lessons Use geometric shapes, their measures, and their Students may use simulation software and modeling HS.G.MG.A properties to describe objects (e.g., modeling a software to explore which model best describes a set of Prentice Hall tree trunk or a human torso as a cylinder). data or situation. • PH G o Ch 1.2 o Ch 11.1 MARS Shell Center • Rolling Cups Tasks Activities Practice Assessments HS.G.C.A.1 Prove that all circles are similar. HS.G.C.A: Understand and apply theorems about circles. HS.G.C.A.1 Students may use geometric simulation software to model transformations and demonstrate a sequence of transformations to show congruence or similarity of figures. Lessons HS.G.C.A.1 Howard County • Lesson Similar Circles Page | 35 Curriculum Guide 2014-2015 Common Core State Standards HS.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. High School Geometry Unit 4: Circles With and Without Coordinates March 2, 2015 – March 27, 2015 Explanations/Examples HS.G.C.A.2 • Given the circle below with radius of 10 and chord length of 12, find the distance from the chord to the center of the circle. • HS.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. HS.G.C.A.4 Construct a tangent line from a point outside a given circle to the circle. Find the unknown length in the picture below. HS.G.C.A.3 Students may use geometric simulation software to make geometric constructions. HS.G.C.A.4 Students may use geometric simulation software to make geometric constructions. Poudre School District Resources Lessons (continued) HS.G.C.A.2/HS.G.C.A.4 Prentice Hall • PH G o p. 677-678 HS.G.C.A.2 Prentice Hall • PH G o Ch 10.6 o Ch 12.1-12.4 o p. 669 Howard County • Angles in a Circle • Inscribed and Central Angles NCTM Illuminations • Power of Points HS.G.C.A.3 Prentice Hall • PH G o Ch 5.3 o p. 271 HS.G.C.A.4 Prentice Hall • PH G o Ch 12.1 o p. 693 Page | 36 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 4: Circles With and Without Coordinates March 2, 2015 – March 27, 2015 Explanations/Examples Poudre School District Resources Tasks HS.G.C.A.1 Illustrative Mathematics • Similar Circles HS.G.C.A.2 Illustrative Mathematics • Right Triangles Inscribed in Circles 1 • Right Triangles Inscribed in Circles 2 • Tangent Lines and the Radius of a Circle Inside Mathematics • What's Your Angle: Level E HS.G.C.A.3 Howard County • Task Your Trip to Paris Illustrative Mathematics • Circles and Triangles • Inscribing a Circle in a Triangle I • Inscribing a Circle in a Triangle II • Inscribing a Triangle in a Circle • Locating a Warehouse • Placing a Fire Hydrant Page | 37 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 4: Circles With and Without Coordinates March 2, 2015 – March 27, 2015 Explanations/Examples Poudre School District Resources Tasks (continued) HS.G.C.A.4 Illustrative Mathematics • Neglecting the Curvature of the Earth: How Far Can You See From the Top of Mt. Whitney • Tangent to a Circle at a Point Activities Practice HS.G.C.A.2 Shmoop • Circles Worksheet 2 HS.G.C.A.3 Shmoop • Functions Worksheet 3 HS.G.C.A.4 Schmoop • Circles Worksheet 4 Assessments HS.G.C.A.2 Inside Mathematics • Circle and Squares • Circle in Triangle Page | 38 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 4: Circles With and Without Coordinates March 2, 2015 – March 27, 2015 Explanations/Examples Poudre School District Resources Assessments (continued) HS.G.C.A.3 MARS Shell Center • Inscribing and Circumscribing Right Triangles HS.G.C.B: Find arc lengths and areas of sectors of circles. Emphasize the similarity of all circles. Note that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. HS.G.C.B.5 HS.G.C.B.5 Lessons Derive using similarity the fact that the length of Students can use geometric simulation software to HS.G.C.B the arc intercepted by an angle is proportional to explore angle and radian measures and derive the Prentice Hall the radius, and define the radian measure of the formula for the area of a sector. • PH G angle as the constant of proportionality; derive o Ch 10.4 the formula for the area of a sector. o Ch 10.6-10.7 MARS Shell Center • Sectors of Circles Tasks HS.G.C.B Illustrative Mathematics • Mutually Tangent Circles • Setting Up Sprinklers • Two Wheels and a Belt Activities Page | 39 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 4: Circles With and Without Coordinates March 2, 2015 – March 27, 2015 Explanations/Examples Poudre School District Resources Practice HS.G.C.B Prentice Hall • PH G o p. 557 #35 Assessments HS.G.GPE.A: Translate between the geometric description and the equation for a conic section. HS.G.GPE.A.1 HS.G.GPE.A.1 Lessons Derive the equation of a circle of given center and Students may use geometric simulation software to HS.G.GPE.A radius using the Pythagorean Theorem; complete explore the connection between circles and the Prentice Hall the square to find the center and radius of a circle Pythagorean Theorem. • PH A2 • Write an equation for a circle with a radius of 2 given by an equation. o Ch 10.3 units and center at (1, 3). • PH G • Write an equation for a circle given that the o Ch 12.5 endpoints of the diameter are (-2, 7) and (4, -8). • Find the center and radius of the circle Tasks 4x2 + 4y2 - 4x + 2y – 1 = 0. HS.G.GPE.A Illustrative Mathematics • Explaining the Equation of a Circle • Slopes and Circles Activities Practice Assessments Page | 40 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 4: Circles With and Without Coordinates March 2, 2015 – March 27, 2015 Common Core State Standards Explanations/Examples Resources HS.G.GPE.B: Use coordinates to prove simple geometric theorems algebraically. Include simple proofs involving circles. HS.G.GPE.B.4 HS.G.GPE.B.4 Lessons Use coordinates to prove simple geometric Students may use geometric simulation software to HS.G.GPE.B theorems algebraically. For example, prove or model figures and prove simple geometric theorems. Prentice Hall disprove that a figure defined by four given points Use slope and distance formula to verify the polygon • PH G in the coordinate plane is a rectangle; prove or formed by connecting the points o Ch 6.1 (-3, -2), (5, 3), (9, 9), (1, 4) is a parallelogram. o Ch 6.7 disprove that the point �1, √3�lies on the circle centered at the origin and containing the point MARS Shell Center (0, 2). • Equations of Circles 1 NCTM Illuminations • Pi Line Tasks Activities Practice HS.G.GPE.B Prentice Hall • PH G o p. 309 #13-18 o p. 311 #64 Assessments Page | 41 Curriculum Guide 2014-2015 High School Geometry Unit 5: Extending to Three Dimensions Poudre School District Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line. HS.G.MG.A.1 HS.G.GMD.B.4 HS.G.GMD.A.1 HS.G.GMD.A.3 HS.G.MG.A.3 HS.G.MG.A.2 Page | 42 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 5: Extending to Three Dimensions March 30, 2015 – April 24, 2015 Common Core State Standards Explanations/Examples Resources HS.G.MG.A: Apply geometric concepts in modeling situations. Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles. HS.G.MG.A.1 HS.G.MG.A.1 Lessons Use geometric shapes, their measures, and their Students may use simulation software and modeling HS.G.MG.A.1/HS.G.MG.A.3 properties to describe objects (e.g., modeling a software to explore which model best describes a set of MARS Shell Center tree trunk or a human torso as a cylinder). data or situation. • Modeling: Rolling Cups • Solving Quadratic Equations: Cutting HS.G.MG.A.2 HS.G.MG.A.2 Corners Apply concepts of density based on area and Students may use simulation software and modeling volume in modeling situations (e.g., persons per software to explore which model best describes a set of HS.G.MG.A.1 square mile, BTUs per cubic foot). data or situation. Prentice Hall • PH G HS.G.MG.A.3 HS.G.MG.A.3 o Ch 11.1 Apply geometric methods to solve design Students may use simulation software and modeling o p. 605 problems (e.g., designing an object or structure to software to explore which model best describes a set of data or situation. satisfy physical constraints or minimize cost; HS.G.MG.A.3 working with typographic grid systems based on Prentice Hall ratios). • PH G o p. 657 Tasks HS.G.MG.A Illustrative Mathematics • Coins in a Circular Pattern Page | 43 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 5: Extending to Three Dimensions March 30, 2015 – April 24, 2015 Explanations/Examples Poudre School District Resources Tasks (continued) HS.G.MG.A.1/HS.G.MG.A.2 Illustrative Mathematics • How Many Cells are in the Human Body? • How Many Leaves on a Tree? • How Many Leaves on a Tree? (Version 2) • How Thick is a Soda Can I? • How Thick is a Soda Can II? HS.G.MG.A.1/HS.G.MG.A.3 Illustrative Mathematics • Eratosthenes and the Circumference of the Earth • Paper Clip • Regular Tessellations of the Plane • Running Around a Track I • Running Around a Track II Page | 44 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 5: Extending to Three Dimensions March 30, 2015 – April 24, 2015 Explanations/Examples Poudre School District Resources Tasks (continued) HS.G.MG.A.1 Illustrative Mathematics • Global Positioning System II • Hexagonal Pattern of Beehives • Seven Circles III • Solar Eclipse • Tennis Balls in a Can • The Lighthouse Problem • Tilt of Earth's Axis and the Four Seasons • Toilet Roll • Use Cavalieri’s Principle to Compare Aquarium Volumes Activities HS.G.MG.A NCTM Illuminations • Tube Viewer Simulation HS.G.MG.A.1 NCTM Illuminations • Soccer Problem Practice HS.G.MG.A.2 Prentice Hall • PH G o p. 642 #43 Assessments Page | 45 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 5: Extending to Three Dimensions March 30, 2015 – April 24, 2015 Common Core State Standards Explanations/Examples Resources HS.G.GMD.B: Visualize the relation between two-dimensional and three-dimensional objects. HS.G.GMD.B.4 HS.G.GMD.B.4 Lessons Identify the shapes of two-dimensional crossStudents may use geometric simulation software to HS.G.GMD.B sections of three-dimensional objects, and model figures and create cross sectional views. Prentice Hall Identify the shape of the vertical, horizontal, and other identify three-dimensional objects generated by • PH G cross sections of a cylinder. rotations of two-dimensional objects. o Ch 11.1 o Ch 11.4-11.6 MARS Shell Center • 2D Representations of 3D Objects • Modeling: Rolling Cups Tasks HS.G.GMD.B Illustrative Mathematics • Global Positioning System I • Global Positioning System II • Tennis Balls in a Can Activities Practice Assessments Page | 46 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 5: Extending to Three Dimensions March 30, 2015 – April 24, 2015 Common Core State Standards Explanations/Examples Resources HS.G.GMD.A: Explain volume formulas and use them to solve problems. Informal arguments for area and volume formulas can make use of the way in which area an volume scale under similarity transformation: when one figure in the plane results from another by applying a similarity transformation with scale fact k, its area is k2 times the area of the first. Similarity, volumes of solid figures scale by k3 under a similarity transformation with scale factor k. HS.G.GMD.A.1 HS.G.GMD.A.1 Lessons Give an informal argument for the formulas for Cavalieri’s principle is if two solids have the same HS.G.GMD.A the circumference of a circle, area of a circle, height and the same cross-sectional area at every level, Prentice Hall volume of a cylinder, pyramid, and cone. Use then they have the same volume. • PH G dissection arguments, Cavalieri’s principle, and o Ch 11.4-11.6 informal limit arguments. HS.G.GMD.A.1 HS.G.GMD.A.3 HS.G.GMD.A.3 Prentice Hall Use volume formulas for cylinders, pyramids, Missing measures can include but are not limited to • PH G slant height, altitude, height, diagonal of a prism, edge o Ch 1.9 cones, and spheres to solve problems. ★ length, and radius. o Ch 10.6-10.7 o p. 575 o p. 581 MARS Shell Center • Equations of Circles 2 HS.G.GMD.A.3 MARS Shell Center • Calculating Volumes of Compound Objects • Evaluating Statements About Enlargements (2D and 3D) Page | 47 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 5: Extending to Three Dimensions March 30, 2015 – April 24, 2015 Explanations/Examples Poudre School District Resources Tasks HS.G.GMD.A.3 Illustrative Mathematics • Centerpiece • Doctor's Appointment Activities HS.G.GMD.A.3 NCTM Illuminations • Dynamic Paper • Geometric Solids Practice Assessments Page | 48 Curriculum Guide 2014-2015 High School Geometry Unit 6: Applications of Probability Poudre School District Building on probability concepts that began in the middle grades, students use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions. HS.S.CP.A.1 HS.S.CP.A.2 HS.S.CP.B.7 HS.S.CP.A.3 HS.S.CP.B.8 HS.S.CP.A.5 HS.S.CP.B.6 HS.S.CP.A.4 HS.S.CP.B.9 HS.S.MD.B.6 HS.S.MD.B.7 Page | 49 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 6: Applications of Probability April 27, 2015 - May 15, 2015 Common Core State Standards Explanations/Examples Resources HS.S.CP.A: Understand the independence and conditional probability and use them to interpret data. Build on work with two-way tables from Algebra 1 Units 3 (S.ID.5) to develop understanding of conditional probability and independence. HS.S.CP.A.1 HS.S.CP.A.1 Lessons Describe events as subsets of a sample space (the Intersection: The intersection of two sets A and B is HS.S.CP.A.1/HS.S.CP.A.2 set of outcomes) using characteristics (or the set of elements that are common to both set A and Prentice Hall categories) of the outcomes, or as unions, set B. It is denoted by A ∩ B and is read ‘A intersection • PH A2 intersections, or complements of other events B’. o Ch 9.7 (“or,” “and,” “not”). a. A ∩ B in the diagram is {1, 5} HS.S.CP.A.1/HS.S.CP.A.4 b. this means: BOTH/AND Prentice Hall • PH A2 o Ch 12.1 U A 2 3 1 5 B 4 7 8 Union: The union of two sets A and B is the set of elements, which are in A or in B or in both. It is denoted by A ∪ B and is read ‘A union B’. a. A ∪ B in the diagram is {1, 2, 3, 4, 5, 7} b. this means: EITHER/OR/ANY c. could be both Complement: The complement of the set A ∪B is the set of elements that are members of the universal set U but are not in A ∪B. It is denoted by (A ∪ B )’ (A ∪ B )’ in the diagram is {8} HS.S.CP.A.1 NCTM Illuminations • Exploration with Chance HS.S.CP.A.2 Interactivate • Replacement Probability HS.S.CP.A.3/HS.S.CP.A.5 Prentice Hall • PH A2 o Ch 12.2 HS.S.CP.A.3 Interactivate • Conditional Probability and Simultaneous Events Page | 50 Curriculum Guide 2014-2015 Common Core State Standards HS.S.CP.A.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. High School Geometry Unit 6: Applications of Probability April 27, 2015 - May 15, 2015 Explanations/Examples HS.S.CP.A.2 HS.S.CP.A.3 HS.S.CP.A.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. HS.S.CP.A.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the twoway table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. HS.S.CP.A.4 Students may use spreadsheets, graphing calculators, and simulations to create frequency tables and conduct analyses to determine if events are independent or determine approximate conditional probabilities. Poudre School District Resources Lessons (continued) HS.S.CP.A.3 (continued) MARS Shell Center • Medical Testing • Modeling Conditional Probabilities 1 • Modeling Conditional Probabilities 2 HS.S.CP.A.4 Stat Trek • Two Way Tables HS.S.CP.A.5 Alabama Learning Exchange • Dartboard Probability Tasks HS.S.CP.A.1 Illustrative Mathematics • Return to Fred's Fun Factory • The Titanic I Inside Mathematics • Friends That You Can Count On • Got Your Number • Party Time • Rod Trains Math Forum @ Drexel • Monty Hall Problem Page | 51 Curriculum Guide 2014-2015 Common Core State Standards HS.S.CP.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. High School Geometry Unit 6: Applications of Probability April 27, 2015 - May 15, 2015 Explanations/Examples HS.S.CP.A.5 • What is the probability of drawing a heart from a standard deck of cards on a second draw, given that a heart was drawn on the first draw and not replaced? Are these events independent or dependent? • At Johnson Middle School, the probability that a student takes computer science and French is 0.062. The probability that a student takes computer science is 0.43. What is the probability that a student takes French given that the student is taking computer science? Poudre School District Resources Tasks (continued) HS.S.CP.A.2 Illustrative Mathematics • Cards and Independence • Rain and Lightning • The Titanic II HS.S.CP.A.3 Illustrative Mathematics • Lucky Envelopes Math Forum @ Drexel • Boy or Girl HS.S.CP.A.4 Illustrative Mathematics • How Do you Get to School • The Titanic 3 HS.S.CP.A.5 Illustrative Mathematics • Breakfast Before School • But Mango is My Favorite... Activities HS.S.CP.A.1 Interactivate • Venn Diagram Shape Sorter Page | 52 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 6: Applications of Probability April 27, 2015 - May 15, 2015 Explanations/Examples Poudre School District Resources Activities (continued) HS.S.CP.A.1 (continued) NCTM Illuminations • Venn Diagram Applet HS.S.CP.A.3 Cut-the-Knot • Conditional Probability and Independent Events Practice HS.S.CP.A.2 Shmoop • Conditional Probability Worksheet 2 HS.S.CP.A.3 Shmoop • Conditional Probability Worksheet 3 HS.S.CP.A.4 Shmoop • Conditional Probability Worksheet 4 HS.S.CP.A.5 Shmoop • Conditional Probability Worksheet 5 Page | 53 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 6: Applications of Probability April 27, 2015 - May 15, 2015 Explanations/Examples Poudre School District Resources Assessments HS.S.CP.A.1 Inside Mathematics • Marble Game HS.S.CP.B: Use the rules of probability to compute probabilities of compound events in a uniform probability model. HS.S.CP.B.6 HS.S.CP.B.6 Lessons Find the conditional probability of A given B as Students could use graphing calculators, simulations, HS.S.CP.B.6 the fraction of B’s outcomes that also belong to A, or applets to model probability experiments and Prentice Hall and interpret the answer in terms of the model. interpret the outcomes. • PH A2 o Ch 12.2 HS.S.CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. HS.S.CP.B.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. HS.S.CP.B.7 Students could use graphing calculators, simulations, or applets to model probability experiments and interpret the outcomes. In a math class of 32 students, 18 are boys and 14 are girls. On a unit test, 5 boys and 7 girls made an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student? HS.S.CP.B.8 Students could use graphing calculators, simulations, or applets to model probability experiments and interpret the outcomes. Learnist • Bayes Theorem Links HS.S.CP.B.7 Learnist • Webpage Links to Addition Rule HS.S.CP.B.8 Learnist • Webpage Links to Multiplication Rule HS.S.CP.B.9 Prentice Hall • PH A1 o Ch 12.7-12.8 Page | 54 Curriculum Guide 2014-2015 Common Core State Standards HS.S.CP.B.9 Use permutations and combinations to compute probabilities of compound events and solve problems. High School Geometry Poudre School District Unit 6: Applications of Probability April 27, 2015 - May 15, 2015 Explanations/Examples Resources HS.S.CP.B.9 Lessons (continued) Students may use calculators or computers to HS.S.CP.B.9 (continued) determine sample spaces and probabilities. Math Forum @ Drexel • Choosing Computer Disks • You and two friends go to the grocery store and each buys a soda. If there are five different kinds of • Combinations for License Plates soda, and each friend is equally likely to buy each • Random Card Shuffling Probabilities variety, what is the probability that no one buys the • Seating People in a Row same kind? • Telephone Number Possibility • Ways to List 1 Through 10 Out of Order • Winning UK Lottery Tasks HS.S.CP.B.7 Illustrative Mathematics • Coffee at Mom's Diner HS.S.CP.B.9 Illustrative Mathematics • Alex Mel and Chelsea Play a Game • Random Walk 3 • Random Walk 4 Activities HS.S.CP.B.6 Interactivate • Fire HS.S.CP.B.8 Interactivate • Crazy Choices Game Page | 55 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 6: Applications of Probability April 27, 2015 - May 15, 2015 Explanations/Examples Poudre School District Resources Activities (continued) HS.S.CP.B.9 Better Explained • Navigating a Grid Using Combinations and Permutations IXL • Combinations Online Quiz • Permutation and Combination Notation Quiz • Permutations Online Quiz Practice HS.S.CP.B.6 Shmoop • Functions Worksheet 6 HS.S.CP.B.7 Shmoop • Functions Worksheet 7 HS.S.CP.B.8 Shmoop • Conditional Probability Worksheet 8 HS.S.CP.B.9 Shmoop • Conditional Probability Worksheet 9 Assessments Page | 56 Curriculum Guide 2014-2015 High School Geometry Poudre School District Unit 6: Applications of Probability April 27, 2015 - May 15, 2015 Common Core State Standards Explanations/Examples Resources HS.S.MD.B: Use probability to evaluate outcomes of decisions. This unit sets the stage for work in Algebra II, where the ideas of statistical inference are introduced. Evaluating the risks associated with conclusions drawn from sample data (i.e. incomplete information) required an understanding of probability concepts. HS.S.MD.B.6 HS.S.MD.B.6 Lessons Use probabilities to make fair decisions (e.g., Students may use graphing calculators or programs, HS.S.MD.B.6 drawing by lots, using a random number spreadsheets, or computer algebra systems to model Prentice Hall generator). and interpret parameters in linear, quadratic or • PH G exponential functions. o Ch 10.8 HS.S.MD.B.7 HS.S.MD.B.7 Analyze decisions and strategies using Students may use graphing calculators or programs, Learnist probability concepts (e.g., product testing, spreadsheets, or computer algebra systems to model • Videos and Web Links for Game medical testing, pulling a hockey goalie at the end and interpret parameters in linear, quadratic or Fairness of a game). exponential functions. Math Forum @ Drexel • Drawing Prizes • Lottery Strategy and Odds of Winning • Randomness of a Shuffled Deck • Selecting a Student Council HS.S.MD.B.7 Learnist • Videos and Web links for Decision Making with Probability MARS Shell Center • Evaluating Statements About Probability Tools for the Common Core Standards • Medical Testing Details Page 19 Page | 57 Curriculum Guide 2014-2015 Common Core State Standards High School Geometry Unit 6: Applications of Probability April 27, 2015 - May 15, 2015 Explanations/Examples Poudre School District Resources Tasks HS.S.MD.B.7 Illustrative Mathematics • Fred's Fun Factory Activities HS.S.MD.B.6 Interactivate • Adjustable Spinner HS.S.MD.B.7 Interactivate • The Game of Life Practice Assessments HS.S.MD.B.6 Shmoop • Quiz on How to Make Fair Decisions Page | 58 Performance Level Descriptors – Geometry Congruence Transformations HS.G.CO.B.6 HS.G.CO.C Similarity HS.G.SRT.A.1a HS.G.SRT.A.1b HS.G.SRT.A.2 HS.G.SRT.B.5 Similarity in Trigonometry HS.G.SRT.C.6 HS.G.SRT.C.7-2 HS.G.SRT.C.8 Geometry: Sub-Claim A The student solves problems involving the Major Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Determines and uses appropriate geometric theorems and properties of rigid motions, lines, angles, triangles and parallelograms to solve non-routine problems and prove statements about angle measurement, triangles, distance, line properties and congruence. Uses transformations and congruence and similarity criteria for triangles and to prove relationships among composite geometric figures and to solve multistep problems. Determines and uses appropriate geometric theorems and properties of rigid motions, lines, angles, triangles and parallelograms to solve routine problems and prove statements about angle measurement, triangles, distance, line properties and congruence. Uses given geometric theorems and properties of rigid motions, lines, angles, triangles and parallelograms to solve routine problems and prove statements about angle measurement, triangles, distance, line properties and congruence. Uses given geometric theorems and properties of rigid motions, lines, angles, triangles and parallelograms to solve routine problems and reason about angle measurement, triangles, distance, line properties and congruence. Uses transformations and congruence and similarity criteria for triangles to prove relationships among geometric figures and to solve problems. Uses transformations to determine relationships among geometric figures and to solve problems. Identifies transformation relationships in geometric figures. Uses trigonometric ratios, the Pythagorean Theorem and the relationship between sine and cosine to solve right triangles in applied non-routine problems. Uses trigonometric ratios, the Pythagorean Theorem and the relationship between sine and cosine to solve right triangles in applied problems. Uses trigonometric ratios, the Pythagorean Theorem and the relationship between sine and cosine to solve right triangles in applied problems. Uses trigonometric ratios and the Pythagorean Theorem to determine the unknown side lengths and angle measurements of a right triangle. Uses similarity transformations with right Uses similarity transformations with right triangles to define Page | 59 Performance Level Descriptors – Geometry Geometry: Sub-Claim A The student solves problems involving the Major Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Modeling and Applying HS.G.SRT.C.7-2 HS.G.SRT.C.8 HS.G.GPE.B.6 G-Int.1 triangles to define trigonometric ratios for acute angles. trigonometric ratios for acute angles. Uses geometric relationships in the coordinate plane to solve problems involving area, perimeter and ratios of lengths. Uses geometric relationships in the coordinate plane to solve problems involving area, perimeter and ratios of lengths. Uses geometric relationships in the coordinate plane to solve problems involving area, perimeter and ratios of lengths. Applies geometric concepts and trigonometric ratios to describe, model and solve applied problems (including design problems) related to the Pythagorean theorem, density, geometric shapes, their measures and properties. Applies geometric concepts and trigonometric ratios to describe, model and solve applied problems related to the Pythagorean theorem, density, geometric shapes, their measures and properties. Applies geometric concepts to describe, model and solve applied problems related to the Pythagorean theorem, geometric shapes, their measures and properties. Uses provided geometric relationships in the coordinate plane to solve problems involving area and perimeter. Applies geometric concepts to describe, model and solve applied problems related to the Pythagorean theorem, geometric shapes, their measures and properties. Page | 60 Performance Level Descriptors – Geometry Transformations HS.G.CO.A.1 HS.G.CO.A.3 HS.G.CO.A.5 Geometric Constructions HS.G.CO.D Geometry: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Given a figure and a sequence of transformations, draws the transformed figure. Uses precise geometric terminology to specify more than one sequence of transformations that will carry a figure onto itself or another. Makes geometric constructions: copying a segment, copying an angle, bisecting an angle, bisecting a segment, including the perpendicular bisector of a line segment. Given a line and a point not on the line, uses a variety of tools and methods to construct perpendicular and parallel lines, equilateral triangles, squares and regular hexagons inscribed in circles to prove geometric theorems. Given a figure and a transformation, draws the transformed figure. Given a figure and a transformation, draws the transformed figure. Uses precise geometric terminology to specify a sequence of transformations that will carry a figure onto itself or another. Specifies a sequence of transformations that will carry a figure onto another. Makes geometric constructions: copying a segment, copying an angle, bisecting an angle, bisecting a segment, including the perpendicular bisector of a line segment. Makes geometric constructions: copying a segment, copying an angle, bisecting an angle, bisecting a segment, including the perpendicular bisector of a line segment. Given a line and a point not on the line, uses a variety of tools and methods to construct perpendicular and parallel lines, equilateral triangles, squares and regular hexagons inscribed in circles. Given a line and a point not on the line, constructs perpendicular and parallel lines. Given a figure and a transformation, draws the transformed figure. Makes basic geometric constructions: copying a segment, copying an angle, bisecting an angle, bisecting a segment, including the perpendicular bisector of a line segment. Page | 61 Performance Level Descriptors – Geometry Applying Geometric Properties and Theorems HS.G.C.A.Int.1 HS.G.C.B.Int.1 HS.G.GPE.A.1-2 Geometric Formulas HS.G.GMD.A.1 HS.G.GMD.A.3 HS.G.GMD.A.4 Geometry: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Applies properties and theorems of angles, segments and arcs in circles to solve problems, model relationships and formulate generalizations. Completes the square to find the center and radius of a circle given by an equation. Applies properties and theorems of angles, segments and arcs in circles to solve problems and model relationships. Completes the square to find the center and radius of a circle given by an equation. Uses volume formulas to solve mathematical and contextual problems that involve cylinders, pyramids, cones and spheres. Uses volume formulas to solve mathematical and contextual problems that involve cylinders, pyramids, cones and spheres. Uses dissection arguments, Cavalieri’s principle and informal limit arguments to support the formula for the circumference of a circle, area of a circle, volume of a cylinder, pyramid and cone. Gives an informal argument for the formula for the circumference of a circle, area of a circle and volume of a cylinder, including dissection arguments. Identifies the shapes of twodimensional cross-sections of three-dimensional Identifies the shapes of twodimensional cross-sections of three-dimensional objects and identifies Applies properties and theorems of angles, segments and arcs in circles to solve problems. Applies provided properties and theorems of angles, segments and arcs in circles to solve problems. Completes the square to find the center and radius of a circle given by an equation. Using formulas, determines the volume of cylinders, pyramids, cones and spheres. Using formulas, determines the volume of cylinders, pyramids, cones and spheres. Gives an informal argument for the formula for the circumference of a circle and area of a circle, including dissection arguments. Identifies the shapes of twodimensional cross-sections of three-dimensional objects. Identifies the shapes of twodimensional cross-sections of three-dimensional objects. Page | 62 Performance Level Descriptors – Geometry Geometry: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command objects and identifies threedimensional objects generated by rotations of two-dimensional objects. three-dimensional objects generated by rotations of two-dimensional objects. Page | 63 Performance Level Descriptors – Geometry Geometry: Sub-Claim C The student expresses course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Distinguished Level 4: Strong Level 3: Moderate Level 2: Partial Command Command Command Command Reasoning HS.C.13.1 HS.C.13.2 HS.C.13.3 HS.C.14.1 HS.C.14.2 HS.C.14.3 HS.C.14.5 HS.C.14.6 HS.C.15.14 HS.C.18.2 Clearly constructs and communicates a complete response based on: Clearly constructs and communicates a complete response based on: Constructs and communicates a response based on: • • • • • • a chain of reasoning to justify or refute algebraic and/or geometric propositions or conjectures geometric reasoning in a coordinate setting, OR a response to a multistep problem, • • by: by: • • a chain of reasoning to justify or refute algebraic and/or geometric propositions or conjectures geometric reasoning in a coordinate setting, OR a response to a multistep problem, using a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate) providing an efficient and logical progression of steps or chain of reasoning with appropriate justification • • • • a chain of reasoning to justify or refute algebraic and/or geometric propositions or conjectures geometric reasoning in a coordinate setting, OR a response to a multistep problem, by : by: using a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate) providing a logical progression of steps or chain of reasoning with appropriate justification • • • • Constructs and communicates an incomplete response based on: • a chain of reasoning to justify or refute algebraic and/or geometric propositions or conjectures • geometric reasoning in a coordinate setting, OR • a response to a multistep problem, using a logical approach based on a conjecture and/or stated assumptions providing a logical, but incomplete, progression of steps or chain of reasoning performing minor calculation errors using some grade-level vocabulary, symbols • • • • using an approach based on a conjecture and/or stated or faulty assumptions providing an incomplete or illogical chain of reasoning, or progression of steps making an intrusive calculation error using limited gradelevel vocabulary, Page | 64 Performance Level Descriptors – Geometry Geometry: Sub-Claim C The student expresses course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Distinguished Level 4: Strong Level 3: Moderate Level 2: Partial Command Command Command Command and labels symbols and labels • performing precise • performing precise • providing a partial • providing a partial calculations calculations justification of a justification of a • using correct grade• using correct grade• • • level vocabulary, symbols and labels providing a justification of a conclusion determining whether an argument or conclusion is generalizable evaluating, interpreting and critiquing the validity and efficiency of others’ responses, approaches and reasoning – utilizing mathematical connections (when appropriate) – and providing a counterexample where applicable • • level vocabulary, symbols and labels providing a justification of a conclusion evaluating, interpreting and critiquing the validity of others’ responses, approaches and reasoning – utilizing mathematical connections (when appropriate) • conclusion based on own calculations evaluating the validity of others’ approaches and conclusions conclusion based on own calculations Page | 65 Performance Level Descriptors – Geometry Modeling HS.D.1-2 HS.D.2-1 HS.D.2-2 HS.D.2-11 HS.D.3-2 HS.D.3-4 Geometry: Sub-Claim D The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems, knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure and/or looking for and expressing regularity in repeated reasoning. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Devises and enacts a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: Devises and enacts a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: Devises and enacts a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: Devises a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: • • • • • • • • using stated assumptions and making assumptions and approximations to simplify a real-world situation (includes micro-models) mapping relationships between important quantities selecting appropriate tools to create models analyzing relationships mathematically between important quantities to draw conclusion analyzing and/or • • • using stated assumptions and making assumptions and approximations to simplify a real-world situation (includes micro-models) mapping relationships between important quantities selecting appropriate tools to create models analyzing relationships mathematically between important quantities to draw conclusions • • • using stated assumptions and approximations to simplify a real-world situation illustrating relationships between important quantities using provided tools to create models analyzing relationships mathematically between important quantities to draw conclusions • • • • • using stated assumptions and approximations to simplify a real-world situation identifying important quantities using provided tools to create models analyzing relationships mathematically to draw conclusions writing an algebraic expression or equation to describe a situation applying proportional reasoning and percentages Page | 66 Performance Level Descriptors – Geometry Geometry: Sub-Claim D The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems, knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure and/or looking for and expressing regularity in repeated reasoning. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command creating constraints, • interpreting • interpreting • applying common • • • • • • relationships and goals interpreting mathematical results in the context of the situation reflecting on whether the results make sense improving the model if it has not served its purpose writing a complete, clear and correct algebraic expression or equation to describe a situation applying proportional reasoning and percentages justifying and defending models which lead to a conclusion applying geometric principles and theorems • • • • • • mathematical results in the context of the situation reflecting on whether the results make sense improving the model if it has not served its purpose writing a complete, clear and correct algebraic expression or equation to describe a situation applying proportional reasoning and percentages applying geometric principles and theorems writing and using functions in any form to describe how one quantity of interest depends on another • • • • • • • • mathematical results in a simplified context reflecting on whether the results make sense modifying the model if it has not served its purpose writing an algebraic expression or equation to describe a situation applying proportional reasoning and percentages applying geometric principles and theorems writing and using functions to describe how one quantity of interest depends on another using statistics using reasonable estimates of known • • • geometric principles and theorems using functions to describe how one quantity of interest depends on another using statistics using estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity Page | 67 Performance Level Descriptors – Geometry Geometry: Sub-Claim D The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems, knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure and/or looking for and expressing regularity in repeated reasoning. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command quantities in a chain of • writing and using • using statistics reasoning that yields an functions in any form to • using reasonable • • describe how one quantity of interest depends on another using statistics using reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity estimate of an unknown quantity Page | 68