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Geometry Curriculum Mapping 8/2/2011 Quarter 1 1 Geometry Standards Deconstructed Standards I Can Unit 1: Congruence, Proof, and Constructions Describe the undefined terms: point, line, and distance along a line in a plane. (Knowledge) I can describe the terms: point, line, and distance along a line in a plane. Unit 1: Congruence, Proof, and Constructions Identify and use properties of: Vertical angles, Parallel lines with transversals, All angle relationships, Corresponding angles, Alternate interior angles, Perpendicular bisector, Equidistant from endpoint (Knowledge) I can identify and use properties of: vertical angles, parallel lines with transversals, all angle relationships, corresponding angles, alternate interior angles, perpendicular bisector, equidistant from endpoint. 1 Vocabulary Point Circle Distance G.CO.1 Know precise definitions of angle, circle, Define perpendicular lines, parallel lines, line Ray perpendicular line, segments, and angles. (Knowledge) I can define perpendicular lines, parallel lines, line Plane parallel line, and line segment, based on the segments, and angles. Line undefined notions of point, line, distance along a Define circle and the distance around a circular arc. Perpendicular Lines line, and distance around a circular arc. (Knowledge) I can define circle and the distance around a circular Parallel Lines arc. *Skew Lines Line Segment Angle *Circular Arc * Theorem * Alternate Exterior Angles * Vertical Angles G.CO.9 Prove theorems about lines and angles. * Transversals Theorems include: * Corresponding Angles vertical angles are congruent; when a * Alternate Interior Angles transversal crosses parallel lines, Prove vertical angles are congruent. (Reasoning) * Perpendicular Bisector alternate interior angles are congruent and I can prove vertical angles are congruent. * Equidistant corresponding angles are congruent; points on a Prove corresponding angles are congruent when * Proof (2-column, paragraph, flow perpendicular bisector of a line segment are two parallel lines are cut by a transversal and chart) exactly those equidistant from the segment’s converse. (Reasoning) I can prove corresponding angles are congruent * Linear Pair endpoints. when two parallel lines are cut by a transversal and * Consecutive Interior Angles Prove alternate interior angles are congruent when converse. two parallel lines are cut by a transversal and converse. (Reasoning) I can prove alternate interior angles are congruent when two parallel lines are cut by a transversal and Prove points are on a perpendicular bisector of a converse. line segment are exactly equidistant from the segments endpoint. (Reasoning) I can prove points are on a perpendicular bisector of a line segment are exactly equidistant from the segments endpoint. *Bolded word Use Marzano 6 Step Process Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 1: Congruence, Proof, and Constructions Deconstructed Standards I Can G.CO.9 (Standard continued) From Appendix A: 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. Unit 1: Congruence, Proof, and Constructions Explain the construction of geometric figures using I can explain the construction of geometric figures a variety of tools and methods. (Knowledge) using a variety of tools and methods. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Apply the definitions, properties and theorems about line segments, rays and angles to support geometric constructions. (Reasoning) Unit 1: Congruence, Proof, and Constructions Perform geometric constructions including: Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line, using a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). 1 G.CO.12 (Standard continues) 1 *Bolded word Use Marzano 6 Step Process Apply properties and theorems about parallel and perpendicular lines to support constructions. (Reasoning) I can apply the definitions, properties and theorems about line segments, rays and angles to support geometric constructions. I can apply properties and theorems about parallel and perpendicular lines to support constructions. From Appendix A: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these I can emphasize the ability to formalize and explain constructions result in the desired objects. Some of how these constructions result in the desired these constructions are closely related to previous objects. standards and can be introduced in conjunction with them. 2 Vocabulary Ray * Geometric Construction Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 1: Congruence, Proof, and Constructions G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Deconstructed Standards Note: Underpinning performance, reasoning, and knowledge targets, if applicable, are addressed in G.CO.12 I Can 3 Vocabulary I can emphasize the ability to formalize and explain Equilateral Triangle how these constructions result in the desired * Inscribed Polygon objects. From Appendix A: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. (Knowledge) 1 Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Construct an equilateral triangle, a square and a regular hexagon inscribed in a circle. (Product) 1 Unit 4: Connecting Algebra and Geometry Through Coordinates Recognize that slopes of parallel lines are equal. (Knowledge) G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses 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). Recognize that slopes of perpendicular lines are opposite reciprocals (i.e, the slopes of perpendicular lines have a product of -1) (Knowledge) Find the equation of a line parallel to a given line that passes through a given point. (Knowledge) I can recognize that slopes of parallel lines are equal. Opposite Reciprocals Parallel Lines Perpendicular Lines I can recognize that slopes of perpendicular lines are Slope opposite reciprocals . Equations of Lines Slope-Intercept Form Point-Slope Form Standard Form I can find the equation of a line parallel to a given line that passes through a given point. From Appendix A: Relate work on parallel lines in G.GPE.5 to work on A.REI.5 in High School I can find the equation of a line perpendicular to a Algebra 1 involving systems of equations having Find the equation of a line perpendicular to a given given line that passes through a given point. no solution or infinitely many solutions. line that passes through a given point. (Knowledge) I can prove the slope criteria for parallel and perpendicular lines and use them to solve geometric Prove the slope criteria for parallel and problems. perpendicular lines and use them to solve geometric problems. (Reasoning) *Bolded word Use Marzano 6 Step Process Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Deconstructed Standards Unit 4: Connecting Algebra and Geometry Through Coordinates Recall the definition of ratio. (Knowledge) G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Recall previous understandings of coordinate geometry. (Knowledge) I Can I can find the point on a directed line segment between two given points that partitions the segment in a given ratio. 4 Vocabulary * Partition Ratio * Coordinate Geometry I can recall the definition of a ratio. Given a line segment (including those with positive I can recall previous understandings of coordinate and negative slopes) and a ratio, find the point on geometry. the segment that partitions the segment into the given ratio. (Reasoning) I can find the point on the segment that partitions the segment into a given ratio. 1 Unit 1: Congruence, Proof, and Constructions G.CO.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. 1,2 *Bolded word Use Marzano 6 Step Process Classify types of quadrilaterals. (Knowledge) Explain theorems for parallelograms and relate to figure. (Knowledge) Use the principle that corresponding parts of congruent triangles are congruent to solve problems. (Reasoning) Use properties of special quadrilaterals in a proof. (Reasoning) From Appendix A: 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. I can classify types of quadrilaterals. I can explain theorems for parallelograms and relate to figure. I can use the principle that corresponding parts of congruent triangles are congruent to solve problems. I can use properties of special quadrilaterals in a proof. I can write proofs in multiple ways, such as in narrative paragraphs, using flow diagrams, in twocolumn format, and using diagrams without words. I can focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. * Proof (2-column, paragraph, flow chart) * CPCTC (Corresponding Parts of Congruent Triangles are Congruent) Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Deconstructed Standards I Can Unit 1: Congruence, Proof, and Constructions Identify the hypothesis and conclusion of a theorem. (Knowledge) I can identify the hypothesis and conclusion of a theorem. G.CO.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. Design an argument to prove theorems about triangles. (Reasoning) I can design an argument to prove theorems about triangles. Analyze components of the theorem. (Reasoning) I can analyze components of the theorem. Prove theorems about triangles. (Reasoning) I can prove theorems about triangles. From Appendix A: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. . 5 Vocabulary Midpoint Median of a Triangle * Exterior Angle Theorem Triangle Sum Theorem Isosceles Triangle Theorem (Base Angle Theorem) * Points of Concurrency Centroid In center Circumcenter Orthocenter I can write proofs in multiple ways, such as in narrative paragraphs, using flow diagrams, in twocolumn 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. I can focus on the validity of the underlying Implementations of G.CO.10 may be extended to reasoning while exploring a variety of formats for include concurrence of perpendicular bisectors and expressing that reasoning. angle bisectors as preparation for G.C.3 in Unit 5. Unit 2: Similarity, Proof, and Trigonometry 2 G.SRT.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. *Bolded word Use Marzano 6 Step Process Recall postulates, theorems, and definitions to prove theorems about triangles. (Knowledge) I can recall postulates, theorems, and definitions to * Postulate prove theorems about right triangles. * Axiom Triangle Similarity Prove theorems involving similarity about triangles. I can prove theorems involving similarity about (Reasoning) 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.) Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 2: Similarity, Proof, and Trigonometry G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 2 Deconstructed Standards 2 I can recall congruence and similarity criteria for triangles. Use congruency and similarity theorems for triangles to solve problems. (Reasoning) I can use congruency and similarity theorems for triangles to solve problems. Use congruency and similarity theorems for triangles to prove relationships in geometric figures. (Reasoning) I can use congruency and similarity theorems for triangles to prove relationships in geometric figures. Recall previous understandings of coordinate I can recall previous understandings of coordinate geometry (including, but not limited to: distance, geometry. G.GPE.4 Use coordinates to prove simple midpoint and slope formula, equation of a line, geometric theorems algebraically. For example, definitions of parallel and perpendicular lines, etc.) prove or disprove that a figure defined by four (Knowledge) given points in the coordinate plane is a rectangle; prove or disprove that the point (1, Use coordinates to prove simple geometric √3) lies on the circle centered at the origin and theorems algebraically. (Reasoning) I can use coordinates to prove simple geometric containing the point (0, 2). 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). From Appendix A: Include simple proofs involving circles. *Bolded word Use Marzano 6 Step Process Vocabulary I Can Recall congruence and similarity criteria for triangles. (Knowledge) Unit 5: Circles With and Without Coordinates 6 Congruence Coordinate Geometry Proof Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Deconstructed Standards Informally use rigid motions to take angles to I can informally use rigid motions to take angles to angles and segments to segments (from 8th grade). angles and segments to segments (from 8th grade). (Knowledge) G.CO.8 Explain how the criteria for triangle I can formally use dynamic geometry software or congruence (ASA, SAS, and SSS) follow from the Formally use dynamic geometry software or straightedge and compass to take angles to angles definition of congruence in terms of rigid straightedge and compass to take angles to angles and segments to segments. motions. and segments to segments. (Knowledge) 2,3 Explain how the criteria for triangle congruence (ASA, SAS, SSS) follows from the definition of congruence in terms of rigid motions (i.e. if two angles and the included side of one triangle are transformed by the same rigid motion(s) then the triangle image will be congruent to the original triangle). (Reasoning) Unit 4: Connecting Algebra and Geometry Through Coordinates 2,3 I can explain how the criteria for triangle congruence (ASA, SAS, SSS) follows from the definition of congruence in terms of rigid motions. *Bolded word Use Marzano 6 Step Process Rigid Motion ASA SAS AAS SSS HL HA LL LA I can reason from the basic properties of rigid motions . Recall previous understandings of coordinate I can recall previous understandings of coordinate geometry (including, but not limited to: distance, geometry. midpoint and slope formula, equation of a line, G.GPE.4 Use coordinates to prove simple definitions of parallel and perpendicular lines, etc.) geometric theorems algebraically. For example, (Knowledge) prove or disprove that a figure defined by four given points in the coordinate plane is a Use coordinates to prove simple geometric rectangle; prove or disprove that the point (1, theorems algebraically. I can use coordinates to prove simple geometric √3) lies on the circle centered at the origin and For example, prove or disprove that a figure theorems algebraically. containing the point (0, 2). 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). (Reasoning) e.g., derive the equation of a line through 2 points using similar right triangles. (Reasoning) Vocabulary I Can Unit 1: Congruence, Proof, and Constructions From Appendix A: 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. 7 Distance Formula Midpoint Formula Slope Formula Parallel Line Perpendicular Line * Theorem Coordinate Geometry * Proof Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 5: Circles With and Without Coordinates Deconstructed Standards Define inscribed and circumscribed circles of a triangle. (Knowledge) G.C.3 Construct the inscribed and circumscribed circles of a triangle, Recall midpoint and bisector definitions. and prove properties of angles for a (Knowledge) quadrilateral inscribed in a circle. Define a point of concurrency. (Knowledge) Prove properties of angles for a quadrilateral inscribed in a circle. (Reasoning) 2,3 Construct inscribed circles of a triangle. (Performance) Construct circumscribed circles of a triangle. (Performance) Unit 1: Congruence, Proof, and Constructions 3 Vocabulary I can define inscribed and circumscribed circles of a triangle. I can recall midpoint and bisector definitions. I can define a point of concurrency. I can prove properties of angles for a quadrilateral inscribed in a circle. I can construct inscribed circles of a triangle. I can construct circumscribed circles of a triangle. I can describe the different types of transformations * Transformation including translations, reflections, rotations and * Translation dilations. * Rotation G.CO.2 Represent transformations in the plane * Reflection using, e.g., transparencies and geometry Describe transformations as functions that take * Dilation software; describe transformations as functions points in the coordinate plane as inputs and give I can describe transformations as functions that use * Rigid Motion that take points in the plane as inputs and give other points as outputs (Knowledge) points in the coordinate plane as inputs and produce * Non-Rigid Motion other points as outputs. Compare points as outputs. * Scale Factor transformations that preserve distance and Represent transformations in the plane using, e.g., * Composition of Transformations angle to those that do not (e.g., translation transparencies and geometry software. * Vectors versus horizontal stretch). (Reasoning) I can represent transformations in the plane using technology. Write functions to represent transformations. (Reasoning) I can write functions to represent transformations. *Bolded word Use Marzano 6 Step Process Describe the different types of transformations including translations, reflections, rotations and dilations. (Knowledge) I Can 8 Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 1: Congruence, Proof, and Constructions Deconstructed Standards Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch) I Can 9 Vocabulary I can compare transformations that preserve distance and angle to those that do not. G.CO.2 Standard (continued) From Appendix A: Build on student experience with I can build on my experiences with rigid motions rigid motions from earlier grades. Point out the from earlier grades and point out the basis of rigid basis of rigid motions in geometric concepts, e.g, motions in geometric concepts. translations move points a specific distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle. 3 Unit 1: Congruence, Proof, and Constructions Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and/or reflections that carry it onto itself. (Knowledge) G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the From Appendix A: Build on student experience with rotations and reflections that carry it onto itself. rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, e.g, translations move points a specific distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle. 3 *Bolded word Use Marzano 6 Step Process I can describe the rotations and reflections that carries a polygon, e.g. a rectangle, parallelogram, trapezoid, or regular shape, onto itself. I can point out the basis of rigid motions in geometric concepts, e.g, translations move points a specific distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle. Regular Polygon Rectangle Trapezoid Parallelogram Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 1: Congruence, Proof, and Constructions G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 3 Unit 1: Congruence, Proof, and Constructions 3 Deconstructed Standards Recall definitions of angles, circles, perpendicular and parallel lines and line segments. (Knowledge) Vocabulary * Transformation * Translation * Rotation Develop definitions of rotations, reflections and I can develop definitions of rotations, reflections and * Reflection translations in terms of angles, circles, translations in terms of angles, circles, perpendicular * Dilation perpendicular lines, parallel lines and line lines, parallel lines and line segments. * Rigid Motion segments. (Reasoning) * Non-Rigid Motion I can point out the basis of rigid motions in * Scale Factor From Appendix A: Build on student experience with geometric concepts, e.g., translations move points a * Composition of Transformations rigid motions from earlier grades. Point out the specific distance along a line parallel to a specified * Vectors basis of rigid motions in geometric concepts, e.g., line; rotations move objects along a circular arc with translations move points a specific distance along a a specified center through a specified angle. line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle. Given a geometric figure and a rotation, reflection or translation, draw the transformed figure using, e.g. graph paper, tracing paper or geometry G.CO.5 Given a geometric figure and a rotation, software. (Knowledge) reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or Draw a transformed figure and specify the geometry software. Specify a sequence of sequence of transformations that were used to transformations that will carry a given figure carry the given figure onto the other. (Reasoning) onto another. From Appendix A: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, e.g., translations move points a specific distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle. *Bolded word Use Marzano 6 Step Process I Can 10 I can recall definitions of angles, circles, perpendicular and parallel lines and line segments. I can draw the transformed figure in multiple ways. * Transformation * Translation * Rotation * Reflection * Dilation I can draw a transformed figure and specify the * Rigid Motion sequence of transformations that were used to carry * Non-Rigid Motion the given figure onto the other. * Scale Factor * Composition of Transformations I can point out the basis of rigid motions in * Vectors geometric concepts. Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Deconstructed Standards I Can Unit 1: Congruence, Proof, and Constructions Use geometric descriptions of rigid motions to transform figures. G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Predict the effect of a given rigid motion on a given I can predict the effect of a given rigid motion on a figure. given figure. 3 Unit 1: Congruence, Proof, and Constructions G.CO.6 Standard (continued) 3 *Bolded word Use Marzano 6 Step Process 11 Vocabulary I can use geometric descriptions of rigid motions to * Congruence transform figures. * Composition of Transformations Define congruence in terms of rigid motions (i.e. two figures are congruent if there exists a rigid motion, or composition of rigid motions, that can take one figure to the second). I can define congruence in terms of rigid motions Decide if two figures are congruent in terms of rigid motions (it is not necessary to find the precise transformation(s) that took one figure to a second, only to understand that such a transformation or composition exists). I can decide if two figures are congruent in terms of rigid motions (it is not necessary to find the precise transformation(s) that took one figure to a second, only to understand that such a transformation or composition exists). From Appendix A: 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 are their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. I can reason from the basic properties of rigid motions, which can then be used to prove other theorems. Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Deconstructed Standards I Can 12 Vocabulary Unit 1: Congruence, Proof, and Constructions Identify corresponding angles and sides of two triangles. (Knowledge) I can identify corresponding angles and sides of two * Corresponding Parts triangles. G.CO.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. Identify corresponding pairs of angles and sides of congruent triangles after rigid motions. (Knowledge) I can identify corresponding pairs of angles and sides of congruent triangles after rigid motions. Unit 1: Congruence, Proof, and Constructions From Appendix A: 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. 3 G.CO.7 (Standard continue) 3 *Bolded word Use Marzano 6 Step Process I can use the definition of congruence in terms of Use the definition of congruence in terms of rigid rigid motions to show that two triangles are motions to show that two triangles are congruent if congruent if corresponding pairs of sides and corresponding pairs of sides and corresponding corresponding pairs of angles are congruent. pairs of angles are congruent. (Reasoning) I can use the definition of congruence in terms of Use the definition of congruence in terms of rigid rigid motions to show that if the corresponding pairs motions to show that if the corresponding pairs of of sides and corresponding pairs of angles of two sides and corresponding pairs of angles of two triangles are congruent then the two triangles are triangles are congruent then the two triangles are congruent. congruent. (Reasoning) I can justify congruency of two triangles using transformations. I can 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. Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter 3 Geometry Standards Deconstructed Standards Unit 2: Similarity, Proof, and Trigonometry Define image, pre-image, scale factor, center, and similar figures as they relate to transformations. (Knowledge) Unit 2: Similarity, Proof, and Trigonometry Explain that the scale factor represents how many times longer or shorter a dilated line segment is than its pre-image. (Knowledge) I Can I can define image, pre-image, scale factor, center, * Dilation and similar figures as they relate to transformations. Center G.SRT.1 Verify experimentally the properties of Scale Factor dilations given by a I can identify a dilation stating its scale factor and center and a scale factor. Identify a dilation stating its scale factor and center. center. a. A dilation takes a line not passing through the (Knowledge) center of the dilation I can verify experimentally that a dilated image is to a parallel line, and leaves a line passing Verify experimentally that a dilated image is similar similar to its pre-image by showing congruent through the center unchanged. to its pre-image by showing congruent corresponding angles and proportional sides. b. The dilation of a line segment is longer or corresponding angles and proportional sides. shorter in the ratio given by the scale factor. (Reasoning) I can verify experimentally that a dilation takes a line not passing through the enter of the dilation to a Verify experimentally that a dilation takes a line not parallel line by showing that the lines are parallel. passing through the center of the dilation to a parallel line by showing the lines are parallel. I can verify experimentally that dilation leaves a line (Reasoning) passing through the center of the dilation unchanged by showing that it is the same line. Verify experimentally that dilation leaves a line passing through the center of the dilation unchanged by showing that it is the same line. (Reasoning) G.SRT.1 Standard (continued) Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor. (Reasoning) 3 *Bolded word Use Marzano 6 Step Process I can explain that the scale factor represents how many times longer or shorter a dilated line segment is than its pre-image. I can verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor. 13 Vocabulary Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 2: Similarity, Proof, and Trigonometry 3 G.SRT.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. Unit 2: Similarity, Proof, and Trigonometry G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 3 Unit 2: Similarity, Proof, and Trigonometry G.SRT.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. 3 *Bolded word Use Marzano 6 Step Process Deconstructed Standards By using similarity transformations, explain that triangles are similar if all pairs of corresponding angles are congruent and all corresponding pairs of sides are proportional. (Knowledge) Given two figures, decide if they are similar by using the definition of similarity in terms of similarity transformations. (Reasoning) I Can 14 Vocabulary I can use similarity transformations to explain that * Similarity Transformation triangles are similar if all pairs of corresponding angles are congruent and all corresponding pairs of sides are proportional. I can decide if two figures are similar by using the definition of similarity in term sof similarity transformations. Recall the properties of similarity transformations. I can recall the properties of similarity of (Knowledge) transformations. * Similarity Transformation Establish the AA criterion for similarity of triangles by extending the properties of similarity transformations to the general case of any two similar triangles. (Reasoning) I can establish the AA criterion for similarity of triangles by extending the properties for similarity transformations to the general case of any two similar triangles. Names the sides of right triangles as related to an acute angle. (Knowledge) Recognize that if two right triangles have a pair of acute, congruent angles that the triangles are similar. (Knowledge) I can name the sides of right triangles as they relate Opposite Leg to an acute angle. Adjacent Leg Hypotenuse I recognize that if two right triangles have a pair of Trigonometric Ratios acute, congruent angles, then the triangles are similar. Compare common ratios for similar right triangles and develop a relationship between the ratio and the acute angle leading to the trigonometry ratios. (Reasoning) I can compare common ratios for similar right triangles and develop a relationship between the ratio and the acute angle leading to the trigonometric ratios. Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 2: Similarity, Proof, and Trigonometry 3 Deconstructed Standards I Can Use the relationship between the sine and cosine of I can use the relationship between the sine and complementary angles. (Knowledge) cosine of complementary angles. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary Explain how the sine and cosine of complementary angles. angles are related to each other. (Reasoning) I can explain how the sine and cosine of complementary angles are related to each other. Unit 2: Similarity, Proof, and Trigonometry G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ Recognize which methods could be used to solve right triangles in applied problems. (Knowledge) I can recognize which methods could be used to solve right triangles in applied problems. Solve for an unknown angle or side of a right triangle using sine, cosine, and tangent. (Knowledge) I can solve for an unknown angle or side of a right triangle using sine, cosine, and tangent. Apply right triangle trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. (Reasoning) 3 Describe a topographical grid system. (Knowledge) I can describe a topographical grid system. G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with topographic grid systems based on ratios).* Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy I can apply geometric methods to solve design physical constraints or minimize cost; working with problems. topographic grid systems based on ratios). (Reasoning) *Bolded word Use Marzano 6 Step Process Vocabulary Sine Cosine Complementary Angles * Angle of Elevation * Angle of Depression Pythagorean Theorem I can apply right triangle trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Unit 2: Similarity, Proof, and Trigonometry 3 15 From Appendix A: Focus on situations well modeled by trigonometric ratios for acute angles. I can identify situations well-modeled by (Reasoning) trigonometric ratios for acute angles. * Angle of Elevation * Angle of Depression Constraints Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 2: Similarity, Proof, and Trigonometry Deconstructed Standards Recall right triangle trigonometry to solve mathematical problems (Knowledge) I Can I can recall right triangle trigonometry to solve mathematical problems. 16 Vocabulary * Auxiliary Line G.SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary Derive the formula A = 1/2 ab sin(C) for the area of I can derive the formula A = 1/2 ab sin(C) for the line from a vertex perpendicular to the opposite a triangle by drawing an auxiliary line from a vertex area of a triangle by drawing an auxiliary line from side. perpendicular to the opposite side. (Reasoning) the vertex perpendicular to the opposite side. 3 Unit 2: Similarity, Proof, and Trigonometry Use the Laws of Sines and Cosines this to find missing angles or side length measurements. G.SRT.10 (+) Prove the Laws of Sines and Cosines (Knowledge) and use them to solve problems. Prove the Law of Sines (Reasoning) I can use the Law of Sines and Cosines to find missing angles or side length measurements. I can prove the Law of Sines. I can prove the Law of Cosines. Prove the Law of Cosines (Reasoning) 3 *Bolded word Use Marzano 6 Step Process I can recognize when the Law of Sines or Law of Recognize when the Law of Sines or Law of Cosines Cosines can be applied to a problem. can be applied to a problem and solve problems in context using them. (Reasoning) I can extend the general case of the Laws of Sines and Cosines to obtuse angles. From Appendix A: With respect to the general case of Laws of Sines and Cosines, the definition of sine and cosine must be extended to obtuse angles. * Law of Sines * Law of Cosines Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 2: Similarity, Proof, and Trigonometry 3 Deconstructed Standards I Can Vocabulary Determine from given measurements in right and I can determine whether it is appropriate to use the * Law of Sines non-right triangles whether it is appropriate to use Law of Sines or Cosines given the measurements in a * Law of Cosines the Law of Sines or Cosines. (Knowledge) right or non-right triangle. G.SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right Apply the Law of Sines and the Law of Cosines to I can apply the Law of Sines and Law of Cosines to and non-right triangles find unknown measurements in right and non-right find unknown measurements in right and non-right (e.g., surveying problems, resultant forces). triangles (e.g., surveying problems, resultant triangles. forces). (Reasoning) I can extend the definition of sine and cosine to From Appendix A: With respect to the general case obtuse angles with respect to the general case of of the Laws of Sines and Cosines, the definition of the Laws of Sines and Cosines. sine and cosine must be extended to obtuse angles. Unit 4: Connecting Algebra and Geometry Through Coordinates Use the coordinates of the vertices of a polygon to I can use the coordinates of the vertices of a polygon find the necessary dimensions for finding the to find the necessary dimensions for finding the perimeter (i.e., the distance between vertices). perimeter. G.GPE.7 Use coordinates to compute perimeters (Knowledge) of polygons and areas of triangles and rectangles, e.g., using the Use the coordinates of the vertices of a triangle to I can use the coordinates of the vertices of a triangle distance formula.★ find the necessary dimensions (base, height) for or rectangle to find the necessary dimensions for finding the area (i.e., the distance between vertices finding the area. by counting, distance formula, Pythagorean Theorem, etc.). (Knowledge) 3 17 From Appendix A: G.GPE.7 provides practice with the distance formula and its connection with the Pythagorean theorem Use the coordinates of the vertices of a rectangle to find the necessary dimensions (base, height) for finding the area (i.e., the distance between vertices by counting, distance formula). (Knowledge) Formulate a model of figures in contextual problems to compute area and/or perimeter. (Reasoning) . *Bolded word Use Marzano 6 Step Process I can formulate a model of figures in contextual problems to compute area and/or perimeter. * Coordinate Geometry Perimeter Area Formulas Vertices Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 2: Similarity, Proof, and Trigonometry G.MG.1 Use geometric shapes, their measures, and their properties to Given a real world object, classify the object as a describe objects (e.g., modeling a tree trunk or a known geometric shape – use this to solve human torso as a problems in context. (Reasoning) cylinder).* From Appendix A: Focus on situations well modeled by trigonometric ratios for acute angles. Vocabulary I Can Deconstructed Standards Use measures and properties of geometric shapes to describe real world objects. (Knowledge) 18 I can use measures and properties of geometric shapes to describe real world objects. * Oblique Polyhedron I can classify real world objects as a known geometric shape and use this to solve problems in context. I can identify situations well-modeled by trigonometric ratios for acute angles. 4 Unit 2: Similarity, Proof, and Trigonometry Define density. (Knowledge) I can define density. G.MG.2 Apply concepts of density based on area Apply concepts of density based on area and I can apply concepts of density based on area and and volume in volume to model real-life situations (e.g., persons volume to model real-life situations. modeling situations (e.g., persons per square per square mile, BTUs per cubic foot). (Reasoning) mile, BTUs per cubic foot).* 4 *Bolded word Use Marzano 6 Step Process Density Volume Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 3: Extending to Three Dimensions 4 Deconstructed Standards Recognize cross-sections of solids as twodimensional shapes I Can I can decompose volume formulas into area formulas using cross-sections. 19 Vocabulary Volume Circumference G.GMD.1 Give an informal argument for the Cylinder formulas for the Recognize formulas for area and circumference of a I can recognize cross sections of solids as twoPyramid circumference of a circle, area of a circle, circle and volume of a cylinder, pyramid, and cone. dimensional shapes. Cone volume of a cylinder, pyramid, * Dissection and cone. Use dissection arguments, Cavalieri’s Recognize Cavalieri's principle. I can recognize formulas for area and circumference Cavalieri's Principle principle, and informal of a circle and volume of a cylinder, pyramid, and a * Limit limit arguments. Decompose volume formulas into area formulas cone. using cross-sections. I can use the techniques of dissection and limit Apply dissections and limit arguments (e.g. arguments. Archimedes' inscription and circumscription of polygons about a circle and as a component of the I can apply Cavalier's Principle as a component of informal argument for the formulas for the the informal argument for the formulas for the circumference and area of a circle.) volume of a cylinder, pyramid, and a cone. Apply Cavalieri's Principle as a component of the I can make informal arguments for area and volume informal argument for the formulas for the volume formulas and can make use of the way in which area of a cylinder, pyramid, and cone. and volume scale under similarity transformations. Unit 3: Extending to Three Dimensions G.GMD.1 (Standard continued) 4 *Bolded word Use Marzano 6 Step Process From Appendix A: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor K, its area is K^2 times the area of the first. Similarly, volumes of solid figures scale by K^3 under a similarity transformations with scale factor K. Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 3: Extending to Three Dimensions G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems Deconstructed Standards 20 Vocabulary I Can Utilize the appropriate formula for volume depending on the figure. (Knowledge) I can utilize the appropriate formula for volume depending on the figure. Use volume formulas for cylinders, pyramids, cones, and spheres to solve contextual problems. (Reasoning) I can use volume formulas for cylinders, pyramids, cones, and spheres to solve contextual problems. Sphere Volume Cylinders Pyramids Cones I can make informal arguments for area and volume From Appendix A: Informal arguments for area and formulas using the way area and volume scale under volume formulas can make use of the way in which similarity transformations. area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor K, its area is K2 times the area of the first. Similarly, volumes of solid figures scale by K3 under a similarity transformations with scale factor K. 4 Unit 3: Extending to Three Dimensions Use strategies to help visualize relationships between two-dimensional and three dimensional G.GMD.4 Identify the shapes of two-dimensional objects. (Knowledge) cross-sections of three-dimensional objects, and identify three-dimensional objects generated Relate the shapes of two-dimensional crossby rotations of two-dimensional objects. sections to their three-dimensional objects. (Reasoning) 4 I can identify the shapes of two-dimensional cross- * Cross-section sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. I can use strategies to help visualize relationships between two-dimensional and three-dimensional objects. Discover three-dimensional objects generated by rotations of two-dimensional objects. (Reasoning) I can relate the shapes of two-dimensional crosssections to their three-dimensional objects. I can discover three-dimensional objects generated by rotations of two-dimensional objects. *Bolded word Use Marzano 6 Step Process Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 3: Extending to Three Dimensions 4 Deconstructed Standards Use measures and properties of geometric shapes to describe real world objects. (Knowledge) I Can 21 Vocabulary I can use geometric shapes, their measures, and their properties to describe objects. G.MG.1 Use geometric shapes, their measures, and their properties to Given a real world object, classify the object as a describe objects (e.g., modeling a tree trunk or a known geometric shape; use this to solve problems I can be given a real world object, classify the object human torso as a cylinder).* in context. (Reasoning) as a known geometric shape and use this to solve problems in context. From Appendix A: Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles. Unit 4: Connecting Algebra and Geometry Through Coordinates Define a parabola including the relationship of the focus and the equation of the directrix to the parabolic shape. (Knowledge) I can derive the equation of a parabola given a focus Parabola and directix. * Focus * Directrix I can define a parabola including the relationship of Axis of Symmetry the focus and the equation of the directix to the * Latus Rectum parabolic shape. From Appendix A: The directrix should be parallel to a coordinate axis. I can understand that the directix should be parallel to a coordinate axis. G.GPE.2 Derive the equation of a parabola given a focus and directrix. 4 *Bolded word Use Marzano 6 Step Process Derive the equation of parabola given the focus and I can derive the equation of parabola given the focus directrix. (Reasoning) and directix. Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 5: Circles With and Without Coordinates G.C.1 Prove that all circles are similar. 4 Deconstructed Standards Recognize when figures are similar. (Two figures are similar if one is the image of the other under a transformation from the plane into itself that multiplies all distances by the same positive scale factor, k. That is to say, one figure is a dilation of the other. ) (Knowledge) I can recognize when figures are similar. Compare the ratio of the circumference of a circle to the diameter of the circle. (Reasoning) I can compare the ratio of the circumference of a circle to the diameter of the circle. Discuss, develop and justify this ratio for several circles. (Reasoning) Determine that this ratio is constant for all circles. (Reasoning) Unit 5: Circles With and Without Coordinates 4 Identify inscribed angles, radii, chords, central angles, circumscribed angles, diameter, tangent. G.C.2 Identify and describe relationships among (Knowledge) inscribed angles, radii, and chords. Include the relationship between central, inscribed, and Recognize that inscribed angles on a diameter are circumscribed angles; inscribed angles on a right angles. (Knowledge) diameter are right angles; the radius of a circle is perpendicular to the Recognize that radius of a circle is perpendicular to tangent where the radius the radius at the point of tangency. (Knowledge) intersects the circle. Examine the relationship between central, inscribed and circumscribed angles by applying theorems about their measures. (Reasoning) *Bolded word Use Marzano 6 Step Process I Can 22 Vocabulary Circle Circumference Diameter Ratio Pi * Concentric Circles I can discuss, develop and justify this ratio for several circles. I can determine that this ratio is constant for all circles. I can examine the relationship between central, inscribed and circumscribed angles by applying theorems about their measures. I can identify inscribed angles, radii, chords, central angles, circumscribed angles, diameter, tangent. I can recognize that inscribed angles on a diameter are right angles. I can recognize that radius of a circle is perpendicular to the radius at the point of tangency. * Inscribed Angle * Central Angle Tangent Circumscribed Angle * Tangent Line * Tangent Segment * Secant Line * Point of Tangency * Chord Diameter Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 5: Circles With and Without Coordinates G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle. Deconstructed Standards I Can Recall vocabulary: Tangent Radius Perpendicular bisector Midpoint (Knowledge) I can recall vocabulary: Tangent, Radius, Perpendicular bisector, and Midpoint. Identify the center of the circle (Knowledge) I can identify the center of the circle. 23 Vocabulary *Common Tangents Tangent Radius Perpendicular Bisector Midpoint Center of a Circle Point of Tangency Synthesize theorems that apply to circles and tangents, such as: I can synthesize theorems that apply to circles and *Tangents drawn from a common external point tangents. are congruent and *A radius is perpendicular to a tangent at the point of tangency. (Reasoning) 4 Construct the perpendicular bisector of the line segment between the center C to the outside point I can construct the perpendicular bisector of the line P. (Performance) segment between the center C to the outside point P. Construct arcs on circle C from the midpoint Q, having length of CQ. (Performance) I can construct arcs on circle C from the midpoint Q, having length of CQ. Construct the tangent line. (Performance) I can construct the tangent line. Unit 5: Circles With and Without Coordinates 4 G.C.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 *Bolded word Use Marzano 6 Step Process Recall how to find the area and circumference of a I can recall how to find the area and circumference circle. (Knowledge) of a circle. Explain that 1° = Π/180 radians (Knowledge) I can explain that 1° = Π/180 radians Recall from G.C.1, that all circles are similar. (Knowledge) I can recall that all circles are similar. Determine the constant of proportionality (scale factor). (Knowledge) I can determine the constant of proportionality (scale factor). * Major Arc * Minor Arc * Intercepted Arc * Radian Constant of Proportionality Scale Factor Central Angle Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 5: Circles With and Without Coordinates G.C.5 (Standard continued) Deconstructed Standards Justify the radii of any two circles (r1 and r2) and the arc lengths (s1 and s2) determined by congruent central angles are proportional, such that r1 /s1 = r2/s2 (Reasoning) Vocabulary I can justify the radii of any two circles and the arc lengths determined by congruent central angles are proportional. Verify that the constant of a proportion is the same as the radian measure, Θ, of the given central I can verify that the constant of a proportion is the angle. Conclude s = r Θ (Reasoning) same as the radian measure, Θ, of the given central angle. From Appendix A: 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. 4 Unit 5: Circles With and Without Coordinates 4 I Can 24 Define a circle. (Knowledge) G.GPE.1 Derive the equation of a circle of given Use Pythagorean Theorem. (Knowledge) center and radius using the Pythagorean Theorem; complete the square Complete the square of a quadratic equation. to find the center and (Knowledge) radius of a circle given by an equation. Derive equation of a circle using the Pythagorean From Appendix A: Emphasize the similarity of all Theorem – given coordinates of the center and circles. Note that by similarity of sectors with length of the radius. (Reasoning) the same central angle, arc lengths are proportional to the radius. Use this as a basis for Determine the center and radius by completing the introducing radian as a unit of measure. It is not square. (Reasoning) intended that it be applied to the development of circular trigonometry in this course. *Bolded word Use Marzano 6 Step Process I can define a circle. I can use Pythagorean Theorem. I can complete the square of a quadratic equation. I can derive equation of a circle using the Pythagorean Theorem – given coordinates of the center and length of the radius. I can determine the center and radius by completing the square. Completing the Square Pythagorean Theorem Quadratic Equation Equation of a Circle Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 5: Circles With and Without Coordinates Deconstructed Standards Use measures and properties of geometric shapes to describe real world objects. (Knowledge) I Can I can use measures and properties of geometric shapes to describe real world objects. G.MG.1 Use geometric shapes, their measures, and their properties to Given a real world object, I can classify the object as describe objects (e.g., modeling a tree trunk or a Given a real world object, classify the object as a a known geometric shape - use this to solve human torso as a known geometric shape - use this to solve problems problems in context. cylinder).* in context. (Reasoning) 4 From Appendix A: Focus on situations in which the analysis of circles is required. (Reasoning) Unit 6: Applications of Probability 4 S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). Unit 6: Applications of Probability 4 S.CP.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. *Bolded word Use Marzano 6 Step Process Define unions, intersections and complements of events. (Knowledge) Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or”, “and”, “not”). (Reasoning) Categorize events as independent or not using the characterization that two events A and B are independent when the probability of A and B occurring together is the product of their probabilities. (Knowledge) From Appendix A: Build on work from 2-way tables from Algebra 1 Unit 3 (S.ID.5) to develop understanding of conditional probability and independence. (Knowledge) I can define unions, intersections and complements Union of events. Intersection Complement I can describe events as subsets of a sample space Sample Space using characteristics of the outcomes, or as unions, intersections, or complements of other events. I can categorize events as independent or not using Independent the characterization that two events A and B are Dependent independent when the probability of A and B occurring together is the product of their probabilities. 25 Vocabulary Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 6: Applications of Probability 4 S.CP.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. Unit 6: Applications of Probability 4 S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way 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. *Bolded word Use Marzano 6 Step Process Deconstructed Standards I Can Know the conditional probability of A given B as P(A I know the conditional probability of A given B as and B)/P(B) (Knowledge) P(A and B)/P(B) 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. (Knowledge) Vocabulary Conditional Probability I can 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. Use the two-way table as a sample space to decide I can use the two-way table as a sample space to if events are independent and to approximate decide if events are independent and to conditional probabilities. approximate conditional probabilities. (Knowledge) I can interpret two-way frequency tables of data From Appendix A: Build on work with two-way when two categories are associated with each tables from Algebra 1 Unit 3 (S.ID.5) to develop object being classified. understanding of conditional probability and independence. Interpret two-way frequency tables of data when two categories are associated with each object being classified. (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 10th grade. Do the same for other subjects and compare the results.) (Reasoning) 26 Sample Space Two-Way Frequency Table Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 6: Applications of Probability 4 S.CP.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. Deconstructed Standards Recognize the concepts of conditional probability and independence in everyday language and everyday situations. (Knowledge) I Can 27 Vocabulary I can recognize the concepts of conditional Theoretical Probability probability and independence in everyday language Conditional Probability and everyday situations. Experimental Probability Explain the concepts of conditional probability and I can explain the concepts of conditional probability independence in everyday language and everyday and independence in everyday language and situations. (For example, compare the chance of everyday situations. having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.) (Reasoning) Unit 6: Applications of Probability 4 Find the conditional probability of A given B as the I can find the conditional probability of A given B as Conditional Probability fraction of B’s outcomes that also belong to A. the fraction of B’s outcomes that also belong to A. Compound Event S.CP.6 Find the conditional probability of A given (Knowledge) B as the fraction of B’s I can interpret the answer in terms of the model. outcomes that also belong to A, and interpret Interpret the answer in terms of the model. the answer in terms of the (Reasoning) model. Unit 6: Applications of Probability Use the Additional Rule, P(A or B) = P(A) + P(B) – P(A and B) (Knowledge) S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), Interpret the answer in terms of the model. and interpret the answer in terms of the model. (Reasoning) 4 *Bolded word Use Marzano 6 Step Process I can use the Additional Rule, P(A or B) = P(A) + P(B) Addition Rule of Probability – P(A and B) I can interpret the answer in terms of the model. Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 6: Applications of Probability 4 S.CP.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. Unit 6: Applications of Probability S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems. 4 Unit 6: Applications of Probability S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 4 *Bolded word Use Marzano 6 Step Process Deconstructed Standards I Can Use the multiplication rule with correct notation. (Knowledge) I can use the multiplication rule with correct notation. Apply the general Multiplication Rule in a uniform probability model P(A and B) = P(A)P(B|A) = P(B)P(A|B). (Reasoning) I can apply the general Multiplication Rule in a uniform probability model P(A and B) = P(A)P(B|A) = P(B)P(A|B). Interpret the answer in terms of the model. (Reasoning) I can interpret the answer in terms of the model. Identify situations that are permutations and those I can identify situations that are permutations and that are combinations. those that are combinations. (Knowledge) Use permutations and combinations to compute probabilities of compound events and solve problems. (Reasoning) I can use permutations and combinations to compute probabilities of compound events and solve problems. Compute Theoretical and Experimental Probabilities. (Knowledge) I can compute theoretical and experimental probabilities. Use probabilities to make fair decisions (e.g. drawing by lots, using a random number generator.) (Reasoning) I can use probabilities to make fair decisions. From Appendix A: 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) requires an understanding of probability concepts. 28 Vocabulary Multiplication Rule of Probability Permutations Combinations Compound Event Theoretical Probability Experimental Probability Resources Technology Resources Assessments Geometry Curriculum Mapping 8/2/2011 Quarter Geometry Standards Unit 6: Applications of Probability S.MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). 4 *Bolded word Use Marzano 6 Step Process Deconstructed Standards Recall prior understandings of probability. (Knowledge) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game.) (Reasoning) From Appendix A: 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) requires an understanding of probability concepts. I Can I can recall prior understandings of probability. I can analyze decisions and strategies using probability concepts. 29 Vocabulary Resources Technology Resources Assessments