Math CCSS Pacing Guide

Pojoaque Valley Schools Math CCSS Pacing Guide Geometry

*Skills adapted from Kentucky Department of Education Math Deconstructed Standards ** Evidence of attainment/assessment, Vocabulary, Knowledge, Skills and Essential Elements adapted from Wisconsin Department of Education and Standards Insights Computer-Based Program

Pojoaque Valley Schools ELA Common Core Pacing Guide Introduction

The Pojoaque Valley Schools pacing guide documents are intended to guide teachers’ use of Common Core State Standards (CCSS) over the course of an instructional school year. The guides identify the focus standards by quarter. Teachers should understand that the focus standards emphasize deep instruction for that timeframe. However, because a certain quarter does not address specific standards, it should be understood that previously taught standards should be reinforced while working on the focus standards for any designated quarter. Some standards will recur across all quarters due to their importance and need to be addressed on an ongoing basis.

The CCSS are not intended to be a check-list of knowledge and skills but should be used as an integrated model of literacy instruction to meet end of year expectations.

The English Language Arts CCSS pacing guides contain the following elements:  College and Career Readiness (CCR) Anchor Standard  Strand: Identify the type of standard  Cluster: Identify the sub-category of a set of standards.  Grade Level: Identify the grade level of the intended standards

1Version 3 2015-2016  Standard: Each grade-specific standard (as these standards are collectively referred to) corresponds to the same-numbered CCR anchor standard. Put another way, each CCR anchor standard has an accompanying grade-specific standard translating the broader CCR statement into grade-appropriate end-of-year expectations.  Standards Code: Contains the strand, grade, and number (or number and letter, where applicable), so that RI.4.3, for example, stands for Reading, Informational Text, grade 4, standard 3  Skills and Knowledge: Identified as subsets of the standard and appear in one or more quarters. Define the skills and knowledge embedded in the standard to meet the full intent of the standard itself.

Version 2 of the Pojoaque Valley School District Pacing guides for Reading Language Arts and Mathematics are based on the done by staff and teachers of the school district using the Kentucky model, and a synthesis of the excellent work done by Wisconsin Cooperative Educational Service Agency 7 (CESA 7) School Improvement Services, Green Bay, WI. (2010), Standards Insight project. Standards Insight was developed to give educators a tool for in depth investigation of the Common Core State Standards (CCSS). The CCSS are “unpacked” or dissected, identifying specific knowledge, skills, vocabulary, understandings, and evidence of student attainment for each standard. Standards Insight may be used by educators to gain a thorough grasp of the CCSS or as a powerful collaborative tool supporting educator teams through the essential conversations necessary for developing shared responsibility for student attainment of all CCSS. . . . serves as a high-powered vehicle to help educators examine the standards in a variety of ways. The Version 2 Pojoaque Valley School District Pacing guides present the standard with levels of detail and then the necessary skills by quarter based on the Kentucky model. On the second page for each standard, the synthesis of the Standards Insight project is presented in a way that further defines and refines the standard such that teachers may use the information to refine their teaching practices. Based on this synthesis of work and the purpose for the unpacking, the following fields were selected as most helpful to aid in understanding of the Common Core Standards that will lead to shifts in instruction: 1. Evidence of Student Attainment: “What could students do to show attainment of the standard?” 2. Vocabulary: “What are key terms in the standard that are essential for interpretation and understanding in order for students to learn the content?”

2Version 3 2015-2016 3. Knowledge: “What does the student need to know in order to aid in attainment of this standard?” 4. Skills and Understanding: “What procedural skill(s) does the student need to demonstrate for attainment of this standard?”, and “What will students understand to attain the standard?” The following fields are included in Version 2: Evidence of Student Attainment: This field describes what the standard may look like in student work. Specific expectations are listed in performance terms showing what students will say or do to demonstrate attainment of the standard.

Standards Vocabulary: This field lists words and phrases specific to each standard. Shared interpretation and in depth understanding of standards vocabulary are essential for consistent instruction across and within grade levels and content areas.

Knowledge: The knowledge field lists what students will need to know in order to master each standard (facts, vocabulary, definitions).

Skills and Understanding: The skills field identifies the procedural knowledge students apply in order to master each standard (actions, applications, strategies), as well as the overarching understanding that connects the standard, knowledge, and skills. Understandings included in Standards Insight synthesize ideas and have lasting value.

Instructional Achievement Level Descriptors: This field lists, by level what a teacher can expect to see in a student who achieves at a particular level. Additionally teachers can use this filed to differentiate instruction to provide further growth for student’s in moving from one level to another. This field can be used to provide specific teaching approaches to the standard in question.

A Note About High School Standards: The high school standards are listed in conceptual categories. Conceptual categories portray a coherent view of high school instruction that crosses traditional course boundaries. We have done everything possible, with teacher input, to link individual standards to the appropriate pacing guides, References to Tables: References to tables within the standards in the Standards Insight tool refer to Tables 1-5 found in the glossary of the Mathematics Common Core State Standards document found at www.corestandards.org.

Quarterly View of Standards Geometry Pacing Guide

G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of a point, line and distance along a line and distance around a circular arc. G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch) G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G.CO.4 Develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments. G.CO.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. 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. G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS) follow from the definition of congruence in terms of rigid motions. G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 3Version 3 2015-2016 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. 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. 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. Quarter G.CO.13 Construct an equilateral triangle, a square and a regular hexagon inscribed in a circle. G.SRT.1a Verify experimentally the properties of dilations given by a center and a scale factor.a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. 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. G.SRT.1b Verify experimentally the properties of dilations given by a center and a scale factor. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 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. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 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. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* (*Modeling Standard) G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* (*Modeling Standard) 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 typographic grid systems based on ratios).*(*Modeling Standard) G.SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G.SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. Quarter G.SRT.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).

G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*(*Modeling Standard) G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two- dimensional objects. G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*(*Modeling Standard) G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). G.GPE.7 Use coordinates to compute perimeters of polygons and area of triangles and rectangles, e.g., using the distance formula.*(*Modeling Standard) G.GPE.2 Derive the equation of a parabola given a focus and directrix. G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). G.C.1 Prove that all circles are similar. G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G.C.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.

4Version 3 2015-2016 G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G.C. 4 (+) Construct a tangent line from a point outside a given circle to the circle. G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0,2). 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.

Quarter 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”).Statistics and Probability is a Modeling Conceptual Category. G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*(*Modeling Standard) 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.Statistics and Probability is a Modeling Conceptual Category. 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. Modeling Conceptual Category. 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 th given that the student is in 10 grade. Do the same for other subjects and compare the results. Statistics and Probability is a Modeling Conceptual Category. S.CP.5 Recognizeandexplaintheconceptsofconditionalprobabilityandindependenceineverydaylanguageand everyday situations. having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Statistics and Probability is a Modeling Conceptual Category. S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A and interpret the answer in terms of the model. and Probability is a Modeling Conceptual Category. S.CP.7 Apply the Additional Rule, P(A or B) = P(A) + P(B) – P(A and B) and interpret the answer in terms of the model. Conceptual Category. 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.Statistics and Probability is a Modeling Conceptual Category.

Quarter S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Statistics and Probability is a Modeling Conceptual Category. S.MD.6 (+) Use probabilities to make fair decisions (e.g. drawing by lots, using a random number generator.) 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. Statistics and Probability is a Modeling Conceptual Category. 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.) Statistics and Probability is a Modeling Conceptual Category.

5Version 3 2015-2016 CCSS Math Pacing Guide

Grade Level/ Course: Geometry Unit 1

Standard: G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of a point, line and distance along a line and distance around a circular arc.

Domain: Cluster: Experiment with transformations in the plane. Congruence

Quarter 1: Quarter 2: Quarter 4:

Describe the Define circle and the distance around a circular arc. undefined terms: point, line, and distance along a line in a plane.

Define perpendicular lines, parallel lines, line segments, and angles.

Make sense of Reason Construct viable arguments Model with Use appropriate tools problems and abstractly and critique the reasoning mathematics. strategically.

6Version 3 2015-2016 persevere in solving and of others. them. quantitativel y.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are Given undefined notions of Undefined notions of point, line, able to: point, line, distance along a distance along a line, and distance Use known and developed line, and distance around a around a circular arc, definitions and logical circular arc, Properties of a mathematical connections to develop new Develop precise definitions of definition, i.e. the smallest amount definitions. angle, circle, perpendicular of information and properties that Geometric definitions are line, parallel line, and line are enough to determine the developed from a few segment, concept. (Note: may not include all undefined notions by a Identify examples and non- information related to concept). logical sequence of examples of angles, circles, connections that lead to a perpendicular lines, parallel precise definition, lines, and line segments. A precise definition should allow for the inclusion of all examples of the concept, and require the exclusion of all non-examples.

Grade Level/ Course: Geometry Unit 1

7Version 3 2015-2016 Standard: G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch)

Domain: Congruence Cluster: Experiment with transformations in the plane.

Quarter 1: Quarter 2: Quarter 3: Describe the different types of transformations including translations, reflections, rotations and dilations.

Describe transformations as functions that take points in the coordinate plane as inputs and give other points as outputs

Represent transformations in the plane using, e.g., transparencies and geometry software.

Write functions to represent transformations.

Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

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. Make sense of Reason abstractly Construct viable Model with Use appropriate problems and and quantitatively. arguments and mathematics. tools strategically. persevere in solving critique the them. reasoning of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are Given a variety of Characteristics of transformations able to: transformations (translations, (translations, rotations, reflections, Accurately perform rotations, reflections, and and dilations), dilations, rotations, dilations), Methods for representing reflections, and translations Represent the transformations transformations, on objects in the coordinate in the plane using a variety of plane with and without methods (e.g., technology, Characteristics of functions, technology, transparencies, semi- Conventions of functions with Communicate the results of transparent mirrors (MIRAs), mapping notation. performing transformations patty paper, compass), on objects and their Describe transformations as corresponding coordinates functions that take points in in the coordinate plane, 8Version 3 2015-2016 the plane as inputs and give including when the other points as outputs, transformation preserves explain why this satisfies the distance and angle, definition of a function, and Use the language and adapt function notation to that notation of functions as of a mapping [e.g. F(x,y) → mappings to describe F(x+a, y+b)], transformations. Compare transformations that Mapping one point to preserve distance and angle to another through a series of those that do not. transformations can be recorded as a function, Some transformations (translations, rotations, and reflections) preserve distance and angle measure, and the image is then congruent to the pre-image, while dilations preserve angle but not distance, and the pre-image is similar to the image, Distortions, such as only a horizontal stretch, preserve neither.

Grade Level/ Course (HS): Geometry Unit 1

Standard: G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Domain: Cluster: Experiment with transformations in the plane. Congruence

9Version 3 2015-2016 Quarter 1: Quarter 2: Quarter 4:

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and/or reflections that carry it onto itself.

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. Make sense of Reason abstractly and Construct viable Model with mathematics. Use problems and quantitatively. arguments and appropriate persevere in solving critique the tools them. reasoning of strategically. others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are able Given a collection of figures Characteristics of transformations to: that include rectangles, (translations, rotations, Accurately perform dilations, parallelograms, trapezoids, or reflections, and dilations), rotations, reflections, and regular polygons, Characteristics of rectangles, translations on objects in the Identify which figures that parallelograms, trapezoids, and coordinate plane with and have rotations or reflections regular polygons. without technology, that carry the figure onto Communicate the results of itself, 10Version 3 2015-2016 Perform and communicate performing transformations rotations and reflections that on objects and their map the object to itself, corresponding coordinates in Distinguish these the coordinate plane. transformations from those Mapping one point to another which do not carry the object through a series of back to itself, transformations can be Describe the relationship of recorded as a function, these findings to symmetry. Since rotations and reflections preserve distance and angle measure, the image is then congruent.

Grade Level/ Course: Geometry Unit 1

Standard: G.CO.4 Develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments.

Domain: Cluster: Experiment with transformations in a plane. Congruence

Quarter 1: Quarter 2: Quarter 3:

Recall definitions of angles, circles, perpendicular and parallel lines and line segments.

Develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments.

From Appendix A: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions 11Version 3 2015-2016 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. Make sense of Reason abstractly Construct viable arguments Model with Use appropriate tools problems and and quantitatively. and critique the reasoning of mathematics. strategically. persevere in solving others. them.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are able Use geometric terminology Characteristics of transformations to: (angles, circles, (translations, rotations, reflections, Accurately perform perpendicular lines, parallel and dilations), rotations, reflections, and lines, and line segments) to Properties of a mathematical translations on objects with describe the series of steps definition, i.e., the smallest and without technology, necessary to produce a amount of information and Communicate the results of rotation, reflection, or properties that are enough to performing transformations translation, determine the concept. (Note: may on objects, Use these descriptions to not include all information related Use known and developed communicate precise to concept). definitions and logical definitions of rotation, connections to develop new reflection, and translation. definitions. Geometric definitions are developed from a few undefined notions by a logical sequence of connections that lead to a precise definition, A precise definition should allow for the inclusion of all examples of the concept and require the exclusion of all non-examples.

12Version 3 2015-2016 Grade Level/ Course: Geometry Unit 1

Standard: G.CO.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.

Domain: Cluster: Experiment with transformations in the plane. Congruence

Quarter 1: Quarter 2: Quarter 4:

Given a geometric figure and a rotation, reflection or translation, draw the transformed figure using, e.g. graph paper, tracing paper or geometry software.

Draw a transformed figure and specify the sequence of transformations that were used to carry the given figure onto the other.

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.

13Version 3 2015-2016 Make sense of Reason abstractly Construct viable Model with Use appropriate problems and and quantitatively. arguments and mathematics. tools strategically. persevere in solving critique the reasoning them. of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are Given a geometric figure, Characteristics of transformations able to: Produce the image of the (translations, rotations, reflections, Accurately perform figure under a rotation, and dilations), rotations, reflections, and reflection, or translation using Techniques for producing images translations on objects graph paper, tracing paper, or under transformations using graph using graph paper, tracing geometry software, paper, tracing paper, or geometry paper, or geometry Describe and justify the software. software, sequence of transformations Communicate the results of that will carry a given figure performing transformations onto another. on objects. The same transformation may be produced using a variety of tools, but the geometric sequence of steps that describe the transformation is consistent, Any distance preserving transformation is a combination of rotations, reflections, and translations.

Grade Level/ Course: Geometry Unit 1

Standard: 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.

Domain: Cluster: Understand congruence in terms of rigid motions Congruence

Quarter 1: Quarter 2: Quarter 3: Identify corresponding angles and sides of two triangles.

14Version 3 2015-2016 Identify corresponding pairs of angles and sides of congruent triangles after rigid motions.

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if corresponding pairs of sides and corresponding pairs of angles are congruent.

Use the definition of congruence in terms of rigid motions to show that if the corresponding pairs of sides and corresponding pairs of angles of two triangles are congruent then the two triangles are congruent.

Justify congruency of two triangles using transformations.

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 Make sense of Reason abstractly Construct viable arguments Model with Use problems and and quantitatively. and critique the reasoning of mathematics. appropriate persevere in solving others. tools them. strategicall

15Version 3 2015-2016 y.

Evidence of Student Vocabulary Knowledge Attainment/Assessment Students: If and only if Students know: Given a triangle and its image under Characteristics of translations, a sequence of rigid motions rotations, and reflections including (translations, reflections, and the definition of congruence, translations), Techniques for producing images Verify that corresponding sides and under transformations, corresponding angles are congruent. Geometric terminology which Given two triangles that have the describes the series of steps same side lengths and angle necessary to produce a rotation, measures, - Find a sequence of rigid reflection, or translation. motions that will map one onto the other.

Grade Level/ Course: Geometry Unit 1 16Version 3 2015-2016 Standard: G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS) follow from the definition of congruence in terms of rigid motions.

Domain: Cluster: Understand congruence in terms of rigid motions Congruence

Quarter 1: Quarter 2: Quarter 4:

Informally use rigid motions to take angles to angles and segments to segments (from 8th grade).

Formally use dynamic geometry software or straightedge and compass to take angles to angles and segments to segments.

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).

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

17Version 3 2015-2016 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. Make sense of Reason abstractly Construct viable Model with Use appropriate tools problems and and quantitatively. arguments and mathematics. strategically. persevere in solving critique the them. reasoning of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Triangle congruence Students know: Students understand/are able to: Use rigid motions and the ASA Basic properties of rigid motions basic properties of rigid (that they preserve distance and Use logical reasoning to motions (that they preserve SAS angle), connect geometric ideas to distance and angle), which are SSS Methods for presenting logical justify other results, assumed without proof to reasoning using assumed Perform rigid motions of establish that the usual understandings to justify subsequent geometric figures, triangle congruence criteria results. make sense and can then be Determine whether two used to prove other theorems. plane figures are congruent by showing whether they coincide when superimposed by means of a sequence of rigid motions (translation, reflection, or rotation), Identify two triangles as congruent if the lengths of corresponding sides are equal (SSS criterion), if the lengths of two pairs of corresponding sides and the measures of the corresponding angles between them are equal (SAS criterion), or if two pairs of corresponding angles are congruent and the lengths of the corresponding sides between them are equal

18Version 3 2015-2016 (ASA criterion), Apply the SSS, SAS, and ASA criteria to verify whether or not two triangles are congruent. It is beneficial to have minimal sets of requirements to justify geometric results (e.g., use ASA, SAS, or SSS instead of all sides and all angles for congruence).

Grade Level/ Course: Geometry Unit 1

Standard: G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

Domain: Cluster: Prove Geometric Theorems Congruence

Quarter 1: Quarter 2: Quarter 3: Identify and use properties of:  Vertical angles  Parallel lines with transversals  All angle relationships  Corresponding angles  Alternate interior angles  Perpendicular bisector  Equidistant from endpoint

19Version 3 2015-2016 Prove vertical angles are congruent.

Prove corresponding angles are congruent when two parallel lines are cut by a transversal and converse.

Prove alternate interior angles are congruent when two parallel lines are cut by a transversal and converse.

Prove points are on a perpendicular bisector of a line segment are exactly equidistant from the segments endpoint.

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. Make sense of Reason abstractly Construct viable Model with Use appropriate problems and and quantitatively. arguments and critique the mathematics. tools strategically. persevere in solving reasoning of others. them.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Prove Students know: Students understand/are able to: Make, explain, and justify (or Transversal Requirements for a mathematical refute) conjectures about proof, Communicate logical Alternate interior 20Version 3 2015-2016 geometric relationships with angles Techniques for presenting a reasoning in a systematic way and without technology, Corresponding proof of geometric theorems. to present a mathematical Explain the requirements of a angles proof of geometric theorems, mathematical proof, Generate a conjecture about Present a complete geometric relationships that mathematical proof of calls for proof. geometry theorems including Proof is necessary to establish the following: vertical angles that a conjecture about a are congruent; when a relationship in mathematics is transversal crosses parallel always true, and may also lines, alternate interior angles provide insight into the are congruent and mathematics being addressed. corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints, Critique proposed proofs made by others.

Grade Level/ Course: Geometry Unit 1

Standard: 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.

Domain: Cluster: Prove Geometric Theorems Congruence

Quarter 1: Quarter 2: Quarter 4:

Identify the hypothesis and conclusion of a theorem.

Design an argument to prove theorems about triangles.

Analyze components of the theorem.

Prove theorems about triangles.

From Appendix A: 21Version 3 2015-2016 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. Implementations of G.CO.10 may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for G.C.3 in Unit 5. Make sense of Reason abstractly Construct viable Model with Use appropriate problems and and quantitatively. arguments and mathematics. tools strategically. persevere in solving critique the reasoning them. of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are able Make, explain, and justify Requirements for a mathematical to: (or refute) conjectures about proof, Communicate logical geometric relationships with Techniques for presenting a proof reasoning in a systematic and without technology, of geometric theorems. way to present a Explain the requirements of mathematical proof of a mathematical proof, geometric theorems, Present a complete Generate a conjecture about mathematical proof of geometric relationships that geometry theorems about calls for proof. triangles, including the Proof is necessary to following: measures of establish that a conjecture interior angles of a triangle about a relationship in sum to 180°; base angles of mathematics is always true isosceles triangles are and may also provide insight 22Version 3 2015-2016 congruent; the segment into the mathematics being joining midpoints of two addressed. sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point, Critique proposed proofs made by others.

Grade Level/ Course: Geometry Unit 1

Standard: 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.

Domain: Cluster: Prove Geometric Theorems Congruence

Quarter 1: Quarter 2: Quarter 4:

Classify types of quadrilaterals.

Explain theorems for parallelograms and relate to figure.

Use the principle that corresponding parts of congruent triangles are congruent to solve problems.

Use properties of special quadrilaterals in a proof.

Make sense of Reason abstractly Construct viable Model with Use appropriate problems and and quantitatively. arguments and mathematics. tools strategically. persevere in solving critique the reasoning them. of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are Make, explain, and justify (or Requirements for a mathematical able to: 23Version 3 2015-2016 refute) conjectures about proof, Communicate logical geometric relationships with Techniques for presenting a proof reasoning in a systematic and without technology, of geometric theorems. way to present a Explain the requirements of a mathematical proof of mathematical proof, geometric theorems, Present a complete Generate a conjecture about mathematical proof of geometric relationships that geometry theorems about calls for proof. parallelograms, including the Proof is necessary to following: opposite sides are establish that a conjecture congruent, opposite angles are about a relationship in congruent, the diagonals of a mathematics is always true parallelogram bisect each and may also provide other, and conversely, insight into the mathematics rectangles are parallelograms being addressed. with congruent diagonals, Critique proposed proofs made by others.

Grade Level/ Course: Geometry Unit 1 Standard: 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. Domain: Cluster: Prove Geometric Theorems Congruence Quarter 1: Quarter 2: Quarter 3: Explain the construction of geometric figures using Perform geometric Perform geometric constructions including: Copying a segment; copying an angle; bisecting a segment; bisecting an a variety of tools and constructions including: angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line methods Copying a segment; parallel to a given line through a point not on the line, using a variety of tools and methods (compass and straightedge, copying an angle; string, reflective devices, paper folding, dynamic geometric software, etc.). Apply the definitions, bisecting a segment; properties and theorems bisecting an angle; about line segments, rays constructing perpendicular and angles to support lines, including the geometric constructions. perpendicular bisector of a line segment; and Apply properties and constructing a line parallel theorems about parallel to a given line through a From Appendix A: Build on prior student experience with simple constructions. Emphasize the ability to formalize and and perpendicular lines to point not on the line, using explain how these constructions result in the desired objects. Some of these constructions are closely related to support constructions. a variety of tools and previous standards and can be introduced in conjunction with them. methods (compass and Perform geometric straightedge, string, constructions including: reflective devices, paper Copying a segment; folding, dynamic copying an angle; geometric software, etc.). 24Version 3 2015-2016 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 From Appendix A: Build point not on the line, using on prior student a variety of tools and experience with simple methods (compass and constructions. Emphasize straightedge, string, the ability to formalize reflective devices, paper and explain how these folding, dynamic constructions result in the geometric software, etc.). desired objects. Some of these constructions are From Appendix A: Build closely related to previous on prior student standards and can be experience with simple introduced in conjunction constructions. Emphasize with them. 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. Make sense of problems Reason abstractly and Construct viable Model with mathematics. Use appropriate tools and persevere in solving quantitatively. arguments and critique strategically. them. the reasoning of others.

Evidence of Student Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Attainment/Assessme nt Students: Students know: Students EEG-CO.12-13. Make and justify Methods for accurately understand/are able to: formal geometric using tools to perform Choose and use constructions with a geometric appropriate variety of tools and constructions, construction tools methods (e.g., including compass and strategically to compass and straightedge, string, perform geometric straightedge, string, reflective devices, constructions, reflective devices, paper folding, and Use logical reasoning paper folding, dynamic dynamic geometric and properties of and geometric software, software, relationships between etc.) including the Methods for justifying geometric figures to following: Copying a a geometric justify geometric segment; copying an construction using constructions. angle; bisecting a geometric properties. segment; bisecting an Limiting oneself to a angle; constructing specific tool or set of perpendicular lines, tools to perform a including the geometric construction perpendicular bisector illuminates important 25Version 3 2015-2016 of a line segment; and mathematical features constructing a line of the object being parallel to a given line constructed, through a point not on Different tools for the line, geometric Compare and contrast constructions may different methods for offer different levels of doing the same precision in the construction, and construction and the identify geometric purpose of the properties that justify construction should steps in the help determine the tool constructions. of choice, Properties of geometric figures can and should be used to verify the correctness of geometric constructions regardless of the construction tool or method used.

Grade Level/ Course: Geometry Unit 1

Standard: G.CO.13 Construct an equilateral triangle, a square and a regular hexagon inscribed in a circle.

Domain: Cluster: Make geometric constructions Congruence

Quarter 1: Quarter 2: Quarter 3:

Make sense of Reason abstractly Construct viable Model with Use appropriate tools problems and and quantitatively. arguments and mathematics. strategically. persevere in solving critique the them. reasoning of others.

26Version 3 2015-2016 Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Construct Students know: Students understand/are able to: Use tools (e.g., compass, Inscribed Properties of regular polygons, straight edge, geometry Characteristics of inscribed figures, Choose and use appropriate software) and geometric construction tools relationships to construct Methods for accurately using tools strategically to perform regular polygons inscribed in to perform geometric constructions. geometric constructions, circles, Communicate with logical Explain and justify the reasoning the series of steps sequence of steps taken to necessary for constructing complete the construction. an inscribed figure and the justification for each step. Properties of geometric figures can and should be used to verify the correctness of geometric constructions regardless of the construction tool or method used.

Grade Level/ Course: Geometry Unit 2

Standard: G.SRT.1a Verify experimentally the properties of dilations given by a center and a scale factor. parallel line, and leaves a line passing through the center unchanged.

Domain: Similarity, Cluster: Understand similarity in terms of similarity transformations Right Triangles, and Trigonometry

Quarter 1: Quarter 2: Quarter 3:

Define image, pre- image, scale factor, 27Version 3 2015-2016 center, and similar figures as they relate to transformations.

Identify a dilation stating its scale factor and center.

Verify experimentally that a dilated image is similar to its pre- image by showing congruent corresponding angles and proportional sides.

Verify experimentally that a dilation takes a line not passing through the center of the dilation to a parallel line by showing the lines are parallel.

Verify experimentally that dilation leaves a line passing through the center of the dilation unchanged by showing that it is the same line. Make sense of Reason abstractly Construct Model with Use appropriate tools problems and and quantitatively. viable mathematics. strategically. persevere in solving arguments them. and critique the reasoning of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Dilations Students know: Students understand/are able Given a center of dilation, a to: Center Methods for finding the length scale factor, and a polygonal of line segments (both in a Accurately create a new image image, Scale factor coordinate plane and through from a center of dilation, a Create a new image by 28Version 3 2015-2016 extending a line segment from measurement), scale factor, and an image, the center of dilation through Dilations may be performed on Accurately find the length of each vertex of the original polygons by drawing lines line segments and ratios of line figure by the scale factor to through the center of dilation segments, find each new vertex, and each vertex of the polygon Communicate with logical Present a convincing argument then marking off a line segment reasoning a conjecture of that line segments created by changed from the original by the generalization from the dilation are parallel to their scale factor. experimental results. pre-images unless they pass through the center of dilation, A dilation uses a center and in which case they remain on line segments through vertex the same line, points to create an image which is similar to the original Find the ratio of the length of image but in a ratio specified the line segment from the by the scale factor, center of dilation to each vertex in the new image and The ratio of the line segment the corresponding segment in formed from the center of the original image and dilation to a vertex in the new compare that ratio to the scale image and the corresponding factor, vertex in the original image is equal to the scale factor. Conjecture a generalization of these results for all dilations.

Grade Level/ Course: Geometry Unit 2

Standard: 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.

Domain: Similarity, Cluster: Understand similarity in terms of similarity transformations Right Triangles, and Trigonometry

Quarter 1: Quarter 2: Quarter 3:

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.

Given two figures, decide if they are similar by using the definition of similarity in terms of 29Version 3 2015-2016 similarity transformations.

Make sense of Reason abstractly Construct Model with Use appropriate tools problems and and quantitatively. viable mathematics. strategically. persevere in solving arguments and them. critique the reasoning of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Similarity Students know: Students understand/are able Given two figures, transformation Properties of rigid motions and to: Determine if they are similar dilations, Apply rigid motion and by demonstrating whether one Definition of similarity in terms of dilation to a figure, figure can be obtained from similarity transformations, Explain and justify whether or the other through a dilation not one figure can be obtained and a combination of Techniques for producing images under a dilation and rigid motions. from another through a translations, reflections, and combination of rigid motion rotations. and dilation. A figure that may be obtained Given a triangle, from another through a Produce a similar triangle dilation and a combination of through a dilation and a translations, reflections, and combination of translations, rotations is similar to the rotations, and reflections, original, Demonstrate that a dilation and When a figure is similar to a combination of translations, another the measures of all reflections, and rotations corresponding angles are maintain the measure of each equal, and all of the angle in the triangles and all corresponding sides are in the corresponding pairs of sides of same proportion. the triangles are proportional.

Grade Level/ Course: Geometry Unit 2

30Version 3 2015-2016 Standard: G.SRT.1b Verify experimentally the properties of dilations given by a center and a scale factor. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Domain: Similarity, Cluster: Understand similarity in terms of similarity transformations Right Triangles, and Trigonometry

Quarter 1: Quarter 2: Quarter 4:

Define image, pre-image, scale factor, center, and similar figures as they relate to transformations.

Identify a dilation stating its scale factor and center.

Explain that the scale factor represents how many times longer or shorter a dilated line segment is than its pre-image.

Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Dilations Students know: Students understand/are able Given a center of dilation, a to: Center Methods for finding the length of scale factor, and a polygonal line segments (both in a coordinate Accurately create a new image image, Scale factor plane and through measurement), from a center of dilation, a Create a new image by Dilations may be performed on scale factor, and an image, extending a line segment from polygons by drawing lines through Accurately find the length of the center of dilation through the center of dilation and each line segments and ratios of line each vertex of the original vertex of the polygon then segments, figure by the scale factor to marking off a line segment find each new vertex, Communicate with logical changed from the original by the reasoning a conjecture of Present a convincing scale factor. generalization from argument that line segments experimental results. created by the dilation are parallel to their pre-images A dilation uses a center and unless they pass through the line segments through vertex center of dilation, in which points to create an image case they remain on the same which is similar to the original

31Version 3 2015-2016 line, image but in a ratio specified Find the ratio of the length of by the scale factor, the line segment from the The ratio of the line segment center of dilation to each formed from the center of vertex in the new image and dilation to a vertex in the new the corresponding segment in image and the corresponding the original image and vertex in the original image is compare that ratio to the scale equal to the scale factor. factor, Conjecture a generalization of these results for all dilations.

Grade Level/ Course: Geometry Unit 2

Standard: G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Domain: Similarity, Cluster: Understand similarity in terms of similarity transformations Right Triangles, and Trigonometry

Quarter 1: Quarter 2: Quarter 3:

Recall the properties of similarity transformations.

Establish the AA criterion for similarity of triangles by extending the properties of similarity transformations to the general case of any two similar triangles.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are able criterion Given two triangles, The sum of the measures of the to: Explain why if the measures angles of a triangle is 180 degrees, Explain and justify why the of two angles from one Properties of rigid motions and third pair of corresponding triangle are equal to the dilations. angles of two triangles must measures of two angles from be equal if each of the other another triangle, then two corresponding pairs are measures of the third angles equal, must be equal to each other, Justify through the use of Use this established fact and rigid motion and dilation the properties of a similarity why corresponding sides of transformation to demonstrate triangles are in the same that the Angle-Angle () proportion if the measures of criterion for similar triangles two pairs of corresponding is sufficient. angles are equal. It is beneficial to have minimal sets of requirements to justify geometric results (i.e., use instead of all sides proportional and all angles congruent for similarity), If the measures of two angles of one triangle are equal to the measures of two angles of another triangle, then the triangles are similar and the similarity of the triangles can be justified through similarity transformations.

Grade Level/ Course: Geometry Unit 2

Standard: 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.

Domain: Similarity, Cluster: Prove theorems involving similarity. Right Triangles, and Trigonometry

Quarter 1: Quarter 2: Quarter 4:

33Version 3 2015-2016

Recall postulates, theorems, and definitions to prove theorems about triangles.

Prove theorems involving similarity 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.)

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Theorem Students know: Students understand/are able to: Given a triangle and a line parallel Properties of similar triangles and methods Apply properties of similar to one of the sides, of showing that triangles are similar, triangles to justify relationships of Prove the other two sides are Properties of equality (Table 4), the sides of a triangle, divided proportionally by using , Explain and justify that a line similarity properties, previously Previously proven theorems including those concerning parallel lines. passing through the triangle proven theorems and properties of divides the sides proportionally, if equality (Table 4). and only if, the line is parallel to a side of the triangle, Given a triangle with two of the Justify the Pythagorean Theorem sides divided proportionally, through the use of similar Prove the line dividing the sides is triangles. parallel to the third side of the Triangle similarity may be used to triangle. justify theorems involving the connection between the Given a right triangle, proportion of sides and whether Use similar triangles and properties or not the line dividing the sides of equality (Table 4) to prove the is parallel to the other side of the Pythagorean Theorem. triangle, Through the use of similar triangles, a right triangle may be divided into two right triangles which are similar to the original right triangle; therefore the corresponding sides must be proportional and may be used to prove the Pythagorean Theorem, The same theorem may be proven in many different ways (i.e., the

34Version 3 2015-2016 Pythagorean Theorem).

Grade Level/ Course: Geometry Unit 2

Standard: G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Domain: Similarity, Cluster: Prove theorems involving similarity Right Triangles, and Trigonometry

Quarter 1: Quarter 2: Quarter 3:

Recall congruence and similarity criteria for triangles.

Use congruency and similarity theorems for triangles to solve problems.

Use congruency and similarity theorems for triangles to prove relationships in geometric figures.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Congruence and Students know: Students understand/are able Given a contextual situation similarity criteria Criteria for congruent (SAS, ASA, to: involving triangles, for triangles S, SSS) and similar () triangles and Accurately solve a contextual Determine solutions to transformation criteria, problem by applying the problems by applying Techniques to apply criteria of criteria of congruent and 35Version 3 2015-2016 congruence and similarity congruent and similar triangles for similar triangles, criteria for triangles to assist in solving a contextual problem. Provide justification for the solving the problem, solution process, Justify solutions and critique Analyze the solutions of the solutions of others. others and explain why their solutions are valid or invalid, Given a geometric figure, Justify relationships in Establish and justify geometric figures through the relationships in the figure use of congruent and similar through the use of congruence triangles. and similarity criteria for Congruence and similarity triangles. criteria for triangles may be used to find solutions of contextual problems, Relationships in geometric figures may be proven through the use of congruent and similar triangles.

Grade Level/ Course: Geometry Unit 2

Standard: 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.

Domain: Similarity, Cluster: Define trigonometric ratios and solve problems involving right triangles Right Triangles, and Trigonometry

Quarter 1: Quarter 2: Quarter 3:

Names the sides of right triangles as related to an acute angle.

Recognize that if two right triangles have a pair of acute, congruent angles that the triangles are similar.

Compare common ratios for similar right triangles and develop a relationship between the ratio and 36Version 3 2015-2016 the acute angle leading to the trigonometry ratios.

Make sense of Reason abstractly Construct viable arguments Model with Use appropriate problems and and quantitatively. and critique the reasoning of mathematics. tools strategically. persevere in solving others. them.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Side ratios Students know: Students understand/are Given a collection of right Techniques to construct similar able to: triangles, Trigonometric ratios triangles, Accurately find the side Construct similar right Properties of similar triangles. ratios of triangles, triangles of various sizes for Explain and justify each right triangle given, relationships between the Compare the ratios of the sides side ratios of a right triangle of the original triangles to the and the angles of a right ratios of the sides of the triangle. similar triangles, The ratios of the sides of Communicate observations right triangles are dependent made about changes (or no on the size of the angles of change) to such ratios as the the triangle. length of the side opposite an angle to the hypotenuse, or the side opposite the angle to the side adjacent, as the size of the angle changes or in the case of similar triangles, remains the same, Summarize these observations by defining the six trigonometric ratios.

Grade Level/ Course: Geometry Unit 2

37Version 3 2015-2016 Standard: G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.

Domain: Similarity, Cluster: Define trigonometric ratios and solve problems involving right triangles Right Triangles, and Trigonometry

Quarter 1: Quarter 2: Quarter 3:

Use the relationship between the sine and cosine of complementary angles.

Explain how the sine and cosine of complementary angles are related to each other.

Make sense of Reason abstractly Construct viable Model with Use appropriate problems and and quantitatively. arguments and critique mathematics. tools strategically. persevere in solving the reasoning of others. them.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Sine Students know: Students understand/are able Given a right triangle, to: Cosine Methods for finding sine and cosine Explain why the two smallest ratios in a right triangle (e.g., use of Accurately solve a angles must be complements, Complementary triangle properties: similarity; contextual problem by using angles Compare the side ratios of Pythagorean Theorem; isosceles the sine and cosine ratios, opposite/hypotenuse and and equilateral characteristics for Justify solutions and discuss adjacent/hypotenuse for each 45-45-90 and 30-60-90 triangles other possible solutions of these angles and discuss and technology for others). through the use of conclusions. complementary angles and the sine or cosine ratios. Given a contextual situation The sine of an angle is equal involving right triangles, to the cosine of the complement of the angle, 38Version 3 2015-2016 Compare solutions to the Switching between using a situation using the sine of the given angle or its given angle and the cosine of complement and between its complement. sine or cosine ratios may be used when solving contextual problems.

Grade Level/ Course: Geometry Unit 2

Standard: G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Domain: Similarity, Cluster: Define trigonometric ratios and solve problems involving right triangles Right Triangles, and Trigonometry

Quarter 1: Quarter 2: Quarter 4:

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.

Apply right triangle trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Make sense of Reason abstractly Construct viable Model with Use appropriate problems and and arguments and mathematics. tools strategically. persevere in solving quantitatively. critique the reasoning them. of others.

39Version 3 2015-2016 Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are able Given a contextual situation Methods of using the to: involving right triangles, trigonometric ratios to solve for Create an accurate diagram to Create a drawing to model the sides or angles in a right model a contextual situation situation, triangle, involving right triangles and Find the missing sides and The Pythagorean Theorem and use it to solve the right angles using trigonometric its use in solving for unknown triangles, ratios and the Pythagorean parts of a right triangle. Identify the trigonometric Theorem, ratio useful to solve for a Use the above information to particular unknown part of a interpret results in the context right triangle and use that of the situation. ratio to accurately solve for the unknown part. use the Pythagorean Theorem to find unknown sides of a right triangle explain the solution in terms of the given contextual situation Unknown parts of right triangles may be found through the use of trigonometric ratios, Pythagorean Theorem, or a combination of both, Right triangles may be used to model and solve contextual situations.

Grade Level/ Course: Geometry Unit 2

Standard: G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* (*Modeling Standard)

Domain: Modeling Cluster: Apply geometric concepts in modeling situations with Geometry

Quarter 1: Quarter 2: Quarter 3:

Use measures and properties of geometric shapes to describe real world objects.

Given a real world object, classify the object as a known 40Version 3 2015-2016 geometric shape – use this to solve problems in context.

From Appendix A: Focus on situations well modeled by trigonometric ratios for acute angles.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are Given a real-world object, Techniques to find measures of able to: Select an appropriate geometric shapes, Model a real-world object geometric shape to model the Properties of geometric shapes. through the use of a object, geometric shape, Provide a description of the Justify the model by object through the measures connecting its measures and and properties of the properties to the object. geometric shape which is Geometric shapes may be modeling the object, used to model real-world Explain and justify the model objects, which was selected. Attributes of geometric figures help us identify the figures and find their measures, therefore matching these figures to real world objects allows the application of geometric techniques to real world problems.

41Version 3 2015-2016 Grade Level/ Course: Geometry Unit 2

Standard: G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* (*Modeling Standard)

Domain: Modeling Cluster: Apply geometric concepts in modeling situations with Geometry

Quarter 1: Quarter 2: Quarter 3:

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Density Students know: Students understand/are Given a contextual situation Geometric concepts of area and able to: involving density, volume, Accurately model a Model the situation by Properties of rates, situation involving density, creating an average per unit of Modeling techniques. Justify how the model is an area or unit of volume, accurate representation of Generate questions raised by the given situation.

42Version 3 2015-2016 the model and defend answers Situations involving density they produce to the generated may be modeled through a questions (e.g., should representation of a population density be given concentration per unit of per square mile or per acre? area or unit of volume. What insights might one yield over the other?), Explain and justify the model in terms of the original context.

Grade Level/ Course: Geometry Unit 2

Standard: 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 typographic grid systems based on ratios).*(*Modeling Standard)

Domain: Modeling Cluster: Apply geometric concepts in modeling situations with Geometry

Quarter 1: Quarter 2: Quarter 3:

Describe a typographical grid system.

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on

43Version 3 2015-2016 ratios).

From Appendix A: Focus on situations well modeled by trigonometric ratios for acute angles.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Geometric methods Students know: Students understand/are Given a contextual situation Properties of geometric shapes, able to: involving design problems, Design problems Characteristics of a mathematical Accurately model and solve Create a geometric method to model. a design problem, model the situation and solve Justify how their model is the problem, an accurate representation Explain and justify the model of the given situation. which was created to solve Design problems may be the problem. modeled with geometric methods, Geometric models may have physical constraints, Models represent the mathematical core of a situation without extraneous information, for the benefit in a problem solving situation.

44Version 3 2015-2016 Grade Level/ Course: Geometry Unit 2

Standard: G.SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Domain: Similarity, right triangles, and Cluster: Apply trigonometry to general triangles trigonometry

Quarter 1: Quarter 2: Quarter 3:

Recall right triangle trigonometry to solve mathematical problems.

Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Make sense of Reason abstractly and Construct viable Model with Use appropriate problems and quantitatively. arguments and mathematics. tools strategically. persevere in solving critique the them. reasoning of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Auxiliary line Students know: Students understand/are able Given any triangle, to: Vertex The auxiliary line drawn from the Derive the given formula for vertex perpendicular to the opposite Properly label a triangle the area using the lengths of side forms an altitude of the according to convention. two sides of the triangle and triangle. Perform algebraic the included angle. The formula for the area of a manipulations. triangle (A = 1/2 bh). Given the lengths of the Properties of the sine ratio. sides and included angle of any triangle the area can be determined. 45Version 3 2015-2016 There is more than one formula to find the area of a triangle.

Grade Level/ Course: Geometry Unit 2

Standard: G.SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.

Domain: Similarity, Right Triangles, and Cluster: Apply trigonometry to general triangles Trigonometry

Quarter 1: Quarter 2: Quarter 3:

Use the Laws of Sines and Cosines this to find missing angles or side length measurements.

Prove the Law of Sines

Prove the Law of Cosines

Recognize when the Law of Sines or Law of Cosines can be applied to a problem and solve problems in context using them.

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.

Make sense of problems Reason abstractly and Construct viable Model with Use appropriate and persevere in solving quantitatively. arguments and mathematics. tools strategically. them. critique the reasoning of others.

46Version 3 2015-2016 Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Law of Sines Students know: Students understand/are able Given any triangle, to: Law of Cosines The auxiliary line drawn from the Derive the Law of Sines and vertex perpendicular to the Properly label a triangle the Law of Cosines. opposite side forms an altitude of according to convention. the triangle. Perform algebraic Given any triangle, Properties of the Sine and Cosine manipulations. Use the Law of Sines or the ratios. The given information will Law of Cosines to find Pythagorean Theorem. determine whether it is unknown lengths of sides and Pythagorean Identity. appropriate to use the Law of measures of angles. Sines or the Law of Cosines. Proof is necessary to establish that a conjecture about a relationship in mathematics is always true and may provide insight into the mathematics being addressed.

Grade Level/ Course: Geometry Unit 2

Standard: G.SRT.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).

Domain: Similarity, Cluster: Apply trigonometry to general triangles Right Triangles, and Trigonometry

Quarter 1: Quarter 2: Quarter 3:

Determine from given measurements in right and non-right triangles whether it is appropriate to use the Law of Sines or Cosines.

Apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right 47Version 3 2015-2016 triangles (e.g., surveying problems, resultant forces).

From Appendix A: With respect to the general case of the Laws of Sines and Cosines, the definition of sine and cosine must be extended to obtuse angles.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Law of Sines Students know: Students understand/are able Given a contextual situation, to: Law of Cosines The Laws of Sines and Cosines Choose the appropriate law can apply to any triangle, right or Label triangles in context and and apply it to determine the non-right. by convention. measures of unknown Laws of Sines and Cosines. Perform algebraic quantities. Vector quantities can represent manipulations. lengths of sides and angles in a Find inverse sine and cosine triangle. values. Values of the sin(90°) and Proven laws allow us to solve cos(90°). problems in contextual situations.

48Version 3 2015-2016 Grade Level/ Course: Geometry Unit 3

Standard: G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*(*Modeling Standard)

Domain: Geometric Cluster: Explain volume formulas and use them to solve problems. Measurement and Dimension

Quarter 1: Quarter 2: Quarter 4:

Make sense of Reason abstractly Construct viable Model with Use appropriate problems and and arguments and critique mathematics. tools strategically. persevere in solving quantitatively. the reasoning of others. them.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are Given a contextual situation Volume formulas for cylinders, able to: that requires finding the pyramids, cones, and spheres, Accurately model a volume of a cylinder, Techniques for modeling 3-D contextual situation with a pyramid, cone, or sphere as cylinder, pyramid, cone, part of its solution, objects with shapes or 2-D drawings, and for using these sphere, or a 2-D drawing, Use an appropriate shape or models to identify specific values Use the model or drawing to 2-D drawing to model the for use in volume formulas. find values needed for use situation, in the volume formula, Solve using the appropriate Accurately find a solution to formula, the given situation, and Justify and explain the explain the solution in the solution and solution path in context of the situation. the context of the given A contextual situation situation. involving cylinders, pyramids, cones, and spheres may be modeled by 49Version 3 2015-2016 shapes or 2-D drawings, and the model may provide insight into the solution of the problem, Formulas are useful for efficiency when many problems of the same type need to be solved.

Grade Level/ Course (high school): Geometry Unit 3

Standard: G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two- dimensional objects.

Domain: Geometric Cluster: Visualize relationships between two-dimensional and three-dimensional objects Measurement & Dimension

Quarter 1: Quarter 2: Quarter 4:

Make sense of Reason Construct viable Model with Use appropriate tools problems and abstractly and arguments and critique mathematics. strategically. persevere in solving quantitatively. the reasoning of others. them.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Two-dimensional Students know: Students understand/are Given 3-D objects, cross-sections Characteristics of 2-D and 3-D able to: Conjecture about the Conjecture about the 50Version 3 2015-2016 characteristics of geometric Two-dimensional geometric objects, characteristics of geometric shapes formed if a cross- objects Techniques for finding a cross- shapes formed from taking a section of a 3-D shape is Three-dimensional section of a 3-D object, cross-section of a 3-D taken, objects shape, or rotating a 2-D Techniques for rotating a 2-D shape about a line, Take 2-D cross-sections at Rotations object about a line. different angles of cut, Accurately determine the Explain the shape formed by geometric shapes formed taking 2-D cross-sections, from taking a cross-section of a 3-D shape, or rotating a Compare and contrast the 2-D shape about a line. figures formed when the angle of the cut changes. 3-D objects can be created from 2-D plane figures through transformations Given 2-D objects, such as rotations, Conjecture about the Cross-sections of 3-D characteristics of geometric objects can be formed in a shapes formed from rotating a variety of ways, depending 2-D shape about a line, on the angle of the cut with Rotate the object about given the base of the object. lines, Explain the 3-D objects formed if the 2-D object is rotated about a line.

Grade Level/ Course: Geometry Unit 3

Standard: G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*(*Modeling Standard)

Domain: Modeling Cluster: Apply geometric concepts in modeling situations with Geometry

51Version 3 2015-2016 Quarter 1: Quarter 2: Quarter 4:

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are able Given a real-world object, Techniques to find measures of to: Select an appropriate geometric shapes, Model a real-world object geometric shape to model the Properties of geometric shapes. through the use of a object, geometric shape, Provide a description of the Justify the model by object through the measures connecting its measures and and properties of the properties to the object. geometric shape which is Geometric shapes may be modeling the object, used to model real-world Explain and justify the model objects, which was selected. Attributes of geometric figures help us identify the figures and find their measures, therefore matching these figures to real world objects allows the application of geometric techniques to real world problems.

52Version 3 2015-2016 Grade Level/ Course (high School): Geometry Unit 4

Standard: G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

Domain: Cluster: Use coordinates to prove simple geometric theorems algebraically. Expressing Geometric Properties With Equations

Quarter 1: Quarter 2: Quarter 3: Recall previous understandings of coordinate geometry (including, but not limited to: distance, midpoint and slope formula, equation of a line, definitions of parallel and perpendicular lines, etc.)

Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). e.g., derive the equation of a line through 2 points using similar right triangles.

From Appendix A: This unit has a close connection with the next unit. For example, a curriculum might merge G.GPE.1 and the unit 5 treatment of G.GPE.4 with the standards in this unit. Reasoning with triangles in this unit is limited to right triangles. Make sense of Reason abstractly Construct viable arguments Model with Use appropriate problems and and quantitatively. and critique the reasoning mathematics. tools strategically. persevere in solving of others. them.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Simple geometric Students know: Students understand/are able Given coordinates and theorems Relationships (e.g. distance, slope to: geometric theorems and of line) between sets of points, Accurately determine what statements defined on a Properties of geometric shapes, information is needed to coordinate system, prove or disprove a statement Use the coordinate system Coordinate graphing rules and or theorem, techniques, and logical reasoning to Accurately find the needed justify (or deny) the statement 53Version 3 2015-2016 or theorem, and to critique Techniques for presenting a proof information and explain and arguments presented by of geometric theorems. justify conclusions, others. Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.

Grade Level/ Course: Geometry Unit 4

Standard: G.GPE.7 Use coordinates to compute perimeters of polygons and area of triangles and rectangles, e.g., using the distance formula.*(*Modeling Standard)

Domain: Cluster: Use coordinates to prove simple geometric theorems algebraically Expressing Geometric Properties with Equations

Quarter 1: Quarter 2: Quarter 3:

Use the coordinates of the vertices of a polygon to find the necessary dimensions for finding the perimeter (i.e., the distance between vertices).

Use the coordinates of the vertices of a triangle to find the necessary dimensions (base, height) for finding the area (i.e., the distance between vertices by counting, distance formula, Pythagorean Theorem, etc.).

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).

Formulate a model of figures in contextual problems to compute area and/or perimeter.

From Appendix A: G.GPE.7 provides practice with the distance formula and its connection with the Pythagorean theorem. Make sense of Reason abstractly Construct Model with Use appropriate tools problems and and quantitatively. viable mathematics. strategically. persevere in solving arguments and them. critique the reasoning of others.

54Version 3 2015-2016 Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are able Given a contextual situation The distance formula and its to: that requires the perimeter applications, Create geometric figures on a and/or area of a polygon as coordinate system from a part of its solution, Techniques for coordinate graphing. contextual situation, Find the solution to the Accurately find the perimeter situation through the use of of polygons and the area of coordinates and the distance triangles and rectangles from formula as appropriate, the coordinates of the shapes, through modeling the situation in a Cartesian Explain and justify solutions coordinate system and in the original context of the explain and justify the situation. solution. Contextual situations may be modeled in a Cartesian coordinate system, Coordinate modeling is frequently useful to visualize a situation and to aid in solving contextual problems.

Grade Level/ Course (high school): Geometry Unit 4

Standard: G.GPE.2 Derive the equation of a parabola given a focus and directrix.

Domain: Cluster: Translate between the geometric description and the equation for a conic section. Expressing Geometric Properties with Equations

Quarter 1: Quarter 2: Quarter 3:

55Version 3 2015-2016 Make sense of Reason abstractly Construct viable Model with Use appropriate problems and and quantitatively. arguments and mathematics. tools strategically. persevere in solving critique the them. reasoning of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Parabola Students know: Students understand/are able Given the focus and directrix to: Focus The distance formula for points of a parabola, in the coordinate plane and Create and justify a formula Use the distance formula to Directrix methods of finding the distance through algebraic write an equation for all between a point and a line, manipulation after setting the points which are equidistant Properties of equality (Table 4). distances between two points from a given focus and and between a point and a line directory, equal to each other. Use logical repeated The points that satisfy the reasoning to generalize to a equation of a parabola general point and general represent all of the points that directrix, developing the are equidistant from a given formula for the equation of a point and a given line. parabola, Compare and contrast parabola equations for horizontal and vertical directrices.

Grade Level/ Course: Geometry Unit 4

56Version 3 2015-2016 Standard: G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Domain: Cluster: Use coordinates to prove simple geometric theorems algebraically Expressing Geometric Properties with Equations

Quarter 1: Quarter 2: Quarter 3: Recognize that slopes of parallel lines are equal.

Recognize that slopes of perpendicular lines are opposite reciprocals (i.e, the slopes of perpendicular lines have a product of -1)

Find the equation of a line parallel to a given line that passes through a given point.

Find the equation of a line perpendicular to a given line that passes through a given point.

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.

From Appendix A: Relate work 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. Make sense of Reason abstractly Construct Model with Use appropriate tools problems and and quantitatively. viable mathematics. strategically. persevere in solving arguments and them. critique the reasoning of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Slope criteria for Students know: Students understand/are Given a line, parallel and Techniques to find the slope of a able to: Create lines parallel to the perpendicular lines line, Explain and justify given line and compare the Key features needed to solve conclusions reached slopes of parallel lines by geometric problems, regarding the slopes of examining the rise/run ratio of parallel and perpendicular each line, Techniques for presenting a proof lines, of geometric theorems. Create lines perpendicular to Apply slope criteria for the given line by rotating the parallel and perpendicular line 90 degrees and compare lines to accurately find the the slopes by examining the solutions of geometric rise/run ratio of each line, problems and justify the Use understandings of similar solutions, triangles and logical Communicate logical 57Version 3 2015-2016 reasoning to prove that reasoning in a systematic parallel lines have equal way to present a slopes and the slopes of mathematical proof of perpendicular lines are geometric theorems. negative reciprocals. Modeling geometric figures or relationships on Given a geometric problem a coordinate graph assists involving parallel or in determining truth of a perpendicular lines, statement or theorem, Apply the appropriate slope Geometric theorems may criteria to solve the problem be proven or disproven by and justify the solution examining the properties of including finding equations of the geometric shapes in the lines parallel or perpendicular theorem through the use of to a given line. appropriate algebraic techniques.

Grade Level/ Course (HS): Geometry Unit 5

Standard: G.C.1 Prove that all circles are similar.

Domain: Circles Cluster: Understand and apply theorems about circles

Quarter 1: Quarter 2: Quarter 3:

Make sense of Reason abstractly Construct viable Model with Use appropriate tools problems and and arguments and mathematics. strategically. persevere in solving quantitatively. critique the them. reasoning of others.

58Version 3 2015-2016 Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are Given a collection of circles, Techniques to create dilations, able to: Show that in each case there Similar figures have the same Accurately create circles by exists a transformation that shape, making dilations of a given consists of a dilation and a circle, combination of rigid motions A figure transformed by a dilation and any combination of rigid Communicate logical that will take one of the reasoning in a systematic circles to any of the others, motions will be similar to the image. way to present a Verify that the ratio of the mathematical proof of circumference of the circles geometric theorems. created through dilations is Any circle can be created equal to the ratio of the radii through a dilation of another of the circles, wh circle and a combination of ich is the same as the scale rigid motions, factor of the dilation, Any geometric figure that is Explain with logical fully defined by a single reasoning how a circle is fully parameter will be similar to defined by a single parameter all other figures in that class "r" so the only non- (squares, equilateral translational changes that can triangles, circles, parabola, be made is alteration of "r", etc.) which changes the size and not the shape and therefore the circles are similar.

Grade Level/ Course (high school): Geometry Unit 4

Standard: G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Domain: Cluster: Use coordinates to prove simple geometric theorems algebraically Expressing Geometric Properties with Equations

Quarter 1: Quarter 2: Quarter 3:

Recall the definition of ratio.

Recall previous understandings of coordinate geometry.

Given a line segment (including those with positive and negative slopes) and a ratio, find the

59Version 3 2015-2016 point on the segment that partitions the segment into the given ratio.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Directed line segment Students know: Students understand/are Given two points and a ratio able to: Partitions Techniques for finding the that partitions the segment distance between two points and Accurately find the distance between the points, the equation of a line passing between two points and the Construct a circle using one through two points, equation of a line passing of the given points as the Forms for writing the equation of a through two points, center and the distance circle dependent on the Accurately find the equation between the points as the information given to find the of a dilation of a circle, radius, equation of the dilation of a circle, Find the intersection Construct a dilation of the Techniques to find the intersection point(s) of a line and a circle using the given ratio as between a line and a circle. circle. the scale factor and find the intersection between the A radius of a circle may be dilation and the equation of used to show the distance the line passing through the between two points, given points, A dilation of a circle may be Justify and explain the used to partition a line reasons for each step in the segment by making it the process of finding a point that radius, in a given ratio. partitions a segment in a given ratio.

Grade Level/ Course (HS): Geometry Unit 5

60Version 3 2015-2016 Standard: G.C.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.

Domain: Circles Cluster: Understand and apply theorems about circles

Quarter 1: Quarter 2: Quarter 3:

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Central angles Students know: Students understand/are able Given circles with two points to: Inscribed angles Definitions and characteristics of on the circle, central, inscribed, and Explain and justify possible Compare the measures of the Circumscribed angles circumscribed angles in a circle, relationships among central, angles (with and without Techniques to find measures of inscribed, and circumscribed technology) formed by angles including using technology angles sharing intersection creating radii to the given (dynamic geometry software). points on the circle, points, creating chords from Accurately find measures of a third point on the circle to angles (including using the given points, and creating technology (dynamic tangents from a third point geometry software)) formed outside the circle to the given from inscribed angles, radii, points, and conjecture about chords, central angles, possible relationships among circumscribed angles, and the angles, tangents Use logical reasoning to Relationships that exist 61Version 3 2015-2016 justify (or deny) the among inscribed angles, conjectures (in particular radii, and chords may be justify that an inscribed used to find the measures of angle is one half the central other angles when angle cutting off the same appropriate conditions are arc, and the circumscribed given, angle cutting off that arc is Identifying and justifying supplementary to the central relationships exist in angle relating all three). geometric figures.

Given circles with chords from a point on the circle to the endpoints of a diameter, Find the measure of the angles (with and without technology), conjecture about and explain possible relationships, Use logical reasoning to justify (or deny) the conjectures (in particular justify that an inscribed angle on a diameter is a right angle).

Given a circle with a tangent and radius intersecting at a point on the circle, Find the measure of the angle at the intersection point (with and without technology), conjecture about and explain possible relationships, Use logical reasoning to justify (or deny) the conjectures (in particular justify that the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Grade Level/ Course (HS): Geometry Unit 5

Standard: G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

62Version 3 2015-2016 Domain: Circles Cluster: Understand and apply theorems about circles.

Quarter 1: Quarter 2: Quarter 3:

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Construct Students know: Students understand/are Given a triangle, able to: Inscribed and Techniques to find circumcenter Use tools (e.g., compass, circumscribed circles and in center of a triangle, and to Use appropriate tools to straight edge, geometry of a triangle use this in inscribing or accurately construct software) to construct circumscribing the triangle, inscribed and circumscribed inscribed and circumscribed Quadrilateral circles of a triangle, inscribed in a circle Properties of inscribed angles. circles, Explain and justify the steps Explain and justify the that are used when creating sequence of steps taken to the construction, complete the construction. Apply the properties of inscribed angles to reach a Given a quadrilateral conclusion. inscribed in a circle, Every triangle has a point Conjecture possible which is equidistant from relationships among the each vertex of the triangle angles through the use of and a point which is 63Version 3 2015-2016 inscribed angles use logical equidistant from each side reasoning to justify (or deny) of the triangle, the conjectures (in particular, Opposite angles of a justify that diagonally quadrilateral inscribed in a opposite angles of a circle are supplementary. quadrilateral are supplementary).

Grade Level/ Course (HS): Geometry Unit 5

Standard: G.C. 4 (+) Construct a tangent line from a point outside a given circle to the circle.

Domain: Circles Cluster: Understand and Apply Theorems about circles

Quarter 1: Quarter 2: Quarter 3:

64Version 3 2015-2016 Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Tangent line Students know: Students understand/are Given a circle and an external Definition of a tangent line. able to: point, Tangent lines are perpendicular to Perform basic geometric Use tools to construct a the radius of the circle at the point constructions using tools. tangent line to the circle from of tangency. Any external point has two the point. An angle inscribed in a semi-circle tangent lines to the circle. is a right angle.

Grade Level/ Course (HS): Geometry Unit 5

Domain: Cluster: Use coordinates to prove simple geometric theorems algebraically Expressing Geometric Properties with Equations

Quarter 1: Quarter 2: Quarter 3:

65Version 3 2015-2016 Make sense of Reason abstractly Construct Model with Use appropriate problems and and quantitatively. viable mathematics. tools strategically. persevere in solving arguments and them. critique the reasoning of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Simple geometric Students know: Students understand/are Given coordinates and theorems Relationships (e.g. distance, slope able to: geometric theorems and of line) between sets of points, Accurately determine what statements defined on a Properties of geometric shapes, information is needed to coordinate system, prove or disprove a Use the coordinate system Coordinate graphing rules and statement or theorem, techniques, and logical reasoning to Accurately find the needed justify (or deny) the statement Techniques for presenting a proof information and explain and or theorem, and to critique of geometric theorems. justify conclusions, arguments presented by others. Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.

Grade Level/ Course (HS): Geometry Unit 5

Standard: 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.

Domain: Circles Cluster: Find arc lengths and areas of sectors of circles.

Quarter 1: Quarter 2: Quarter 4: Recall how to find the area and circumference of a circle.

Explain that 1° = Π/180 radians

66Version 3 2015-2016 Recall from G.C.1, that all circles are similar.

Determine the constant of proportionality (scale factor).

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

Verify that the constant of a proportion is the same as the radian measure, Θ, of the given central angle. Conclude s = r Θ

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. Make sense of Reason abstractly Construct viable Model with Use appropriate tools problems and and quantitatively. arguments and mathematics. strategically. persevere in solving critique the reasoning them. of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Similarity Students know: Students understand/are Given an arc intercepted by Techniques to use dilations able to: an angle, Constant of proportionality (including using dynamic Reason from progressive Use dilations to create arcs geometry software) to create examples using dynamic intercepted by the same Sector circles with arcs intercepted by geometry software to form central angle with radii of same central angles, conjectures about various sizes (including using Techniques to find arc length, relationships among arc dynamic geometry software), length, central angles, and and use the ratios of the arc Formulas for area and the radius, circumference of a circle. lengths and radii to make Use logical reasoning to conjectures regarding possible justify (or deny) these relationship between the arc conjectures and critique the length and the radius, reasoning presented by Justify the conjecture for the others, formula for any arc length Interpret a sector as a (i.e., since 2πr is the portion of a circle, and use circumference of the whole the ratio of the portion to circle, a piece of the circle is the whole circle to create a reduced by the ratio of the arc formula for the area of a angle to a full angle (360)), sector. Find the ratio of the arc length Radians measure the ratio of to the radius of each the arc length to the radius intercepted arc and use the for an intercepted arc, ratio to name the angle calling this the radian measure of the The ratio of the area of a angle by extending the sector to the area of a circle definition of one radian as the is proportional to the ratio angle which intercepts an arc of the central angle to a 67Version 3 2015-2016 of the same length as the complete revolution. radius, Develop the formula for the area of a sector by interpreting a circle as a complete revolution and a sector as a fractional part of a revolution.

Grade Level/ Course: Geometry Unit 6

Standard: 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”).Statistics and Probability is a Modeling Conceptual Category.

Domain: Cluster: Understand independence and conditional probability and use them to interpret data. Conditional Probability and the Rules of Probability

Quarter 1: Quarter 2: Quarter 3:

Make sense of Reason abstractly Construct viable arguments Model with Use appropriate problems and and quantitatively. and critique the reasoning of mathematics. tools persevere in solving others. strategically. them.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Subsets Students know: Students understand/are able Given scenarios involving to: Sample space Methods for describing events chance, from a sample space using set Interpret the given Determine the sample space Unions language (subset, union, information in the problem, and a variety of simple and 68Version 3 2015-2016 compound events that may be Intersections intersection, complement). Accurately determine the defined from the sample space, Complements probability of the scenario. Use the language of union, Set language can be useful to intersection, and complement define events in a probability appropriately to define events. situation and to symbolize relationships of events.

Grade Level/ Course (HS): Geometry Unit 5

Standard: G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Domain: Cluster: Translate between the geometric description and the equation for a conic section Expressing Geometric Properties with Equations

Quarter 1: Quarter 2: Quarter 3:

69Version 3 2015-2016 Make sense of Reason abstractly Construct viable Model with Use appropriate tools problems and and quantitatively. arguments and mathematics. strategically. persevere in solving critique the reasoning them. of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are able Given the center (h, k) and Key features of a circle, to: radius (r) of a circle, The Pythagorean Theorem, Create a right triangle in a Explain and justify that every circle using the horizontal point on the circle is a The technique of completing the and vertical shifts from the combination of a horizontal square. center as the legs and the and vertical shift from the radius of the circle as the center with a length equal to hypotenuse, the radius, Convert an equation of a Create a right triangle from circle from general form to the center of a circle to a standard form using the general point on the circle, method of completing the and show that the legs of the square. right triangle are the absolute Circles represent a fixed values of x-h and y-k, and the distance in all directions in a hypotenuse is r, then apply plane from a given point, and Pythagorean theorem to show 2 2 2 a right triangle may be that r = (x - h) + (y - k) . created to show the relationship of the horizontal Given an equation of a circle and vertical shift to the in general form, distance, Complete the square to Rewriting algebraic rewrite the equation in the expressions or equations in form r2 = (x - h)2 + (y - k)2 equivalent forms often and determine the center and reveals significant features of radius. the expression, (i.e., circles written in standard form are 70Version 3 2015-2016 useful for recognizing the center and radius of a circle).

Grade Level/ Course: Geometry Unit 5

Standard: G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*(*Modeling Standard)

Domain: Modeling Cluster: Apply geometric concepts in modeling situations with Geometry

Quarter 1: Quarter 2: Quarter 3:

71Version 3 2015-2016 Explain and justify the model objects, which was selected. Attributes of geometric figures help us identify the figures and find their measures, therefore matching these figures to real world objects allows the application of geometric techniques to real world problems.

Grade Level/ Course: Geometry Unit 6

Standard: 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.Statistics and Probability is a Modeling Conceptual Category.

Domain: Cluster: Understand independence and conditional probability and use them to interpret data. Conditional Probability and the Rules of Probability

Quarter 1: Quarter 2: Quarter 3:

Grade Level/ Course: Geometry Unit 6

Standard: 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.Statistics and Probability is a Modeling Conceptual Category.

73Version 3 2015-2016 Domain: Cluster: Understand independence and conditional probability and use them to interpret data Conditional Probability and the Rules of Probability

Quarter 1: Quarter 2: Quarter 4:

Know the conditional probability of A given B as P(A and B)/P(B)

Interpret independence of A and B as saying that the conditional probability of as the probability of B.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Conditional Students know: Students understand/are Given scenarios involving probability Methods to find probability of able to: two events A and B both simple and compound events, Accurately determine the when A and B are Independence Techniques to find conditional probability of simple and independent and when A compound events, and B are dependent, probability. Accurately determine the Determine the probability of conditional probability P(A each individual event, then given B)from a sample limit the sample space to space or from the those outcomes where B has knowledge of P(A∩B) and occurred and calculate the the P(B). probability of A, compare the P(A) and the P(A given The independence of two B), and explain the equality events is determined by the or difference in the original effect that one event has on context of the problem, the outcome of another event, Justify that P(A given B) = P(A∩B)/P(B). The occurrence of one event may or may not influence the likelihood 74Version 3 2015-2016 that another event occurs (e.g., successive flips of a coin - the first toss exerts no influence on whether a head occurs on the second, drawing an ace from a deck changes the probability that the next card drawn is an ace).

Grade Level/ Course: Geometry Unit 6 Standard: 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 10 compare the results. Statistics and Probability is a Modeling Conceptual Category. Domain: Cluster: Understand independence and conditional probability and use them to interpret data Conditional Probability and the Rules of Probability Quarter 1: Quarter 2: Quarter 3:

Make sense of Reason abstractly Construct viable Model with Use appropriate tools

75Version 3 2015-2016 problems and and quantitatively. arguments and critique mathematics. strategically. persevere in solving the reasoning of others. them. Evidence of Student Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Attainment/Assessme nt Students: Two way frequency Students know: Students Level IV Students will: Given a situation in tables Techniques to understand/are able EES-CP.1-5. Find the probability of an event after another event has occurred. which it is to: Ex. Find the probability of the next coin flip after a succession of coin flips (e.g., If Joe flipped Sample space construct two-way meaningful to collect frequency tables, Accurately construct a coin four times in row and got heads each time, what is the probability of getting heads on the categorical data for Independent a two-way frequency next flip?). two categories Techniques to find Ex. Find the probability of drawing a particular color after a succession of draws (e.g., If Sam Conditional table, simple and had three die in a bag - one red, one blue, and one green, what is the probability of drawing and Collect data and probabilities conditional Accurately find rolling a blue?). create two-way probability in two- simple and frequency tables, Ex. Find the probability of drawing a particular color after the color has been withdrawn (e.g., way frequency tables. conditional A bag contains four blue, three red, two yellow, and one black balls. Wes randomly selected the Determine probability from a black ball. What is the probability he will select a yellow ball next if the black ball is not probabilities of two-way frequency replaced in the bag?). simple events and table. conditional events Two-way frequency Level III Students will: from the table, tables show EES-CP.1-5. Identify when events are independent or dependent. explain whether the conditional Ex. When asked if winning the lottery depends on the weather, reply no. events are probability and can be Ex. When asked if the basketball game is likely to be canceled if it rains, reply no. independent based on used to test for Ex. When asked if the baseball game is likely to will be canceled if it rains, indicate likely. the context and the independence. Ex. When asked whether catching the bus depends upon whether you get up on time, reply yes. probability calculations. Level II Students will: EES-CP.1-5. Identify the outcomes of an event. Ex. What happens when an egg falls off the table? Ex. Two red and two blue balls are in a bag, two balls are taken out, what colors (two red, two blue, or red and blue) could the balls be?

Level I Students will: EES-CP.1-5. Determine which event occurs first in a sequence. Ex. Which is put on first - socks or shoes? Ex. Using a daily schedule, what activity would come next? Grade Level/ Course: Geometry Unit 6 Standard: S.CP.5 Recognizeandexplaintheconceptsofconditionalprobabilityandindependenceineverydaylanguageand everyday situations. cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Statistics and Probability is a Modeling Conceptual Category. Domain: Cluster: Understand independence and conditional probability and use them to interpret data Conditional Probability and Rules of Probability Quarter 1: Quarter 2: Quarter 3: Quarter 4:

Recognize the concepts of conditional probability and independence in everyday language and everyday situations.

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.)

76Version 3 2015-2016 Make sense of Reason abstractly Construct viable Model with Use appropriate problems and and quantitatively. arguments and mathematics. tools strategically. persevere in solving critique the them. reasoning of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessme nt Students: Conditional Students know: Students understand/are Given a contextual probability Possible relationships and differences able to: situation and Independence between the simple probability of an Communicate the concepts scenarios involving event and the probability of an event of conditional probability two events, under a condition. and independence using Explain the meaning everyday language by of independence from discussing the impact of a formula perspective the occurrence of one event P(A∩B) = P(A) x on the likelihood of the P(B) and from the other occurring. intuitive notion that A The occurrence of one occurring has no event may or may not impact on whether B influence the likelihood occurs or not, that another event occurs Compare these two (e.g., successive flips of a interpretations within coin - first toss exerts no the context of the influence on whether a scenario. head occurs on the second, drawing an ace from a deck changes the probability that the next card drawn is an ace), Events are independent if the occurrence of one does not affect the probability of the other occurring.

77Version 3 2015-2016 Grade Level/ Course: Geometry Unit 6 Standard: S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A and interpret the answer in terms of the model. Modeling Conceptual Category. Domain: Cluster: Use rules of probability to compute probabilities of compound events in a uniform probability model. Conditional Probability and Rules of Probability Quarter 1: Quarter 2: Quarter 3: Quarter 4:

Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A.

Interpret the answer in terms of the model.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Conditional Students know: Students understand/are Given a contextual situation probability Possible relationships and able to: consisting of two events, differences between the simple Accurately determine the Determine the probability of probability of an event and the probability of simple and each individual event, then probability of an event under a compound events, limit the sample space to condition. Accurately determine the those outcomes where B has conditional probability P(A occurred and calculate the given B) from a sample probability of A, compare the space or from the P(A) and the P(A given B), knowledge of P(A∩B) and and explain the equality or the P(B). difference in the original context of the problem, Conditional probability is the probability of an event Determine the probability of occurring given that another each individual event, then 78Version 3 2015-2016 limit the sample space to event has occurred. those outcomes where B has occurred and calculate the P(A and B), compare the ratio of P(A and B) and P(B) to P(A given B), and explain the equality or difference in the original context of the problem.

Grade Level/ Course: Geometry Unit 6

Standard: S.CP.7 Apply the Additional Rule, P(A or B) = P(A) + P(B) – P(A and B) and interpret the answer in terms of the model.

Domain: Cluster: Use rules of probability to compute probabilities of compound events in a uniform probability model. Conditional Probability and Rules of Probability

Quarter 1: Quarter 2: Quarter 3:

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment

79Version 3 2015-2016 Students: Addition Rule Students know: Students understand/are Given a contextual situation Techniques for finding able to: consisting of two events, probabilities of simple and Accurately determine the Determine the simple compound events. probability of simple and probability of each event, compound events. Determine the P(A or B) and Formulas are useful to P(A and B), generalize regularities, but Interpret the Addition Rule must be justified, by counting outcomes in the The Addition Rule may be four events A, B, A and B, A used for finding compound or B and showing the probability. relationship to P(A or B) = P(A) + P(B) - P(A and B), Interpret the Addition Rule in the case that the P(A and B) = 0.

Grade Level/ Course: Geometry Unit 6

Standard: 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. and Probability is a Modeling Conceptual Category.

Domain: Conditional Probability and Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model. Rules of Probability

Quarter 1: Quarter 2: Quarter 3:

80Version 3 2015-2016 Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Uniform probability Students know: Students understand/are able Given a contextual situation model Techniques for finding to: consisting of two events A probabilities of simple and Determine the probability of a and B, General Multiplication Rule conditional events. single event. Use the definition of Determine the probability of a conditional probability P(B| conditional event. A) = P(A and B)/P(A) to determine the probability of The general Multiplication the compound event (A and Rule for probability is a B) when the P(A|B) and the manipulation of the formula P(A) are known or may be for conditional probability. determined. The formula P(A and B) = Interpret the probability as it P(A)P(B|A) will always apply relates to the context. regardless of whether the events are independent or dependent.

Grade Level/ Course: Geometry Unit 6

Standard: S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Statistics and Probability is a Modeling Conceptual Category.

Domain: Cluster: Use rules of probability to compute probabilities of compound events in a uniform probability model. Conditional Probability and Rules of Probability

Quarter 1: Quarter 2: Quarter 3:

81Version 3 2015-2016 Make sense of Reason abstractly Cons Model with mathematics. Use appropriate tools problems and and quantitatively. truct strategically. persevere in solving viabl them. e argu ment s and criti que the reas onin g of othe rs.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Permutations Students know: Students understand/are Given a contextual able to: Combinations Order is the determining factor situation, in whether an event requires a Evaluate factorial Choose the appropriate Compound events permutation or a combination to expressions. counting technique count the number of possible Apply the multiplication (permutation or outcomes of the event. and addition rules to combination), find the Techniques for finding determine probabilities. number of ways an event(s) probabilities of simple and Interpret and apply the can occur, and use these compound events. counts to determine different notations for probabilities of the event, Techniques for finding the combinations and including compound events. number of permutations or permutations. combinations of an event. Perform procedures to evaluate expressions involving the number of combinations and permutations of n things taken r at a time. There are contextual situations that can be

82Version 3 2015-2016 interpreted through the use of combinations and permutations. The contextual situation determines whether combinations or permutations must be utilized. Mathematics is a coherent whole. Structure within mathematics allows for procedures or models from one concept to be applied elsewhere (e.g., Pascal's triangle as it applies to the number of combinations).

Grade Level/ Course (HS): Geometry Unit 6

Standard: S.MD.6 (+) Use probabilities to make fair decisions (e.g. drawing by lots, using a random number generator.)

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. Statistics and Probability is a Modeling Conceptual Category.

Domain: Using Cluster: Use probability to evaluate outcomes of decisions Probability to Make Decisions

Quarter 1: Quarter 2: Quarter 3:

83Version 3 2015-2016 Make sense of Reason abstractly Construct viable Model with Use appropriate tools problems and and quantitatively. arguments and mathematics. strategically. persevere in solving critique the reasoning them. of others.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Fair decisions Students know: Students understand/are Given a contextual situation The characteristics of a random able to: in which a decision needs to sample. Randomly select a sample be made, from a population (using Use a random probability technology when selection model to produce appropriate). unbiased decisions. Multiple factors may ultimately determine the decision one makes other than the probability of events, such as ethical constraints, social policy, or feelings of others. Probabilities can be used to explain why a decision was considered to be fair or objective.

84Version 3 2015-2016 Grade Level/ Course (HS): Geometry Unit 6

Standard: 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.) Statistics and Probability is a Modeling Conceptual Category.

Domain: Using Cluster: Use probability to evaluate outcomes of decisions Probability to Make Decisions

Quarter 1: Quarter 2: Quarter 3:

Make sense of Reason abstractly Construct viable Model with Use problems and and quantitatively. arguments and critique mathematics. appropriate persevere in solving the reasoning of others. tools them. strategically.

Evidence of Student Vocabulary Knowledge Skills Attainment/Assessment Students: Students know: Students understand/are Given a contextual situation Techniques for finding able to: in which a decision needs to probabilities of simple, compound, Choose the appropriate be made, and conditional events and from probability concept for the Use probability concepts to probability distributions. given situation. analyze, justify, and make Use and apply the selected objective decisions. probability rule. Communicate the reasoning behind decisions. Objective decision making can be mathematically 85Version 3 2015-2016 based, often using analysis involving probability concepts. Multiple factors may ultimately determine the decision one makes other than the probability of events, such as ethical constraints, social policy, or feelings of others.

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