Curriculum Guide - Geometry Introduction

Transcription

1 Curriculum Guide - Geometry Introduction Geometry Curriculum Guide Appropriate Common Core State Standards and Clusters are followed by one of the following symbols. Major Clusters/Standards Supporting Clusters/Standards o Additional Clusters/Standards High school mathematical modeling standards FS Fluency Standard All testable standards (SPIs) from the 'TCAP-EOC Geometry Framework' have been embedded within this guide. Common Core Mathematical Practice Standards The CCSS for Mathematical Practices are expected to be integrated into every mathematics lesson for all students grades K Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Last updated on 5/27/2014 Additional resources, CRAs, Instructional Tasks, etc. listed at the bottom of each unit will be found on the sharing server. The file name will be in parentheses after a brief description and will be in: Sharing/Consulting Teachers/Math/Geometry/Geometry Resources Questions or comments should be directed to Karl Bittinger, Math Curriculum Consulting Teachers. Page 1 of 42

2 Curriculum Guide - Geometry Introduction Clarifications, Evidence, and Assessment PBA - Performance Based Assessment (PBA) evidence statements, clarifications, math practice standards, and calculator usage were taken from the following location: EOY - End of Year (EOY) evidence statements, clarifications, math practice standards, and calculator usage were taken from the following location: PLD - Performance Level Descriptors (PLD) Level 5 descriptors were taken from the following location: Calculator - When we begin Common Core assessments in , students will only be permitted to use the online calculator for state assessments, which will be similar to the TI-84. While this calculator policy will not be enacted for the school year, it has been left in this document to help teachers prepare for the upcoming Common Core changes. Yes indicates a calculator will be accessible through the computer for the indicated assessment No indicates a calculator will not be accessible through the computer for the indicated assessment Neutral indicates a calculator will be accessible through the computer for the indicated assessment but may not be needed Item specific indicates the standard will only have a calculator accessible for certain items on the assessment A balanced use of calculators continues to be encouraged. Limitations - Assessment limits for standards assessed on more than one end-of-course (EOC) test: Algebra I, Geometry, and Algebra II. Limitations can be found on pages 56 through 59 of the Model Content Frameworks, Mathematics, Grades 3-11 at: - For the purposes of CMCSS pacing, Learning Targets are written in teacher friendly language. The Learning Targets in our pacing guides should be completely aligned to the content standards and exhaust the meaning of the standards. The Learning Targets may lead to clear targets. Page 2 of 42

3 Curriculum Guide - Geometry Unit Schedule 1st Semester Unit Title Dates Days 1 Constructions August 6 - September 4, Geometry in the Coordinate Plane September 5 - October 3, Congruence and Transformations October 6 - November 6, Similarity and Dilations November 7 - December 10, First Semester Review and Exam December 11 - December 19, Total Days 86 2nd Semester 6 Trigonometry January 6 - January 30, Circles February 2 - February 27, Working with Circles March 2 - March 30, D Geometry March 31 - April 17, Modeling April 20 - May 4, Second Semester Review and Exam May 5 - May 21, Total Days 90 Assessments Dates Page 3 of 42

4 Unit 1 Unit 1: Constructions 20 Days: August 6 - September 4, 2014 Standard Clarifications, Evidence, and Assessment G.CO.C.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. PBA - Construct, autonomously, chains of reasoning that will justify or refute geometric propositions or conjectures - MP 3 Calculator - Yes EOY - Prove geometric theorems as detailed in G.CO.C - About 75% of tasks align to G.CO.C.9 or G.CO.C.10 - MP 3 and 6 - Calculator - Neutral PLD (PBA and EOY) - Determine and use appropriate geometric theorems and properties of rigid motions, lines, angles, triangles, and parallelograms to solve non-routine problems and prove statements about angle measurement, triangles, distance, line properties, and congruence. - Prove vertical angles are congruent. - Prove corresponding angles and alternate interior angles are congruent when two parallel lines are cut by a transversal and converse. - Prove points on a perpendicular bisector of a line segment are exactly equidistant from the segments endpoint. - Use properties of congruence and equality in various types of proofs as in two-column proofs, flow-chart proofs, and paragraph proofs. - Encourage students to use precise language to build logical arguments when developing these proofs (MP 3 & MP 6). Page 4 of 42

5 Unit 1 G.CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. PBA - Construct, autonomously, chains of reasoning that will justify or refute geometric propositions or conjectures - MP 3 Calculator - Yes EOY - Prove geometric theorems as detailed in G.CO.C - About 75% of tasks align to G.CO.C.9 or G.CO.C.10 - MP 3 and 6 - Calculator - Neutral PLD (PBA and EOY) - Determine and use appropriate geometric theorems and properties of rigid motions, lines, angles, triangles, and parallelograms to solve non-routine problems and prove statements about angle measurement, triangles, distance, line properties, and congruence. - Prove the measures of interior angles of a triangle sum up to 180⁰. - Prove that base angles of an isosceles triangle are congruent. - Prove the medians of a triangle meet at a point. - Explore properties of triangles by using tools such as tracing paper, compass & straightedge, flow charts, and geometric software for given situations. Page 5 of 42

6 Unit 1 G.CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. EOY - Prove geometric theorems as detailed in G.CO.C - About 75% of tasks align to G.CO.C.9 or G.CO.C.10 - MP 3 and 6 - Calculator - Neutral PLD (PBA and EOY) - Determine and use appropriate geometric theorems and properties of rigid motions, lines, angles, triangles, and parallelograms to solve non-routine problems and prove statements about angle measurement, triangles, distance, line properties, and congruence. - Prove opposite sides and opposite angles of a parallelogram are congruent. - Prove that diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Page 6 of 42

7 Unit 1 G.CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). 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. FS PBA - Construct, autonomously, chains of reasoning that will justify or refute geometric propositions or conjectures. - MP 3 Calculator - Yes EOY - Make geometric constructions as detailed in G.CO.D - About 75% of tasks align to G.CO.C.12 - MP 3, 5, and 6 Calculator - Neutral PLD (PBA and EOY) - Makes geometric constructions: copying a segment, copying an angle, bisecting an angle, bisecting a segment, including the perpendicular bisector of a line segment. - Given a line and a point non on the line, uses a variety of tools and methods to construct perpendicular and parallel lines, equilateral triangles, squares, and regular hexagons inscribed in circles to prove geometric theorems. - Determine and use appropriate geometric theorems and properties of rigid motions, lines, angles, triangles, and parallelograms to solve non-routine problems and prove statements about angle measurement, triangles, distance, line properties, and congruence. Page 7 of 42

8 Unit 1 Additional Resources: STEM Integration An activity for G.CO.C.11 (Is this a parallelogram) may be found at: (IM_GCOC11_Parallelogram) An activity for G.CO.C.11 (Parallelograms and Translations) (IM_GCOC11_Pgrams&Translations) Activities for G.CO.D.12 and G.CO.D.13 may be found at: (IM_GCOD12_Warehouse) (IM_GCOD12_AngleBisect&Mdpt) Additional resources for G.CO.C.9 and G.CO.C.10 may be found at: ml (IM_GCOC9_EquidistantPts) html (IM_GCOC10_ClassifyingTriangles) Additional resources for G.CO.C.9, G.CO.C.10, G.CO.C.11, G.CO.C.12, and G.CO.C.13 may be found on pages of the HS CCSS Flip Book on the server. (High-School-CCSS-Flip-Book-USD ) Assessments: General formative assessment lessons may be found at: General formative assessment tasks may be found at: Unit Vocabulary: Common Student Misconceptions Prerequisite Skills Page 8 of 42

9 Instructional Tasks: CMCSS Curriculum Guide - Geometry Unit 1 Constructed Response Assessments Page 9 of 42

10 Unit 2 Unit 2: Geometry in the Coordinate Plane: 21 Days: September 5 - October 3, 2014 Standard Clarifications, Evidence, and Assessment G.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). FS PBA - Apply geometric reasoning in a coordinate setting, and/or use coordinates to draw geometric conclusions - MP 3 Calculator - Yes - Recall previous understandings of coordinate geometry (including, but not limited to: distance, midpoint, and slope formulas; equation of a line; definitions of parallel and perpendicular lines; etc.) G.GPE.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). FS PBA - Apply geometric reasoning in a coordinate setting, and/or use coordinates to draw geometric conclusions - MP 3 Calculator - Yes - Using slope, prove that lines are parallel. - Using slope, prove that lines are perpendicular. (Recognize that slopes of perpendicular lines are opposite reciprocals; i.e., the slopes of perpendicular lines have a product of 1.) Page 10 of 42

11 Unit 2 G.GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G.GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. FS PBA - MP 1 and 5 Calculator - Neutral PBA - Apply geometric reasoning in a coordinate setting, and/or use coordinates to draw geometric conclusions - MP 3 Calculator - Yes EOY - MP 1 and 5 Calculator - Neutral - 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 point on the segment that partitions the segment into the given ratio. PBA - Apply geometric reasoning in a coordinate setting, and/or use coordinates to draw geometric conclusions - MP 3 Calculator - Yes Additional Learning Target - Formulate a model of figures in contextual problems to compute area and/or perimeter. Page 11 of 42

12 Unit 2 Additional Resources: Video: Prove if a point is on a circle G.GPE.B.4: Two activities for G.GPE.B.4 and two for G.GPE.B.5 may be found at: (Select Geometry then the standard listed) STEM Integration (IM_GGPEB4_Midpoint) (IM_GGPEB4_UnitSquares&Triangles) (IM_GGPEB5_EqualAreaTriangles) (IM_GGPEB5_InscribedTriangle) Activities for G.GPE.B.6 and G.GPE.B.7 (IM_GGPEB6_TriangleCoordinates) (IM_GGPEB7_SquaresOnCoordinateGrid) Additional resources for G.GPE.B.6 and G.GPE.B.7 may be found on pages of the HS CCSS Flip Book on the server. (High-School-CCSS-Flip-Book-USD ) General formative assessment lessons may be found at: General formative assessment tasks may be found at: Assessments: Common Student Misconceptions Unit Vocabulary: Prerequisite Skills Instructional Tasks: Compare shapes on the Coordinate Plane (Geom Comparing shapes.pdf) Construct and analyze a quadrilateral (Geom Expanding Tri.pdf) Constructed Response Assessments Geom Congruent Tri SG Geom In Shape SG Page 12 of 42

13 Introduce mid-segment (Geom Midpoint.pdf) TNCore Instructional Task Arc 1 Task 1: Explores the concept of midpoint as a point that partitions a segment into two equal segments. Discusses the strategy of using coordinates to verify lengths. Task 2: Deepens understanding of the use of coordinates to verify conjectures by examining lengths of segments on the coordinate plane. CMCSS Curriculum Guide - Geometry Unit 2 Geom Lucios Ride SG Geom Park City SG Task 3: Solidifies understanding of the use of coordinate geometry to verify conjectures by using coordinate geometry to generalize the midpoint formula and the distance formula. Task 4: Explores the use of coordinate geometry to make and verify conjectures about points equidistant from two points on a coordinate plane. Task 5: Uses coordinate geometry to examine the concept of converse in the case of the perpendicular bisector. Introduces iff and notation. Task 6: Uses coordinate geometry to explore partitions other than in a 1:2 ratio; builds off of midpoint s 1:2 ratio. Task 7: Uses coordinate geometry to make and test conjectures about the midsegments of a triangle and to examine perimeter. Explores ratios other than 1:2. Task 8: Solidifies understanding about the use of coordinate geometry in generalization. Solidifies understanding about partitioning and midsegments. (Geom Task Arc 1 Coordinate.pdf) Page 13 of 42

14 Unit 3 Unit 3: Congruence and Transformations 18 Days: October 6 - November 6, 2014 Standard Clarifications, Evidence, and Assessment G.CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G.CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. EOY Calculator - Neutral - Use descriptions of rigid motion and transformed geometric figures to predict the effects rigid motion has on figures in the coordinate plane. - Knowing that rigid transformations preserve size and shape or distance and angle, use this fact to connect the idea of congruency and develop the definition of congruent. PLD - Determine and use appropriate geometric theorems and properties of rigid motions, lines, angles, triangles, and parallelograms to solve non-routine problems and prove statements about angle measurement, triangles, distance, line properties, and congruence. - Use the definition of congruence, based on rigid motion, to show two triangles are congruent if and only if their corresponding sides and corresponding angles are congruent. - Use the definition of congruence, based on rigid motion, to develop and explain the triangle congruence criteria; ASA, SSS, and SAS. Page 14 of 42

15 Unit 3 G.CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. PBA - MP 6 Calculator - Neutral EOY - MP 6 Calculator - Neutral - Understand and use definitions of angles, circles, perpendicular lines, parallel lines, and line segments based on the undefined term of a point, a line, the distance along a line, and the length of an arc. G.CO.A.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.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. - Use various technologies such as transparencies, geometry software, interactive whiteboards, and digital visual presenters to represent and compare rigid and size transformations of figures in a coordinate plane. - Compare transformations that preserve distance and angle to those that do not. - Describe and compare function transformations on a set of points as inputs to produce another set of points as outputs, to include translations and horizontal and vertical stretching G.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. - Using previous comparisons and descriptions of transformations, develop and understand the meaning of rotations, reflections, and translations based on angles, circles, perpendicular lines, parallel lines, and line segments. Page 15 of 42

16 Unit 3 G.CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. EOY - MP 5, 6, and 7 Calculator - Neutral - Transform a geometric figure given a rotation, reflection, or translation using graph paper, tracing paper, or geometric software. - Create sequences of transformations that map a geometric figure on to itself and another geometric figure. G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. FS EOY - For example, find a missing angle or side in a triangle - MP 7 Calculator - Neutral - Apply triangle congruence and similarity postulates to solve problem situations (i.e. indirect measurement, missing side measures, missing angle measures, side splitting). Page 16 of 42

17 Unit 3 Additional Resources: Activities for G.CO.B.6, G.CO.B.7, and G.CO.B.8 can be found at: On the server: (IM_GCOB6_TilePatternHexagons) (IM_GCOB6_TilePatternOctagons) (IM_GCOB7_PropertiesOfCongTriangles) (IM_GCOB8_SAS) (IM_GCOB8_SSS) (IM_GCOB8_SSA) (IM_GCOB8_SSA) Activities for G.SRT.B.5 can be found on the server: (IM_GSRTB5_BankShot) (IM_GSRTB5_Rectangles) Additional resources for G.CO.B.6, G.CO.B.7, and G.CO.B.8 may be found on pages of the HS CCSS Flip Book on the server. Additional resources for G.CO.A.1, G.CO.A.2, G.CO.A.3, G.CO.A.4, and G.CO.A.5 may be found on pages of the HS CCSS Flip Book on the server. Activities for G.CO.A.1, G.CO.A.2, G.CO.A.3, G.CO.A.4, and G.CO.A.5 can be found at: (IM_GCOA1_ParallelLines) (IM_GCOA1_PerpendicularLines) (IM_GCOA2_Dilations) (IM_GCOA2_RigidMotion) (IM_GCOA3_Quadrilateral) STEM Integration Page 17 of 42

18 (IM_GCOA3_Rectangles) (IM_GCOA4_Reflections) IM_GCOA4_Rotations) (IM_GCOA5_ReflectedTriangles) Assessments: CMCSS Curriculum Guide - Geometry Unit 3 Common Student Misconceptions Unit Vocabulary: Prerequisite Skills Instructional Tasks: TNCore Instructional Task Arc 2 Constructed Response Assessments Geom Harolds Trans SG Task 1: Explores the translation of shapes in the coordinate plane and introduces both notation and vector as a way to represent that translation. Examines congruence of shapes that have undergone the rigid motion of translation. Task 2: Explores the rotation of shapes in the coordinate plane and represents rotation in terms of center of rotation and angle of rotation. Examines multiple ways to accomplish the same transformation by performing a sequence of rigid motions. Verifies the congruence of shapes that have been rotated. Geom Hexagon Art SG Geom Skate Park SG Task 3: Examines the reflection of shapes in the coordinate plane and represents reflection in terms of the line of reflection. Examines multiple ways to accomplish the same transformation by performing a sequence of rigid motions. Verifies the congruence of shapes that have been reflected. Task 4: Solidifies understanding of the rigid motions translations, rotations, and reflections, and the congruence of corresponding parts of congruent figures. Requires precision in the description of a rigid motion, or sequence of rigid motions. Page 18 of 42

19 Task 5: Informally explores the SSS, SAS, and ASA criteria for triangle congruence by determining whether or not triangles with specified dimensions are congruent. Task 6: Uses rigid motion and logical reasoning to prove that two triangles are congruent if two angles and an included side are congruent. Rigid motion is also used to develop a justification for the SAS criteria for triangle congruence. Task 7: Informally explores triangles where two sides and the non-included angle are congruent to determine whether this condition is sufficient for the triangles to be congruent. Task 8: Solidifies understanding of the triangle congruence criteria, and how they can be used to determine that two triangles are congruent. (Geom Task Arc 2 Motion.pdf) CMCSS Curriculum Guide - Geometry Unit 3 Page 19 of 42

20 Unit 4 Unit 4: Similarity and Transformations 20 Days: November 7 - December 10, 2014 Standard Clarifications, Evidence, and Assessment G.SRT.A.1a 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. PBA - MP 1, 3, 5, and 8 Calculator - Neutral EOY - MP 1, 3, 5, and 8 Calculator - Neutral - Given a center and a scale factor, verify experimentally, that when dilating a figure in a coordinate plane, a segment of the preimage that does not pass through the center of the dilation, is parallel to its image when the dilation is performed. However, a segment that passes through the center remains unchanged. - Verify experimentally that a dilated image is similar to its perimage by showing congruent corresponding angels 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. - Define image, pre-image, scale factor, center, and similar figures as they relate to transformations. - Identify a dilation stating its scale factor and center. Page 20 of 42

21 Unit 4 G.SRT.A.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. PBA - MP 1, 3, 5, and 8 Calculator - Neutral EOY - MP 1, 3, 5, and 8 Calculator - Neutral - Given a center and a scale factor, verify experimentally, that when performing dilations of a line segment, the pre-image, the segment which becomes the image is longer or shorter based on the ratio given by the scale factor. - Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor. - Explain that the scale factor represents how many times longer or shorter a dilated line segment is than its pre-image. Page 21 of 42

22 Unit 4 G.SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. PBA - The "explain" part of standard G.SRT.A.2 is not assessed here. See sub-claim C aspect. - MP 7 Calculator - Neutral EOY - The "explain" part of standard G.SRT.A.2 is not assessed here but is assessed on the PBA. - MP 7 Calculator - Neutral - Use the idea of dilation transformations to develop the definition of similarity. - Given two figures determine whether they are similar and explain their similarity based on the equality of corresponding angles and the proportionality of corresponding sides. - Given two figures, decide if they are similar by using the definition of similarity in terms of similarity transformations. - 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. G.SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. - Use the properties of similarity transformations to develop the criteria for proving similar triangles; AA. - Establish the AA criterion for similarity of triangles by extending the properties of similarity transformations to the general case of any two similar triangles. Page 22 of 42

23 Unit 4 G.SRT.B.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. - Use AA, SAS, SSS similarity theorems to prove triangles are similar. - Use triangle similarity to prove other theorems about triangles a) Prove a line parallel to one side of a triangle divides the other two proportionally, and it s converse b) Prove the Pythagorean Theorem using triangle similarity. - 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. - Recall postulates, theorems, and definitions to prove theorems about triangles. G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. FS EOY - For example, find a missing angle or side in a triangle - MP 7 Calculator - Neutral - Using similarity theorems prove that two triangles are congruent. - Prove geometric figures, other than triangles, are similar and/or congruent. - 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. - Find a missing angle or side in a triangle. Page 23 of 42

24 Unit 4 G.C.B. 5 has been identified as a standard which may be assessed on the middle of the year PBA. A basic introduction of this standard is recommended during this unit. However, full mastery of the standard will be expected in unit 6. (G.C.B.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector) Additional Resources: For these standards, additional resources may be found at: STEM Integration Select Geometry (orange box at the top of the page), then select the second line (Similarity, Right Triangles, and Trigonometry, G- SRT), then select A. Understand similarity in terms of similarity transformations, last select (see illustrations) at the end of sentence 1. This will open a pdf document/activity which walks the teacher through this standard. Two activities for G.SRT.A.2 and two for G.SRT.B.5 may be found at (IM_GSRTA1_DilatingALine) (IM_GSRTA2_Similar) (IM_GSRTA3_SimilarTriangles) (IM_GSRTB4_MdptsOfTriangleSides) (IM_GSRTB5_PythThm) Page 24 of 42

25 Unit 4 Additional resources for these standards may be found on pages of the HS CCSS Flip Book on the server. Assessments: General formative assessment lessons may be found at General formative assessment tasks may be found at Unit Vocabulary: Common Student Misconceptions Prerequisite Skills Instructional Tasks: Constructed Response Assessments Page 25 of 42

26 Unit 5 Unit 5: First Semester Review and Exam 7 Days: December 11 - December 19, 2014 Additional Resources: Assessments: Page 26 of 42

27 Unit 6 Unit 6: Trigonometry 18 Days: January 6 - January 30, 2015 Standard Clarifications, Evidence, and Assessment G.SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. PBA - Trigonometric ratios include sine, cosine, tangent, cotangent, secant, and cosecant. - MP 7 Calculator - Neutral - Name 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. Page 27 of 42

28 Unit 6 G.SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. PBA - Use the relationship between the sine and cosine of complementary angles. - MP 7 Calculator - Neutral PBA - Tasks have multiple steps. - Tasks have context. - For rational solutions, exact values are required. For irrational solutions, exact or decimal approximations may be required. Simplifying or rewriting radicals is not required. - MP 1, 2, 4, 5, and 6 Calculator - Item specific EOY - Use the relationship between sine and cosine of complementary angles. - The "explain" part of standard G.SRT.C.7 is not assessed here, but is assessed on the PBA. - MP 7 Calculator - Neutral - Explore the sine of an acute angle and the cosine of its complement and determine their relationship. Page 28 of 42

29 Unit 6 G.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. PBA - Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of courselevel knowledge and skills articulated in G.SRT.C.8 involving right triangles in an applied setting - Tasks may or may not require the student to autonomously make an assumption or simplification in order to apply techniques of right triangles. For example, a configuration of three buildings might form a triangle that is nearly but not quite a right triangle, so that a good approximate result can be obtained if the student autonomously approximates the triangle as a right triangle - MP 2 and 4 Calculator - Yes EOY - Tasks have multiple steps - Tasks have context - MP 1, 2, 5, and 6 Calculator - Item specific Page 29 of 42

30 Unit 6 Additional Resources: Six activities for G.SRT.C.6 may be found at STEM Integration Select Geometry then the standard listed. One activity each for G.SRT.C.6, G.SRT.C.7, and G.SRT.C.8 may be found at Assessments: General formative assessment lessons may be found at General formative assessment tasks may be found at Unit Vocabulary: Instructional Tasks: Sine, Cosine, Tangent (Geom Interstate.pdf) Find missing points using Trig (Geom Making Right Tri.pdf) Common Student Misconceptions Prerequisite Skills Constructed Response Assessments Geom Skate Park SG Geom TV Size SG Apply similar figures to Trig (Geom Ratios and Proportions.pdf) Geom Ratios and Proportions.pdf Compare sine and cosine (Geom Relating Trig.pdf) Page 30 of 42

31 Unit 7 Unit 7: Circles 19 Days: February 2 - February 27, 2015 Standard Clarifications, Evidence, and Assessment G.C.A.1 Prove that all circles are similar. o - Recognize when figures are similar. (Two figures are similar if one is the image of the other under a transformation from the plane into itself that multiplies all distances by the same positive scale fact, k. That is to say, one figure is a dilation of the other.) - Compare the ratio of the circumference of a circle to the diameter of the circle. - Discuss, develop, and justify this ratio for several circles. - Determine that this ratio is constant for all circles. (MP 5, 8) G.C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. o - Identify inscribed angles, radii, chords, central angles, circumscribed angles, diameter, and tangent. - Recognize that inscribed angles on a diameter are right angles. - Recognize that inscribed angles on a diameter are right angles. - Recognize that radius of a circle is perpendicular to the radius at the point of tangency. - Examine the relationship between central, inscribed and circumscribed angles by applying theorems about their measures. (MP 2, 7) Page 31 of 42

32 Unit 7 G.C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. o Additional Resources: Assessments: Unit Vocabulary: Instructional Tasks: - Define inscribed and circumscribed circles of a triangle. - Recall midpoint and bisector definitions. - Define a point of concurrency. - Prove properties of angles for a quadrilateral inscribed in a circle. - Construct inscribed circles of a triangle. - Construct circumscribed circles of a triangle. (MP 4, 5, 6) STEM Integration Common Student Misconceptions Prerequisite Skills Constructed Response Assessments Page 32 of 42

33 Unit 8 Unit 8: Working with Circles 16 Days: March 2 - March 30, 2015 Standard Clarifications, Evidence, and Assessment G.GPE.A.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. o EOY - Complete the square to find the center and radius of a circle given by an equation. - The "derive" part of standard G.GPE.A.1 is not assessed here. - MP 6 Calculator - Neutral - Define a circle. - Use the Pythagorean Theorem. - Complete the square of a quadratic equation. - Derive equation of a circle using the Pythagorean Theorem, given coordinates of the center and length of the radius. - Determine the center and radius by completing the square. (MP 1, 2) G.C.B.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. o - Recall how to find the area and circumference of a circle. - Explain that 1⁰ = π/180 radians. - 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 proportionate, 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*theta. (MP 2, 3, 5) Page 33 of 42

34 Unit 8 Additional Resources: Assessments: Unit Vocabulary: Instructional Tasks: STEM Integration Common Student Misconceptions Prerequisite Skills Constructed Response Assessments Page 34 of 42

35 Unit 9 Unit 9: 3D Geometry 13 Days: March 31 - April 17, 2015 Standard Clarifications, Evidence, and Assessment G.GMD.B.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. o EOY - MP 7 Calculator - Neutral - Use strategies to help visualize relationships between twodimensional and three dimensional objects G.GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri s principle, and informal limit arguments. o G.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. o EOY - MP 3, 6, and 7 Calculator - Neutral - Explain the formulas for the circumference of a circle and the area of a circle by determining the meaning of each term or factor. - Explain the formulas for the volume of a cylinder, pyramid and cone by determining the meaning of each term or factor. EOY - MP 4 Calculator - Item specific - Solve problems using volume formulas for cylinders, pyramids, cones, and spheres. Page 35 of 42

36 Unit 9 Additional Resources: Assessments: Unit Vocabulary: Instructional Tasks: STEM Integration Common Student Misconceptions Prerequisite Skills Constructed Response Assessments Page 36 of 42

37 Unit 10 Unit 10: Modeling 11 Days: April 20 - May 4, 2015 Standard Clarifications, Evidence, and Assessment G.MG.A.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). G.MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). G.MG.A.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). - Given a real world object, classify the object as a known geometric shape-use this to solve problems in context. - Use the concept of density when referring to situations involving area and volume models, such as persons per square mile. - Solve design problems by designing an object or structure that satisfies certain constraints, such as minimizing cost or working with a grid system based on ratios (i.e., The enlargement of a picture using a grid and ratios and proportions). Additional Resources: STEM Integration Assessments: Common Student Misconceptions Unit Vocabulary: Prerequisite Skills Instructional Tasks: Constructed Response Assessments Page 37 of 42

38 Unit 11 Unit 11: Second Semester Review and Assessment 13 Days: May 5 - May 21, 2015 Additional Resources: Assessments: Page 38 of 42

39 Curriculum Guide - Geometry Integrated Targets The following are evidence statements and clarifications which should be integrated throughout the school year. PLD (PBA) Reasoning Clearly constructs and communicates a complete response based on: - a chain of reasoning to justify or refute algebraic and/or geometric propositions or conjectures - geometric reasoning in a coordinate setting, or - a response to a multi-step problem by: - using a logical approach based on conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate) - providing an efficient and logical progression of steps or chain of reasoning with appropriate justification - performing precise calculations - using correct grade-level vocabulary, symbols, and labels - providing a justification of a conclusion - determining whether an argument or conclusion is generalizable - evaluating, interpreting, and critiquing the validity and efficiency of others' responses, approaches and reasoning - utilizing mathematical connections (when appropriate) - and providing a counter example where applicable Page 39 of 42

40 Curriculum Guide - Geometry Integrated Targets PLD (PBA and EOY) Modeling - Devises and enacts a plan to apply mathematics in solving problems arising in everyday life, society, and the workplace by: --- using stated assumptions and making assumptions and approximations to simplify a real-world situation (includes micro-models) --- mapping relationships between important quantities --- selecting appropriate tools to create models --- analyzing relationships mathematically between important quantities to a conclusion --- analyzing and/or creating constraints, relationships, and goals --- interpreting mathematical results in the context of the situation --- reflecting on whether the results make sense --- improving the model if it has not served its purpose --- writing a complete, clear, and correct algebraic expression or equation to describe a situation --- applying proportional reasoning and percentages justifying and defending models which lead to a conclusion --- applying geometric principles and theorems --- writing and using functions in any form to describe how on quantity of interest depends on another --- using statistics --- using reasonable estimates of known quantities in a chain of reasoning that yields an estimate of unknown quantity G.CO.C PLD (PBA and EOY) - Determine and use appropriate geometric theorems and properties of rigid motions, lines, angles, triangles, and parallelograms to solve non-routine problems and prove statements about angle measurement, triangles, distance, line properties, and congruence. G.CO.A PBA - Construct, autonomously, chains of reasoning that will justify or refute geometric propositions or conjectures - MP 3 Calculator - Yes Page 40 of 42

41 Curriculum Guide - Geometry Integrated Targets G.CO.B PBA - Construct, autonomously, chains of reasoning that will justify or refute geometric propositions or conjectures - MP 3 Calculator - Yes G.SRT.A.1a, G.SRT.A.1b, G.SRT.A.2, and G.SRT.B.5 PLD (PBA and EOY) - Uses transformations and congruence and similarity criteria for triangles and to prove relationships among composite geometric figures and to solve multi-step problems. G.CO.A.1, G.CO.A.3, and G.CO.A.5 PLD (PBA and EOY) - Given a figure and a sequence of transformations, draw the transformed figure - Use precise geometric terminology to specify more than one sequence of transformations that will carry a figure onto itself or another. G.SRT.C PBA - Present solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equals signs approximately (for example, rubrics award less than full credit for the presence of nonsense statements such as 1+4=5+7=12 even if the final answer is correct), or identify or describe errors in solutions to multi-step problems and present corrected solutions. - MP 6 Calculator - Yes G.SRT.C.6, G.SRT.C.7, and G.SRT.C.8 PLD (PBA and EOY) - Use trigonometric ratios, the Pythagorean Theorem and the relationship between sine and cosine to solve right triangles in applied nonroutine problems. - Use similarity transformations with right triangles to define trigonometric ratios for acute angles. Page 41 of 42

42 Curriculum Guide - Geometry Integrated Targets G.SRT.C.7, G.SRT.C.8, and G.GPE.B.6 PLD (PBA and EOY) - Use geometric relationships in the coordinate plane to solve problems involving area, perimeter, and ratios of lengths. - Apply geometric concepts and trigonometric ratios to describe, model, and solve applied problems (including design problems) related to the Pythagorean theorem, density, geometric shapes, their measures and properties. G.C.A EOY - Identify and describe relationships among inscribed angles, radii, and chords and apply these concepts in problem solving situations - MP 1 and 5 Calculator - Item Specific G.C.B EOY - Find arc lengths and areas of sectors of circles. - Tasks involve computing are lengths or areas of sectors given the radius and the angle subtended; or vice versa. Calculator - Item specific G.C.A, G.C.B, and G.GPE.A.1 PLD (PBA and EOY) - Apply properties and theorems of angles, segments and arcs in circles to solve problems, model relationships and formulate generalizations. - Complete the square to find the center and radius of a circle given by an equation. G.GMD.A.1, G.GMD.A.3, and G.GMD.B.4 PLD (PBA and EOY) - Use volume formulas to solve mathematical and contextual problems that involve cylinders, pyramids, cones, and spheres. - Use dissection arguments, Cavalieri's principle, and informal limit arguments to support the formula for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. - Identify the shapes of two-dimensional cross sections of three-dimensional objects and identifies three-dimensional objects generated by rotations of two-dimensional objects. Page 42 of 42