Power Standards Cover Page for the Curriculum Guides. Geometry

Transcription

1 Power Standards Cover Page for the Curriculum Guides Geometry Quarter 1 G.CO.B.6 G.CO.C.9 G.GPE.B.3 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 determine informally if they are congruent. Prove theorems about lines and angles. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. Quarter 2 G.CO.C.10 G.CO.C.11 G.CO.B.7 G.CO.B.8 G.GPE.B.2 Prove theorems about triangles. Prove theorems about parallelograms. 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. Explain how the criteria for triangle congruence (ASA, SAS, AAS, and SSS) follow from the definition of congruence in terms of rigid motions. Use coordinates to prove simple geometric theorems algebraically. Quarter 3 G.SRT.A.2 G.SRT.A.3 G.SRT.B.4 G.SRT.B.5 G.SRT.C.6 G.GPE.A.1 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. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems about similar triangles. (this may need to be a power standard but we need more info.) Use congruence and similarity criteria for triangles to solve problems and to justify relationships in geometric figures. 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. Know and write the equation of a circle of given center and radius using the Pythagorean Theorem. Quarter 4 G.GMD.A.2 G.MG.A.1 G.MG.A.2 Know and use volume and surface area formulas for cylinders, cones, prisms, pyramids, and spheres to solve problems. Use geometric shapes, their measures, and their properties to describe objects. Apply geometric methods to solve real-world problems.

2 Listing of Standards and an overview of when they are taught Standard Quarter(s) Unit(s) G.CO.A.1 1, 2 1, 2, 6 G.CO.A G.CO.A G.CO.A G.CO.A G.CO.B G.CO.B G.CO.B G.CO.C G.CO.C G.CO.C G.CO.D.12 1, 2 1, 2, 3, 6 G.SRT.A G.SRT.A G.SRT.A G.SRT.B.4 3 8, 9 G.SRT.B.5 2, 3 7, 8 G.SRT.C G.SRT.C G.SRT.C G.C.A G.C.A G.C.A G.C.B G.GPE.A G.GPE.B.2 2, 3 5,10 G.GPE.B G.GPE.B G.GPE.B G.GMD.A G.GMD.A G.MG.A.1 1, 4 1, 11 G.MG.A Clusters highlighted green indicate the major content of the grade

3 Unit 1: The Language of Geometry First Nine Weeks Instructional Guide Weeks: 1-3 Domain Cluster Standard Student Outcomes (LEQ) Resources Weekly Pacing 1. Know precise definitions of What are the three angle, circle, perpendicular line, undefined terms of point, line, plane, collinear, coplanar, parallel line, and line segment, geometry and how can noncollinear, noncoplanar, skew, parallel based on the undefined notions they be modeled? planes, intersecting planes of point, line, distance along a line, and distance around a circular arc. Congruence (G-CO) A. Experiment with transformations in the plane How do I use symbolic notation to describe points, lines, planes, and line segments? I can describe possible intersections of lines and planes. Section 1.1 and page Geometry pages 1-2 Section 1.1 Section 1.1 and page 13, 2.5 HMH Resource Locker (click the link) for Differentiated Instruction, RTI, Math on the Spot Videos. y-module-1 Week 1 1

4 Expressing Geometric Properties with Equations (G GPE) B. Use coordinates to prove simple geometric theorems algebraically 4. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Also addresses G.CO.D.12: Make formal geometric constructions with a variety of tools and methods How do I apply the concepts of betweeness of points and midpoint to solve problems? How can we represent congruent segments and midpoints with constructions? How can we calculate the distance and midpoint in the coordinate plane (distance formula and Pythagorean Theorem)? line segment, between, betweeness of points, congruent segments, construction, distance, midpoint, segment bisector, Pythagorean Theorem. Segment Addition Postulate, Midpoint Theorem, Distance Formula, Midpoint Formula. Copy a segment, add segments, bisect a segment Section 1.1 Geometry pages Section Section , page 142 Task 6: _file/geo_guide_arc.pdf Week 1 2

5 Congruence (G-CO) A. Experiment with transformations in the plane 2. Represent transformations in the plane in multiple ways, including technology. Describe transformations as functions that take points in the plane (preimage) as inputs and give other points (image) as outputs. Compare transformations that preserve distance and angle measure to those that do not (e.g., translation versus horizontal stretch). 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. How do I describe transformations in the coordinate plane using algebraic representations and using words? How can I define translation, reflection, and rotation in terms of vocabulary that I have already studied? preimage, image, rigid motion, coordinate notation, transformation notation Section 1.3, Geometry pages 3-4, 7-12 Section 1.2, 7.1 Section 4.7, reflection, rotation, translation, Section 1.3, Geometry pages 3-4, 7-12 Week 2 Section 7.1 Week 2 Section 4.7, HMH Resource Locker (click on the link) for Differentiated Instruction, Math on the Spot Videos, and RTI. 3

6 5. Given a geometric figure and a rigid motion, draw the image of the figure in multiple ways, including technology. Specify a sequence of rigid motions that will carry a given figure onto another. Rigid motions include rotations, reflections, and translations. How do I draw the image of a figure under a translation? How do I draw the image of a figure under a reflection? How do I draw the image of a figure under a rotation? I can explain in my own words how a directed line segment (or vector) is the distance and direction of the translation. vector, initial point, terminal point (vector can also be defined as directed line segment) Section 1.3, Geometry pages 3-4, 7-12 Section 1.2, 7.1 Section 4.7, Week 2 4

7 Congruence (G-CO) B. Understand congruence in terms of rigid motions 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 determine informally if they are congruent. How do I use rigid motions to transform and predict the image given the preimage? How do I use rigid motions to determine if two figures are congruent? Section 1.3, , Geometry pages 3-4, 7-12 Section 7.2 Section 4.7, _file/0geomhexagonart.pdf. Week 2 D. Make geometric constructions 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). constructing the following objects inscribed in a circle: an equilateral triangle, square, and a regular hexagon. I can construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. I can explain how a construction can be used to verify the properties of a regular polygon. Inscribed, Circumscribed, equilateral triangle, square, regular hexagon Section 6.1 Geometry pages Sections 1.2 Week 3 5

8 Expressing Geometric Properties with Equations (G GPE) B. Use coordinates to prove simple geometric theorems algebraically 5. Know and use coordinates to compute perimeters of polygons and areas of triangles and rectangles. For example, use the distance formula. Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ). How do I find the perimeter and area of polygons in the coordinate plane? Polygon, concave, convex, regular polygon, perimeter, circumference, area Section 10.5 Geometry pages Sections Section _file/geo_guide_arc.pdf Week 3 6

9 Modeling with Geometry (G MG) A. Apply geometric concepts in modeling situations 1. Use geometric shapes, their measures, and their properties to describe objects. For example, modeling a tree trunk or a human torso as a cylinder). Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ). How do I describe 3- dimensional shapes using everyday objects to model them? Prism, Pyramid, cylinder, cone, Sphere Units: 16, 18, 19 Geometry pages Sections Units: 10, 11, 12, 13, 14 Section 1.7 Week 3 Assess Unit 1 7

10 Unit 2: Angles and Angle Relationships First Nine Weeks Instructional Guide Weeks: 4-5 Domain Cluster Standard Student Outcomes (LEQ) Resources Weekly Pacing 1. Know precise definitions of What is an angle and what angle, circle, perpendicular line, are the ways to classify Complementary, Supplementary, Linear Pair, parallel line, and line segment, angles according to their Adjacent Angles, Vertical Angles. based on the undefined notions measures? of point, line, distance along a line, and distance around a Sections 1.2, 1.4, 4.1 circular arc. Congruence (G-CO) A. Experiment with transformations in the plane What are the special angle pairs and their relationships? What are the names of the angle pairs formed by a transversal intersecting parallel lines and what relationships exist between those pairs of angles? Sections 1.4, 2.2 Section , 2.8 alternate interior angles, alternate exterior angles, corresponding angles, consecutive (same-side) interior, and consecutive (sameside) exterior Sections 4.2, 4.3 Sections 2.4 Section 3.2 Week 4 8

11 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). How do I use construction tools to copy angles and construct angle bisectors? Betweeness of rays, angle bisector Section 1.2 Geometry pages Congruence (G-CO) D. Make geometric constructions Constructions include but are not limited to: copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; constructing a line parallel to a given line through a point not on the line, and constructing the following objects inscribed in a circle: an equilateral triangle, square, and a regular hexagon. How can constructing parallel lines help me to understand and recall the relationships between the special angle pairs? How does knowing the relationship between the angles help me to show that the lines are parallel? Section 1.4 Section 1.4 Parallel lines, transversal Section 4.3 Geometry pages Section 1.6 Week 5 Assess Unit 2 Section 3.5 9

12 Unit 3: Parallel and Perpendicular Lines First Nine Weeks Instructional Guide Weeks: 6-7 Domain Cluster Standard Student Outcomes (LEQ) Resources Weekly Pacing 12. Make formal geometric How can this construction constructions with a variety of help me to show that the Perpendicular, Perpendicular Bisector tools and methods (compass and points on a perpendicular straightedge, string, reflective bisector of a line segment devices, paper folding, dynamic are equidistant from the geometric software, etc.). segment s endpoints? Congruence (G-CO) D. Make geometric constructions constructing a line perpendicular to a given line through a point not on the line. Section 4.4 Geometry pages Section 1.6, 7.7 Section 5.1 (page 322) and page 55 Geometry Lab Week 6 10

13 Expressing Geometric Properties with Equations (G GPE) B. Use coordinates to prove simple geometric theorems algebraically 3. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. Also G.CO.D.12 for the constructions What is the parallel postulate and how can I represent it geometrically (construction) and algebraically (equation)? What is the perpendicular postulate and how can I represent it geometrically (construction) and algebraically (equation)? Sections Geometry pages Sections 1.5 Sections , 3.6 Week 6 Week 7 Assess Unit 3 11

14 Unit 4: Logic and Proof First Nine Weeks Instructional Guide Weeks: 8-9 Domain Cluster Standard Student Outcomes (LEQ) Resources Weekly Pacing 9. Prove theorems about lines and angles. Congruence (G-CO) C. Prove Geometric Theorems Proving includes, but is not limited to, completing partial proofs; constructing two-column or paragraph proofs; using transformations to prove theorems; analyzing proofs; and critiquing completed proofs. Theorems include but are not limited to: 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. What are the two types of reasoning used in geometry? Truth value, inductive reasoning, deductive reasoning, conjecture, counterexample, hypothesis, conclusion, Conditional, Converse, Inverse, Contrapositive, Biconditional, Section 1.4 Geometry pages Section 2.1 Sections Task Arcs: Task 3, 4, and 6 andard/7110 Week 8: one block day 12

15 What is the structure of a conditional statement, its converse, and biconditional? How can writing algebraic proofs help me understand two-column proofs about theorems of geometry? I can complete a partial proof involving lines and angles. I can construct a twocolumn or paragraph proof about lines and angles. I can critique completed proofs. Properties of Equality: (i.e. reflexive, symmetric, transitive, addition, subtraction, multiplication, division, substitution) Congruent Supplements Theorem, Congruent Complements Theorem, Vertical angles Theorem, Alternate Interior Angles Theorem Converse, et.al. Section 1.4 Geometry pages Section 2.1 Week 8-9 Sections

16 Unit 5: Parallelograms Second Nine Weeks Weeks: 1-2 Domain Cluster Standard Student Outcomes (LEQ) Resources Weekly Pacing 3. Given a rectangle, parallelogram, I can translate geometric trapezoid, or regular polygon, figures on a coordinate plane. quadrilateral, parallelogram, rectangle, describe the rotations and I can rotate geometric figures rhombus, square, trapezoid, diagonal, reflections that carry it onto itself. on a coordinate plane. consecutive angles, and opposite angles. I can reflect geometric figures on a coordinate plane. Congruence (G CO) A. Experiment with transformations in the plane Sections 2.4 Geometry pages 5-6 Sections 7.1 and use the Skills Practice book. Section Week 1, 1 block day Congruence (G CO) C. Prove geometric theorems 11. Prove theorems about parallelograms. Proving includes, but is not limited to, completing partial proofs; constructing two-column or paragraph proofs; using transformations to prove theorems; analyzing proofs; and critiquing completed proofs. Theorems include but are not limited to: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, What can I conclude about the sides, angles, and diagonals of a parallelogram? What criteria can I use to prove that a quadrilateral is a parallelogram? What are the properties of rectangles, rhombi, and squares? I can compare and contrast the properties of a trapezoid and parallelogram. Sections Geometry pages Sections 10.1 and 10.2, 10.6 and use the Skills Practice book. Section _file/geopii-parallelogramanchorset_final.pdf Week 1 and week 2 14

17 and conversely, rectangles are parallelograms with congruent diagonals. _file/geometrytworectanglestaskanchorset. pdf Task 5, 6, 8 are aligned with this standard: _file/g_co_9thru11arcfinal.pdf Expressing Geometric Properties with Equations (G GPE) B. Use coordinates to prove simple geometric theorems algebraically 2. 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 G.GPE.B.3 is also utilized here. (can also use G.GPE.B.5 if you use the tasks) How can I use given coordinates to show that a quadrilateral is a parallelogram? How can I use given coordinates to show that a parallelogram is a rectangle, rhombus, or square? Sections 10.1, 10.2, 10.4 Geometry pages Section 10.7 and use the skills practice book pages Section Assessment Tasks: _file/0geomgettinginshape.pdf _file/0geomluciosride.pdf Week 2 Assess Unit 5 15

18 _file/0geomparkcity.pdf _file/geometrytworectanglestaskanchorset. pdf Unit 6: Triangles Second Nine Weeks Weeks: 3-5 Domain Cluster Standard Student Outcomes Resources Weekly Pacing 1. Know precise definitions of What can you say about the Prior Knowledge / angle, circle, perpendicular line, interior angles of a triangle How to classify a triangle according to its parallel line, and line segment, AND other polygons? angles and sides. 7 th and 8 th grade standards. based on the undefined notions of What can you say about the point, line, distance along a line, exterior angles of a triangle and distance around a circular arc. AND other polygons? Vertex angle, Base angles, legs, hypotenuse, Congruence (G CO) A. Experiment with transformations in the plane Section 7.1 Geometry pages 1-2 Sections , 5.2 Section 4.1, 4.2, 4.6, Week 3

19 Congruence (G CO) C. Prove geometric theorems 10. Prove theorems about triangles. Proving includes, but is not limited to, completing partial proofs; constructing two-column or paragraph proofs; using transformations to prove theorems; analyzing proofs; and critiquing completed proofs. Theorems include but are not limited to: 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. What are the special relationships between angles and sides in isosceles and equilateral triangles? How can I prove the Triangle Angle Sum Theorem? How can I prove the Base Angle Theorem (Isosceles Triangle Theorem) How can I prove the Midsegment Theorem? Isosceles Triangle Theorem (and converse), Equilateral Triangle Theorem (and converse) Midsegment Theorem Sections 7.1, 7.2 and 8.4 Geometry pages Section and 1.7 and page 476 Lesson 4-6; pages and Lesson 7-4 page 485. Instructional Resources: Task Arcs: Task # 1, 2, 4, 7, and 8 _file/g_co_9thru11arcfinal.pdf Week 4 Congruence (G CO) D. Make geometric constructions 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Constructions include but are not limited to: copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector How can I use perpendicular bisectors to find the point that is equidistant from the vertices of a triangle? How can I use angle bisectors to find the point that is equidistant from the sides of a triangle? How can I find the balance point or center of gravity of a triangle? Concurrent, point of concurrency, circumcenter, incenter, centroid, orthocenter, perpendicular bisector, angle bisector, median, altitude, midsegment. Sections Geometry pages Week 5 Assess Unit 6 17

20 of a line segment; constructing a line parallel to a given line through point not on the line, and constructing the following objects inscribed in a circle: an equilateral triangle, square, and a regular hexagon. How are the segments that join the midpoints of a triangle s sides related to the triangle s sides? Section 1.7 Sections Circles (G.C) A. Understand and apply theorems about circles. (G.GPE.B.4 and 5 can also be used with finding the orthocenter of a triangle algebraically. G.CO.C.10 can be used when solving problems with the Centroid.) 3 Construct the incenter and circumcenter of a triangle and use their properties to solve problems in context. I can construct an incenter using the angle bisectors in a triangle. I can construct a circumcenter using the perpendicular bisectors in a triangle. I can use the properties of incenter and circumcenter to solve conceptual problems. Sections Geometry pages (#1-3) Section 1.7 Sections 5.1 Week 5 Assess Unit 6 18

21 Unit 7: Proving Triangles Second Nine Weeks Weeks: 6-7 Domain Cluster Standard Student Outcomes Resources Weekly Pacing 7. Use the definition of congruence How can I show that two in terms of rigid motions to show triangles are congruent using CPCTC that two triangles are congruent if transformations? and only if corresponding pairs of sides and corresponding pairs of Section 5.1 angles are congruent. Geometry pages Congruence (G CO) B. Understand congruence in terms of rigid motions 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. What does the ASA triangle congruence postulate tell you about triangles? What does the SAS triangle postulate tell you about triangles? What does the SSS triangle congruence postulate tell you about triangles? What does the AAS triangle congruence theorem tell you about two triangles? Sections Sections Task Arc uses CO.B.6, CO.B.7, and CO.C.11: _file/geometrytaskarc.pdf ASA Congruence Postulate SAS Congruence Postulate SSS Congruence Postulate AAS Congruence Theorem HL, HA, LL, LA Congruence Theorems Sections and Geometry pages Week 6, 1 Block Day Week 6 and Week 7 Assess Unit 7

22 How are the four triangle congruence theorems associated with right triangles related to those for all triangles? Sections (for ASA, SAS, SSS, AAS) Sections 8.1 (for HL, HA, LL, LA) Lesson 4-5 extension; page 282. Instructional Resources: _file/geometrytaskarc.pdf Academic Lesson Drawing Triangles : =astandard/7109 Assessment Tasks (covers CO.B.8 and CO.C.10): _file/0geomharoldstransformation.pdf 20

23 Similarity, Right Triangles, and Trigonometry (G SRT) B. Prove theorems involving similarity 5. Use congruence (and similarity) criteria for triangles to solve problems and to prove relationships in geometric figures. How can I apply the congruence criteria for triangles to solve problems? Geometry pages Section 7.1 Explore 2 21

24 Unit 8: Similarity Weeks: 1-3 Third Nine Weeks Domain Cluster Standard Student Outcomes (LEQ) Resources Similarity, Right Triangles, and Trigonometry (G SRT) A. Understand similarity in terms of similarity transformations 1. Verify informally the properties of dilations given by a center and a scale factor. Properties include but are not limited to: 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 of the dilation unchanged; the dilation of a line segment is longer or shorter in the ratio given by the scale factor. 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. (Also, see G.C.A.1 Prove that all circles are similar. ) What is a dilation? How does a dilation transform a figure? How does the dilation of a figure effect the perimeter? How does the dilation of a figure effect the area? How does the dilation of a figure effect the volume? How do I determine if two figures are similar? How can similarity transformations be used to show that two figures are similar? If you know two figures are similar, what can you determine about the measures of corresponding angles and side lengths? Prerequisite Skill: Ratio and Proportion (6.PR and 7.PR) Dilation, center of dilation, scale factor, similarity transformation, enlargement, reduction Section 11.1 Geometry pages Section 6.1 Section 7-6 and 9-6; Section 7-2 example 4; Section 11-5 Section Geometry pages Section 6-1 Section 7-6 and 9-6 Weekly Pacing Week 1, 2 block days Week 1,Friday; Week 2, 1 block day 22

25 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. How can I show that two triangles are similar by AA? What are other methods to show that two triangles are similar? included side, included angle AA Similarity Postulate SSS Similarity Theorem SAS Similarity Theorem Section 11.4 Geometry pages Week 2, 1 block day Section 6.2 Similarity, Right Triangles, and Trigonometry (G SRT) B. Prove theorems involving similarity 4. Prove theorems about similar triangles. Proving includes, but is not limited to, completing partial proofs; constructing twocolumn or paragraph proofs; using transformations to prove theorems; analyzing proofs; and critiquing completed proofs. Theorems include but are not limited to: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. I can prove two triangles are similar by AA, SSS, or SAS Similarity. When a line parallel to one side of a triangle intersects the other two sides, how does it divide those sides? If three parallel lines intersect with two transversals, how are the transversals divided? How does an angle bisector of a triangle divide the side opposite that angle? Section 7-3 Triangle Proportionality Theorem Triangle Proportionality Th. Converse Triangle Proportionality Th. Corollaries (e.g. Proportional Segments Theorem) Triangle Angle Bisector Theorem Section 12.1 Geometry pages Section 6.3 Section 7.4, 7.5 Week 2, Friday; Week 3 1 block day 23

26 5. Use (congruence and) similarity criteria for triangles to solve problems and to justify relationships in geometric figures. How can I use similar triangles to solve problems? **further clarification is needed about justifying from the state/state assessments. Indirect Measurement, Section 12.3 Geometry pages Lessons 6.6 Week 3, 1 block day Expressing Geometric Properties with Equations (G GPE) B. Use coordinates to prove simple geometric theorems algebraically 4. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. How do I find a point on a directed line segment that partitions the given segment in a given ratio? Section 7.3 Example 5 Partition, directed line segment Section 12.2 Geometry pages College Board (SpringBoard Geometry) 2015 Lesson 26-4 Assessment Task: ry_congruenttrianglesanchor_final.pdf Week 3, Friday Review and Assess Unit 8 Week 4, 1 block day 24

27 Unit 9: Similarity in Right Triangles Weeks: 4-6 Third Nine Weeks Domain Cluster Standard Student Outcomes Resources Similarity, Right Triangles, and Trigonometry (G SRT) B. Prove theorems involving similarity 4. Prove theorems about similar triangles. Proving includes, but is not limited to, completing partial proofs; constructing twocolumn or paragraph proofs; using transformations to prove theorems; analyzing proofs; and critiquing completed proofs. Theorems include but are not limited to: the Pythagorean Theorem proved using triangle similarity. 5. Use similarity criteria for triangles to solve problems and to prove relationships in geometric figures. How does the altitude to the hypotenuse of a right triangle help you use similar right triangles to solve problems? How do I prove the Pythagorean Theorem (and its converse) using similar triangles? Geometric Mean Geometric Means Theorems (Leg and Altitude) Pythagorean Theorem Converse Reference sections 12.4 Geometry pages Lessons 6.4, 6.5, Section 8.1 (*NOTE: there doesn t appear to be any mention of triangle inequalities in the standards for one triangle or two. Monitor the test resources that are forthcoming for this topic. Also, there does not appear any reference to using the Pythagorean Theorem Converse to classify triangles or to use Pythagorean Triples.) Weekly Pacing Mid-week 4, 1 block day and Friday 25

28 Similarity, Right Triangles, and Trigonometry (G SRT) C. Define trigonometric ratios and solve problems involving right triangles 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. 7. Explain and use the relationship between the sine and cosine of complementary angles. How can I use the Pythagorean Theorem to explore the relationship between the side lengths of a Triangle and a Triangle? I can use the sine/cosine/ tangent ratio in a right triangle to solve for unknown side lengths. I can use the inverse sine/ inverse cosine/ inverse tangent ratio in a right triangle to solve for unknown angle measures. How can I find trigonometric ratios of the acute angles of a right triangle using AA Similarity? I can explain the relationship between the sine (cosine) of an angle and the cosine (sine) of its complement. How can the relationship between the sine and cosine of complementary angles help you solve problems? sine, cosine, tangent, opposite side, adjacent side, inverse sine, inverse cosine, inverse tangent Sections Geometry pages Sections and Section 8.3 Assessment Task: eomwhatsyoursine.pdf Section 13.2 Geometry pages Section 9.5 Assessment Task: eomtelevisionsize.pdf Week 5 Week 5 Instructional Resource: ometry%20-relatingtrigfunctions.pdf 26

29 8 Solve triangles. a. Know and use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. b. Know and use the Law of Sines and Law of Cosines to solve problems in real life situations. Recognize when it is appropriate to use each. Ambiguous cases will not be included in assessment. Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ). How can I solve a right triangle? I can state the Law of Sines and Law of Cosines and determine when each is appropriate to solve a problem. I can use the Law of Sines to solve contextual problems. I can use the Law of Cosines to solve contextual problems. solve a triangle Section 13.4 Section /resource_locker_se_aga_ _/index.html?lesson=53 Section /resource_locker_se_aga_ _/index.html?lesson=54 Geometry pages Lesson 9.5 Assessment Tasks: eomplayingcatch.pdf Week 6 Review and Assess Unit 9 on Week 7, 1 block day o-skateparkanchorset_final.pdf Instructional Resource: ometry-interstate.pdf 27

30 Unit 10: Circles Third Nine Weeks Weeks: 7-9 Domain Cluster Standard Student Outcomes Resources Weekly Pacing I can prove that all circles page Recognize that all circles are similar. Geometry are similar. pages Circles (G C) A. Understand and apply theorems about circles Page Mid-Week 7, ½ block day 28

31 Circles (G C) A. Understand and apply theorems about circles 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, and properties of angles for a quadrilateral inscribed in a circle. (Also: G.CO.A.1) Also go back and review G.CO.D.12 standard here. Specifically: constructing the following objects inscribed in a circle: an equilateral triangle, square, and a regular hexagon. How can I determine the measures of the central angle or inscribed angle of a circle? I can recognize and use relationships between arcs and chords in a circle. What are the key theorems about tangents to a circle? What are the relationships between angles formed by lines (secant and tangent) that intersect in, on, or outside the circle? What are the relationships between the segments formed by two secants and/or tangents that intersect in or outside the circle? What is the relationship between the opposite angles of a quadrilateral inscribed in a circle? Circle, center, radius, diameter, chord, secant, tangent, point of tangency, central angle, inscribed angle, intercepted arc, adjacent arcs Section 15.1, 15.3, 15.4, 15.5 Geometry pages Section Section Section 15.2 Geometry pages Section 12.1 Remainder of Week 7 Week 8 29

32 Circles (G C) Modeling with Geometry (G.MG) B. Find areas of sectors of circles A. Apply geometric concepts in modeling situations 4. Know the formula and find the area of a sector of a circle in a real-world context. For example, use proportional relationships and angles measured in degrees or radians. 2. Apply geometric methods to solve real-world problems. Geometric methods may include but are not limited to using geometric shapes, the probability of a shaded region, density, and design problems. I can distinguish between arc measure and arc length. How do I find the length of an arc? How do I find the area of a sector? I can distinguish between degree measure and radian measure. I can find the probability of a shaded sector in a circle. Pre-requisite Skills: Circumference and Area of a circle (7.G.B.3; Power Standard) arc, arc length, area of a sector, radian Section Geometry pages Section Week 9 30

33 Expressing Geometric Properties with Equations (G GPE) A. Translate between the geometric description and the equation for a circle. 1. Know and write the equation of a circle of given center and radius using the Pythagorean Theorem. I can write the equation of a circle given: o center and radius o endpoints of the diameter Given the equation of a circle, I can identify the center and the radius. How do I use completing the square to rewrite an equation in general form to standard form? Pre-requisite Skills: Completing the Square (A1.A.SSE.B.3b; Algebra I) Equation of a circle (in standard form), Equation of a circle (in general form), completing the square Section 17.1 Geometry pages Section 13.2 Section 10.8 Quarter 4 Week 1 Expressing Geometric Properties with Equations (G GPE) B. Use coordinates to prove simple geometric theorems algebraically 2. Use coordinates to prove simple geometric theorems algebraically. For example prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). Given a point in the coordinate plane, I can determine whether it lies inside, outside, or on a given circle. Section 17.1 Geometry page 65 #3 Section 13.3 Quarter 4 Week 1 Review and Asses week 2, 1 block day 31

34 Unit 11: Volume Weeks: 2-4 Geometric Measurement and Dimension (G GMD) A. Explain volume and surface area formulas and use them to solve problems. Fourth Nine Weeks Domain Cluster Standard Student Outcomes (LEQ) Resources Weekly Pacing 1 Give an informal argument for the I can give an informal Should have been introduced in Q1. Can formulas for the circumference of a argument for the formulas be explored more in depth (or reviewed) circle and the volume and surface for the circumference of a here. area of a cylinder, cone, prism, and circle and the area of a pyramid. Informal arguments may include but are not limited to using the dissection argument, applying Cavalieri s principle, and constructing informal limit arguments. 2 Know and use volume and surface area formulas for cylinders, cones, prisms, pyramids, and spheres to solve problems. Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ). circle. I can give an informal argument for the volume of a cylinder, pyramid, and cone. I can use the volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Section and 16.1 HMH Common Core Assessment Readiness: Geometry pages 73-74, Section 4.2, 4.3, 4.4 page 782 Section HMH Common Core Assessment Readiness: Geometry pages Section 4.6 Remainder of Week 2 Week 3 Review and Assess Week 4

35 1. Use geometric shapes, their measures, and their properties to describe objects. For example, modeling a tree trunk or a human torso as a cylinder. This concept will be modeled throughout the study of volume. Modeling with Geometry (G MG) A. Apply geometric concepts in modeling situations Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ). 2. Apply geometric methods to solve problems. Geometric methods may include but are not limited to using geometric shapes, the probability of a shaded region, density, and design problems. HMH Common Core Assessment Readiness page This concept will be modeled throughout the study of volume. Throughout the course Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ). HMH Common Core Assessment Readiness page 81-82

36 Unit 12: EOC REVIEW Weeks: 4-5 Fourth Nine Weeks Domain Cluster Standard Student Outcomes (LEQ) Resources # of items CAB (classroom assessment builder) will be available beginning Sept 2017 where teachers can build their own assessments. The CAB will include all item types with 50 math items available. This is estimated to be April 9-20 in Assuming that Geometry will be tested in the week April in % of test Use the mini practice test forms available on Nextera. The mini practice test forms are intended to be approximately 50% of the full test length. Be aware of materials that are labeled TNReady in the title or within the contents of their instructional materials. The department does not endorse any of these materials as official TNReady products.