geometry nr 2015 -...

MOUNT VERNON CITY SCHOOL DISTRICT

CCLS Geometry NRCurriculum Guide

THIS HANDBOOK IS FOR THE IMPLEMENTATION OF THE CCLSGEOMETRY NR CURRICULUM IN MOUNT VERNON.

2015-2016

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Mount Vernon City School District

Board of Education

Adriane SaundersPresident

Serigne GningueVice President

Board TrusteesCharmaine FearonRosemarie Jarosz

Micah J.B. McOwenOmar McDowell

Darcy MillerWanda WhiteLesly Zamor

Superintendent of SchoolsDr. Kenneth Hamilton

Deputy SuperintendentDr. Jeff Gorman

Assistant Superintendent of BusinessKen Silver

Assistant Superintendent of Human ResourcesDenise Gagne-Kurpiewski

Administrator of Mathematics and Science (K-12)Dr. Satish Jagnandan

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TABLE OF CONTENTS

I. COVER …..……………………………………....... 1

II. MVCSD BOARD OF EDUCATION …..……………………………………....... 2

III. TABLE OF CONTENTS …..……………………………………....... 3

IV. IMPORTANT DATES …..……………………………………....... 4

V. VISION STATEMENT …..……………………………………....... 5

VI. PHILOSOPHY OF MATHEMATICS CURRICULUM ……………. 6

VII. NYS P-12 COMMON CORE LEARNING STANDARDS ……………..7

VII. MVCSD CCLS GEOMTRY NR PACING GUIDE …………... 12

VIII. WORD WALL …………... 24

IX. SETUP OF A MATHEMATICS CLASSROOM …………... 25

X. SECONDARY GRADING POLICY …………... 26

XI. SAMPLE NOTEBOOK RUBRIC …………... 27

XII. CLASSROOM AESTHETICS …………... 28

XIII. SYSTEMATIC DESIGN OF A MATHEMATICS LESSON …………... 29

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IMPORTANT DATES 2015-16

REPORT CARD – 10 WEEK PERIOD

MARKINGPERIOD

MARKINGPERIODBEGINS

INTERIMPROGRESSREPORTS

MARKINGPERIOD

ENDS

DURATION REPORT CARDDISTRIBUTION

MP 1 September 8,2015

October 9,2015

November 13,2015

10 weeks Week ofNov. 23, 2015

MP 2 November 16,2015

December 18,2015

January 29,2016

10 weeks Week ofFebruary 8, 2016

MP 3 February 1,2016

March 11,2016

April 15,2016

9 weeks Week ofApril 25, 2016

MP 4 April 18,2016

May 20,2016

June 23,2016

10 weeks Last Day of SchoolJune 23, 2016

The Parent Notification Policy states “Parent(s) / guardian(s) or adult students are

to be notified, in writing, at any time during a grading period when it is apparent -

that the student may fail or is performing unsatisfactorily in any course or grade

level. Parent(s) / guardian(s) are also to be notified, in writing, at any time during

the grading period when it becomes evident that the student's conduct or effort

grades are unsatisfactory.”

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VISION STATEMENT

True success comes from co-accountability and co-responsibility. In a coherentinstructional system, everyone is responsible for student learning and studentachievement. The question we need to constantly ask ourselves is, "How are ourstudents doing?"

The starting point for an accountability system is a set of standards andbenchmarks for student achievement. Standards work best when they are welldefined and clearly communicated to students, teachers, administrators, andparents. The focus of a standards-based education system is to provide commongoals and a shared vision of what it means to be educated. The purposes of aperiodic assessment system are to diagnose student learning needs, guideinstruction and align professional development at all levels of the system.

The primary purpose of this Instructional Guide is to provide teachers andadministrators with a tool for determining what to teach and assess. Morespecifically, the Instructional Guide provides a "road map" and timeline forteaching and assessing the Common Core Learning Standards.

I ask for your support in ensuring that this tool is utilized so students are able tobenefit from a standards-based system where curriculum, instruction, andassessment are aligned. In this system, curriculum, instruction, and assessment aretightly interwoven to support student learning and ensure ALL students have equalaccess to a rigorous curriculum.

We must all accept responsibility for closing the achievement gap and improvingstudent achievement for all of our students.

Dr. Satish Jagnandan

Administrator for Mathematics and Science (K-12)

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PHILOSOPHY OF MATHEMATICS CURRICULUM

The Mount Vernon City School District recognizes that the understanding of mathematics is

necessary for students to compete in today’s technological society. A developmentally

appropriate mathematics curriculum will incorporate a strong conceptual knowledge of

mathematics through the use of concrete experiences. To assist students in the understanding and

application of mathematical concepts, the mathematics curriculum will provide learning

experiences which promote communication, reasoning, and problem solving skills. Students will

be better able to develop an understanding for the power of mathematics in our world today.

Students will only become successful in mathematics if they see mathematics as a whole, not as

isolated skills and facts. As we develop mathematics curriculum based upon the standards,

attention must be given to both content and process strands. Likewise, as teachers develop their

instructional plans and their assessment techniques, they also must give attention to the

integration of process and content. To do otherwise would produce students who have temporary

knowledge and who are unable to apply mathematics in realistic settings. Curriculum,

instruction, and assessment are intricately related and must be designed with this in mind. All

three domains must address conceptual understanding, procedural fluency, and problem solving.

If this is accomplished, school districts will produce students who will

1. 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.

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New York State P-12 Common Core Learning Standards for

Mathematics

Mathematics - High School Geometry: Introduction

An understanding of the attributes and relationships of geometric objects can be applied indiverse contexts—interpreting a schematic drawing, estimating the amount of wood needed toframe a sloping roof, rendering computer graphics, or designing a sewing pattern for the mostefficient use of material.

Although there are many types of geometry, school mathematics is devoted primarily to planeEuclidean geometry, studied both synthetically (without coordinates) and analytically (withcoordinates). Euclidean geometry is characterized most importantly by the Parallel Postulate,that through a point not on a given line there is exactly one parallel line. (Spherical geometry, incontrast, has no parallel lines.)

During high school, students begin to formalize their geometry experiences from elementary andmiddle school, using more precise definitions and developing careful proofs. Later in collegesome students develop Euclidean and other geometries carefully from a small set of axioms.

The concepts of congruence, similarity, and symmetry can be understood from the perspective ofgeometric transformation. Fundamental are the rigid motions: translations, rotations, reflections,and combinations of these, all of which are here assumed to preserve distance and angles (andtherefore shapes generally). Reflections and rotations each explain a particular type of symmetry,and the symmetries of an object offer insight into its attributes—as when the reflective symmetryof an isosceles triangle assures that its base angles are congruent.

In the approach taken here, two geometric figures are defined to be congruent if there is asequence of rigid motions that carries one onto the other. This is the principle of superposition.For triangles, congruence means the equality of all corresponding pairs of sides and allcorresponding pairs of angles. During the middle grades, through experiences drawing trianglesfrom given conditions, students notice ways to specify enough measures in a triangle to ensurethat all triangles drawn with those measures are congruent. Once these triangle congruencecriteria (ASA, SAS, and SSS) are established using rigid motions, they can be used to provetheorems about triangles, quadrilaterals, and other geometric figures.

Similarity transformations (rigid motions followed by dilations) define similarity in the sameway that rigid motions define congruence, thereby formalizing the similarity ideas of "sameshape" and "scale factor" developed in the middle grades. These transformations lead to thecriterion for triangle similarity that two pairs of corresponding angles are congruent.

The definitions of sine, cosine, and tangent for acute angles are founded on right triangles andsimilarity, and, with the Pythagorean Theorem, are fundamental in many real-world andtheoretical situations. The Pythagorean Theorem is generalized to non-right triangles by the Lawof Cosines. Together, the Laws of Sines and Cosines embody the triangle congruence criteria for

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the cases where three pieces of information suffice to completely solve a triangle. Furthermore,these laws yield two possible solutions in the ambiguous case, illustrating that Side-Side-Angleis not a congruence criterion.

Analytic geometry connects algebra and geometry, resulting in powerful methods of analysis andproblem solving. Just as the number line associates numbers with locations in one dimension, apair of perpendicular axes associates pairs of numbers with locations in two dimensions. Thiscorrespondence between numerical coordinates and geometric points allows methods fromalgebra to be applied to geometry and vice versa. The solution set of an equation becomes ageometric curve, making visualization a tool for doing and understanding algebra. Geometricshapes can be described by equations, making algebraic manipulation into a tool for geometricunderstanding, odeling, and proof. Geometric transformations of the graphs of equationscorrespond to algebraic changes in their equations.

Dynamic geometry environments provide students with experimental and modeling tools thatallow them to investigate geometric phenomena in much the same way as computer algebrasystems allow them to experiment with algebraic phenomena.

Connections to Equations.

The correspondence between numerical coordinates and geometric points allows methods fromalgebra to be applied to geometry and vice versa. The solution set of an equation becomes ageometric curve, making visualization a tool for doing and understanding algebra. Geometricshapes can be described by equations, making algebraic manipulation into a tool for geometricunderstanding, modeling, and proof.

Mathematical Practices1. Make sense of problems and persevere in solvingthem.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoningof 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.

Geometry OverviewCongruence• Experiment with transformations in the plane• Understand congruence in terms of rigidmotions• Prove geometric theorems• Make geometric constructions

Similarity, Right Triangles, and Trigonometry• Understand similarity in terms of similaritytransformations• Prove theorems involving similarity• Define trigonometric ratios and solve problemsinvolving right triangles• Apply trigonometry to general triangles

Circles• Understand and apply theorems about circles• Find arc lengths and areas of sectors of circles

Expressing Geometric Properties with Equations• Translate between the geometric descriptionand the equation for a conic section• Use coordinates to prove simple geometrictheorems algebraically

Geometric Measurement and Dimension• Explain volume formulas and use them to solveproblems• Visualize relationships between twodimensionaland three-dimensional objects

Modeling with Geometry• Apply geometric concepts in modelingsituations

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Congruence G-CO

Experiment with transformations in the plane1. 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.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. Comparetransformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carryit onto itself.

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

5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graphpaper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figureonto another.

Understand congruence in terms of rigid motions6. 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 theyare congruent.

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

8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruencein terms of rigid motions.

Prove geometric theorems9. 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; pointson a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; baseangles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallelto the third side and half the length; the medians of a triangle meet at a point.

11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles arecongruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelogramswith congruent diagonals.

Make geometric constructions12. 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 bisectorof a line segment; and constructing a line parallel to a given line through a point not on the line.

13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Similarity, Right Triangles, & Trigonometry G-SRT

Understand similarity in terms of similarity transformations1. 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 linepassing through the center unchanged.

b. 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 allcorresponding pairs of angles and the proportionality of all corresponding pairs of sides.

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

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Prove theorems involving similarity4. 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.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric

figures.

Define trigonometric ratios and solve problems involving right triangles6. 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.

8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

Apply trigonometry to general triangles9. (+) 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.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.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).

Circles G-C

Understand and apply theorems about circles1. Prove that all circles are similar.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 acircle is perpendicular to the tangent where the radius intersects the circle.

3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateralinscribed in a circle.

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

Find arc lengths and areas of sectors of circles5. 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 asector.

Expressing Geometric Properties with Equations G-GPE

Translate between the geometric description and the equation for a conic section1. 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.2. Derive the equation of a parabola given a focus and directrix.3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of

distances from the foci is constant.

Use coordinates to prove simple geometric theorems algebraically4. 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).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).6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance

formula.★

Geometric Measurement & Dimension G-GMD

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Explain volume formulas and use them to solve problems1. 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.2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other

solid figures.

3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★

Visualize relationships between two-dimensional and three-dimensional objects4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Modeling with Geometry G-MG

Apply geometric concepts in modeling situations1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a

human torso as a cylinder).★

2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile,

BTUs per cubic foot).★

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).★

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CCLS GEOMETRY NR PACING GUIDE

This guide using NYS CCLS Geometry Modules was created to provide teachers with a time frame to complete the New York StateMathematics Geometry Curriculum.

Module Unit Title CCLS Standards Days Month

1Congruence, Proof, and

Constructions

G-CO.1, G-CO.2, G-CO.3, G-CO.4, G-CO.5G-CO.6, G-CO.7, G-CO.8, G-CO.9, G-CO.10

G-CO.11, G-CO.12, G-CO.1345 Sept. 8 – Nov. 16

2 Similarity, Proof, and TrigonometryG-SRT.A.1, G-SRT.B.4, G-SRT.A.2, G-SRT.A.3,

G-SRT.B.5, G-MG.A.1G-CO.12, G-SRT.C.6G-SRT.C.7, G-SRT.C.8

45 Nov. 17 – Feb. 9

3 Extending to Three DimensionsG-GMD.A.1, G-GMD.A.3, G-GMD.B.4, G.MG.A.1,

G.MG.A.2, G.MG.A.317 Feb 10 – Mar. 11

4Connecting Algebra and Geometry

through CoordinatesG-SRT.A.1, G-SRT.B.4, G-SRT.C.6,

G-SRT.C.7 G-SRT.C.827 Mar. 14 – Apr. 26

5Circles with and Without

CoordinatesG-C.A.1, G-C.A.2, G-C.A.3, , G-C.B.5, G-GPE.A.1, G-

GPE.B.428 Apr. 28 – June 7

Red – End of Module Assessment Period

Note that the curriculum assumes that each school day includes 40-45 minutes of math. Although pacing will vary somewhat inresponse to variations in school calendars, needs of students, your school's years of experience with the curriculum, and other localfactors, following the suggested pacing and sequence will ensure that students benefit from the way mathematical ideas areintroduced, developed, and revisited across the year.

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1Congruence, Proof, and

Constructions

G-CO.1, G-CO.2, G-CO.3, G-CO.4, G-CO.5G-CO.6, G-CO.7, G-CO.8, G-CO.9, G-CO.10

G-CO.11, G-CO.12, G-CO.1345 Sept. 8 – Nov. 16

Module 1 embodies critical changes in Geometry as outlined by the Common Core. The heart of the module is the study oftransformations and the role transformations play in defining congruence. In previous grades, students were asked to draw trianglesbased on given measurements. They also have prior experience with rigid motions—translations, reflections, and rotations—and havestrategically applied a rigid motion to informally show that two triangles are congruent. In this module, students establish trianglecongruence criteria, based on analyses of rigid motions and formal constructions. They build upon this familiar foundation of trianglecongruence to develop formal proof techniques. Students make conjectures and construct viable arguments to prove theorems—usinga variety of formats—and solve problems about triangles, quadrilaterals, and other polygons. They construct figures by manipulatingappropriate geometric tools (compass, ruler, protractor, etc.) and justify why their written instructions produce the desired figure.

Standards Topics Days

G-CO.1G-CO.12G-CO.13

A Basic ConstructionsLesson 1: Construct an Equilateral TriangleLesson 2: Construct an Equilateral Triangle IILesson 3: Copy and Bisect an AngleLesson 4: Construct a Perpendicular BisectorLesson 5: Points of Concurrencies

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G-CO.9 B Unknown AnglesLesson 6: Solve for Unknown Angles—Angles and Lines at a PointLesson 7: Solve for Unknown Angles—TransversalsLesson 8: Solve for Unknown Angles—Angles in a TriangleLesson 9: Unknown Angle Proofs—Writing ProofsLesson 10: Unknown Angle Proofs—Proofs with ConstructionsLesson 11: Unknown Angle Proofs—Proofs of Known Facts

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G-CO.2 C Transformations/Rigid Motions 10

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G-CO.3G-CO.4G-CO.5G-CO.6G-CO.7

G-CO.12

Lesson 12: Transformations—The Next LevelLesson 13: RotationsLesson 14: ReflectionsLesson 15: Rotations, Reflections, and SymmetryLesson 16: TranslationsLesson 17: Characterize Points on a Perpendicular BisectorLesson 18: Looking More Carefully at Parallel LinesLesson 19: Construct and Apply a Sequence of Rigid MotionsLesson 20: Applications of Congruence in Terms of Rigid Motions

Mid-Module Assessment: Topic A through C (assessment 1 day, remediation or further applications1 day)

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G-CO.7G-CO.8

D CongruenceLesson 22: Congruence Criteria for Triangles—SASLesson 23: Base Angles of Isosceles TrianglesLesson 24: Congruence Criteria for Triangles—ASA and SSSLesson 25: Congruence Criteria for Triangles—AAS and HLLesson 26: Triangle Congruency Proofs—Part ILesson 27: Triangle Congruency Proofs—Part II

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G-CO.9G-CO.10G-CO.11

E Proving Properties of Geometric FiguresLesson 28: Properties of ParallelogramsLessons 29-30: Special Lines in Triangles

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G-CO.13 F Advanced ConstructionsLesson 31: Construct a Square and a Nine-Point CircleLesson 32: Construct a Nine-Point Circle

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G-CO.1G-CO.2G-CO.3G-CO.4G-CO.5G-CO.6

G Axiomatic SystemsLessons 33-34: Review of the Assumptions

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G-CO.7G-CO.8G-CO.9

G-CO.10G-CO.11G-CO.12G-CO.13

End-of-Module Assessment: Topics A through G (assessment 1 day, remediation or furtherapplications 1 day)

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Total Number of Instructional Days 45

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2 Similarity, Proof, and TrigonometryG-SRT.A.1, G-SRT.B.4, G-SRT.A.2, G-SRT.A.3,

G-SRT.B.5, G-MG.A.1G-CO.12, G-SRT.C.6G-SRT.C.7, G-SRT.C.8

45 Nov. 17 – Feb. 9

Just as rigid motions are used to define congruence in Module 1, so dilations are added to define similarity in Module 2. To be ableto define similarity, there must be a definition of similarity transformations and consequently a definition for dilations. Students areintroduced to the progression of terms beginning with scale drawings, which they first study in Grade 7 (Module 1, Topic D), but in amore observational capacity than in Grade 10: Students determine the scale factor between a figure and a scale drawing or predict thelengths of a scale drawing, provided a figure and a scale factor.

Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. Theyidentify criteria for similarity of triangles, make sense of and persevere in solving similarity problems, and apply similarity to righttriangles to prove the Pythagorean Theorem. Students attend to precision in showing that trigonometric ratios are well defined, andapply trigonometric ratios to find missing measures of general (not necessarily right) triangles. Students model and make sense out ofindirect measurement problems and geometry problems that involve ratios or rates.


G-SRT.A.1G-SRT.B.4

A Scale DrawingsLesson 1: Scale DrawingsLesson 2: Making Scale Drawings Using the Ratio MethodLesson 3: Making Scale Drawings Using the Parallel MethodLesson 4: Comparing the Ratio Method with the Parallel MethodLesson 5: Scale Factors

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G-SRT.A.1G-SRT.B.4

B DilationsLesson 6: Dilations as Transformations of the PlaneLesson 7: How Do Dilations Map Segments?Lesson 8: How Do Dilations Map Lines, Rays, and Circles?Lesson 9: How Do Dilations Map Angles?Lesson 10: Dividing the King’s Foot into 12 Equal Pieces

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Lesson 11: Dilations from Different Centers

G-SRT.A.2G-SRT.A.3G-SRT.B.5G-MG.A.1G-CO.12

C Similarity and DilationsLesson 12: What Are Similarity Transformations, and Why Do We Need Them?Lesson 13: Properties of Similarity TransformationsLesson 14: SimilarityLesson 15: The Angle-Angle (AA) Criterion for Two Triangles to be SimilarLesson 16: Between-Figure and Within-Figure RatiosLesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be SimilarLesson 18: Similarity and the Angle Bisector TheoremLesson 19: Families of Parallel Lines and the Circumference of the EarthLesson 20: How Far Away Is the Moon?

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Mid-Module Assessment: Topic A through C (assessment 1 day, remediation or further applications 1 day) 2

G-SRT.B.4 D Applying Similarity to Right TrianglesLesson 21: Special Relationships Within Right Triangles—Dividing into Two Similar Sub-TrianglesLesson 22: Multiplying and Dividing Expressions with RadicalsLesson 23: Adding and Subtracting with Expressions with RadicalsLesson 24: Prove the Pythagorean Theorem Using Similarity

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G-SRT.C.6G-SRT.C.7 G-

SRT.C.8

E TrigonometryLesson 25: Incredibly Useful RatiosLesson 26: The Definition of Sine, Cosine, and TangentLesson 27: Sine and Cosine of Complementary Angles and Special AnglesLesson 28: Solving Problems Using Sine and CosineLesson 29: Applying TangentsLesson 30: Trigonometry and the Pythagorean TheoremLesson 31: Using Trigonometry to Determine AreaLesson 32: Using Trigonometry to Find Side Lengths of an Acute TriangleLesson 33: Applying the Laws of Sines and Cosines

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End-of-Module Assessment: Topics A through E (assessment 1 day, remediation or further applications 1day)

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3 Extending to Three DimensionsG-GMD.A.1, G-GMD.A.3, G-GMD.B.4, G.MG.A.1,

G.MG.A.2, G.MG.A.317 Feb 10 – Mar. 11

Module 3, Extending to Three Dimensions, builds on students’ understanding of congruence in Module 1 and similarity in Module 2to prove volume formulas for solids. Students’ experience with two-dimensional and three-dimensional objects is extended to includeinformal explanations of circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensionalshapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line. They reason abstractlyand quantitatively to model problems using volume formulas.


G-GMD.A.1 A AreaLesson 1: What is Area?Lesson 2: Properties of AreaLesson 3: The Scaling Principle for AreaLesson 4: Proving the Area of a Disk

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G-GMD.A.1G-GMD.A .3G-GMD.B.4G.MG.A.1G.MG.A.2G.MG.A.3

B VolumeLesson 5: Three-Dimensional SpaceLesson 6: General Prisms and Cylinders and Their Cross-SectionsLesson 7: General Pyramids and Cones and Their Cross-SectionsLesson 8: Definition and Properties of VolumeLesson 9: Scaling Principle for VolumesLesson 10: The Volume of Prisms and Cylinders and Cavalieri’s PrincipleLesson 11: The Volume Formula of a Pyramid and ConeLesson 12: The Volume Formula of a SphereLesson 13: How do 3D Printers Work?

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End-of-Module Assessment: Topics A through B (assessment 1 day, remediation or further applications 1day)

2


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4Connecting Algebra and Geometry

through CoordinatesG-SRT.A.1, G-SRT.B.4, G-SRT.C.6,

G-SRT.C.7 G-SRT.C.827 Mar. 14 – Apr. 26

Building on their work with the Pythagorean Theorem in 8th grade to find distances, students analyze geometric relationships in thecontext of a rectangular coordinate system, including properties of special triangles and quadrilaterals and slopes of parallel andperpendicular lines, relating back to work done in the first module. Students attend to precision as they connect the geometric andalgebraic definitions of parabola. They solve design problems by representing figures in the coordinate plane, and in doing so, theyleverage their knowledge from synthetic geometry by combining it with the solving power of algebra inherent in analytic geometry.

In this module, students explore and experience the utility of analyzing algebra and geometry challenges through the framework ofcoordinates. The module opens with a modeling challenge (G-MG.A.1, G-MG.A.3), one that reoccurs throughout the lessons, to usecoordinate geometry to program the motion of a robot that is bound within a certain polygonal region of the plane—the room in whichit sits. To set the stage for complex work in analytic geometry (computing coordinates of points of intersection of lines and linesegments or the coordinates of points that divide given segments in specific length ratios, and so on), students will describe the regionvia systems of algebraic inequalities (A-REI.D.12) and work to constrain the robot motion along line segments within the region (A-REI.C.6, G-GPE.B.7).


G-SRT.A.1G-SRT.B.4

A Rectangular and Triangular Regions Defined by Inequalities (G-GPE.B.7)Lesson 1: Searching a Region in the PlaneLesson 2: Finding Systems of Inequalities That Describe Triangular and Rectangular RegionsLesson 3: Lines That Pass Through RegionsLesson 4: Designing a Search Robot to Find a Beacon

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G-SRT.A.1G-SRT.B.4

B Perpendicular and Parallel Lines in the Cartesian Plane (G-GPE.B.4, G-GPE.B.5)Lesson 5: Criterion for PerpendicularityLesson 6: Segments That Meet at Right AnglesLesson 7: Equations for Lines Using Normal SegmentsLesson 8: Parallel and Perpendicular Lines

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Mid-Module Assessment: Topic A through C (assessment 1 day, remediation or further applications1 day)

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G-SRT.B.4 C Perimeters and Areas of Polygonal Regions in the Cartesian Plane (G-GPE.B.7)Lesson 9: Perimeter and Area of Triangles in the Cartesian PlaneLesson 10: Perimeter and Area of Polygonal Regions in the Cartesian PlaneLesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities

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G-SRT.C.6G-SRT.C.7G-SRT.C.8

D Partitioning and Extending Segments and Parameterization of Lines (G-GPE.B.4, G-GPE.B.6)Lesson 12: Dividing Segments ProportionatelyLesson 13: Analytic Proofs of Theorems Previously Proved by Synthetic MeansLesson 14: Motion Along a Line—Search Robots Again (Optional)Lesson 15: The Distance from a Point to a Line

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End-of-Module Assessment: Topics A through E (assessment 1 day, remediation or furtherapplications 1 day)

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5Circles with and Without

CoordinatesG-C.A.1, G-C.A.2, G-C.A.3, , G-C.B.5, G-GPE.A.1, G-

GPE.B.4,28 Apr. 28 – June 7

This module’s focus is on the possible geometric relationships between a pair of intersecting lines and a circle drawn on the page. Ifthe lines are perpendicular and one passes through the center of the circle, then the relationship encompasses the perpendicularbisectors of chords in a circle and the association between a tangent line and a radius drawn to the point of contact. If the lines meet ata point on the circle, then the relationship involves inscribed angles. If the lines meet at the center of the circle, then the relationshipinvolves central angles. If the lines meet at a different point inside the circle or at a point outside the circle, then the relationshipincludes the secant angle theorems and tangent angle theorems.

Topic A, through a hands-on activity, leads students first to Thales’ theorem (an angle drawn from a diameter of a circle to a point onthe circle is sure to be a right angle), then to possible converses of Thales’ theorem, and finally to the general inscribed-central angletheorem. Students use this result to solve unknown angle problems. Through this work, students construct triangles and rectanglesinscribed in circles and study their properties (G-C.A.2, G-C.A.3).

Topic B defines the measure of an arc and establishes results relating chord lengths and the measures of the arcs they subtend.Students build on their knowledge of circles from Module 2 and prove that all circles are similar. Students develop a formula for arclength in addition to a formula for the area of a sector and practice their skills solving unknown area problems (G-C.A.1, G-C.A.2, G-C.B.5).

In Topic C, students explore geometric relations in diagrams of two secant lines, or a secant and tangent line (possibly even twotangent lines), meeting a point inside or outside of a circle. They establish the secant angle theorems and tangent-secant angletheorems. By drawing auxiliary lines, students also notice similar triangles and thereby discover relationships between lengths of linesegments appearing in these diagrams (G-C.A.2, G-C.A.3, G-C.A.4).Topic D brings in coordinate geometry to establish the equation of a circle. Students solve problems to find the equations of specifictangent lines or the coordinates of specific points of contact. They also express circles via analytic equations (G-GPE.A.1, G-GPE.B.4).

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The module concludes with Topic E focusing on the properties of quadrilaterals inscribed in circles and establishing Ptolemy’stheorem. This result codifies the Pythagorean theorem, curious facts about triangles, properties of the regular pentagon, andtrigonometric relationships. It serves as a final unifying flourish for students’ year-long study of geometry (G-C.A.3).


G-C.A.2G-C.A.3

A Topic A: Central and Inscribed Angles (G-C.A.2, G-C.A.3)Lesson 1: Thales’ TheoremLesson 2: Circles, Chords, Diameters, and Their RelationshipsLesson 3: Rectangles Inscribed in CirclesLesson 4: Experiments with Inscribed AnglesLesson 5: Inscribed Angle Theorem and Its ApplicationsLesson 6: Unknown Angle Problems with Inscribed Angles in Circles

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G-C.A.1G-C.A.2G- C.B.5

B Arcs and Sectors (G-C.A.1, G-C.A.2, G- C.B.5)Lesson 7: The Angle Measure of an ArcLesson 8: Arcs and ChordsLesson 9: Arc Length and Areas of SectorsLesson 10: Unknown Length and Area Problems

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Mid-Module Assessment: Topics A through B (assessment 1 day, return, remediation, or furtherapplications 1 day)

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G-C.A.2G-C.A.3

C Secants and Tangents (G-C.A.2, G-C.A.3)Lesson 11: Properties of TangentsLesson 12: Tangent SegmentsLesson 13: The Inscribed Angle Alternate—A Tangent AngleLesson 14: Secant Lines; Secant Lines That Meet Inside a CircleLesson 15: Secant Angle Theorem, Exterior CaseLesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams

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G-GPE.A.1G-GPE.B.4

D Equations for Circles and Their Tangents (G-GPE.A.1, G-GPE.B.4)Lesson 17: Writing the Equation for a CircleLesson 18: Recognizing Equations of CirclesLesson 19: Equations for Tangent Lines to Circles

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G-C.A.3 E Topic E: Cyclic Quadrilaterals and Ptolemy’s Theorem (G-C.A.3)Lesson 20: Cyclic QuadrilateralsLesson 21: Ptolemy’s Theorem

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End-of-Module Assessment: Topics C through E (assessment 1 day, remediation or furtherapplications 1 day)

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WORD WALLS ARE DESIGNED …

to promote group learning support the teaching of important general principles about words and how they work Foster reading and writing in content area Provide reference support for children during their reading and writing Promote independence on the part of young students as they work with words Provide a visual map to help children remember connections between words and the

characteristics that will help them form categories Develop a growing core of words that become part of their vocabulary

Important Notice A Mathematics Word Wall must be present in every mathematics classroom.

Math Word Wall

Create a math wordwall

Place math words onyour current wordwall but highlightthem in some way.

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SETUP OF THE MATHEMATICS CLASSROOM

I. Prerequisites for a Mathematics Classroom Teacher Schedule Class List Seating Chart Code of Conduct / Discipline Grade Level Common Core Learning Standards (CCLS) Updated Mathematics Student Work Mathematics Grading Policy Mathematics Diagrams, Charts, Posters, etc. Grade Level Number Line Grade Level Mathematics Word Wall Mathematics Portfolios Mathematics Center with Manipulatives (Grades K - 12)

II. Updated Student WorkA section of the classroom must display recent student work. This can be of anytype of assessment, graphic organizer, and writing activity. Teacher feedback mustbe included on student’s work.

III. Board Set-UpEvery day, teachers must display the Lesson # and Title, Objective(s), CommonCore Learning Standard(s), Opening Exercise and Homework. At the start ofthe class, students are to copy this information and immediately begin on theFluency Activity or Opening Exercise.

IV. Spiraling HomeworkHomework is used to reinforce daily learning objectives. The secondary purposeof homework is to reinforce objectives learned earlier in the year. Theassessments are cumulative, spiraling homework requires students to reviewcoursework throughout the year.

Student’s Name: School:

Teacher’s Name: Date:

Lesson # and Title:

Objective(s)

CCLS:

Opening Exercise:

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SECONDARY MATHEMATICS GRADING POLICY

This course of study includes different components, each of which are assigned the

following percentages to comprise a final grade. I want you--the student--to understand

that your grades are not something that I give you, but rather, a reflection of the work

that you give to me.

COMPONENTS

1. Common Assessments → 35%

2. Quizzes → 20%

3. Homework → 20%

4. Notebook and/or Journal → 10%

5. Classwork / Class Participation → 15%

o Class participation will play a significant part in the determination of your

grade. Class participation will include the following: attendance, punctuality

to class, contributions to the instructional process, effort, contributions during

small group activities and attentiveness in class.

Important Notice

As per MVCSD Board Resolution 06-71, the Parent Notification Policy states

“Parent(s) / guardian(s) or adult students are to be notified, in writing, at any time during

a grading period when it is apparent - that the student may fail or is performing

unsatisfactorily in any course or grade level. Parent(s) / guardian(s) are also to be

notified, in writing, at any time during the grading period when it becomes evident that

the student's conduct or effort grades are unsatisfactory.”

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SAMPLE NOTEBOOK SCORING RUBRIC

Student Name:________________________________________________

Teacher Name:___________________________________________

Criteria 4 3 2 1 Points

Completion ofRequired Sections

All requiredsections arecomplete.

One requiredsection ismissing.

Two or threerequired sections

are missing.

More than threerequired sections

are missing.

Missing SectionsNo sections of

the notebook aremissing.

One sections ofthe notebook is

missing.

Two sections of thenotebook are

missing.

Three or moresections of thenotebook are

missing.

Headers / Footers

No requiredheader(s) and/or

footer(s) aremissing within

notebook.

One or tworequired

header(s) and/orfooter(s) are

missing withinnotebook.

Three or fourrequired header(s)and/or footer(s) are


More than fourrequired header(s)and/or footer(s) are


Organization

All assignmentand/or notes arekept in a logical

or numericalsequence.

One or twoassignments

and/or notes arenot in a logical or

numericalsequence.

Three or Fourassignments and/ornotes are not in a

logical ornumericalsequence.

More than fourassignments and/ornotes are not in a

logical ornumericalsequence.

NeatnessOverall notebookis kept very neat.

Overall notebookis kept in asatisfactorycondition.

Overall notebook iskept in a below

satisfactorycondition.

Overall notebook isunkept and very

disorganized.

Total

Teacher’s Comments:

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CLASSROOM AESTHETICS

“PRINT–RICH” ENVIRONMENT CONDUCIVE TO LEARNING

TEACHER NAME: _________________________________________________________

COURSE / PERIOD: _________________________________________________________

ROOM: _________________________________________________________

CHECKLISTYES NO

Teacher Schedule

Class List

Seating Chart

Code of Conduct / Discipline

Grade Level Mathematics CCLS

Mathematics Grading Policy

Mathematics Diagrams, Posters, Displays, etc.

Grade Level Number Line

Updated Student Work (Projects, Assessments, Writing, etc.)

Updated Student Portfolios

Updated Grade Level Mathematics Word-Wall

Mathematics Centers with Manipulatives

Organization of Materials

Cleanliness

Principal Signature: _________________________________________ Date: ____________

Asst. Pri. Signature: _________________________________________ Date: ____________

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SYSTEMATIC DESIGN OF A MATHEMATICS LESSON

What are the components of a Mathematics Block?

ComponentFluency Practice Information processing theory supports the view that automaticity in math facts is

fundamental to success in many areas of higher mathematics. Without the ability to retrievefacts directly or automatically, students are likely to experience a high cognitive load as theyperform a range of complex tasks. The added processing demands resulting from inefficientmethods such as counting (vs. direct retrieval) often lead to declarative and procedural errors.Accurate and efficient retrieval of basic math facts is critical to a student’s success inmathematics.

Opening Exercise - Whole Group This can be considered the motivation or Do Now of the lesson It should set the stage for the day's lesson Introduction of a new concept, built on prior knowledge Open-ended problemsConceptual Development - Whole Group (Teacher Directed, Student Centered) Inform students of what they are going to do. Refer to Objectives. Refer to the Key Words

(Word Wall) Define the expectations for the work to be done Provide various demonstrations using modeling and multiple representations (i.e. model a

strategy and your thinking for problem solving, model how to use a ruler to measure items,model how to use inch graph paper to find the perimeter of a polygon,)

Relate to previous work Provide logical sequence and clear explanations Provide medial summaryApplication Problems - Cooperative Groups, Pairs, Individuals, (Student Interaction &Engagement, Teacher Facilitated) Students try out the skill or concept learned in the conceptual development Teachers circulate the room, conferences with the students and assesses student work (i.e.

teacher asks questions to raise the level of student thinking) Students construct knowledge around the key idea or content standard through the use of

problem solving strategies, manipulatives, accountable/quality talk, writing, modeling,technology applied learning

Student Debrief - Whole Group (Teacher Directed, Student Centered) Students discuss their work and explain their thinking Teacher asks questions to help students draw conclusions and make references Determine if objective(s) were achieved Students summarize what was learned Allow students to reflect, share (i.e. read from journal)Homework/Enrichment - Whole Group (Teacher Directed, Student Centered) Homework is a follow-up to the lesson which may involve skill practice, problem solving

and writing

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Homework, projects or enrichment activities should be assigned on a daily basis. SPIRALLING OF HOMEWORK - Teacher will also assign problems / questions pertaining to

lessons taught in the past

Remember: Assessments are on-going based on students’ responses.Assessment: Independent Practice (It is on-going! Provide formal assessment whennecessary / appropriate) Always write, use and allow students to generate Effective Questions for optimal learning Based on assessment(s), Re-teach the skill, concept or content using alternative strategies

and approaches

Important Notice

All lessons must be numbered with corresponding homework. For example, lesson #1 will

corresponded to homework #1 and so on.

Writing assignments at the end of the lesson (closure) bring great benefits. Not only do they

enhance students' general writing ability, but they also increase both the understanding of

content while learning the specific vocabulary of the disciplines.

Spiraling Homework

o Homework is used to reinforce daily learning objectives. The secondary purpose of

homework is to reinforce objectives learned earlier in the year. The assessments are

cumulative, spiraling homework requires students to review coursework throughout the

year.

Manipulative must be incorporated in all lessons. With students actively involved in

manipulating materials, interest in mathematics will be aroused. Using manipulative

materials in teaching mathematics will help students learn:

a. to relate real world situations to mathematics symbolism.

b. to work together cooperatively in solving problems.

c. to discuss mathematical ideas and concepts.

d. to verbalize their mathematics thinking.

e. to make presentations in front of a large group.

f. that there are many different ways to solve problems.

g. that mathematics problems can be symbolized in many different ways.

that they can solve mathematics problems without just following teachers' directions.

geometry nr 2015 -...

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