MIDDLETOWN PUBLIC SCHOOLS GEOMETRY CURRICULUM. High School. Curriculum Writers: Stephen G. Fagan, MaryBeth Murphy, and Gus Steppen.

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1 MIDDLETOWN PUBLIC SCHOOLS CURRICULUM High Schl January 2012

2 2/20/2012 Middletwn Public Schls 1

3 T he Middletwn Public Schls Mathematics Curriculum fr grades K-12 was cmpleted in January 2012 by a K-12 team f teachers. The team, identified as the Mathematics Task Frce and Mathematics Curriculum Writers referenced extensive resurces t design the dcument that included: Cmmn Cre State Standards fr Mathematics Cmmn Cre State Standards fr Mathematics, Appendix A Understanding Cmmn Cre State Standards, Kendall PARCC Mdel Cntent Framewrks Numerus state curriculum Cmmn Cre framewrks, e.g. Ohi Department f Educatin High Schl Traditinal Plus Mdel Curse Sequence, Achieve, Inc. Grade Level and Grade Span Expectatins (GLEs/GSEs) fr Mathematics Third Internatinal Mathematics and Science Test (TIMSS) Best Practice, New Standards fr Teaching and Learning in America s Schls; Differentiated Instructinal Strategies Instructinal Strategies That Wrk, Marzan Gals fr the district Missin Statement Our missin is t prvide a sequential and cmprehensive K-12 mathematics curriculum in a cllabrative student centered learning envirnment that develps critical thinkers, skillful prblem slvers, and effective cmmunicatrs f mathematics. The Middletwn Public Schls Mathematics Curriculum identifies what students shuld knw and be able t d in mathematics. Each grade r curse includes Cmmn Cre State Standards (CCSS), Grade Level Expectatins (GLEs), Grade Span Expectatins (GSEs), grade level supprtive tasks, teacher ntes, best practice instructinal strategies, resurces, a map (r suggested timeline), rubrics, checklists, and cmmn frmative and summative assessments. COMMON CORE STATE STANDARDS The Cmmn Cre State Standards (CCSS): Are fewer, higher, deeper, and clearer. Are aligned with cllege and wrkfrce expectatins. Include rigrus cntent and applicatins f knwledge thrugh high-rder skills. Build upn strengths and lessns f current state standards (GLEs and GSEs). Are internatinally benchmarked, s that all students are prepared fr succeeding in ur glbal ecnmy and sciety. Are research and evidence-based. Cmmn Cre State Standards cmpnents include: Standards fr Mathematical Practice (K-12) Standards fr Mathematical Cntent: Categries (high schl nly): e.g. numbers, algebra, functins, data Dmains: larger grups f related standards Clusters: grups f related standards Standards: define what students shuld understand and are able t d The Middletwn Public Schls Cmmn Cre Mathematics Curriculum prvides all students with a sequential cmprehensive educatin in mathematics thrugh the study f: Standards fr Mathematical Practice (K-12) Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics* Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning 2/20/2012 Middletwn Public Schls 2

4 Standards fr Mathematical Cntent: K 5 Grade Level Dmains f Cunting and Cardinality Operatins and Algebraic Thinking Number and Operatins in Base Ten Number and Operatins Fractins Measurement and Data Gemetry 6-8 Grade Level Dmains f Ratis and Prprtinal Relatinships The Number System Expressins and Equatins Functins Gemetry 9-12 Grade Level Cnceptual Categries f Number and Quantity Algebra Functins Mdeling Gemetry Statistics and Prbability RESEARCH-BASED The Middletwn Public Schls Cmmn Cre Mathematics Curriculum prvides a list f research-based best practice instructinal strategies that the teacher may mdel and/r facilitate. It is suggested the teacher: Use frmative assessment t guide instructin Prvide pprtunities fr independent, partner and cllabrative grup wrk Differentiate instructin by varying the cntent, prcess, and prduct and prviding pprtunities fr: anchring cubing jig-sawing pre/pst assessments tiered assignments Address multiple intelligences instructinal strategies, e.g. visual, bdily kinesthetic, interpersnal Prvide pprtunities fr higher level thinking: Webb s Depth f Knwledge, 2,3,4, skill/cnceptual understanding, strategic reasning, extended reasning Facilitate the integratin f Mathematical Practices in all cntent areas f mathematics Facilitate integratin f the Applied Learning Standards (SCANS): cmmunicatin critical thinking prblem slving reflectin/evaluatin research Emply strategies f best practice (student-centered, experiential, hlistic, authentic, expressive, reflective, scial, cllabrative, demcratic, cgnitive, develpmental, cnstructivist/heuristic, and challenging) Prvide rubrics and mdels 2/20/2012 Middletwn Public Schls 3

5 Address multiple intelligences and brain dminance (spatial, bdily kinesthetic, musical, linguistic, intrapersnal, interpersnal, mathematical/lgical, and naturalist) Emply mathematics best practice strategies e.g. using manipulatives facilitating cperative grup wrk discussing mathematics questining and making cnjectures justifying f thinking writing abut mathematics facilitating prblem slving apprach t instructin integrating cntent using calculatrs and cmputers facilitating learning using assessment t mdify instructin COMMON S The Middletwn Public Schls Cmmn Cre Mathematics Curriculum includes cmmn assessments. Required (red ink) indicates the assessment is required f all students e.g. cmmn tasks/perfrmance-based tasks, standardized mid-term exam, standardized final exam. Required Assessments Benchmark Prblems Mid-Term Assessment Final Exam Cmmn Prtfli Tasks (2 Anchr Tasks Per Year, HS) NWEA Test Cmmn Instructinal Assessments (I) - used by teachers and students during the instructin f CCSS. Cmmn Frmative Assessments (F) - used t measure hw well students are mastering the cntent standards befre taking state assessments teacher and student use t make decisins abut what actins t take t prmte further learning n-ging, dynamic prcess that invlves far mre frequent testing serves as a practice fr students Cmmn Summative Assessment (S)- used t measure the level f student, schl, r prgram success make sme srt f judgment, e.g. what grade prgram effectiveness e.g. state assessments (AYP), mid-year and final exams Additinal assessments include: Anecdtal recrds Oral presentatins Cnferencing Prblem/Perfrmance based/cmmn tasks Exhibits Rubrics/checklists (mathematical practice, mdeling) Interviews Tests and quizzes Graphic rganizers Technlgy Jurnals Think-aluds Mathematical Practices Writing genres Mdeling Arguments/ pinin Multiple Intelligences assessments, e.g. Infrmative Rle playing - bdily kinesthetic Narrative Graphic rganizing - visual Research Cllabratin - interpersnal 2/20/2012 Middletwn Public Schls 4

6 FOR Textbks Gemetry, McDugal Littell, 2007 Supplementary Bk 1- Gemetry fr Enjyment and Challenge Bk 2 - Gemetry (McDugal Littell) Bk 3 -Gemetry (Prentice Hall) Bk 4 -Gemetry: A Mving Experience (UH CRDG) Gemetry Tls fr a Changing Wrld, Prentice Hall Classrm Instructin That Wrks, Marzan Gemetry: A Mving Experience, Curriculum Research and Develpment Grup, University f Hawaii MCAS Released Tasks NAEP Released Tasks NECAP Released Tasks NWEA MAP Assessments Technlgy Classrm mnitrs with remte tablet wifi Cmputers ELMO GeGebra Graphing Calculatr Interactive bards LCD prjectrs MIMEO Overhead graphing calculatr Plat Smart bard TI Navigatr Websites (Gizm ) Materials Clred pencils/markers Cmpasses Exp markers MIRAs Prtractrs Rulers Student white bards 2/20/2012 Middletwn Public Schls 5

7 Cngruence (G-CO) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Middletwn Public Schls Students experiment with transfrmatins in the plane. G-CO.1 Knw precise definitins f angle, circle, perpendicular line, parallel line, and line segment, based n the undefined ntins f pint, line, distance alng a line, and distance arund a circular arc. G-CO.2 Represent transfrmatins in the plane using, e.g., transparencies and gemetry sftware; describe transfrmatins as functins that take pints in the plane as inputs and give ther pints as utputs. Cmpare transfrmatins that preserve distance and angle t thse that d nt (e.g., translatin versus hrizntal stretch). G-CO.3 Given a rectangle, parallelgram, trapezid, r regular plygn, describe the rtatins and reflectins that carry it nt itself. TEACHER (G-CO1-5) Build n student experience with rigid mtins frm earlier grades. Pint ut the basis f rigid mtins in gemetric cncepts, e.g., translatins mve pints a specified distance alng a line parallel t a specified line; rtatins mve bjects alng a circular arc with a specified center thrugh a specified angle. ODE RESOURCE S G-CO.4 Develp definitins f rtatins, reflectins, and translatins in terms f angles, circles, perpendicular lines, parallel lines, and line segments. G-CO.5 Given a gemetric figure and a rtatin, reflectin, r translatin, draw the transfrmed figure using, e.g., graph paper, tracing paper, r gemetry sftware. Specify a sequence f transfrmatins that will carry a given figure nt anther. BENCHMARK PROBLEMS remved fr public viewing Students understand cngruence in terms f rigid mtins. Cngruence (G-CO) G-CO.6 Use gemetric descriptins f rigid (cngruent) mtins t transfrm figures and t predict the effect f a given rigid mtin n a given figure; given tw figures, TEACHER Understand cngruence in terms f rigid mtins. RESOURCE 2/20/2012 Middletwn Public Schls 6

8 Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Middletwn Public Schls use the definitin f cngruence in terms f rigid mtins t decide if they are cngruent. Applies the cncepts f cngruency by slving prblems n r ff a crdinate plane invlving reflectins translatins rtatins (state assessment) (G&M) 10 4 (2.3.1) G-CO.7 Use the definitin f cngruence in terms f rigid mtins t shw that tw triangles are cngruent if and nly if crrespnding pairs f sides and crrespnding pairs f angles are cngruent. GSEs same as abve (G&M) 10 4 (2.3.1) Rigid mtins are at the fundatin f the definitin f cngruence. Students reasn frm the basic prperties f rigid mtins (that they preserve distance and angle), which are assumed withut prf. Rigid mtins and their assumed prperties can be used t establish the usual triangle cngruence criteria, which can then be used t prve ther therems. ODE TEACHER Same as abve RESOURCE S G-CO.8 Explain hw the criteria fr triangle cngruence (ASA, SAS, and SSS) fllw frm the definitin f cngruence in terms f rigid mtins. TEACHER Same as abve RESOURCE BENCHMARK PROBLEMS remved fr public viewing Students prve gemetric therems. Cngruence (G-CO ) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning G-CO.9 Prve therems abut lines and angles. Therems include: vertical angles are cngruent; when a transversal crsses parallel lines, alternate interir angles are cngruent and crrespnding angles are cngruent; pints n a perpendicular bisectr f a line segment are exactly thse equidistant frm the segment s endpints. Makes and defends cnjectures, cnstructs gemetric arguments, uses gemetric prperties, r uses therems t slve prblems, within mathematics r acrss disciplines r cntexts, invlving angles lines (state assessment) (G&M) 10 2 (2.2.1) Slves prblems using cngruency invlving prblems within mathematics r acrss disciplines r cntexts (state assessment) (G&M) 10 4 (2.3.1) TEACHER Prve gemetric therems. Encurage multiple ways f writing prfs, such as in narrative paragraphs, using flw diagrams, in tw-clumn frmat, and using diagrams withut wrds. Students shuld be encuraged t fcus n the validity f the underlying reasning while explring a variety f frmats fr expressing that reasning. Implementatin f G.CO.10 may be extended t include cncurrence f perpendicular bisectrs and angle bisectrs as preparatin fr G.C.3 in Unit 5. ODE RESOURCE 2/20/2012 Middletwn Public Schls 7

9 Middletwn Public Schls special angles, e.g. vertically ppsite angles angles frmed by a transversal intersecting parallel lines S G-CO.10 Prve therems abut triangles. Therems include: measures f interir angles f a triangle sum t 180 ; base angles f issceles triangles are cngruent; the segment jining midpints f tw sides f a triangle is parallel t the third side and half the length; the medians f a triangle meet at a pint. TEACHER Same as abve RESOURCE G-CO.11 Prve therems abut parallelgrams. Therems include: ppsite sides are cngruent, ppsite angles are cngruent, the diagnals f a parallelgram bisect each ther, and cnversely, rectangles are parallelgrams with cngruent diagnals. Makes and defends cnjectures, cnstructs gemetric arguments, uses gemetric prperties, r uses therems t slve prblems, within mathematics r acrss disciplines r cntexts, invlving angles lines plygns circles area vlume right triangle ratis (sine, csine, tangent) Pythagrean Therem Triangle Inequality Therem (state assessment) (G&M) 10 2 (2.2.1) TEACHER Same as abve RESOURCE BENCHMARK PROBLEMS remved fr public viewing Cngruence (G-CO ) Practices t Students make gemetric cnstructins. G-CO.12 Make frmal gemetric cnstructins with a variety f tls and methds (cmpass and straightedge, string, reflective devices, paper flding, dynamic gemetric sftware, etc.). cpying a segment TEACHER Make gemetric cnstructins. Build n prir student experience with simple cnstructins. Emphasize the ability t frmalize and explain hw these cnstructins RESOURCE 2/20/2012 Middletwn Public Schls 8

10 Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Middletwn Public Schls cpying an angle bisecting a segment bisecting an angle cnstructing perpendicular lines, including the perpendicular bisectr f a line segment cnstructing a line parallel t a given line thrugh a pint nt n the line G-CO.13 Cnstruct an equilateral triangle, a square, and a regular hexagn inscribed in a circle. BENCHMARK PROBLEMS result in the desired bjects. Sme f these cnstructins are clsely related t previus standards and can be intrduced in cnjunctin with them. ODE TEACHER Same as abve RESOURCE S Students understand similarity in terms f similarity transfrmatins. Similarity, Right Triangles, and Trignmetry (G- SRT) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning G-SRT.1 Verify experimentally the prperties f dilatins given by a center and a scale factr: a. A dilatin takes a line nt passing thrugh the center f the dilatin t a parallel line, and leaves a line passing thrugh the center unchanged. (G-SRT.1a) b. The dilatin f a line segment is lnger r shrter in the rati given by the scale factr. (G-SRT.1b) Applies cncepts f similarity by slving prblems within mathematics r acrss disciplines r cntexts. (state assessment) (G&M) 10 5 (2.4.1) dilatins G-SRT.2 Given tw figures, use the definitin f similarity in terms f similarity transfrmatins t decide if they are similar; explain using similarity transfrmatins the meaning f similarity fr triangles as the equality f all crrespnding pairs f angles and the prprtinality f all crrespnding pairs f sides. Applies cncepts f similarity by slving prblems within mathematics r acrss disciplines r cntexts. (state assessment) (G&M) 10 5 (2.4.1) ratis/prprtins TEACHER Understand similarity in terms f similarity transfrmatins. TEACHER Understand similarity in terms f similarity transfrmatins. TEACHER Understand similarity in terms f similarity transfrmatins. RESOURCE RESOURCE RESOURCE G-SRT.3 Use the prperties f similarity transfrmatins t establish the AA criterin fr tw triangles t be similar. BENCHMARK PROBLEMS remved fr public viewing TEACHER Understand similarity in terms f similarity transfrmatins. RESOURCE 2/20/2012 Middletwn Public Schls 9

11 Middletwn Public Schls S Students prve therems invlving similarity. Similarity, Right Triangles, and Trignmetry (G- SRT) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning G-SRT.4 Prve therems abut triangles. Therems include: a line parallel t ne side f a triangle divides the ther tw prprtinally, and cnversely; the Pythagrean Therem prved using triangle similarity. Makes and defends cnjectures, cnstructs gemetric arguments, uses gemetric prperties, r uses therems t slve prblems, within mathematics r acrss disciplines r cntexts, invlving angles lines plygns circles area vlume right triangle ratis (sine, csine, tangent) Pythagrean Therem Triangle Inequality Therem (state assessment) (G&M) 10 2 (2.2.1) TEACHER Prve therems invlving similarity. RESOURCE G-SRT.5 Use cngruence and similarity criteria fr triangles t slve prblems and t prve relatinships in gemetric figures. TEACHER Prve therems invlving similarity RESOURCE BENCHMARK PROBLEMS remved fr public viewing Similarity, Right Triangles, and Trignmetry (G- SRT) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Students define trignmetric ratis and slve prblems invlving right triangles. G-SRT.6 Understand that by similarity, side ratis in right triangles are prperties f the angles in the triangle, leading t definitins f trignmetric ratis fr acute angles. Applies the cncepts f similarity f right triangles with the trignmetric functins defined as ratis f sides f triangles (D) M(G&M) 12 5 (2.4.2) G-SRT.7 Explain and use the relatinship between the sine and csine f cmplementary angles. TEACHER Define trignmetric ratis and slve prblems invlving right triangles. TEACHER Define trignmetric ratis and slve prblems invlving right triangles. TEACHER RESOURCE RESOURCE 2/20/2012 Middletwn Public Schls 10

12 Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Middletwn Public Schls G-SRT.8 Use trignmetric ratis and the Pythagrean Therem t slve right triangles in applied prblems. Applies the cncepts f similarity f right triangles with the trignmetric functins defined as ratis f sides f triangles (D) M(G&M) 12 5 (2.4.2) BENCHMARK PROBLEMS remved fr public viewing Define trignmetric ratis and slve prblems invlving right triangles. RESOURCE S Similarity, Right Triangles, and Trignmetry (G- SRT) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Students apply trignmetry t general triangles. G-SRT.9 (+) Derive the frmula A = 1/2 ab sin(c) fr the area f a triangle by drawing an auxiliary line frm a vertex perpendicular t the ppsite side. (Trig and Pre-Cal) G-SRT.10 (+) Prve the Laws f Sines and Csines and use them t slve prblems. (Hnrs Algebra II and Trig) G-SRT.11 (+) Understand and apply the Law f Sines and the Law f Csines t find unknwn measurements in right and nn-right triangles (e.g., surveying prblems, resultant frces). (Hnrs Algebra II and Trig) BENCHMARK PROBLEMS TEACHER Apply trignmetry t general triangles. With respect t the general case f the Laws f Sines and Csines, the definitins f sine and csine must be extended t btuse angles. TEACHER Same as abve TEACHER Same as abve RESOURCE RESOURCE RESOURCE Students understand and apply therems abut circles. Circles (G-C) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f G-C.1 Prve that all circles are similar. Creates frmal prfs f prpsitins e.g. circles M(G&M) 10 2 (2.1.1) G-C.2 Identify and describe relatinships amng inscribed angles, radii, and chrds. Include the relatinship between central, inscribed, and circumscribed angles; inscribed angles n a diameter are right angles; the radius f a circle is perpendicular t the tangent where the radius intersects the circle. TEACHER Understand and apply therems abut circles. TEACHER Understand and apply therems abut circles. RESOURCE RESOURCE 2/20/2012 Middletwn Public Schls 11

13 Lk fr and express reasning Middletwn Public Schls Makes and defends cnjectures, cnstructs gemetric arguments, uses gemetric prperties, r uses therems t slve prblems, within mathematics r acrss disciplines r cntexts, invlving circles (state assessment) (G&M) 10 2 (2.2.1) S G-C.3 Cnstruct the inscribed and circumscribed circles f a triangle, and prve prperties f angles fr a quadrilateral inscribed in a circle. TEACHER Understand and apply therems abut circles. RESOURCE G-C.4 (+) Cnstruct a tangent line frm a pint utside a given circle t the circle. BENCHMARK PROBLEMS remved fr public viewing TEACHER Understand and apply therems abut circles. RESOURCE Circles (G-C) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Students find arc lengths and areas f sectrs f circles. G-C.5 Derive using similarity the fact that the length f the arc intercepted by an angle is prprtinal t the radius, and define the radian measure f the angle as the cnstant f prprtinality; derive the frmula fr the area f a sectr. BENCHMARK PROBLEMS TEACHER Radian intrduced nly as unit f measure Find arc lengths and areas f sectrs f circles. Emphasize the similarity f all circles. Nte that by similarity f sectrs with the same central angle, arc lengths are prprtinal t the radius. Use this as a basis fr intrducing radian as a unit f measure. It is nt intended that it be applied t the develpment f circular trignmetry in this curse. ODE RESOURCE Expressing Gemetric Prperties with Equatins (G-GPE) Students translate between the gemetric descriptin and the equatin fr a cnic sectin. G-GPE.1 Derive the equatin f a circle f given center and radius using the Pythagrean Therem; cmplete the square t find the center and radius f a circle given by an equatin. TEACHER Translate between the gemetric descriptin and the equatin fr a cnic sectin. RESOURCE G-GPE.2 Derive the equatin f a parabla given a fcus and directrix. TEACHER RESOURCE 2/20/2012 Middletwn Public Schls 12

14 Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning BENCHMARK PROBLEMS Middletwn Public Schls Translate between the gemetric descriptin and the equatin fr a cnic sectin. The directrix shuld be parallel t a crdinate axis. S Expressing Gemetric Prperties with Equatins (G-GPE) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Students use crdinates t prve simple gemetric therems algebraically. G-GPE.4 Use crdinates t prve simple gemetric therems algebraically. Fr example, prve r disprve that a figure defined by fur given pints in the crdinate plane is a rectangle; prve r disprve that the pint (1, 3) lies n the circle centered at the rigin and cntaining the pint (0, 2). Slves prblems n and ff the crdinate plane invlving distance midpint perpendicular parallel lines slpe (state assessment) (G&M) 10 9 (2.7.1) G-GPE.5 Prve the slpe criteria fr parallel and perpendicular lines and use them t slve gemetric prblems (e.g., find the equatin f a line parallel r perpendicular t a given line that passes thrugh a given pint). GSEs same as abve (G&M) 10 9 (2.7.1) G-GPE.6 Find the pint n a directed line segment between tw given pints that partitins the segment in a given rati. Slves prblems using cngruency invlving prblems within mathematics r acrss disciplines r cntexts (state assessment) (G&M) 10 4 (2.3.2) segments distance frmula midpint frmula G-GPE.7 Use crdinates t cmpute perimeters f plygns and areas f triangles and rectangles, e.g., using the distance frmula. TEACHER Use crdinates t prve simple gemetric therems algebraically. This unit has a clse cnnectin with the next unit. Fr example, a curriculum might merge G.GPE.1 and the Unit 5 treatment f G.GPE.4 with the standards in this unit. Reasning with triangles in this unit is limited t right triangles; e.g., derive the equatin fr a line thrugh tw pints using similar right triangles. Use crdinates t prve simple gemetric therems algebraically. Include simple prfs invlving circles. ODE TEACHER Relate wrk n parallel lines in G.GPE.5 t wrk n A.REI.5 in High Schl Algebra I invlving systems f equatins having n slutin r infinitely many slutins. TEACHER TEACHER G.GPE.7 prvides practice with the RESOURCE RESOURCE RESOURCE RESOURCE 2/20/2012 Middletwn Public Schls 13

15 Middletwn Public Schls GSEs same as abve (G&M) 10 4 (2.3.2) BENCHMARK PROBLEMS remved fr public viewing distance frmula and its cnnectin with the Pythagrean therem. S Gemetric Measurement and Dimensin (G-GMD) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Students explain vlume frmulas and use them t slve prblems. G-GMD.1 Give an infrmal argument fr the frmulas fr the circumference f a circle, area f a circle, vlume f a cylinder, pyramid, and cne. Use dissectin arguments, Cavalieri s principle, and infrmal limit arguments. Makes and defends cnjectures, cnstructs gemetric arguments, uses gemetric prperties, r uses therems t slve prblems, within mathematics r acrss disciplines r cntexts, invlving area/vlume (state assessment) (G&M) 10 2 (2.2.1) Slves prblems invlving circumference within mathematics r acrss disciplines r cntexts. (state assessment) (G&M) 10 6 (2.5.1) G-GMD.3 Use vlume frmulas fr cylinders, pyramids, cnes, and spheres t slve prblems. GSEs same as abve (G&M) 10 6 (2.5.1) BENCHMARK PROBLEMS remved fr public viewing TEACHER Explain vlume frmulas and use them t slve prblems. Infrmal arguments fr area and vlume frmulas can make use f the way in which area and vlume scale under similarity transfrmatins: when ne figure in the plane results frm anther by applying a similarity transfrmatin with scale factr k, its area is k 2 times the area f the first. Similarly, vlumes f slid figures scale by k 3 under a similarity transfrmatin with scale factr k. ODE TEACHER Same as abve RESOURCE RESOURCE Students visualize relatinships between tw- dimensinal and three-dimensinal bjects. Gemetric Measurement and Dimensin (G-GMD) G-GMD.4 Identify the shapes f tw-dimensinal crss-sectins f three-dimensinal bjects, and identify three-dimensinal bjects generated by rtatins f twdimensinal bjects. TEACHER Visualize the relatin between twdimensinal and three-dimensinal bjects. RESOURCE Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f BENCHMARK PROBLEMS 2/20/2012 Middletwn Public Schls 14

16 Lk fr and express reasning Middletwn Public Schls S Mdeling with Gemetry (G-MG) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Students apply gemetric cncepts in mdeling situatins. G-MG.1 Use gemetric shapes, their measures, and their prperties t describe bjects (e.g., mdeling a tree trunk r a human trs as a cylinder). G-MG.2 Apply cncepts f density based n area and vlume in mdeling situatins (e.g., persns per square mile, BTUs per cubic ft). TEACHER Apply gemetric cncepts in mdeling situatins. Fcus n situatins well mdeled by trignmetric ratis fr acute angles. Fcus n situatins that require relating tw- and three-dimensinal bjects, determining and using vlume, and the trignmetry f general triangles Fcus n situatins in which the analysis f circles is required. ODE TEACHER Same as abve RESOURCE RESOURCE G-MG.3 Apply gemetric methds t slve design prblems (e.g., designing an bject r t satisfy physical cnstraints r minimize cst; wrking with typgraphic grid systems based n ratis). TEACHER Same as abve RESOURCE BENCHMARK PROBLEMS STATISTICS AND PROBABILITY Cnditinal Prbability and the Rules f Prbability (S-CP) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments Students understand independence and cnditinal prbability and use them t interpret data. S-CP.1 Describe events as subsets f a sample space (the set f utcmes) using characteristics (r categries) f the utcmes, r as unins, intersectins, r cmplements f ther events ( r, and, nt ). S-CP.2 Understand that tw events A and B are independent if the prbability f A and B TEACHER Understand independence and cnditinal prbability and use them t interpret data. Build n wrk with tw-way tables frm Algebra I Unit 3 (S.ID.5) t develp understanding f cnditinal prbability and independence. ODE TEACHER RESOURCE RESOURCE 2/20/2012 Middletwn Public Schls 15

17 and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Middletwn Public Schls ccurring tgether is the prduct f their prbabilities, and use this characterizatin t determine if they are independent. (S-CP.3) Understand the cnditinal prbability f A given B as P(A and B)/P(B), and Interpret independence f A and B as saying that the cnditinal prbability f A given B is the same as the prbability f A, and the cnditinal prbability f B given A is the same as the prbability f B. Same as abve TEACHER Same as abve RESOURCE S (S-CP.4) Cnstruct and interpret tw-way frequency tables f data when tw categries are assciated with each bject being classified. Use the tw-way table as a sample space t decide if events are independent and t apprximate cnditinal prbabilities. Fr example, cllect data frm a randm sample f students in yur schl n their favrite subject amng math, science, and English. Estimate the prbability that a randmly selected student frm yur schl will favr science given that the student is in tenth grade. D the same fr ther subjects and cmpare the results. In respnse t a teacher r student generated questin r hypthesis decides the mst effective methd (e.g., survey, bservatin, research, experimentatin) and sampling techniques (e.g., randm sample, stratified randm sample) t cllect the data necessary t answer the questin cllects, rganizes, and apprpriately displays the data; analyzes the data t draw cnclusins abut the questins r hyptheses being tested while cnsidering the limitatins f the data that culd affect interpretatins when apprpriate makes predicatins, asks new questins, r makes cnnectins t real-wrld situatins. (DSP) 10 6 (4.6.1) TEACHER Same as abve RESOURCE (S-CP.5) Recgnize and explain the cncepts f cnditinal prbability and independence in everyday language and everyday situatins. Fr example, cmpare the chance f having lung cancer if yu are a smker with the chance f being a smker if yu have lung cancer. TEACHER Link t data frm simulatins r experiments Same as abve RESOURCE BENCHMARK PROBLEMS STATISTICS AND PROBABILITY Students use the rules f prbability t cmpute prbabilities f cmpund events in a unifrm prbability mdel. TEACHER RESOURCE 2/20/2012 Middletwn Public Schls 16

18 Cnditinal Prbability and the Rules f Prbability (S-CP) Middletwn Public Schls (S-CP.6) Find the cnditinal prbability f A given B as the fractin f B s utcmes that als belng t A, and interpret the answer in terms f the mdel. Use the rules f prbability t cmpute prbabilities f cmpund events in a unifrm prbability mdel. S Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning (S-CP.7) Apply the Additin Rule, P(A r B) = P(A) + P(B) P(A and B), and interpret the answer in terms f the mdel. (S-CP.8) (+) Apply the general Multiplicatin Rule in a unifrm prbability mdel, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms f the mdel. (4 th year curse/hnrs) (S-CP.9) (+) Use permutatins and cmbinatins t cmpute prbabilities f cmpund events and slve prblems. (4 th year curse/hnrs) BENCHMARK PROBLEMS remved fr public viewing TEACHER Same as abve TEACHER Same as abve TEACHER Same as abve RESOURCE RESOURCE RESOURCE STATISTICS AND PROBABILITY Using Prbability t Make Decisins (S-MD) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning MODELING Students use prbability t evaluate utcmes f decisins. (S-MD.6) (+) Use prbabilities t make fair decisins (e.g., drawing by lts, using a randm number generatr). (S-MD.7) (+) Analyze decisins and strategies using prbability cncepts (e.g., prduct testing, medical testing, pulling a hckey galie at the end f a game). Students BENCHMARK PROBLEMS TEACHER Use prbability t evaluate utcmes f decisins. This unit sets the stage fr wrk in Algebra II, where the ideas f statistical inference are intrduced. Evaluating the risks assciated with cnclusins drawn frm sample data (i.e. incmplete infrmatin) requires an understanding f prbability cncepts. ODE TEACHER Same as abve RESOURCE RESOURCE Chsing and using Understand and use descriptive mdeling which simply describes the apprpriate phenmena r summarizes them in a cmpact frm. Graphs f mathematics and bservatins are a familiar descriptive mdel - fr example, graphs f statistics t glbal temperature and atmspheric CO 2 ver time. 2/20/2012 Middletwn Public Schls 17

19 analyze empirical situatins Middletwn Public Schls Understand that analytical mdeling seeks t explain data n the basis f deeper theretical ideas, albeit with parameters that are empirically based; fr example, expnential grwth f bacterial clnies (until cut-ff mechanics such as pllutin r starvatin intervene) fllws a cnstant reprductin rate. Functins are an imprtant tl fr analyzing such prblems. S Use graphing utilities, spreadsheets, cmputer algebra systems, and dynamic gemetry sftware as pwerful tls that can be used t mdel purely mathematical phenmena (e.g. the behavir f plynmials) as well as physical phenmena. Understands and use the basic mdeling cycle : Prblem: Identifying variables in the situatin and selecting thse that represent essential features Frmulate: frmulating a mdel by creating and selecting gemetric, graphical, tabular, algebraic r statistical representatins that describe relatinships between the variables Cmpute: analyzing and perfrming peratins n these relatinships t draw cnclusins Interpret: interpreting the results f the mathematics in terms f the riginal situatin Validate: validating the cnclusins by cmparing them with the situatin, and then either imprving the mdel r, if it is acceptable Reprt: reprting n the cnclusins and the reasning behind them. 2/20/2012 Middletwn Public Schls 18

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