California Common Core Content Standards: Math 9-12 Geometry

Transcription

1 Califrnia Cmmn Cre Cntent Standards: Math 9-12 Gemetry 9-12.G Gemetry 9-12.G-CO Cngruence Experiment with transfrmatins in the plane 9-12.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. Angle vcabulary (G-C.1) 9-12.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). Classify cngruence transfrmatins (G-L.1) Translatins: graph the image (G-L.2) Translatins: find the crdinates (G-L.3) Translatins: write the rule (G-L.4) Reflectins: graph the image (G-L.5) Reflectins: find the crdinates (G-L.6) Rtatins: graph the image (G-L.8) Rtatins: find the crdinates (G-L.9) Cmpsitins f cngruence transfrmatins: graph the image (G-L.10) Cngruence transfrmatins: mixed review (G-L.12) Dilatins: graph the image (G-L.13) Dilatins: find the crdinates (G-L.14) Dilatins: scale factr and classificatin (G-L.15) Identify transfrmatin matrices (A2-G.15) Transfrmatin matrices: write the vertex matrix (A2-G.16) Transfrmatin matrices: graph the image (A2-G.17) 9-12.G-CO.3 Given a rectangle, parallelgram, trapezid, r regular plygn, describe the rtatins and reflectins that carry it nt itself. Transfrmatins that carry a plygn nt itself (G-L.11) Draw lines f symmetry (G-O.3) Cunt lines f symmetry (G-O.4)

2 Califrnia Cmmn Cre Cntent Standards: Math 9-12 Gemetry 9-12.G-CO.4 Develp definitins f rtatins, reflectins, and translatins in terms f angles, circles, perpendicular lines, parallel lines, and line segments. Rtate plygns abut a pint (G-L.7) 9-12.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. Translatins: graph the image (G-L.2) Reflectins: graph the image (G-L.5) Rtatins: graph the image (G-L.8) Cmpsitins f cngruence transfrmatins: graph the image (G-L.10) Understand cngruence in terms f rigid mtins 9-12.G-CO.6 Use gemetric descriptins f rigid mtins t transfrm figures and t predict the effect f a given rigid mtin n a given figure; given tw figures, use the definitin f cngruence in terms f rigid mtins t decide if they are cngruent. Transfrmatins that carry a plygn nt itself (G-L.11) 9-12.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 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 Prve gemetric therems 9-12.G-CO.9 Prve therems abut lines and angles. Prfs invlving angles (G-C.8) Prfs invlving parallel lines I (G-D.6) Prfs invlving parallel lines II (G-D.7) 9-12.G-CO.10 Prve therems abut triangles. SSS Therem in the crdinate plane (G-K.6) Prfs invlving issceles triangles (G-K.10) Prfs invlving triangles I (G-M.8) Prfs invlving triangles II (G-M.9)

3 Califrnia Cmmn Cre Cntent Standards: Math 9-12 Gemetry 9-12.G-CO.11 Prve therems abut parallelgrams. Prfs invlving quadrilaterals I (G-N.11) Prfs invlving quadrilaterals II (G-N.12) Make gemetric cnstructins 9-12.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.). Cnstruct the midpint r perpendicular bisectr f a segment (G-B.9) Cnstruct an angle bisectr (G-C.6) Cnstruct a cngruent angle (G-C.7) Cnstruct a perpendicular line (G-D.2) Cnstruct parallel lines (G-D.5) Cnstruct an equilateral triangle r regular hexagn (G-G.5) Cnstruct a square (G-G.6) 9-12.G-CO.13 Cnstruct an equilateral triangle, a square, and a regular hexagn inscribed in a circle. Cnstruct an equilateral triangle inscribed in a circle (G-U.14) Cnstruct a square inscribed in a circle (G-U.15) Cnstruct a regular hexagn inscribed in a circle (G-U.16) 9-12.G-SRT Similarity, Right Triangles, and Trignmetry Understand similarity in terms f similarity transfrmatins 9-12.G-SRT.1 Verify experimentally the prperties f dilatins given by a center and a scale factr: 9-12.G-SRT.1.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. Dilatins and parallel lines (G-L.16) 9-12.G-SRT.1.b The dilatin f a line segment is lnger r shrter in the rati given by the scale factr. Dilatins and scale factrs (A1-F.8)

4 Califrnia Cmmn Cre Cntent Standards: Math 9-12 Gemetry 9-12.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. Similar figures: side lengths and angle measures (A1-F.6) Similar triangles and similarity transfrmatins (G-P.8) Similarity f circles (G-P.9) 9-12.G-SRT.3 Use the prperties f similarity transfrmatins t establish the AA criterin fr tw triangles t be similar. Similar triangles and indirect measurement (A1-F.7) Prve therems invlving similarity 9-12.G-SRT.4 Prve therems abut triangles. Prfs invlving triangles II (G-M.9) Prfs invlving similarity in right triangles (G-P.14) Prve the Pythagrean therem (G-P.15) 9-12.G-SRT.5 Use cngruence and similarity criteria fr triangles t slve prblems and t prve relatinships in gemetric figures. Similar triangles and indirect measurement (A1-F.7) SSS and SAS Therems (G-K.1) Prving triangles cngruent by SSS and SAS (G-K.2) ASA and AAS Therems (G-K.3) Prving triangles cngruent by ASA and AAS (G-K.4) SSS, SAS, ASA, and AAS Therems (G-K.5) SSS Therem in the crdinate plane (G-K.6) Prving triangles cngruent by SSS, SAS, ASA, and AAS (G-K.7) Prfs invlving crrespnding parts f cngruent triangles (G-K.8) Cngruency in issceles and equilateral triangles (G-K.9) Hyptenuse-Leg Therem (G-K.11) Similarity ratis (G-P.1) Similarity statements (G-P.2) Identify similar figures (G-P.3) Side lengths and angle measures in similar figures (G-P.4) Similar triangles and indirect measurement (G-P.5) Perimeters f similar figures (G-P.6) Similarity rules fr triangles (G-P.7) Triangle Prprtinality Therem (G-P.10) Areas f similar figures (G-P.11) Prve similarity statements (G-P.12) Prve prprtins r angle cngruences using similarity (G-P.13)

5 Califrnia Cmmn Cre Cntent Standards: Math 9-12 Gemetry Define trignmetric ratis and slve prblems invlving right triangles 9-12.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. Trignmetric ratis: sin, cs, and tan (A1-HH.1) Find trignmetric ratis using a calculatr (A1-HH.2) Trignmetric ratis: sin, cs, and tan (G-R.1) Trignmetric ratis: csc, sec, and ct (G-R.2) Trignmetric ratis in similar right triangles (G-R.3) Find trignmetric functins using a calculatr (G-R.6) Trignmetric ratis: sin, cs, and tan (A2-Y.3) Trignmetric ratis: csc, sec, and ct (A2-Y.4) Trignmetric ratis in similar right triangles (A2-Y.5) Find trignmetric functins using a calculatr (A2-Y.9) Find trignmetric ratis using right triangles (PC-M.5) 9-12.G-SRT.7 Explain and use the relatinship between the sine and csine f cmplementary angles. Trignmetric functins f cmplementary angles (G-R.4) Cmplementary angle identities (A2-AA.1) Cmplementary angle identities (PC-O.1) 9-12.G-SRT.8 Use trignmetric ratis and the Pythagrean Therem t slve right triangles in applied prblems. Pythagrean therem: wrd prblems (A1-F.15) Trignmetric ratis: find a side length (A1-HH.4) Trignmetric ratis: find an angle measure (A1-HH.5) Slve a right triangle (A1-HH.6) Pythagrean Therem (G-Q.1) Cnverse f the Pythagrean therem (G-Q.2) Pythagrean Inequality Therems (G-Q.3) Trignmetric ratis: find a side length (G-R.8) Trignmetric ratis: find an angle measure (G-R.9) Slve a right triangle (G-R.10) Pythagrean Therem and its cnverse (A2-Y.1) Trignmetric ratis: find a side length (A2-Y.14) Trignmetric ratis: find an angle measure (A2-Y.15) Slve a right triangle (A2-Y.16) Trignmetric ratis: find a side length (PC-M.10) Trignmetric ratis: find an angle measure (PC-M.11) Slve a right triangle (PC-M.12)

6 Califrnia Cmmn Cre Cntent Standards: Math 9-12 Gemetry 9-12.G-SRT.8.1 Derive and use the trignmetric ratis fr special right triangles (30,60,90 and 45,45,90 ). Special right triangles (A1-F.17) Special right triangles (G-Q.4) Find trignmetric functins f special angles (G-R.5) Special right triangles (A2-Y.2) Csc, sec, and ct f special angles (A2-Y.8) Trignmetric ratis: find a side length (PC-M.10) Trignmetric ratis: find an angle measure (PC-M.11) Slve a right triangle (PC-M.12) Apply trignmetry t general triangles 9-12.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. Area f a triangle: sine frmula (A2-Y.20) Area f a triangle: Law f Sines (A2-Y.21) Area f a triangle: sine frmula (PC-M.16) 9-12.G-SRT.10 Prve the Laws f Sines and Csines and use them t slve prblems. Law f Sines (G-R.11) Law f Csines (G-R.12) Law f Sines (A2-Y.17) Law f Csines (A2-Y.18) Area f a triangle: Law f Sines (A2-Y.21) Law f Sines (PC-M.13) Law f Csines (PC-M.14) 9-12.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). Slve a triangle (G-R.13) Slve a triangle (A2-Y.19) Slve a triangle (PC-M.15)

7 Califrnia Cmmn Cre Cntent Standards: Math 9-12 Gemetry 9-12.G-C Circles Understand and apply therems abut circles 9-12.G-C.1 Prve that all circles are similar. Similarity f circles (G-P.9) 9-12.G-C.2 Identify and describe relatinships amng inscribed angles, radii, and chrds. Arcs and chrds (G-U.6) Inscribed angles (G-U.9) 9-12.G-C.3 Cnstruct the inscribed and circumscribed circles f a triangle, and prve prperties f angles fr a quadrilateral inscribed in a circle. Cnstruct the circumcenter r incenter f a triangle (G-M.6) Perimeter f plygns with an inscribed circle (G-U.8) Angles in inscribed right triangles (G-U.10) Angles in inscribed quadrilaterals I (G-U.11) Cnstruct the inscribed r circumscribed circle f a triangle (G-U.17) 9-12.G-C.4 Cnstruct a tangent line frm a pint utside a given circle t the circle. Tangent lines (G-U.7) Cnstruct a tangent line t a circle (G-U.13) Find arc lengths and areas f sectrs f circles 9-12.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. Cnvert between degrees and radians. Arc measure and arc length (G-U.3) Area f sectrs (G-U.4) Cnvert between radians and degrees (A2-X.1) Radians and arc length (A2-X.2) Cnvert between radians and degrees (PC-M.1) Radians and arc length (PC-M.2)

8 Califrnia Cmmn Cre Cntent Standards: Math 9-12 Gemetry 9-12.G-GPE Expressing Gemetric Prperties with Equatins Translate between the gemetric descriptin and the equatin fr a cnic sectin 9-12.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. Write equatins f circles in standard frm using prperties (A2-U.4) Cnvert equatins f circles frm general t standard frm (A2-U.5) Find prperties f circles (PC-P.4) Write equatins f circles in standard frm (PC-P.5) Graph circles (PC-P.6) 9-12.G-GPE.2 Derive the equatin f a parabla given a fcus and directrix. Write equatins f parablas in vertex frm using prperties (A2-T.6) Find prperties f parablas (PC-P.1) Write equatins f parablas in vertex frm (PC-P.2) 9-12.G-GPE.3 Derive the equatins f ellipses and hyperblas given the fci, using the fact that the sum r difference f distances frm the fci is cnstant. Write equatins f ellipses in standard frm using prperties (A2-V.5) Write equatins f hyperblas in standard frm using prperties (A2-W.7) Find prperties f ellipses (PC-P.7) Find the eccentricity f an ellipse (PC-P.8) Write equatins f ellipses in standard frm (PC-P.9) Find prperties f hyperblas (PC-P.10) Find the eccentricity f a hyperbla (PC-P.11) Write equatins f hyperblas in standard frm (PC-P.12) Cnvert equatins f cnic sectins frm general t standard frm (PC-P.13) 9-12.G-GPE.3.1 Given a quadratic equatin f the frm ax² + by² + cx + dy + e = 0, use the methd fr cmpleting the square t put the equatin int standard frm; identify whether the graph f the equatin is a circle, ellipse, parabla, r hyperbla, and graph the equatin. Graph a quadratic functin (A2-J.3) Cmplete the square (A2-J.7) Graph parablas (A2-T.9) Cnvert equatins f circles frm general t standard frm (A2-U.5) Graph circles (A2-U.7) Cnvert equatins f ellipses frm general t standard frm (A2-V.6)

9 Califrnia Cmmn Cre Cntent Standards: Math 9-12 Gemetry Cnvert equatins f hyperblas frm general t standard frm (A2-W.8) Write equatins f parablas in vertex frm (PC-P.2) Graph parablas (PC-P.3) Graph circles (PC-P.6) Cnvert equatins f cnic sectins frm general t standard frm (PC-P.13) Use crdinates t prve simple gemetric therems algebraically 9-12.G-GPE.4 Use crdinates t prve simple gemetric therems algebraically 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). Slpes f parallel and perpendicular lines (A1-S.23) Write an equatin fr a parallel r perpendicular line (A1-S.24) Slpes f parallel and perpendicular lines (G-E.5) Equatins f parallel and perpendicular lines (G-E.6) Find the distance between a pint and a line (G-E.7) Find the distance between tw parallel lines (G-E.8) 9-12.G-GPE.6 Find the pint n a directed line segment between tw given pints that partitins the segment in a given rati. Midpints (A1-G.2) Midpints (G-B.4) Midpint frmula (G-B.7) 9-12.G-GPE.7 Use crdinates t cmpute perimeters f plygns and areas f triangles and rectangles, e.g., using the distance frmula. Distance between tw pints (A1-G.3) Distance frmula (G-B.8) Area and perimeter in the crdinate plane I (G-S.5) Area and perimeter in the crdinate plane II (G-S.6) 9-12.G-GMD Gemetric Measurement and Dimensin Explain vlume frmulas and use them t slve prblems 9-12.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 G-GMD.2 Give an infrmal argument using Cavalieri's principle fr the frmulas fr the vlume f a sphere and ther slid figures.

10 Califrnia Cmmn Cre Cntent Standards: Math 9-12 Gemetry 9-12.G-GMD.3 Use vlume frmulas fr cylinders, pyramids, cnes, and spheres t slve prblems. Vlume (A1-F.4) Vlume f prisms and cylinders (G-T.4) Vlume f pyramids and cnes (G-T.5) Surface area and vlume f spheres (G-T.6) Visualize relatinships between tw-dimensinal and three-dimensinal bjects 9-12.G-GMD.4 Identify the shapes f tw-dimensinal crss-sectins f three-dimensinal bjects, and identify three-dimensinal bjects generated by rtatins f tw-dimensinal bjects. Crss-sectins f three-dimensinal figures (G-H.4) Slids f revlutin (G-H.5) 9-12.G-GMD.5 Knw that the effect f a scale factr k greater than zer n length, area, and vlume is t multiply each by k, k², and k³, respectively; determine length, area and vlume measures using scale factrs. Scale drawings: wrd prblems (A1-C.7) Similar figures: side lengths and angle measures (A1-F.6) Similar triangles and indirect measurement (A1-F.7) Dilatins and scale factrs (A1-F.8) Area and perimeter f similar figures (A1-F.9) Vlume and surface area f similar slids (A1-F.12) Perimeter, area, and vlume: changes in scale (A1-F.13) Scale drawings: wrd prblems (G-A.2) Dilatins: scale factr and classificatin (G-L.15) Similarity ratis (G-P.1) Side lengths and angle measures in similar figures (G-P.4) Perimeters f similar figures (G-P.6) Areas f similar figures (G-P.11) Area and perimeter f similar figures (G-S.10) Surface area and vlume f similar slids (G-T.8) 9-12.G-GMD.6 Verify experimentally that in a triangle, angles ppsite lnger sides are larger, sides ppsite larger angles are lnger, and the sum f any tw side lengths is greater than the remaining side length; apply these relatinships t slve real-wrld and mathematical prblems. Angle-side relatinships in triangles (G-M.4) Triangle Inequality Therem (G-M.5)

11 Califrnia Cmmn Cre Cntent Standards: Math 9-12 Gemetry 9-12.G-MG Mdeling with Gemetry Apply gemetric cncepts in mdeling situatins 9-12.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). Area and perimeter: wrd prblems (A1-F.3) 9-12.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) G-MG.3 Apply gemetric methds t slve design prblems (e.g., designing an bject r structure t satisfy physical cnstraints r minimize cst; wrking with typgraphic grid systems based n ratis) S Statistics and Prbability 9-12.S-CP Cnditinal Prbability and the Rules f Prbability Understand independence and cnditinal prbability and use them t interpret data 9-12.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 ). Theretical prbability (A1-JJ.1) Cmpund events: find the number f utcmes (A1-JJ.3) Theretical and experimental prbability (G-X.1) Cmpund events: find the number f utcmes (G-X.2) Calculate prbabilities f events (A2-CC.2) Calculate prbabilities f events (PC-X.2) 9-12.S-CP.2 Understand that tw events A and B are independent if the prbability f A and B ccurring tgether is the prduct f their prbabilities, and use this characterizatin t determine if they are independent. Identify independent and dependent events (A1-JJ.4) Prbability f independent and dependent events (A1-JJ.5) Independent and dependent events (G-X.3) Identify independent events (A2-CC.7) Prbability f independent and dependent events (A2-CC.8)

12 Califrnia Cmmn Cre Cntent Standards: Math 9-12 Gemetry Identify independent events (PC-X.6) 9-12.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. Find cnditinal prbabilities (A2-CC.9) Independence and cnditinal prbability (A2-CC.10) Find cnditinal prbabilities (PC-X.7) Independence and cnditinal prbability (PC-X.8) 9-12.S-CP.4 Cnstruct and interpret tw-way frequency tables f data when tw categries are assciated with each bject being classified. Use the twway table as a sample space t decide if events are independent and t apprximate cnditinal prbabilities. Find prbabilities using tw-way frequency tables (A2-CC.6) Find cnditinal prbabilities using tw-way frequency tables (A2-CC.11) Find prbabilities using tw-way frequency tables (PC-X.5) Find cnditinal prbabilities using tw-way frequency tables (PC-X.9) 9-12.S-CP.5 Recgnize and explain the cncepts f cnditinal prbability and independence in everyday language and everyday situatins Use the rules f prbability t cmpute prbabilities f cmpund events in a unifrm prbability mdel 9-12.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. Gemetric prbability (G-X.7) 9-12.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. Find prbabilities using the additin rule (A2-CC.12) Find prbabilities using the additin rule (PC-X.10) 9-12.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. Find cnditinal prbabilities (A2-CC.9) Find cnditinal prbabilities (PC-X.7)

13 Califrnia Cmmn Cre Cntent Standards: Math 9-12 Gemetry 9-12.S-CP.9 Use permutatins and cmbinatins t cmpute prbabilities f cmpund events and slve prblems. Permutatins (A1-JJ.6) Cunting principle (A1-JJ.7) Permutatin and cmbinatin ntatin (A1-JJ.8) Cunting principle (G-X.4) Permutatins (G-X.5) Permutatin and cmbinatin ntatin (G-X.6) Cunting principle (A2-CC.3) Cmbinatins and permutatins (A2-CC.4) Find prbabilities using cmbinatins and permutatins (A2-CC.5) Cmbinatins and permutatins (PC-X.3) Find prbabilities using cmbinatins and permutatins (PC-X.4) 9-12.S-MD Using Prbability t Make Decisins Use prbability t evaluate utcmes f decisins 9-12.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). Chse the better bet (A2-CC.19) Chse the better bet (PC-Y.8)