CPM COLLEGE PREPARATORY MATH (6 th through 12 th Grade)

Transcription

1 CPM COLLEGE PREPARATORY MATH (6 th thrugh 12 th Grade) The Twin Valley Schl District uses Cllege Preparatry Mathematics (CPM) in the middle grades (6th, 7 th and 8 th ) and in ur secndary math prgram (9th thru 12 th grade). CPM was designated ne f five Exemplary Mathematics Prgrams in the cuntry by the U.S. Department f Educatin in Octber The curriculum emphasizes a variety f teaching methdlgies including lecture, class discussins, and structured study teams. During class, students are actively wrking n guided investigatins, math labs, t develp mathematical cncepts and prblem slving skills. The gals fr CPM are: T help mre students learn mre abut mathematics effectively Regenerate student interest and perfrmance in cllege preparatry mathematics Enable students t take mre cllege preparatry mathematics classes Prvide students with a mathematics educatin that will make them cmpetitive and successful in the marketplace. The curses in 6 th and 7 th grade, Fundatins fr Algebra Year 1 (FFA 1) and Fundatins fr Algebra Year 2 (FFA 2), are designed t prepare students fr Algebra 1 in 8 th grade. Students then take Gemetry in 9 th grade, Algebra 2 in 10 th grade, Precalculus r math analysis in 11 th grade and Calculus in 12 th grade. The fllwing is a list f each f the math curses and their specific cntent. Yu can als visit CPM s website at RP/emb 2/06

2 FOUNDATIONS FOR ALGEBRA YEAR 1 (FFA 1) THE BIG PICTURE Cncepts are based n memrable activities r cncrete mdels s mre students will be successful. Fr example the integer arithmetic is intrduced with a cncrete mdel, integer tiles, and practiced in game frmats befre mving t the abstract. The first five chapters cver: Additin, multiplicatin and subtractin f integers. Central tendency, mean, mde, median, and stem-and-leaf plts. Setting up a cmplete graph and scaling axes. Crdinate graphing. Variables and slving tw step algebraic equatins. Abslute value. Using Guess and Check t slve wrd prblems. Fractin, decimal, percent cnversin. Simplifying expressins using the rder f peratins. Basic gemetric vcabulary. Area f rectangles, parallelgrams, triangles, and trapezids. Gemetric mdel fr multiplicatin. Use f subprblems t slve mre cmplex area prblems. Equivalent fractins using fractin bars, rulers, grids, and rati tables. Identity Prperty f Multiplicatin (seen as a Giant 1 ). Slving prprtins, unit csts, and unit rates using a rati tables. Chapters six thrugh 10 cver: Setting up prprtins and slving using crss multiplicatin in real life applicatins including currency cnversin, distance rate and time, percent, sale price, discunt, similar figure, and scale drawings. Least cmmn multiples. Changing fractins t decimals using lng divisin. Calculate prbability f cmplementary events. Divisin f integers. Fractin and decimal arithmetic. Gemetric angle cncepts. Classifying triangles. Cmbining like terms. Number prperties including the Distributive prperty, cmmutative prperty, assciative Prperty and Identity Prperty. Circumference and area f circles. Vlumes f prisms and cylinders. Surface area. Experimental and theretical prbability. Sampling techniques, bias in plls, analyzing the validity f claims, and crrelatin versus causatin. Each chapter includes mental math and reviews the cncepts develped previusly. All chapters cntain a culminating big prblem r summary activity t tie tgether and/r pssibly extend the cncepts.

3 FOUNDATIONS FOR ALGEBRA YEAR 2 (FFA 2) THE BIG PICTURE Cncepts are based n memrable activities r cncrete mdels s mre students will be successful. Fr example multiplicatin and factring f plynmials are intrduced with a cncrete mdel, algebra tiles, befre mving t the abstract. The first five chapters cver: Data display and interpretatin using scatter plts, line graphs, bar graphs, stem-and-leaf plts, and bx-and-whisker plts. Guess and Check tables slve wrd prblems. Measures f central tendency. Graphing rdered pairs, lines and parablas. Patterns and rules in tables. Arithmetic f integers. Prbability. Arithmetic f fractins. Fractin decimal percent cnversin. Order f Operatins. Writing algebraic expressins and equatins frm Guess and Check tables and wrd prblems. Distributive Prperty and factring using algebra tiles. Cmbining like terms. Slving equatins using inverse peratins with the Cver-up methd and with balances and substitutin. Slving systems f equatins t find the pint f intersectin. Chapters six thrugh 10 cver: Writing and simplifying ratis and prprtins. Area f parallelgrams, triangles, trapezid, and circles. Reducing and enlarging figures. Divisin f fractins. Slving equatins with fractinal cefficients. Frmulas t slve simple and cmpund interest prblems. Markup amunts, selling prices, discunts, sale price and percent f increase r decrease. Distance rate, and time. Writing frmulas. Slving literal equatins. Pythagrean Therem. Square rts f square numbers and estimating square numbers. Nets f 2-dimensinal drawings fr 3-dimensinal mdels. Vlume and surface area f cylinders and prisms. Subprblems t break a cmplex prblem int smaller parts. Slpes triangles, rate f change and equatins f lines. Line f Best Fit Inequalities Expnential grwth. Vlume f a cne. Laws f Expnents. Scientific Ntatin. Each chapter includes mental math and reviews the cncepts develped previusly. All chapters cntain a culminating big prblem r summary activity t tie tgether and/r pssibly extend the cncepts.

4 ALGEBRA 1 (CPM 1) THE BIG PICTURE Cvers the expected cntent f any algebra 1 class. Reviews ver 4-5 chapters what was nce traditinal algebra cntent but is nw pre-algebra material in sme states r ffers a six week abridged versin allwing students with strng pre-algebra skills t quickly get t chapter six. Cncepts are based n memrable activities r cncrete mdels s mre students will be successful. Fr example the first experience with graphing equatins is an utside activity, the algebra walk. Multiplicatin and factring f plynmials are intrduced with a cncrete mdel, algebra tiles, befre mving t the abstract. The first five chapters cver: Arithmetic f integers and ratinal numbers. Graph interpretatin and graphing linear and nnlinear functins. Review f plane gemetric frmulas. Cmbining like terms and the distributive prperty. Writing and slving linear equatins. Ratis in the cntext f prprtins, similar figures, direct variatin, and percent. The sixth chapter cnslidates what has been cvered s far by extending graphing and slving equatins t simple systems and extends the distributive prperty t multiplying binmials. Chapters seven thrugh 12 cver: Slpe and the equatin f lines. Factring and slving quadratic equatins by factring r the quadratic frmula. Using the Pythagrean therem t fcus square rts and using diagrams t write equatins. Slving systems by eliminatin. Prperties f expnents. Ratinal expressins and equatins. Functins and abslute value. Prblem slving and inequalities. The final chapter cnslidates and extends using lines and curves f best fit, the prf f the quadratic frmula, and mre abut quadratic functins. Each chapter reviews the cncepts develped previusly and all chapters cntain a culminating big prblem t tie tgether and/r pssibly extend the algebra cncepts.

5 GEOMETRY (MATH 2)--THE BIG PICTURE Cvers the expected cntent in any gemetry curse. Reviews 85% f Algebra 1 tpics explicitly and uses algebra in gemetric applicatins (e.g., supplementary angle prblems cntain numerical and algebraic expressins as angle measures). Fcuses lgical explanatins thrughut the text. Prvides tw and a half chapters specifically fcused n prf. Allws as rigrus r relaxed an apprach t prf as apprpriate fr the students. The first half f the bk intrduces the fundamentals f lines, angles, and plane figures (abut 70% f the cntent). The first half f the bk: -- Starts with familiar, cncrete tpics in an enjyable frmat designed fr early student success. Tpics include the Pythagrean Therem, area f triangles and quadrilaterals, and linear equatins. -- Intrduces prf thrugh lgic games and puzzles. -- Uses the prblem slving strategies f rganizing data, making tables and lists, and lking fr patterns t intrduce the cncepts fr lines and angles. -- Explres three dimensinal visualizatin, studying prisms and pyramids fr the first f tw times in the curse. -- Studies transfrmatins and then triangle cngruence. -- The sixth chapter serves as a review f the first half f the curse. It intrduces several styles f prf and applies them t prve mst f the cnjectures develped inductively in chapters 0-5. The secnd half f the curse cncentrates n bigger ideas. Except fr circles, each chapter fcuses n ne r tw tpics. The secnd half f the curse: -- Starts with right triangle trignmetry s that students may slve mre interesting, cmplex prblems in subsequent chapters. -- Emphasizes similarity fr tw dimensinal and three dimensinal figures. -- Explres plygns, spending the latter prtin f the chapter n prf, including characteristics f quadrilaterals. -- Intrduces the fundamentals f circles, including arcs and angles, fllwed by the secnd study f prisms and pyramids, alng with cylinders and cnes. -- The last tw chapters apply ideas frm the curse. The first chapter (11) uses area mdels t study gemetric prbability. The last chapter ffers applicatins and big prblems that use many f the main ideas frm the curse. Cnstructins are ffered in an appendix that can prvide an interlude unit f tw r three days between the first and secnd parts f the bk.

6 ALGEBRA 2 (MATH 3) THE BIG PICTURE Cvers the expected cntent f any algebra 2 class. Review and extends 80% f algebra 1 tpics and 40% f gemetry tpics. Fr example, tw dimensin systems frm algebra 1 are extended t threedimensinal systems and are slved multiple ways. Right triangle trignmetry frm gemetry is extended t the laws f sines and csines. There is a regular use f technlgy and a lab apprach similar t what is used in science classes. Instead f spending the first three t fur chapters reviewing algebra 1 as many texts d, the curse starts with new material quickly and reviews algebra 1 material in cntext. Fr example, chapter ne starts with investigating functins using the graphing calculatr and chapter tw uses arithmetic sequences t review linear equatins and systems. Eight f the 13 chapters cver the fundamentals f algebra 2: Linear, quadratic, plynmial, expnential, abslute value, simple ratinal, lgarithmic, and square rt equatins are cvered in a functins apprach. Cnics Sequences and series Systems with and withut matrices Cmplex numbers Whenever pssible, tpics are investigated in cntext f real prblems. Fr example, plynmials are cncluded with a typical calculus prblem f maximum vlume. Technlgy is used t speed investigatins and understanding but students are als expected t be able t d mst prblems by hand. The rest f the bk gives chices based n expectatins f the next curse: Tw trignmetry chapters First, a unit circle apprach t sine, csine, tangent functins, and their inverses with applicatins. Secnd, right triangle trignmetry is extended t the laws f sines and csines. Tw discrete mathematics chapters First, a chapter n prbability using cncrete mdels t slve prbability, cnditinal prbability and expected value prblems. Secnd, a chapter n cunting yielding prblems invlving permutatins, cmbinatins, and the binmial therem. One statistics chapter that cvers measures f central tendency and ways f displaying data. Each chapter reviews the cncepts develped previusly and mst chapters cntain a culminating big prblem t tie tgether and/r pssibly extend the algebra cncepts

7 A SUMMARY OF MATH 4 (Math Analysis) This curse cvers bth trignmetry and Pre-Calculus cncepts. Apprximately 95% f the cncepts in a traditinal Trignmetry class are cvered. The Analysis/Pre-Calculus cncepts are similar t ther text including limits, vectrs, plar and parametric equatins, cnic sectins, matrices, and series. Calculus cncepts are studied in mre depth than traditinal Analysis/Pre- Calculus curses. The use f mathematical mdels is a reccurring theme thrughut the curse. The use f statistical cncepts is als applied in the use f the mdels. Unit 1 Intrduces mdels beginning with Linear mdels and using a median-median line t find a line f best fit. Other mdels, particularly expnential, are als intrduced. Unit 3 and Unit 5 lk extensive trignmetric mdels and applicatins. Unit 6 uses statistical methds t investigate nn-linear data by using regressin lines. Use f lgarithms t linearize data is als applied. Plar crdinates are investigated between Units 8 and 9 and Unit 12 s fcus is n parametric equatins. Cncepts f Calculus are investigated with cnsiderably mre depth then ther Pre-Calculus curses. Unit 2 fcus is n area under a curve. Riemann Sums are used t apprximate the area under a curve. Methds include left endpint rectangles, right endpint rectangles, and trapezids. Transfrmatins f graphs including piecewise defined functins are als studied. Unit 8 and 9 fcus n limits and rates f change. Students lk at limits t infinity and at a pint, apply cncepts f cntinuity and extend these ideas t include the definitin f the derivative. The relatinship between rates f change and area under a curve is als explred. Additinal algebraic techniques that are necessary fr calculus and ther advanced mathematics curses are explred including ratinalizatin, prperties f lgarithms, and use f substitutin. Extensive understanding f functins and inverses is als develped thrughut the curse. Use f graphing/prgrammable calculatrs is extensive. Students will write several prgrams at different times during the curse. Units 1-9 are the cre units cving all trig cncepts and key pre-calculus cncepts. Unit 10 is designed t teach students t read a cllege textbk. The material used in n cnic sectins. Units as well as Appendix A and B can be chsen frm depending n the needs f individual schls. These include matrices, parametric equatins, ne variable statistics and series. Extra prblems and skills review are als prvided in the appendices. Earlier cncepts are reviewed and practiced thrughut the curse.

8 A SUMMARY OF CALCULUS Cvers all cntent required fr the AP Calculus Test bth AB and BC. The curse starts with five majr prblems that intrduce the big ideas f calculus: ptimizatin, limits, differential equatins, expnential functins, the relatinship between distance and velcity, piecewise functins, vlumes f revlutin, vlumes by slicing, and the Fundamental Therem f Calculus. Each f these five majr prblems is revisited again later in the curse fr students t slve using new calculus knwledge. Each chapter reviews the cncepts develped previusly and builds n them. The curriculum cntains several key labs and hands-n activities thrughut the curse t intrduce cncepts, such as when students recgnize that the rate f a walker relates t the slpe f a graph in the "Slpe Walk." Labs als develp cnceptual understanding, such as when the students discver instantaneus velcity in the "Ramp Lab." Students learn abut derivatives and integrals simultaneusly during the first fur chapters and bth are presented gemetrically and in cntext. The first fur chapters cver: Pre-calculus tpics, such as trignmetric functins, dmain and range, and cmpsite functins Limits and cntinuity Applicatins f rates f change, such as velcity and acceleratin The difference between average velcity and instantaneus velcity The definitin f a derivative and the Pwer Rule Slpe Functins functins that find the slpe f anther functin fr all values in the dmain Differentiability and nn-differentiability Increasing and decreasing functins and cncavity Estimating the area under a curve with a Riemann Sum Area functins functins that find the area under a curve frm 0 t all values in the dmain The fifth chapter cnnects derivatives and integrals tgether with the Fundamental Therem. Chapters six thrugh nine cver: Hw t find the distance, velcity and acceleratin f an bject when given infrmatin abut its psitin, velcity r acceleratin Optimizatin Related Rates Derivative tls such as the Prduct, Qutient and Chain Rules, as well as implicit differentiatin and finding derivatives f all trignmetric and inverse trignmetric functins The derivative and integral f the natural lgarithm and y = e x

9 The Mean Value Therem Integratin using Substitutin Differential Equatins and Slpe Fields Vlumes f Revlutins and vlumes f knwn crss-sectin Sme material required fr the BC Calculus Exam is intrduced thrughut the curse in ptinal extensins f Chapters 5-9. These tpics include: Newtn's Methd l'hôpital's Rule Imprper Integrals, as well as integrating with partial fractins and integrating by parts Arclength Chapters ten thrugh thirteen cver additinal BC Calculus cntent, including: Cnvergence and divergence f infinite series Differentiatin and integratin f plar functins, as well as parametric functins and vectr functins Lgistic Curves Apprximating functins with plynmials Taylr and Maclaurin Plynmials, as well as the errr