Area of Learning: Mathematics Pre-calculus 12

Transcription

1 Area f Learning: Mathematics Pre-calculus 12 Big Ideas Elabratins Using inverses is the fundatin f slving equatins and can be extended t relatinships between functins. Understanding the characteristics f families f functins allws us t mdel and understand relatinships and t build cnnectins between classes f functins. Transfrmatins f shapes extend t functins and relatins in all f their representatins. Transfrmatins: inverses: und the peratins within an expressin r functin t reduce the expressin t an identity (e.g., x = ) Sample questins t supprt inquiry with students: Hw can the inverse help t slve an equatin? Hw is slving an equatin related t identifying the specific input fr a functin, given a specific utput? Hw are expnential and lgarithmic functins related? Hw are the laws f expnents cnnected t the laws f lgarithms? What are sme ther examples f inversely related functins? Hw are inverses related graphically, and why? Hw is slving an expnential equatin similar t slving a trignmetric equatin? Hw are inverse peratins related t slving a plynmial equatin by factring? What is the value f using trignmetric identities t find equivalent expressins? Why d sme equatins have extraneus rts and ther equatins d nt? functins: Sample questins t supprt inquiry with students: Hw d we decide which kind f functin t use t mdel a given prblem? What d functins and relatins lk like beynd the visible axes? A set f data lks like a parabla, but it is nt. What functin culd be used t mdel this data? What des the number f zers tell us abut a functin? What cnnectins d we see within the characteristics f a particular class f functin? Sample questins t supprt inquiry with students: Hw can we tell whether a transfrmatin will have invariant pints? 1

2 Under what circumstances will different transfrmatins prduce the same result? Hw d graphical transfrmatins affect the tables f values? Hw des a transfrmatin affect a pint fund at the rigin as cmpared t a pint n an axis r a pint in ne f the fur quadrants? Hw can a ratinal functin f the frm y = ax+b be cnsidered as a transfrmatin f cx+d the reciprcal functin y = 1? x Learning Standards Curricular Cmpetencies Elabratins Cntent Elabratins Students are expected t d the fllwing: thinking strategies: Students are expected t knw the using reasn t determine winning fllwing: Reasning and mdelling strategies transfrmatins f functins and Develp thinking strategies t slve generalizing and extending relatins puzzles and play games expnential functins and equatins Explre, analyze, and apply mathematical ideas using reasn, technlgy, and ther tls Estimate reasnably and demnstrate fluent, flexible, and strategic thinking abut number Mdel with mathematics in situatinal cntexts Think creatively and with curisity and wnder when explring prblems Understanding and slving Develp, demnstrate, and apply cnceptual understanding f mathematical ideas thrugh play, stry, inquiry, and prblem slving Visualize t explre and illustrate analyze: examine the structure f and cnnectins between mathematical ideas (e.g., expnential functins t gemetric sequences) reasn: inductive and deductive reasning predictins, generalizatins, cnclusins drawn frm experiences (e.g., with puzzles, games, and cding) technlgy: graphing technlgy, dynamic gemetry, calculatrs, virtual manipulatives, cncept-based apps can be used t fr a wide variety f purpses, including: explring and demnstrating gemetric sequences and series lgarithms: peratins, functins, and equatins plynmial functins and equatins ratinal functins trignmetry: functins, equatins, and identities transfrmatins: f graphs and equatins f parent functins and relatins (e.g., abslute value, radical, reciprcal, cnics, expnential, lgarithmic, trignmetric) vertical and hrizntal translatins, stretches, and reflectins inverses: graphs and equatins extensin: recgnizing cmpsed functins (e.g., y =) peratins n functins expnential: graphing, including transfrmatins slving equatins with same base and with different bases, including base e slving prblems in situatinal cntexts gemetric: cmmn rati, first term, general 2

3 mathematical cncepts and relatinships Apply flexible and strategic appraches t slve prblems Slve prblems with persistence and a psitive dispsitin Engage in prblem-slving experiences cnnected with place, stry, cultural practices, and perspectives relevant t lcal First Peples cmmunities, the lcal cmmunity, and ther cultures Cmmunicating and representing Explain and justify mathematical ideas and decisins in many ways Represent mathematical ideas in cncrete, pictrial, and symblic frms Use mathematical vcabulary and language t cntribute t discussins in the classrm Take risks when ffering ideas in classrm discurse Cnnecting and reflecting Reflect n mathematical thinking Cnnect mathematical cncepts with each ther, ther areas, and persnal interests Use mistakes as pprtunities t advance learning Incrprate First Peples wrldviews, mathematical relatinships rganizing and displaying data generating and testing inductive cnjectures mathematical mdelling ther tls: manipulatives such as algebra tiles and ther cncrete materials Estimate reasnably: be able t defend the reasnableness f an estimated value r a slutin t a prblem r equatin (e.g., the zers f a graphed plynmial functin) fluent, flexible and strategic thinking: includes: using knwn facts and benchmarks, partitining, applying whle number strategies t ratinal numbers and algebraic expressins chsing frm different ways t think f a number r peratin (e.g., Which will be the mst strategic r efficient?) Mdel: use mathematical cncepts and tls t slve prblems and make decisins (e.g., in real-life and/r abstract scenaris) take a cmplex, essentially nnterm gemetric sequences cnnecting t expnential functins infinite gemetric series sigma ntatin lgarithms: applying laws f lgarithms evaluating with different bases using cmmn and natural lgarithms explring inverse f expnential graphing, including transfrmatins slving equatins with same base and with different bases slving prblems in situatinal cntexts plynmial: factring, including the factr therem and the remainder therem graphing and the characteristics f a graph (e.g., degree, extrema, zers, end-behaviur) slving equatins algebraically and graphically ratinal: characteristics f graphs, including asympttes, intercepts, pint discntinuities, dmain, endbehaviur trignmetry: examining angles in standard psitin 3

4 perspectives, knwledge, and practices t make cnnectins with mathematical cncepts mathematical scenari and figure ut what mathematical cncepts and tls are needed t make sense f it situatinal cntexts: including real-life scenaris and penended challenges that cnnect mathematics with everyday life Think creatively: by being pen t trying different strategies refers t creative and innvative mathematical thinking rather than t representing math in a creative way, such as thrugh art r music curisity and wnder: asking questins t further understanding r t pen ther avenues f investigatin inquiry: includes structured, guided, and pen inquiry nticing and wndering determining what is needed t make sense f and slve prblems Visualize: create and use mental images t supprt understanding Visualizatin can be supprted using dynamic materials (e.g., graphical relatinships and simulatins), in bth radians and degrees explring unit circle, reference and cterminal angles, special angles graphing primary trignmetric functins, including transfrmatins and characteristics slving first- and secnd-degree equatins (ver restricted dmains and all real numbers) slving prblems in situatinal cntexts using identities t reduce cmplexity in expressins and slve equatins (e.g., Pythagrean, qutient, duble angle, reciprcal, sum and difference) 4

5 cncrete materials, drawings, and diagrams. flexible and strategic appraches: deciding which mathematical tls t use t slve a prblem chsing an effective strategy t slve a prblem (e.g., guess and check, mdel, slve a simpler prblem, use a chart, use diagrams, rle-play) slve prblems: interpret a situatin t identify a prblem apply mathematics t slve the prblem analyze and evaluate the slutin in terms f the initial cntext repeat this cycle until a slutin makes sense persistence and a psitive dispsitin: nt giving up when facing a challenge prblem slving with vigur and determinatin cnnected: thrugh daily activities, lcal and traditinal practices, ppular media and news events, crss-curricular integratin by psing and slving prblems r asking questins abut place, stries, and cultural practices 5

6 Explain and justify: use mathematical arguments t cnvince includes anticipating cnsequences decisins: Have students explre which f tw scenaris they wuld chse and then defend their chice. many ways: including ral, written, visual, use f technlgy Represent: using mdels, tables, graphs, wrds, numbers, symbls cnnecting meanings amng varius representatins discussins: partner talks, small-grup discussins, teacher-student cnferences discurse: is valuable fr deepening understanding f cncepts can help clarify students thinking, even if they are nt sure abut an idea r have miscnceptins Reflect: share the mathematical thinking f self and thers, including evaluating strategies and slutins, extending, 6

7 psing new prblems and questins Cnnect mathematical cncepts: t develp a sense f hw mathematics helps us understand urselves and the wrld arund us (e.g., daily activities, lcal and traditinal practices, ppular media and news events, scial justice, crsscurricular integratin) mistakes: range frm calculatin errrs t miscnceptins pprtunities t advance learning: by: analyzing errrs t discver misunderstandings making adjustments in further attempts identifying nt nly mistakes but als parts f a slutin that are crrect Incrprate: by: cllabrating with Elders and knwledge keepers amng lcal First Peples explring the First Peples Principles f Learning ( 7

8 LFP-POSTER-Principles-f- Learning-First-Peples-pster- 11x17.pdf; e.g., Learning is hlistic, reflexive, reflective, experiential, and relatinal [fcused n cnnectedness, n reciprcal relatinships, and a sense f place]; Learning invlves patience and time) making explicit cnnectins with learning mathematics explring cultural practices and knwledge f lcal First Peples and identifying mathematical cnnectins knwledge: lcal knwledge and cultural practices that are apprpriate t share and that are nn-apprpriated practices: Bishp s cultural practices: cunting, measuring, lcating, designing, playing, explaining ( ACP.htm_files/abishp.htm) Abriginal Educatin Resurces ( Teaching Mathematics in a First Natins Cntext, FNESC ( Cmment [mw1]: Carpe Diem: Pssible t embed link in FPPL, r des URL have t be visible? 8

9 -first-peples/) 9