Area f Learning: Mathematics Calculus 12 Big Ideas Elabratins The cncept f a limit is fundatinal t calculus. cncept f a limit: Differentiatin and integratin are defined using limits. Sample questins t supprt inquiry with students: Why is a limit useful? Hw can we use histrical examples (e.g., Achilles and the trtise) t describe a limit? Differential calculus develps the cncept f instantaneus rate f change. instantaneus rate f change: develping rate f change frm average t instantaneus Integral calculus develps the cncept f determining a prduct invlving a cntinuusly changing quantity ver an interval. Sample questins t supprt inquiry with students: Hw can a rate f change be instantaneus? When d we use rate f change? cntinuusly changing: area (height x width) under a curve where the height f the regin is changing; vlume f a slid (area x length) where crss-sectinal area is changing; wrk (frce x distance) where frce is changing Finding these prducts requires finding an infinite sum. Sample questins t supprt inquiry with students: What is the value f using rectangles t apprximate the area under a curve? Why is the fundamental therem f calculus s fundamental? Derivatives and integrals are inversely related. inversely related: The fundamental therem f calculus describes the relatinship between integrals and antiderivatives. Sample questins t supprt inquiry with students: Hw are derivatives and integrals related? Why are antiderivatives imprtant? What is the difference between an antiderivative and an integral? Learning Standards Curricular Cmpetencies Elabratins Cntent Elabratins Students are expected t d the fllwing: thinking strategies: Students are expected t knw the functins: 1
Reasning and mdelling Develp thinking strategies t slve puzzles and play games 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 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 using reasn t determine winning strategies generalizing and extending analyze: examine the structure f and cnnectins between mathematical ideas (e.g., limits, derivatives, integrals) reasn: inductive and deductive reasning predictins, generalizatins, cnclusins drawn frm experiences (e.g., in puzzles, games, cding) technlgy: graphing technlgy, dynamic gemetry, calculatrs, virtual manipulatives, cncept-based apps can be used fr a wide variety f purpses, including: explring and demnstrating 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 estimate acrss mathematical cntexts fluent, flexible, and strategic thinking: fllwing: functins and graphs limits: left and right limits limits t infinity cntinuity differentiatin: rate f change differentiatin rules higher rder, implicit applicatins integratin: apprximatins fundamental therem f calculus methds f integratin applicatins parent functins frm Pre-Calculus 12 piecewise functins inverse trignmetric functins limits: frm table f values, graphically, and algebraically ne-sided versus tw-sided end behaviur intermediate value therem differentiatin: histry definitin f derivative ntatin rate f change: average versus instantaneus slpe f secant and tangent lines differentiatin rules: pwer, prduct; qutient and chain transcendental functins: lgarithmic, expnential, trignmetric applicatins: relating graph f f(x) t f (x) and f (x) increasing/decreasing, cncavity differentiability, mean value therem Newtn s methd prblems in cntextual situatins, including related rates and ptimizatin prblems integratin: definitin f an integral ntatin 2
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, perspectives, knwledge, and practices t make cnnectins with mathematical cncepts includes: using knwn facts and benchmarks, partitining, applying number strategies t apprximate limits, derivatives, and integrals 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 nnmathematical scenari and figure ut what mathematical cncepts and tls are needed t make sense f it situatinal cntexts: including real-life scenaris and pen-ended 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 definite and indefinite apprximatins: Riemann sum, rectangle apprximatin methd, trapezidal methd methds f integratin: antiderivatives f functins substitutin by parts applicatins: area under a curve, vlume f slids, average value f functins differential equatins initial value prblems slpe fields 3
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), 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: 4
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 Explain and justify: using 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 cmmunicating effectively accrding t what is being cmmunicated and t whm Represent: using mdels, tables, graphs, wrds, numbers, symbls cnnecting meanings amng varius representatins discussins: partner talks, small-grup discussins, teacher-student 5
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, 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: 6
by: cllabrating with Elders and knwledge keepers amng lcal First Peples explring the First Peples Principles f Learning (http://www.fnesc.ca/wp/wp- cntent/uplads/2015/09/pub- 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 (http://www.csus.edu/indiv//reyd/ ACP.htm_files/abishp.htm) 7 Cmment [mw1]: Carpe Diem: Pssible t embed link in FPPL r des URL have t be visible?
Abriginal Educatin Resurces (www.abriginaleducatin.ca) Teaching Mathematics in a First Natins Cntext, FNESC (http://www.fnesc.ca/resurces/mat h-first-peples/) 8