Area of Learning: Mathematics Pre-calculus 11. Algebra allows us to generalize relationships through abstract thinking.

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1 Area f Learning: Mathematics Pre-calculus 11 Big Ideas Elabratins Algebra allws us t generalize relatinships thrugh abstract thinking. generalize: The meanings f, and cnnectins between, peratins extend t pwers, radicals, and plynmials. Sample questins t supprt inquiry with students: After slving a prblem, can we extend it? Can we generalize it? Hw can we take a cntextualized prblem and turn it int a mathematical prblem that can be slved? Hw d we tell if a mathematical slutin is reasnable? Where can errrs ccur when slving a cntextualized prblem? What are the similarities and differences between quadratic functins and linear functins? Hw are they cnnected? What d we ntice abut the rate f change in a quadratic functin? Hw d the strategies fr slving linear equatins extend t slving quadratic, radical, r ratinal equatins? What is the cnnectin between dmain and extraneus rts? cnnectins: Sample questins t supprt inquiry with students: Hw are the different peratins (+, -, x,, expnents, rts) cnnected? What are the similarities and differences between multiplicatin f numbers, pwers, radicals, plynmials, and ratinal expressins? Hw can we verify that we have factred a trinmial crrectly? Hw can visualizatin supprt algebraic thinking? Hw can patterns in numbers lead t algebraic generalizatins? When wuld we chse t represent a number with a radical rather than a ratinal expnent? Hw d strategies fr factring x 2 + bx + c extend t ax 2 + bx + c, a 1 1

2 Hw d peratins n ratinal numbers extend t peratins with ratinal expressins? Quadratic relatinships are prevalent in the wrld arund us. relatinships: Sample questins t supprt inquiry with students: What are sme examples f quadratic relatinships in the wrld arund us, and what are the similarities and differences between these? Why are quadratic relatinships s prevalent in the wrld arund us? Hw des the predictable pattern f linear functins extend t quadratic functins? Why is the shape f a quadratic functin called a parabla? Hw can we decide which frm f a quadratic functin t use fr a given prblem? What effect des each term f a quadratic functin have n its graph? Trignmetry invlves using prprtinal reasning t slve indirect measurement prblems. prprtinal reasning: cmparisns f relative size r scale instead f numerical difference indirect measurement: using measurable values t calculate immeasurable values (e.g., calculating the width f a river using the distance between tw pints n ne shre and an angle t a pint n the ther shre) Sample questins t supprt inquiry with students: Hw is the csine law related t the Pythagrean therem? Hw can we use right triangles t find a rule fr slving nn-right triangles? Hw d we decide when t use the sine law r csine law? What wuld it mean fr an angle t have a negative measure? Identify a cntext fr making sense f a negative angle. Cmment [mw1]: Carpe Diem: can we make the questins pp up with bth elabratins? 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 real number system pwers: Develp thinking strategies t slve generalizing and extending pwers with ratinal expnents puzzles and play games analyze: radical peratins and equatins real number: classificatin psitive and negative ratinal expnents 2

3 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 Explain and justify mathematical ideas examine the structure f and cnnectins between mathematical ideas (e.g., trinmial factring, rts f quadratic equatins) 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 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 estimated value r a slutin t a prblem r equatin (e.g., the zers f a graphed plynmial functin) fluent, flexible and strategic thinking: plynmial factring ratinal expressins and equatins quadratic functins and equatins linear and quadratic inequalities trignmetry: nn-right triangles and angles in standard psitin financial literacy: cmpund interest, investments, lans expnent laws evaluatin using rder f peratins numerical and variable bases radical: simplifying radicals rdering a set f irratinal numbers perfrming peratins with radicals slving simple (ne radical nly) equatins algebraically and graphically identifying dmain restrictins and extraneus rts f radical equatins factring: greatest cmmn factr f a plynmial trinmials f the frm ax 2 + bx + c difference f squares f the frm a 2 x 2 b 2 y 2 may extend t a(f(x)) 2 + b(f(x)) + c, a 2 (f(x)) 2 b 2 (f(x)) 2 ratinal: simplifying and applying peratins t ratinal expressins identifying nn-permissible values slving equatins and identifying any extraneus rts quadratic: identifying characteristics f graphs (including dmain and range, 3

4 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, with ther areas, and with 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 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 nnmathematical 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 intercepts, vertex, symmetry), multiple frms, functin ntatin, extrema explring transfrmatins slving equatins (e.g., factring, quadratic frmula, cmpleting the square, graphing, square rt methd) cnnecting equatin-slving strategies cnnecting equatins with functins slving prblems in cntext inequalities: single variable (e.g., 3x 7 4, x 2 5x + 6 > 0) dmain and range restrictins frm prblems in situatinal cntexts sign analysis: identifying intervals where a functin is psitive, negative, r zer symblic ntatin fr inequality statements, including interval ntatin trignmetry: use f sine and csine laws t slve nn-right triangles, including ambiguus cases cntextual and nn-cntextual prblems angles in standard psitin: 4

5 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), 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 degrees special angles, as cnnected with the and triangles unit circle reference and cterminal angles terminal arm trignmetric ratis simple trignmetric equatins financial literacy: cmpund interest intrductin t investments/lans with regular payments, using technlgy buy/lease 5

6 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 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 cmmunicating effectively accrding t what is being cmmunicated and t whm 6

7 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, 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 7

8 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 ( 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 Cmment [mw2]: Carpe Diem: Pssible t embed link in FPPL? Or des URL have t be visible? 8

9 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 ( -first-peples/) 9