Area of Learning: Mathematics Foundations of Mathematics and Pre-calculus 10

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1 Area f Learning: Mathematics Fundatins f Mathematics and Pre-calculus 10 Big Ideas Elabratins Algebra allws us t generalize relatinships thrugh abstract thinking. generalize: The meanings f, and cnnectins between, each peratin extend t pwers and plynmials. Cnstant rate f change is an essential attribute f linear relatins and has meaning in different representatins and cntexts. 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 can we tell if a mathematical slutin is reasnable? Where can errrs ccur when slving a cntextualized prblem? What d we ntice when we square binmials? Hw d we decide n a strategy fr slving a system f equatins? cnnected: Sample questins t supprt inquiry with students: Hw are the different peratins (+, -, x,, expnents) cnnected? What are the similarities and differences between multiplicatin f numbers, pwers, and plynmials? Hw is prime factrizatin helpful? Hw des prime factrizatin f numbers extend t algebraic terms? 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? relatins: Sample questins t supprt inquiry with students: Hw can we tell if a relatin is linear? Hw can we use rate f change t make predictins? What cnnectins can we make between arithmetic sequences and linear functins? 1

2 Hw d we decide which frm f linear equatin t use? 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 height f a tree using distance frm the tree and the angle t the tp f the tree) Sample questins t supprt inquiry with students: When might we need t measure a length r angle indirectly? Why is trignmetry defined in reference t right triangles rather than ther types f triangles? Hw can rate f change be cnnected t trignmetry? What is the rigin f the names fr the trignmetric ratis? Representing and analyzing situatins allws us t ntice and wnder abut relatinships. situatins: situatinal cntexts (e.g., relating vlume t height when filling cntainers f different shapes, relating distance t time fr a bike ride) nn-situatinal cntexts (e.g., the graph f a piecewise functin) Sample questins t supprt inquiry with students: Hw des the representatin f a relatin supprt a strategy when slving a prblem? D all data have trends and relatinships? Why are trends imprtant? Curricular Cmpetencies Elabratins Cntent Elabratins Students are expected t d the fllwing: thinking strategies: Students are expected t knw the pwers: using reasn t determine winning fllwing: psitive and negative expnents Reasning and mdelling strategies peratins n pwers with integral expnent laws Develp thinking strategies t slve generalizing and extending expnents evaluatin using rder f peratins puzzles and play games analyze: prime factrizatin numerical and variable bases Explre, analyze, and apply mathematical examine the structure f and functins and relatins: cnnecting data, prime factrizatin: 2

3 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 mathematical understanding 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 and decisins in many ways Represent mathematical ideas in cnnectins between mathematical ideas (e.g., using an area mdel t factr a trinmial) 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 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., estimating the slutin fr a system graphs, and situatins linear functins: slpe and equatins f lines arithmetic sequences systems f linear equatins multiplicatin f plynmial expressins plynmial factring primary trignmetric ratis financial literacy: grss and net pay expressing prime factrizatin f a number using pwers identifying the factrs f a number includes greatest cmmn factr (GCF) and least cmmn multiple (LCM) strategies include using factr trees and factr pairs functins and relatins: cmmunicating dmain and range in bth situatinal and nn-situatinal cntexts cnnecting graphs and cntext understanding the meaning f a functin identifying whether a relatin is a functin using functin ntatin linear functins: slpe: psitive, negative, zer, and undefined types f equatins f lines (pintslpe, slpe intercept, and general) equatins f parallel and perpendicular lines equatins f hrizntal and vertical lines cnnectins between representatins: graphs, tables, equatins 3

4 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 f equatins frm a graph) 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 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 arithmetic sequences: applying frmal language (cmmn difference, first term, general term) t increasing and decreasing linear patterns cnnecting t linear relatins extensin: explring arithmetic series systems: slving graphically slving algebraically by inspectin, substitutin, eliminatin cnnecting rdered pair with meaning f an algebraic slutin slving prblems in situatinal cntexts multiplicatin: applying the distributive prperty between tw plynmials, including trinmials cnnecting the prduct f binmials with an area mdel factring: greatest cmmn factr f a plynmial simpler cases invlving trinmials (y = x 2 + bx + c) and difference f squares trignmetric: sine, csine, and tangent ratis right-triangle prblems: determining 4

5 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), cncrete materials, drawings, and diagrams. flexible and strategic appraches: deciding which mathematical tls t use t slve a prblem chsing an apprpriate 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 missing sides and/r angles using trignmetric ratis and the Pythagrean therem cntexts invlving direct and indirect measurement financial literacy: types f incme incme tax and ther deductins 5

6 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 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 6

7 cmmunicating effectively accrding t what is being cmmunicated and t whm Represent: using mdels, tables, graphs, wrds, numbers, symbls cnnecting meanings amng varius representatins using cncrete materials and dynamic interactive technlgy 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 7

8 traditinal practices, ppular media and news events, scial justice, crsscurricular integratin) mistakes: range frm calculatin errrs t miscnceptins pprtunities t advance learning: by: Incrprate: by: analyzing errrs t discver misunderstandings making adjustments in further attempts identifying nt nly mistakes but als parts f a slutin that are crrect 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 8

9 [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 ( h-first-peples/) 9

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

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