M O R R I S S C H O O L D I S T R I CT M O R R I S T O W N, N J 2011 ALGEBRA I SUPERINTENDENT Dr. Thmas J. Ficarra 1
Bard f Educatin: 2010 2011 Lisa Pllak, President Marie Frnar, Vice President Nancy Bangila, Curriculum Cmmittee (Chair) Christpher Gardner (Mrris Plains Representative) Dr. Peter Gallerstein Lynn Hrwitz Sandra McNeil Teresa Murphy Ann Rhines Dr. Angela Rieck Central Office Dr. Thmas J. Ficarra, Superintendent f Schls Dr. Mary Ann Reilly, Directr f Curriculum, Grades 6 12 Michael Amendla, Supervisr f Mathematics & Science Mrristwn High Schl Administratin Linda D. Murphy, Principal Celeste Hammell, Supervisr f Instructin Frelinghuysen Middle Schl Administratin Ethel Minchell, Principal Janet Rsff, Vice Principal Curriculum Writer: Dr. Debrah L. Ives 2
Table f Cntents Part I Ratinale and Philsphy 4 Gals and Objectives 6 Part II Units f Study 8 Mastery Objectives 9 Teaching and Learning Activities 14 Assessment and Testing strategies 17 Text and Materials 19 Prcedures fr use f Supplemental Materials 20 Part III Curriculum Map 21 Part IV: Appendix References 31 D Nw Prblems xx Unit f Study Summative Prblems xx Rubrics (frthcming) xx Internet Web Sites xx 3
Ratinale and Philsphy In rder t prepare fr glbal cmpetitin and high expectatins fr all, Mrris Schl District students must have increased pprtunities fr mathematical experiences that extend critical thinking and reasning. Specifically, access t higher mathematics is essential. Algebra I is a curse that prvides an imprtant entry pint fr this pathway t success by extending students understanding and applicatin f the skills, cncepts, and language f algebra. Key cnsideratins: State and Natinal Expectatins There has been great activity n the state and natinal level in terms f expectatins fr the skills, knwledge and expertise students shuld master in mathematics t succeed in wrk and life in the 21 st century. NJ is currently mving frward with the Framewrk fr 21 st Century Learning, a partnership with business, educatin, and gvernment t develp a cllective visin t strengthen American educatin (www.21stcenturyskills.rg). The six cmpnents include: Cre Cntent Mathematics, 21 st Century Cntent, Learning and Thinking skills, Infrmatin and Cmmunicatins Technlgy (ICT) Literacy, Life Skills Real Wrld Applicatins, and 21 st Century Assessments. Students will be assessed n their depth f knwledge f algebra n natinal examinatins beginning in 2010, with additinal high schl mathematics curses t fllw. Equity and Access t Higher Mathematics The belief that all students, nt just a select few, have access t mathematical learning envirnments that enable them t meet wrldclass standards fr bth cllege and the wrld f wrk cntinues t be an essential gal f the Mrris Schl District. The Natinal Cuncil f Teachers f Mathematics initially reflected this imprtant cnsideratin in The Equity Principle whereby, Excellence in mathematics educatin requires equity high expectatins and strng supprt fr all students (NCTM 2000). Subsequent research has revealed that It is imprtant that high schls d everything t prmte success amng all students encuraging enrllment by students frm all demgraphics in advanced math curses. NCTM s recent release f the scientifically research-based, Fcus in High Schl Mathematics: Reasning and Sense Making (2009), stressed the rle f educatrs t help students with a wide range f backgrunds develp cnnectins between applicatins f new learning and their existing knwledge, increasing their likelihd f understanding and thereby allwing increased ptins and entry int advanced mathematics. Building n Existing Partnerships The Algebra I curse fills a critical need fr extensive study twards the develpment f abstract algebraic thinking. This gal is aligned with the district s visin f prviding rich pprtunities fr all students t mve frward n Blms Knwledge Taxnmy cntinuum frm Knwledge and Awareness t Cmprehensin, Applicatin, Analysis, Synthesis, and Evaluatin. Specifically, scafflding n a fundatin f algebra skills, this curse mves students frm the acquisitin and assimilatin f cncepts twards increased applicatin and adaptatin. Adaptatin ccurs when students have the cmpetence t think in cmplex ways and t apply their knwledge and skills. 4
Building n the district s existing partnerships, this curse wuld prvide a springbard fr entrance int higher level mathematics curses fr a greater number f Mrris Schl District students. [Rigr and Relevance Framewrk: Internatinal Center fr Leadership in Educatin, Bill Daggett and Ray McNulty, www.leadered.cm] 5
Gals and Objectives (utcmes): Algebra is the study f patterns and functins. In teaching and learning Algebra I, it is imprtant fr teachers and students t cmprehend the fllwing Big Ideas and Enduring Understandings and t establish cnnectins and applicatins f the individual skills and cncepts t these brad principles as the critical gals and bjectives f the curse: Patterns and Functins Algebra prvides language thrugh which we describe and cmmunicate mathematical patterns that arise in bth mathematical and nnmathematical situatins, and in particular, when ne quantity is a functin f a secnd quantity r where the quantities change in predictable ways. Ways f representing patterns and functins include tables, graphs, symblic and verbal expressins, sequences, and frmulas. Equivalence There are many different but equivalent frms f a number, expressin, functin, r equatin, and these frms differ in their efficacy and efficiency in interpreting r slving a prblem, depending n the cntext. Algebra extends the prperties f numbers t rules invlving symbls; when applied prperly, these rules allw us t transfrm and expressin, functin, r equatin int an equivalent frm and substitute equivalent frms fr each ther. Slving prblems algebraically typically invlves transfrming ne equatin t anther equivalent equatin until the slutin becmes clear. Representatin and Mdeling with Variables Quantities can be represented by variables, whether the quantities are unknwn, changing ver time, parameters, r prbabilities. Representing quantities by variables gives us the pwer t recgnize and describe patterns, make generalizatins, prve r explain cnclusins, and slve prblems. Representing quantities with variables als enables us t mdel situatins in all areas f human endeavr and t represent them abstractly. Linearity In many situatins, the relatinship between tw quantities is linear s the graphical representatin f the relatinship is a gemetric line. Linear functins can be used t shw a relatinship between tw variables that has a cnstant rate f change and t represent the relatinship between tw quantities, which vary prprtinately. Linear functins can als be used t mdel, describe, analyze, and cmpare sets f data. Cnnectins Between Algebra & Gemetry Gemetric bjects can be represented algebraically (fr example, lines can be described using crdinates), and algebraic expressins can be interpreted gemetrically (fr example, systems f equatins and inequalities can be slved graphically). 6
Cnnectins Between Algebra & Systematic Cunting, Prbability, and Statistics Algebra prvides a language and techniques fr analyzing situatins that invlve chance and uncertainty, including systematic listing and cunting f all pssible utcmes (as well as infrmal explratins f Pascal s Triangle), the determinatin f their prbabilities, the calculatin f prbabilities f varius events, predicatins based n experimental prbabilities, and crrelatins between tw variables. Applicatins f Algebraic Cncepts Using the mdel, Webb s Depth-f-Knwledge Levels, Algebra I has a primary gal f increasing pprtunities fr students t reinfrce Levels 1 & 2 and t engage in Level 3 Strategic Thinking and Level 4 Extended Thinking: Strategic Thinking requires reasning, planning, using evidence, and a higher level f thinking than the previus tw levels. In mst instances, requiring students t explain their thinking is a Level 3. Activities that require students t make cnjectures are als at this level. The cgnitive demands at Level 3 are cmplex and abstract. The cmplexity des nt result frm the fact that there are multiple answers, a pssibility fr bth Levels 1 and 2, but because the task requires mre demanding reasning. An activity, hwever, that has mre than ne pssible answer and requires students t justify the respnse they give wuld mst likely be a Level 3. Other Level 3 activities include drawing cnclusins frm bservatins; citing evidence and develping a lgical argument fr cncepts; explaining phenmena in terms f cncepts; and using cncepts t slve prblems. Extended Thinking requires cmplex reasning, planning, develping, and thinking mst likely ver an extended perid f time. The extended time perid is nt a distinguishing factr if the required wrk is nly repetitive and des nt require applying significant cnceptual understanding and higher-rder thinking. Fr example, if a student has t take the water temperature frm a river each day fr a mnth and then cnstruct a graph, this wuld be classified as a Level 2. Hwever, if the student is t cnduct a river study that requires taking int cnsideratin a number f variables, this wuld be a Level 4. At Level 4, the cgnitive demands f the task shuld be high and the wrk shuld be very cmplex. Students shuld be required t make several cnnectins relate ideas within the cntent area r amng cntent areas and have t select ne apprach amng many alternatives n hw the situatin shuld be slved, in rder t be at this highest level. Level 4 activities include designing and cnducting experiments; making cnnectins between a finding and related cncepts and phenmena; cmbining and synthesizing ideas int new cncepts; and critiquing experimental designs. (Wiscnsin Center fr Educatinal Research, www.facstaff.wcer.wisc.educ/nrmw) 7
Units f Study: (Nte: emphasis n prblem slving, applicatins, and mdeling) Number Sense and Operatins including: Reasning with Real Numbers Fractins and Fractals/Chas Thery Prprtinal Reasning and Variatin Matrices Data and Statistical Analysis Algebraic Expressins Plynmial expressins Linear Relatinships Linear Functins Linear Equatins and Inequalities, including Systems Abslute Value Nn-Linear Relatinships Expnents and Expnential Mdels Quadratic Functins and Mdels Cubic Functins Transfrmatins Animating with Transfrmatins Transfrmatins t Mdel Data Matrix Transfrmatins Prbability Cnnectins t Gemetry 8
Mastery Objectives: MASTERY OBJECTIVES (NJCCCS) Algebra I is crrelated t the New Jersey Algebra I Standards 2010, as well as the 2010 Cmmn Cre Curriculum Cntent Standards in Mathematics. New Jersey Algebra I Standards 2010 Students will be able t represent and slve prblems in the fllwing areas: N: Nn-linear Relatinships N1. Nn-linear Functins N1.a Representing quadratic functins in multiple ways N1.b Distinguishing between functin types N1.c Using quadratic mdels N2. Nn-linear Equatins N2.a Slving literal equatins N2.b Slving quadratic equatins N2.B1 Slving simple expnential equatins D: Data, Statistics and Prbability D1. Data and Statistical Analysis D1.a Interpreting linear trends in data D1.b Cmparing data using summary statistics D1.c Evaluating data-based reprts in the media D2. Prbability D2.a Using cunting principles D2.b Determining prbability 9
O: Operatins n Numbers and Expressins O1. Number Sense and Operatins O1.a Reasning with real numbers O1.b Using ratis, rates, and prprtins O1.B1 Using variables in different ways O1.B2 Using matrices O1.c Using numerical expnential expressins* O2.a Using algebraic expnential expressins * O1.d Using numerical radical expressins** O2.d Using algebraic radical expressins** O2. Algebraic Expressins (a and d listed abve) O2.b Operating with plynmial expressins O2.c Factring plynmial expressins L: Linear Relatinships L1. Linear Functins L1.a Representing linear functins in multiple ways L1.b Analyzing linear functin L1.c Graphing linear functins invlving abslute value L1.d Using linear mdels L2. Linear Equatins and Inequalities L2.a Slving linear equatins and inequalities L2.b Slving equatins invlving abslute value L2.c Graphing linear inequalities L2.d Slving systems f linear equatins L2.e Mdeling with single variable linear equatins, ne-r tw-variable inequalities r systems f equatins * Tpics have been cmbined int ne indicatr ** Tpics have been cmbined int ne indicatr 10
2010 Cmmn Cre Curriculum Cntent Standards in Mathematics Algebra Overview Seeing Structure in Expressins Interpret the structure f expressins Write expressins in equivalent frms t slve prblems Arithmetic with Plynmials and Ratinal Functins Perfrm arithmetic peratins n plynmials Understand the relatinship between zers and factrs f plynmials Use plynmial identities t slve prblems Rewrite ratinal functins Creating Equatins Create equatins that describe numbers r relatinships Reasning with Equatins and Inequalities Understand slving equatins as a prcess f reasning and explain the reasning Slve equatins and inequalities in ne variable Slve systems f equatins Represent and slve equatins and inequalities graphically Mathematical Practices Make sense f prblems and persevere in slving them. Reasn abstractly and quantitatively. Cnstruct viable arguments and critique the reasning f thers. 11
Mdel with mathematics. Use apprpriate tls strategically. Attend t precisin. Lk fr and make use f structure. Lk fr and express regularity in repeated reasning. Functins Overview Interpreting Functins Understand the cncept f a functin and use functin ntatin Interpret functins that arise in applicatins in terms f the cntext Analyze functins using different representatins Building Functins Build a functin that mdels a relatinship between tw quantities Build new functins frm existing functins Linear, Quadratic, and Expnential Mdels Cnstruct and cmpare linear and expnential mdels and slve prblems Interpret expressins fr functins in terms f the situatin they mdel Mdeling Standards Mdeling is best interpreted nt as a cllectin f islated tpics but rather in relatin t ther standards. Making mathematical mdels is a Standard fr Mathematical Practice, and specific mdeling standards appear thrughut the high schl standards indicated by a star symbl ( ). 12
All curse f study must include the fllwing, which replace the Wrkplace readiness standards: Career Educatin and Cnsumer, Family, and Life Skills Career and Technical Educatin: All students will develp career awareness and planning, emplyability skills, and fundatinal knwledge necessary fr success in the wrkplace. Cnsumer, Family, and Life Skills: All students will demnstrate critical life skills in rder t be functinal members f sciety. Scans Wrkplace Cmpetencies Effective wrkers can prductively use: Resurces: They knw hw t allcate time, mney, materials, space and staff. Interpersnal Skills: They can wrk n teams, teach thers, serve custmers, lead, negtiate, and wrk well with peple frm culturally diverse backgrunds. Infrmatin: They can acquire and evaluate data, rganize and maintain files, interpret and cmmunicate, and use cmputers t prcess infrmatin. Systems: They understand scial, rganizatinal, and technlgical systems; they can mnitr and crrect perfrmance; and they can design r imprve systems. Technlgy: They can select equipment and tls, apply technlgy t specific tasks, and maintain and trublesht equipment. SCANS Fundatins Skills Cmpetent wrkers in the high-perfrmance wrkplace need: Basic Skills: reading, writing, arithmetic, and mathematics, speaking and listening. Thinking Skills the ability t learn, reasn, think creatively, make decisins, and slve prblems. Persnal Qualities individual respnsibility self-esteem and self-management, sciability, integrity, and hnesty. 13
Teaching/Learning Activities The Standards fr Mathematical Practice describe varieties f expertise that mathematics educatrs at all levels shuld seek t develp in their students thrugh teaching and learning activities. A secndary dcument [Curriculum Map: Algebra I], has been prepared t prvide greater detail as t specific teaching/learning activities fr this curse. Teaching/Learning practices rest n imprtant prcesses and prficiencies with lngstanding imprtance in mathematics educatin. The first f these are the NCTM Prcess Standards f prblem slving, reasning and prf, cmmunicatin, representatin, and cnnectins. The secnd are the strands f mathematical prficiency specified in the Natinal Research Cuncil s reprt Adding It Up: adaptive reasning, strategic cmpetence, cnceptual understanding (cmprehensin f mathematical cncepts, peratins and relatins), prcedural fluency (skill in carrying ut prcedures flexibly, accurately, efficiently and apprpriately), and prductive dispsitin (habitual inclinatin t see mathematics as sensible, useful, and wrthwhile, cupled with a belief in diligence and ne s wn efficacy). The teaching/learning activities in Algebra I are fcused n develping the fllwing mathematical prficiencies: 1. Make sense f prblems and persevere in slving them. Mathematically prficient students start by explaining t themselves the meaning f a prblem and lking fr entry pints t its slutin. They analyze givens, cnstraints, relatinships, and gals. They make cnjectures abut the frm and meaning f the slutin and plan a slutin pathway rather than simply jumping int a slutin attempt. They cnsider analgus prblems, and try special cases and simpler frms f the riginal prblem in rder t gain insight int its slutin. They mnitr and evaluate their prgress and change curse if necessary. Older students might, depending n the cntext f the prblem, transfrm algebraic expressins r change the viewing windw n their graphing calculatr t get the infrmatin they need. Mathematically prficient students can explain crrespndences between equatins, verbal descriptins, tables, and graphs r draw diagrams f imprtant features and relatinships, graph data, and search fr regularity r trends. Mathematically prficient students check their answers t prblems using a different methd, and they cntinually ask themselves, Des this make sense? They can understand the appraches f thers t slving cmplex prblems and identify crrespndences between different appraches. 2. Reasn abstractly and quantitatively. Mathematically prficient students make sense f quantities and their relatinships in prblem situatins. They bring tw cmplementary abilities t bear n prblems invlving quantitative relatinships: the ability t decntextualize t abstract a given situatin and represent it symblically and manipulate the representing symbls as if they have a life f their wn, withut necessarily attending t their referents and the ability t cntextualize, t pause as needed during the manipulatin prcess in rder t prbe int the referents fr the symbls invlved. Quantitative reasning entails habits f creating a cherent representatin f the prblem at hand; cnsidering the units invlved; attending t the meaning f quantities, nt just hw t cmpute them; and knwing and flexibly using different prperties f peratins and bjects. 14
3. Cnstruct viable arguments and critique the reasning f thers. Mathematically prficient students understand and use stated assumptins, definitins, and previusly established results in cnstructing arguments. They make cnjectures and build a lgical prgressin f statements t explre the truth f their cnjectures. They are able t analyze situatins by breaking them int cases, and can recgnize and use cunterexamples. They justify their cnclusins, cmmunicate them t thers, and respnd t the arguments f thers. They reasn inductively abut data, making plausible arguments that take int accunt the cntext frm which the data arse. Mathematically prficient students are als able t cmpare the effectiveness f tw plausible arguments, distinguish crrect lgic r reasning frm that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can cnstruct arguments using cncrete referents such as bjects, drawings, diagrams, and actins. Such arguments can make sense and be crrect, even thugh they are nt generalized r made frmal until later grades. Later in algebra, students learn t determine dmains t which an argument applies. Students at all grades can listen r read the arguments f thers, decide whether they make sense, and ask useful questins t clarify r imprve the arguments. 4. Mdel with mathematics. Mathematically prficient students can apply the mathematics they knw t slve prblems arising in everyday life, sciety, and the wrkplace. In early algebraic reasning, this might be as simple as writing an additin equatin t describe a situatin. In middle grades, a student might apply prprtinal reasning t plan a schl event r analyze a prblem in the cmmunity. By high schl, a student might use gemetry t slve a design prblem r use a functin t describe hw ne quantity f interest depends n anther. Mathematically prficient students wh can apply what they knw are cmfrtable making assumptins and apprximatins t simplify a cmplicated situatin, realizing that these may need revisin later. They are able t identify imprtant quantities in a practical situatin and map their relatinships using such tls as diagrams, twway tables, graphs, flwcharts and frmulas. They can analyze thse relatinships mathematically t draw cnclusins. They rutinely interpret their mathematical results in the cntext f the situatin and reflect n whether the results make sense, pssibly imprving the mdel if it has nt served its purpse. 5. Use apprpriate tls strategically. Mathematically prficient students cnsider the available tls when slving a mathematical prblem. These tls might include pencil and paper, cncrete mdels, a ruler, a prtractr, a calculatr, a spreadsheet, a cmputer algebra system, a statistical package, r dynamic gemetry sftware. Prficient students are sufficiently familiar with tls apprpriate fr their grade r curse t make sund decisins abut when each f these tls might be helpful, recgnizing bth the insight t be gained and their limitatins. Fr example, mathematically prficient high schl students analyze graphs f functins and slutins generated using a graphing calculatr. They detect pssible errrs by strategically using estimatin and ther mathematical knwledge. When making mathematical mdels, they knw that technlgy can enable them t visualize the results f varying assumptins, explre cnsequences, and cmpare predictins with data. Mathematically prficient students at varius grade levels are able t identify relevant external mathematical resurces, such as digital cntent lcated n a website, and use them t pse r slve prblems. They are able t use technlgical tls t explre and deepen their understanding f cncepts. 15
6. Attend t precisin. Mathematically prficient students try t cmmunicate precisely t thers. They try t use clear definitins in discussin with thers and in their wn reasning. They state the meaning f the symbls they chse, including using the equal sign cnsistently and apprpriately. They are careful abut specifying units f measure, and labeling axes t clarify the crrespndence with quantities in a prblem. They calculate accurately and efficiently, express numerical answers with a degree f precisin apprpriate fr the prblem cntext. Initially, students give carefully frmulated explanatins t each ther. By the time they reach high schl they have learned t examine claims and make explicit use f definitins. 7. Lk fr and make use f structure. Mathematically prficient students lk clsely t discern a pattern r structure. Students will see 7 8 equals the well remembered 7 5 + 7 3, in preparatin fr learning abut the distributive prperty. In the expressin x 2 + 9x + 14, algebra students can see the 14 as 2 7 and the 9 as 2 + 7. They recgnize the significance f an existing line in a gemetric figure and can use the strategy f drawing an auxiliary line fr slving prblems. They als can step back fr an verview and shift perspective. They can see cmplicated things, such as sme algebraic expressins, as single bjects r as being cmpsed f several bjects. Fr example, they can see 5 3(x y) 2 as 5 minus a psitive number times a square and use that t realize that its value cannt be mre than 5 fr any real numbers x and y. 8. Lk fr and express regularity in repeated reasning. Mathematically prficient students ntice if calculatins are repeated, and lk bth fr general methds and fr shrtcuts. By paying attentin t the calculatin f slpe as they repeatedly check whether pints are n the line thrugh (1, 2) with slpe 3, algebra students might abstract the equatin (y 2)/(x 1) = 3. Nticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x2 + x + 1) might lead them t the general frmula fr the sum f a gemetric series. As they wrk t slve a prblem, mathematically prficient students maintain versight f the prcess, while attending t the details. They cntinually evaluate the reasnableness f their intermediate results. 16
Assessment and Testing Strategies Sund and prductive classrm assessments are built n a fundatin f the fllwing five key dimensins (Stiggins et al, 2006): Key 1: Assessment serves a clear and apprpriate purpse. Did the teacher specify users and uses, and are these apprpriate? Key 2: Assessment reflects valued achievement targets. Has the teacher clearly specified the achievement targets t be reflected in the exercises? D these represent imprtant learning utcmes? Key 3: Design. Des the selectin f the methd make sense given the gals and purpses? Is there anything in the assessment that might lead t misleading results? Key 4: Cmmunicatin. Is it clear hw this assessment helps cmmunicatin with thers abut student achievement? Key 5: Student Invlvement. Is it clear hw students are invlved in the assessment as a way t help them understand achievement targets, practice hitting thse targets, see themselves grwing in their achievement, and cmmunicate with thers abut their success as learners? The Algebra I curse will include a variety f assessment tls fr the effective teaching and learning f mathematics. In additin t classrm and district assessments, students will demnstrate prficiency f algebraic reasning and skills n the New Jersey Algebra I State-wide Assessment as required fr graduatin. Indicatrs f Sund Classrm Assessment Practice will cnsist f bth frmative and summative assessments that may include, but are nt limited t: Observatin Interviews Prtflis (Prject, Grwth, Achievement, Cmpetence, Celebratin) 17
Paper-and-pencil tests/quizzes Perfrmance Tasks Jurnals/Self-Reflectin 18
Texts and Materials Student Text: TBD Teacher Materials and Resurces: Blueprints fr Success Gld Seal Lessns: Successful Practice Netwrk. www.leadered.cm/spn.html Greenes, C., & Rubenstein, R. (2008). Algebra and Algebraic Thinking in Schl Mathematics, Seventieth Yearbk. Restn, VA: Natinal Cuncil f Teachers f Mathematics. Natinal Cuncil f Teachers f Mathematics. (2001). Navigating Thrugh Algebra in Grades 6-8. Restn, VA: NCTM. Natinal Cuncil f Teachers f Mathematics. (2001). Navigating Thrugh Algebra in Grades 9-12. Restn, VA: NCTM. Natinal Cuncil f Teachers f Mathematics. (2006). Navigating Thrugh Mathematical Cnnectins in Grades 9-12. Restn, VA: NCTM. Natinal Cuncil f Teachers f Mathematics. (2004). Navigating Thrugh Prbability in Grades 9-12. Restn, VA: NCTM. New Jersey Cmmn Cre Curriculum Standards 2010. www.njcccs.rg. Technlgy/Cmputer Sftware Gemeter s Sketchpad: Key Curriculum Press Fathm: Key Curriculum Press 19
Prcedures fr Use f Supplemental Instructinal Materials Instructinal materials nt apprved by the Bard f Educatin must be brught t the attentin f the building principal r vice-principal befre use in any instructinal area. Materials that are apprved include all textbks, vides, and ther supplemental material acquired thrugh purchase rders, and/r ther schl funds. Resurces frm the Cunty Educatin Media and Technlgy Center are als acceptable, with age apprpriateness reviewed. All instructinal materials nt explicitly Bard apprved as utlined in abve, which are intended fr use in any instructinal setting must be apprved by the building principal r vice- principal at least 5 schls days prir t use. The principal r vice-principal may request t review a cpy f the materials, vide, etc, prir t use in the classrm. 20
Curriculum Map Cntent/Objective Essential Questins/ Enduring Understandings Suggested Activity/ Apprpriate Materials-Equipment Evaluatin/Ass essment Operatins n Numbers and Expressins: Ratinal Numbers, Irratinal Numbers and Patterns The learner will: Investigate numeric, algebraic, and gemetric patterns review peratins with ratinal numbers use expnents t represent repeated multiplicatin Use variables in different ways use functin ntatin t represent unique relatinships between dependent and independent variables Use numerical and algebraic expressins Add, subtract, multiply plynmial expressins Factr simple plynmial expressins becme familiar with mathematics f the graphing calculatr and resurces, including a Ntatin Ntebk Essential Questins: Hw can numeric peratins be extended t algebraic bjects? Why is it useful t represent real-life situatins algebraically? What makes an algebraic algrithm bth effective and efficient? Hw can we describe and analyze iterative gemetric and algebraic patterns? What are sme ways t represent, describe, and analyze patterns (that ccur in ur wrld)? Hw can we use algebraic representatins and expressins t analyze patterns? Why are number and algebraic patterns imprtant as rules? Unit 1: Incrprates review f peratins with ratinal numbers; use f numerical and algebraic expressins, and intrduces fractals and recursin. (Recursive thinking is applied later t develp rate f change r slpe.) This chapter als intrduces students t the graphing calculatr and teaches them hw t use algebra resurces, ntes, and websites as learning and study tls. Instructinal Strategies - Sample: Cre Mathematical Prcess - Representatins: Cmpare different representatins fr an irratinal number yu wuld expect t encunter in every day life, including a physical representatin, cmmn decimal apprximatins, cmmn fractin apprximatins, and the value prduced by a calculatr. Discuss the relative accuracies f the apprximatins and suggest apprpriate circumstances fr the use f each. Fr example, students may identify and cmpare every day encunters with the rati f the circumference f a circle t its diameter. IMPORTANT NOTE: Vcabulary and Ntatin What is Algebra? An intrductin and nging dialgue thrughut this curse is essential with respect t terminlgy and symblic ntatin. Algebra as a language used t represent relatinships (relatins and functins) shuld be discussed early in the curse. Pre- Assessment, Checkpint exercises, DNws, prtflis, ral questining, clsure, selfreflectin jurnals, prjects, tests, technlgybased assessments * Nte: Infuse Algebra I EOC Released Exam Questins 21
Cntent/Objective Essential Questins/ Enduring Understandings Suggested Activity/ Apprpriate Materials-Equipment Evaluatin/Ass essment Enduring Understandings: Algebraic and numeric prcedures are intercnnected and build n ne anther t prduce a cherent whle. Lgical patterns exist and are a regular ccurrence in mathematics and the wrld arund us. Algebraic representatin can be used t generalize patterns and relatinships. The same pattern can be fund in many different frms. Prperties f iterative gemetric patterns can be analyzed and described mathematically/algebraically. Cntent/Objective: Data Explratin The learner will: interpret and cmpare a variety f graphs find summary values fr a data set Pascal s Triangle can be helpful in slving prblems related t cmbinatins and t fractals. Essential Questins: Hw can the cllectin, rganizatin, interpretatin, and display f data be used t answer questins? Unit 2: Explres data and representatins f data and establishes the data-analytic apprach. Students begin interpreting graphs and rganizing and cmputing data with matrices. Interdisciplinary Cnnectins - Sample: Determine the plitical cmpsitin f the United States Cngress during the 20th Century by decade, first researching Pre- Assessment, Checkpint exercises, DNws, prtflis, ral questining, clsure, self- 22
Cntent/Objective Essential Questins/ Enduring Understandings Suggested Activity/ Apprpriate Materials-Equipment Evaluatin/Ass essment draw cnclusins abut a data set based n graphs and summary values review hw t graph pints n a plane rganize and cmpute data with matrices ** June Hw can the representatin f data influence decisins? Enduring Understandings: The message cnveyed by the data depends n the display. The results f a statistical investigatin can be used t supprt r refute an argument. the types f party affiliatins (e.g. Republican) and the assciated numbers (f party representatives) in the Huse f Representatives and the Senate. Use matrices t represent tabular infrmatin fr each, with a third matrix that displays the numbers f the U.S. Cngress members by party affiliatin fr each decade. Using an histrical timeline, identify significant laws that were enacted during each decade in the activity abve. Technlgy Integratin: Use the internet t research and present the data fr the task abve, as well as a graphing calculatr r a spreadsheet t accmplish the task. reflectin jurnals, prjects, tests, technlgybased assessments * Nte: Infuse Algebra I EOC Exam Questins Cntent/Objective: Prprtinal Reasning and Variatin The learner will: use prprtinal reasning t understand prblem situatins learn what rates are and use them t make predictins study hw quantities vary directly and inversely use equatins and graphs t represent variatin slve real-wrld prblems using variatin review the rules fr rder f peratins slve equatins using inverse peratins Essential Questin: Hw can we use mathematical mdels t describe change r change ver time? Enduring Understandings: Prprtinality invlves a relatinship in which the rati f tw quantities remains cnstant as the crrespnding values f the quantities change. Prprtins invlve multiplicative rather than additive cmparisns. If tw quantities vary prprtinally, that relatinship can be represented as a linear functin. Unit 3: Lks at rati and prprtin and sets the stage fr rates and slpes. It includes dimensinal analysis. The chapter cntinues by lking at linear equatins fr direct-variatin relatinships, then explring inverse variatin. Students explre hw quantities vary, use rates t make predictins, and represent variatin using graphs. Perfrmance Assessment Task - Sample: Suppse that a drug cmpany has established that a patient must have 40 mg f a certain prescriptin drug in the bdy fr the drug t be effective. Mrever, the cmpany s studies indicate that anything in excess f 600 mg is txic, and its research has shwn that the bdy eliminates 10 percent f the drug every fur hurs. Imagine yu are a dctr prescribing this drug fr a patient. Hw ften wuld yu want yur patient t take the drug, and in what quantity, t ensure effectiveness while aviding txicity? (NCTM Navigating thrugh Mathematical Cnnectins in Grades 9-12) Pre- Assessment, Checkpint exercises, DNws, prtflis, ral questining, clsure, selfreflectin jurnals, prjects, tests, technlgybased assessments Nte: Infuse Algebra I EOC Exam Questins 23
Cntent/Objective Essential Questins/ Enduring Understandings Suggested Activity/ Apprpriate Materials-Equipment Evaluatin/Ass essment Cntent/Objective: Linear Equatins The learner will: investigate and write recursive (arithmetic) sequences graph scatter plts f recursive sequences study rate f change explre time-distance graphs write a linear equatin in intercept frm given a recursin rutine, a graph r data slve and graph linear equatins slve literal equatins Essential Questins: Hw can change be best represented mathematically? Hw can technlgy be used t investigate prperties f linear functins and their graphs? Enduring Understanding: Rules f arithmetic and algebra can be used tgether with (the cncept f) equivalence t transfrm equatins and inequalities s slutins can be fund t slve prblems. Unit 4: Builds upn Unit 3 by lking at linear equatins that are nt direct variatin. It ties tgether recursin frm Unit 1, fitting data frm Unit 2, and rate f change and linear variatin frm Unit 3, and als cvers slving equatins with symblic manipulatin. Order f peratins are intrduced/reviewed. Perfrmance Assessment Task - Sample: The Math Club needs t raise mney fr its annual neighbrhd park beautificatin prject. The club members decide t have a ne-day car wash t raise mney fr this prject. After estimating the cst f the activities, determine the ttal cst f spnges, rags, sap, buckets, and ther materials, and investigate the average lcal charge fr washing ne car. Write a general rule t determine hw much mney can be raised fr any number f cars. Realistically, can the car wash raise enugh mney t supprt this activity? Instructinal Strategies - Sample: Interdisciplinary Cnnectin: Investigate the relatinship between stpping distance and speed f travel in a car. Gather data frm the driver s educatin manual r nline thrugh the Mtr Vehicle Cmmissin (Technlgy Integratin), graph the values fund, nte that the relatinship is linear, and lk fr an equatin that fits the data. Pre- Assessment, Checkpint exercises, DNws, prtflis, ral questining, clsure, selfreflectin jurnals, prjects, tests, technlgybased assessments * Nte: Infuse Algebra I EOC Exam Questins Cntent/Objective: Fitting a Line t Data The learner will: define and calculate slpe f a line identify slpes f rising, falling, hrizntal, and vertical lines write an equatin that fits a set f real-wrld data identify, write and use the pint-slpe frm f an Essential Questins: Hw can the cllectin, rganizatin, interpretatin, and display f data be used t answer questins? Hw can the representatin f data influence decisins? Unit 5: Cntinues explratin f fitting a line t data. Students explre the frmula fr slpe, as well as the intercept and pintslpe frms fr the equatin f a line. Students learn t recgnize equivalent equatins in different frms, and write equatins fr real-wrld data. Interdisciplinary Cnnectins - Sample: Make a mdel f the relatinship between Celsius and Fahrenheit temperatures. Represent the relatinship as an equatin, and check the equatin against tw knwn data pints 0 degrees C = 32 degrees F and 100 degrees C = 212 degrees F. Use the equatin t cnvert between Celsius and Fahrenheit Pre- Assessment, Checkpint exercises, DNws, prtflis, ral questining, clsure, selfreflectin jurnals, prjects, tests, technlgy- 24
Cntent/Objective Essential Questins/ Enduring Understandings Suggested Activity/ Apprpriate Materials-Equipment Evaluatin/Ass essment equatin f a line check the equivalence f linear equatins by using algebraic prperties recgnize equivalent equatins written in different frms evaluate the results f finding a line that represents a set f realwrld data pints Hw can technlgy be used t investigate prperties f linear functins and their graphs? Enduring Understanding: Variables are symbls that take the place f numbers r ranges f numbers; they have different meanings depending n hw they are being used. temperatures. based assessments * Nte: Infuse Algebra I EOC Released Exam Questins: ECR, SCR, MC Cntent/Objective: Systems f Equatins and Inequalities The learner will: mdel real-wrld situatins with systems f tw linear equatins in tw variables slve systems f linear equatins by graphing slve systems using the substitutin methd slve systems using the eliminatin methd slve systems using matrices graph inequalities in ne and tw variables slve systems f linear inequalities Cntent/Objective: Expnents and Expnential Mdels The learner will: write recursive rutines fr nnlinear sequences Essential Questin: Hw can systems f equatins be used t slve real-life situatins? Enduring Understanding: Graphs and equatins are alternative (and ften equivalent) ways fr depicting and analyzing patterns f change. Essential Questin: Hw can we mdel situatins using expnents? Unit 6: Cvers slving systems f equatins and inequalities using graphing, substitutin, eliminatin, and matrix methds. Instructinal Strategies - Sample: Technlgy Integratin: The Cape May-Lewes Ferry has space fr cars and buses. Using the internet, investigate hw many f each can be transprted n a single trip. Use variables t represent the unknwns (e.g. x fr cars and y fr buses) and develp the graph f the inequality, using either paper-andpencil r a graphing calculatr. Recgnizing that the slutins have t be whle numbers, students shuld identify the pints whse cefficients are nn-negative integers and in the first quadrant n r belw the line. Unit 7: Ges beynd linear equatins and lks at expnential equatins. It als cvers the prperties f expnents and scientific ntatin. Pre- Assessment, Checkpint exercises, DNws, prtflis, ral questining, clsure, selfreflectin jurnals, prjects, tests, technlgybased assessments * Nte: Infuse Algebra I EOC Exam Questins Pre- Assessment, Checkpint exercises, DNws, prtflis, ral 25
Cntent/Objective Essential Questins/ Enduring Understandings Suggested Activity/ Apprpriate Materials-Equipment Evaluatin/Ass essment write the expnential frm f a sequence generated recursively by a cnstant multiplier use prperties (multiplicatin, divisin, pwer) f expnents t rewrite expressins write expnential equatins that mdel realwrld grwth and decay data write numbers in scientific ntatin Cntent/Objective: Functins The learner will: investigate the cncept, definitin, ntatin, prperties, and graphs f functins in detail learn terminlgy f independent and dependent variables interpret graphs f realwrld situatins using terminlgy (linear, nnlinear, increasing, decreasing, rate f change, cntinuus, discrete) cnstruct and interpret graphs and functins that describe real-wrld situatins Enduring Understanding: Real wrld situatins, invlving expnential relatinships can be slved using multiple representatins. Essential Questins: Hw are patterns f change related t the behavir f functins? Hw are functins and their graphs related? Hw can patterns, relatins, and functins be used as tls t best describe and help explain real-life situatins? Enduring Understandings: Functinal relatinships can be expressed in real cntexts, graphs, algebraic equatins, tables, and wrds; each representatin f a given functin is simply a different way f expressing the same idea. Perfrmance Assessment Task - Sample: T plan fr paying cllege tuitin, yu investigate at least three ptins fr a savings plan, using internet resurces. Explain hw a family might prepare by investing in an annuity in rder t have at least $100,000 in the annuity by the time a child is 18 years ld if they start saving immediately after the child is brn. Unit 8: Cmprehensive study f functin ntatin, abslutevalue functins, and quadratic functins. Interdisciplinary Cnnectins: Using an electrnic spreadsheet, demnstrate algebraic equivalence by demnstrating that the functins m(x) and a(x) frm the identity m(x) = a(x) in an authentic prblem. Fr example, investigate the fllwing request befre a lcal planning bard: A subdivisin is being placed n a piece f land 1000 m by 1500 m. A bulevard f trees and an access rad f unifrm width frm the brder f the subdivisin. The area f the inner rectangle f huses and parks is t be at least 1.35 millin m2 t accmmdate the planned hmes and parks. What is the largest width that can be set aside inside the perimeter fr the brder cmpsed f the bulevard and rad? (NCTM Navigatins thrugh Algebra In Grades 9-12) questining, clsure, selfreflectin jurnals, prjects, tests, technlgybased assessments * Nte: Infuse Algebra I EOC Released Exam Questins: ECR, SCR, MC Pre- Assessment, Checkpint exercises, DNws, prtflis, ral questining, clsure, selfreflectin jurnals, prjects, tests, technlgybased assessments * Nte: Infuse Algebra I EOC Released Exam Questins 26
Cntent/Objective Essential Questins/ Enduring Understandings Suggested Activity/ Apprpriate Materials-Equipment Evaluatin/Ass essment evaluate functins by substitutin, using the graphing calculatr and written graphs wrk with abslute values (abslute value functin and its graph) wrk with the squaring and square rt functins and the parablic graph f the squaring functin The value f a particular representatin depends n its purpse. Cntent/Objective: Transfrmatins The learner will: graph plygns n crdinate plane using grid paper and graphing calculatr technlgy describe, recgnize and graph transfrmatins (translatins, reflectins, dilatins, rtatins) using paper and technlgy tls write equatins f transfrmed functins and graph them explre cncept f a parent functin and its family mdel real-wrld data with transfrmed equatins investigate ratinal functins f the frm f(x) = a/x and explre basic transfrmatins mdel Essential Questins: Hw are similarity, cngruence, and symmetry related? What situatins can be analyzed using transfrmatins and symmetries? Hw can transfrmatins be described mathematically? Enduring Understanding: A variety f families f functins can be used t mdel and slve real wrld situatins. Unit 9: Lks at functin transfrmatins algebraically and gemetrically and explres families f functins. Pre- Assessment, Checkpint exercises, DNws, prtflis, ral questining, clsure, selfreflectin jurnals, prjects, tests, technlgybased assessments * Nte: Infuse Algebra I EOC Released Exam Questins: ECR, SCR, MC 27
Cntent/Objective Essential Questins/ Enduring Understandings Suggested Activity/ Apprpriate Materials-Equipment Evaluatin/Ass essment real-wrld data with ratinal functins use matrices t represent transfrmatins Cntent/Objective: Quadratic Mdels The learner will: use quadratic functins t mdel and slve equatins based n relatinships fr prjectiles (time, height) slve quadratic equatins using graphs, tables, and symblic methds write quadratic equatins that mdel ther realwrld data find the x-intercepts and vertex f a parabla by graphical and symblic methds cnvert quadratic equatins amng vertex, factred, and general frms multiply binmials and factr trinmials explre cubes, cube rts, and transfrmatins f the parent cubic functin Essential Questins: Hw can we use mathematical language t describe nn-linear change? Hw can we mdel situatins using quadratics? Enduring Understandings: Graphs and equatins are alternative (and ften equivalent) ways fr depicting and analyzing patterns f nn-linear change. Mathematical mdels can be used t describe physical relatinships; these relatinships are ften nnlinear. Real wrld situatins, invlving quadratic relatinships, can be slved using multiple representatins. Unit 10: Cntinues explratin f quadratic functins in general, vertex, and factred frm. Students learn t mdel real-wrld data with quadratic functins, and cmbine and factr plynmials. Instructinal Strategies - Samples: Technlgy Integratin: Students investigate what size square t cut frm each crner f a rectangular piece f cardbard in rder t make the largest pssible pen-tp bx. Students make mdels, recrd the size f the square and the vlume fr each mdel, pltting the pints n a graph using spreadsheet sftware. They nte the relatinship is nt linear and make a cnjecture abut maximum vlume. Students als generate an algebraic expressin and equatin describing this situatin Cre Mathematical Prcess - Representatins: Use algebra tiles t mdel expressins and develp an understanding f factring. Interdisciplinary Cnnectins: Use algebra tiles t represent plynmial expressins in multiple ways, cmparing them frm an artistic viewpint. Fr example, while (2 + x + 2)(3+ x + 3), (x + 4)(x + 6), and x2 + 10x + 24 are all algebraically equal, the algebra-tile representatins f the first tw algebraic expressins wuld vary greatly in artistic symmetry. Pre- Assessment, Checkpint exercises, DNws, prtflis, ral questining, clsure, selfreflectin jurnals, prjects, tests, technlgybased assessments * Nte: Infuse Algebra I EOC Released Exam Questins: ECR, SCR, MC Cre Mathematical Prcess - Representatins: Abu Kamil used gemetric mdels many centuries ag t slve algebraic prblems. Use a sheet f paper t mdel (a + b)2 and remve the b2 sectin frm the crner f the paper. After cutting, rearrange the pieces and explain hw this represents the factrizatin f a2 + 2ab + b 2. 28
Cntent/Objective Essential Questins/ Enduring Understandings Suggested Activity/ Apprpriate Materials-Equipment Evaluatin/Ass essment Cntent/Objective: Prbability The learner will: create and interpret relative frequency graphs determine experimental and theretical prbabilities use prbabilities t describe patterns when utcmes are randm cunt numbers f permutatins and cmbinatins t determine prbabilities determine prbabilities f different utcmes fr a multiple stage experiment calculate the expected value f a randm event Cntent/Objective: Intrductin t Gemetry The learner will: extend knwledge abut slpes f parallel lines t perpendicular lines and midpints t crdinates recgnize and use definitins f quadrilaterals distinguish between deductive and inductive reasning use algebra t describe gemetric relatinships Essential Questins: When des rder matter? Hw can experimental and theretical prbabilities be used t make predictins r draw cnclusins? Enduring Understandings: Tables, charts, tree diagrams, and multiplicatin can be used t determine hw many ways an event can ccur. Prbability is abut predictins ver the lng term rather than predictins f individual events. Essential Questin: Hw can gemetric/algebraic relatinships best be represented and verified? Enduring Understanding: An bject s lcatin n a plane r in space can be described quantitatively. Unit 11: Explres relative frequency graphs, prbability utcmes and trials, randm utcmes, cunting techniques, multiple-stage experiments, and expected value. Instructinal Strategies Samples: Emply Venn diagrams t summarize infrmatin cncerning cmpund events. Interdisciplinary Cnnectins: Use prbability t interpret dds and risks f financial investments ptins and recgnize cmmn miscnceptins. Students investigate return and risk fr varius investments, includ-ing certificates f depsit, stcks, bnds, and real estate. Interdisciplinary Cnnectins: Analyze the risks assciated with a particular accident, illness, r curse f treatment expressed as a prbability. Present varius incidents that have a similar prbability f ccurrence. Unit 12: Includes the midpint frmula, parallel and perpendicular lines, the Pythagrean Therem and square rts, calculating distance between tw pints, and an intrductin t trignmetry. Students als use algebra t describe gemetric relatinships. Emphasize the practical use f radicals and prvide ample applicatins and skill-based mdeling. Pre- Assessment, Checkpint exercises, DNws, prtflis, ral questining, clsure, selfreflectin jurnals, prjects, tests, technlgybased assessments * Nte: Infuse Algebra I EOC Exam Questins ECR, SCR, MC Pre- Assessment, Checkpint exercises, DNws, prtflis, ral questining, clsure, selfreflectin jurnals, prjects, tests, technlgybased assessments 29