Algebra 1 /Algebra 1 Honors Curriculum Map

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1 Algebra 1 /Algebra 1 Hnrs Curriculum Map Mathematics Flrida Standards Vlusia Cunty Curriculum Maps are revised annually and updated thrughut the year. The learning gals are a wrk in prgress and may be mdified as needed.

2 Flrida Standards Standards fr Mathematical Practice 1. Make sense f prblems and persevere in slving them. (MAFS.K12.MP.1) Slving a mathematical prblem invlves making sense f what is knwn and applying a thughtful and lgical prcess which smetimes requires perseverance, flexibility, and a bit f ingenuity. 2. Reasn abstractly and quantitatively. (MAFS.K12.MP.2) The cncrete and the abstract can cmplement each ther in the develpment f mathematical understanding: representing a cncrete situatin with symbls can make the slutin prcess mre efficient, while reverting t a cncrete cntext can help make sense f abstract symbls. 3. Cnstruct viable arguments and critique the reasning f thers. (MAFS.K12.MP.3) A well-crafted argument/critique requires a thughtful and lgical prgressin f mathematically sund statements and supprting evidence. 4. Mdel with mathematics. (MAFS.K12.MP.4) Many everyday prblems can be slved by mdeling the situatin with mathematics. 5. Use apprpriate tls strategically. (MAFS.K12.MP.5) Strategic chice and use f tls can increase reliability and precisin f results, enhance arguments, and deepen mathematical understanding. 6. Attend t precisin. (MAFS.K12.MP.6) Attending t precise detail increases reliability f mathematical results and minimizes miscmmunicatin f mathematical explanatins. 7. Lk fr and make use f structure. (MAFS.K12.MP.7) Recgnizing a structure r pattern can be the key t slving a prblem r making sense f a mathematical idea. 8. Lk fr and express regularity in repeated reasning. (MAFS.K12.MP.8) Recgnizing repetitin r regularity in the curse f slving a prblem (r series f similar prblems) can lead t results mre quickly and efficiently.

3 Algebra 1: Flrida Standards The fundamental purpse f this curse is t frmalize and extend the mathematics that students learned in the middle grades. The critical areas, called units, deepen and extend understanding f linear and expnential relatinships by cntrasting them with each ther and by applying linear mdels t data that exhibit a linear trend, and students engage in methds fr analyzing, slving, and using quadratic functins. The Mathematical Practice Standards apply thrughut each curse and, tgether with the cntent standards, prescribe that students experience mathematics as a cherent, useful, and lgical subject that makes use f their ability t make sense f prblem situatins. Relatinships Between Quantities and Reasning with Equatins/Inequalities: By the end f eighth grade students have learned t slve linear equatins in ne variable and have applied graphical and algebraic methds t analyze and slve systems f linear equatins in tw variables. This unit builds n these earlier experiences by asking students t analyze and explain the prcess f slving an equatin. Students analyze and explain the prcess f slving an equatin. Students develp fluency, writing, interpreting, and translating between varius frms f linear equatins and inequalities, and using them t slve prblems. They master the slutin f linear equatins and apply related slutin techniques and the laws f expnents t the creatin and slutin f simple expnential equatins. Linear/Expnential Relatinships and Functins: In earlier grades, students define, evaluate, and cmpare functins, and use them t mdel relatinships between quantities. In this unit, students will learn functin ntatin and develp the cncepts f dmain and range. They explre many examples f functins, including sequences; they interpret functins given graphically, numerically, symblically, and verbally, translate between representatins, and understand the limitatins f varius representatins. Students build n and infrmally extend their understanding f integer expnents t cnsider expnential functins. They cmpare and cntrast linear and expnential functins, distinguishing between additive and multiplicative change. Students explre systems f equatins and inequalities, and they find and interpret their slutins. They interpret arithmetic sequences as linear functins and gemetric sequences as expnential functins. This unit als builds upn students prir experiences with data, prviding students with mre frmal means f assessing hw a mdel fits data. Students use regressin techniques t describe and apprximate linear relatinships between quantities. They use graphical representatins and knwledge f the cntext t make judgments abut the apprpriateness f linear mdels. With linear mdels, they lk at residuals t analyze the gdness f fit. Expressins and Equatins: In this unit, students build n their knwledge frm the unit f Linear and Expnential Relatinships, where they extended the laws f expnents t ratinal expnents. Students apply this new understanding f number and strengthen their ability t see structure in and create quadratic and expnential expressins. They create and slve equatins, inequalities, and systems f equatins invlving quadratic expressins. Quadratic Functins and Mdeling: In this unit, students cnsider quadratic functins, cmparing the key characteristics f quadratic functins t thse f linear and expnential functins. They select frm amng these functins t mdel phenmena. Students learn t anticipate the graph f a quadratic functin by interpreting varius frms f quadratic expressins. In particular, they identify the real slutins f a quadratic equatin as the zeres f a related quadratic functin. Students expand their experience with functins t include mre specialized functins abslute value, step, and thse that are piecewise-defined.

4 Algebra 1: Flrida Standards At A Glance First Quarter Secnd Quarter Third Quarter Furth Quarter DSA Unit 1 Equatins and Inequalities MAFS.912.A-CED.1.1 MAFS.912.A-CED.1.2 MAFS.912.A-CED.1.3 MAFS.912.A-CED.1.4 MAFS.912.A-REI.1.1 MAFS.912.A-REI.1.2 MAFS.912.A-REI.2.3 MAFS.912.A-REI.4.10 MAFS.912.A-SSE.1.1 MAFS.912.N-Q.1.1 MAFS.912.N-Q.1.2 MAFS.912.N-Q.1.3 MAFS.912.F-BF.1.2 DIA 1 Unit 2 Functins MAFS.912.F-IF.1.1 MAFS.912.F-IF.1.2 MAFS.912.F-IF.2.4 MAFS.912.F-IF.2.5 MAFS.912.F-IF.2.6 MAFS.912.F-IF.3.9 MAFS.912.F-LE.1.2 MAFS.912.F-BF.2.4 DIA 2 Unit 3 Linear and Expnential Relatinships MAFS.912.F-BF.1.1a, b MAFS.912.F-BF.2.3 MAFS.912.F-IF.1.3 MAFS.912.F-IF.3.7a, e MAFS.912.F-IF.3.8b MAFS.912.F-LE.1.1 MAFS.912.F-LE.1.3 MAFS.912.A-SSE.2.3 MAFS.912.F-LE.2.5 MAFS.912.N-RN.1.1 MAFS.912.N-RN.1.2 MAFS.912.F-BF.1.2 MAFS.912.F-LE.1.2 MAFS.912.A-SSE.2.4 DIA 3 Unit 4 Systems f Equatins and Inequalities MAFS.912.A-CED.1.3 MAFS.912.A.-REI.3.5 MAFS.912.A.-REI.3.6 MAFS.912.A.-REI.4.12 MAFS.912.A-REI.4.11 SSA *Highlighted standards are Algebra 1 Hnrs ONLY.* Unit 5 Analyzing Univariate Data MAFS.912.S-ID.1.1 MAFS.912.S-ID.1.2 MAFS.912.S-ID.1.3 MAFS.912.S-ID.1.4 Unit 6 Analyzing Bivariate Data MAFS.912.S-ID.2.6 MAFS.912.S-ID.3.7 MAFS.912.S-ID.3.8 MAFS.912.S-ID.3.9 MAFS.912.S-ID.2.5 DIA 4 Unit 7 Plynmials MAFS.912.A-APR.1.1 MAFS.912.A-SSE.1.1 MAFS.912.A-SSE.1.2 Unit 8 Slving Quadratic Equatins MAFS.912.A-CED.1.1 MAFS.912.A-CED.1.2 MAFS.912.A-APR.2.2 MAFS.912.A-APR.2.3 MAFS.912.A-APR.3.4 MAFS.912.A-APR.4.6 MAFS.912.A-REI.2.4 MAFS.912.A-SSE.2.3 MAFS.912.N-RN.2.3 DIA 5 Unit 9 Graphing Quadratic Equatins MAFS.912.A-CED.1.2 MAFS.912.F-BF.2.3 MAFS.912.F-IF.3.7a, c, d MAFS.912.F-IF.3.8a Unit 10 Piecewise and Abslute Value Functins MAFS.912.A-REI.2.3 MAFS.912.A-REI.4.11 MAFS.912.F-IF.3.7b MAFS.912.F-IF.3.9 DIA 6 Unit 11 Using Graphs f Functins MAFS.912.F-BF.1.1b MAFS.912.F-IF.2.4 MAFS.912.F-IF.2.5 MAFS.912.F-IF.3.7e MAFS.912.F-IF.3.9 MAFS.912.F-LE.1.3 MAFS.912.A-REI.3.7 Diagnstic EOC

5 Fluency Recmmendatins A/G- Algebra I students becme fluent in slving characteristic prblems invlving the analytic gemetry f lines, such as writing dwn the equatin f a line given a pint and a slpe. Such fluency can supprt them in slving less rutine mathematical prblems invlving linearity, as well as in mdeling linear phenmena (including mdeling using systems f linear inequalities in tw variables). A-APR.1- Fluency in adding, subtracting, and multiplying plynmials supprts students thrughut their wrk in Algebra, as well as in their symblic wrk with functins. Manipulatin can be mre mindful when it is fluent. A-SSE.1b- Fluency in transfrming expressins and chunking (seeing parts f an expressin as a single bject) is essential in factring, cmpleting the square, and ther mindful algebraic calculatins. The fllwing Mathematics and English Language Arts CCSS shuld be taught thrughut the curse: MAFS.912.N-Q.1.1: Use units as a way t understand prblems and t guide the slutin f multi-step prblems; chse and interpret units cnsistently in frmulas; chse and interpret the scale and the rigin in graphs and data displays. MAFS.912.N-Q.1.2: Define apprpriate quantities fr the purpse f descriptive mdeling. MAFS.912.N-Q.1.3: Chse a level f accuracy apprpriate t limitatins n measurement when reprting quantities. LACC.910.RST.1.3: Fllw precisely a cmplex multistep prcedure when carrying ut experiments, taking measurements r perfrming tasks, attending t special cases r exceptins defined in the text. LACC.910.RST.2.4: Determine the meaning f symbls, key terms, and ther dmain-specific wrds and phrases as they are used in cntext and tpics. LACC.910.RST.3.7: Translate quantitative r technical infrmatin expressed in wrds in a text int visual frm and translate infrmatin expressed visually r mathematically int wrds. LACC.910.SL.1.1: Initiate and participate effectively in a range f cllabrative discussins with diverse partners. LACC.910.SL.1.2: Integrate multiple surces f infrmatin presented in diverse media r frmats evaluating the credibility and accuracy f each surce. LACC.910.SL.1.3: Evaluate a speaker s pint f view, reasning, and use f evidence and rhetric, identifying any fallacius reasning r exaggerated r distrted evidence. LACC.910.SL.2.4: Present infrmatin, findings and supprting evidence clearly, cncisely, and lgically such that listeners can fllw the line f reasning. LACC.910.WHST.1.1: Write arguments fcused n discipline-specific cntent. LACC.910.WHST.2.4: Prduce clear and cherent writing in which the develpment, rganizatin, and style are apprpriate t task, purpse, and audience. LACC.910.WHST.3.9: Draw evidence frm infrmatinal texts t supprt analysis, reflectin, and research.

6 Unit 1 Slving and Applying Equatins and Inequalities Students will analyze and explain the prcess f evaluating, simplifying and writing expressins and slving equatins and inequalities. Students will develp fluency n writing, graphing, interpreting, translating between linear equatins and inequalities, and use them t slve prblems. Essential Questin(s): Hw are algebraic equatins used in real life? Hw can yu use the prperties f equality t supprt yur slutin t a linear equatin? Hw d yu justify the slutin t a linear inequality? Hw d yu slve literal equatins and inequalities? Hw d yu rewrite frmulas? By which methds d we translate expressins, equatins and inequalities int real wrld prducts? Hw can yu use linear equatins and inequalities t mdel the result f real wrld? What is the slpe f a linear functin and hw can yu use it t graph the functin? Hw can yu represent a functin symblically frm a graph, a verbal descriptin, r a table f values? Standard Learning Gals The students will: I can: MAFS.912A-CED.1.1. Understand variables, expressins, terms, factrs, Create equatins and inequalities in ne variable and use them t slve prblems. cefficients, and rder f peratins. define expressin, term, factr, and cefficient. MAFS.912.A-CED.1.2 Create equatins in tw r mre variables t represent relatinships between quantities interpret the real-wrld meaning f the terms, factrs, and cefficients f an expressin in terms f their units. grup the parts f an expressin differently in rder t better interpret their meaning. MAFS.912A.CED.1.3 Represent cnstraints by equatins r inequalities, and by systems f equatins and/r inequalities, and interpret slutins as viable r nn-viable ptins in a mdeling cntext. MAFS.912A.CED.1.4 Rearrange frmulas t highlight a quantity f interest, using the same reasning as in slving equatins. MAFS.912A.REI.1.1 Explain each step in slving a simple equatin as fllwing frm the equality f numbers asserted at the previus step, starting frm the assumptin that the riginal equatin has a slutin. Cnstruct a viable argument t justify a slutin methd. Write expressins, equatins, and inequalities in ne variable. identify the variables and quantities represented in a real-wrld prblem. write the equatin r inequality that best mdels the prblem. Slve equatins and inequalities in ne variable. slve the linear equatin r inequality. interpret the slutin in the cntext f the prblem. determine the best mdel fr the real-wrld prblem. apply rder f peratins and inverse peratins t slve equatins. cnstruct an argument t justify my slutin prcess. interpret slutins in the cntext f the situatin mdeled and decide if they are reasnable. slve linear equatins in ne variable including

7 MAFS.912.A-REI.1.2: Slve simple ratinal and radical equatins in ne variable, and give examples shwing hw extraneus slutins may arise. MAFS.912A.REI.2.3 Slve linear equatins and inequalities in ne variable, including equatins with cefficients represented by letters. MAFS.912.A-REI Understand that the graph f an equatin in tw variables is the set f all its slutins pltted in the crdinate plane, ften frming a curve (which culd be a line). MAFS.912.A-SSE.1.1: Interpret expressins that represent a quantity in terms f its cntext. a. Interpret parts f an expressin, such as terms, factrs and cefficients. b. Interpret cmplicated expressins by viewing ne r mre f their parts as a single entity. MAFS.912.N-Q.1.1 Use units as a way t understand prblems and t guide the slutin f multi-step prblems; chse and interpret units cnsistently in frmulas. MAFS.N-Q.1.2 Define apprpriate quantities fr the purpse f descriptive mdeling. MAFS.912.N-Q.1.3 Chse a level f accuracy apprpriate t limitatins n measurement when reprting quantities. MAFS.912.F-BF.1.2: Write arithmetic and gemetric sequences bth recursively and with an explicit frmula, use them t mdel situatins, and translate between the tw frms. equatins with cefficients represented by letters. slve linear inequalities in ne variable including equatins with cefficients represented by letter. graph linear inequalities in ne variable n a number line. Slve literal equatins. slve frmulas fr a specified variable. Write linear equatins and inequalities in tw variables. understand slpe- intercept frm, pint-slpe frm, standard frm. cnvert between the different frms f linear equatins. interpret the real-wrld meaning f the terms, factrs, and cefficients f an expressin in terms f their units Graph linear equatins and inequalities in tw variables. set up crdinate axes using an apprpriate scale and label the axes graph equatins n crdinate axes with apprpriate labels and scales. chse an apprpriate scale and rigin fr graphs and data displays. interpret the scale and rigin fr graphs and data displays. identify the variables r quantities f significance frm the data prvided. identify r chse the apprpriate unit f measure fr each variable r quantity. explain that every rdered pair n the graph f an equatin represents values that make the equatin true. verify that any pint n a graph will result in a true equatin when their crdinates are substituted int the equatin. Slve simple radical and ratinal equatins in ne variable. define extraneus slutin. slve a ratinal equatin in ne variable. determine which numbers cannt be slutins f a ratinal equatin and explain why they cannt be slutins. generate examples f ratinal equatins with extraneus slutins.

8 slve a radical equatin in ne variable. determine which numbers cannt be slutins a radical equatin and explain why they cannt be slutins. generate examples f radical equatins with extraneus slutins Write arithmetic sequences recursively and explicitly. explain that recursive frmula tells me hw a sequence starts and tells me hw t use the previus value(s) t generate the next element f the sequence. explain that an explicit frmula allws me t find any element f a sequence withut knwing the element befre it. distinguish between explicit and recursive frmulas fr sequences. define an arithmetic sequence as a sequence f numbers that is frmed s that the difference between cnsecutive terms is always the same knwn as a cmmn difference. determine the cmmn difference between tw terms in an arithmetic sequence. explain hw t change a term f an arithmetic sequence int the next term and write a recursive frmula fr the sequence, a n = a n 1 + d. write an explicit frmula fr an arithmetic sequence, a n = a 1 + (n 1)d.. decide when a real wrld prblem mdels an arithmetic sequence and write an equatin t mdel the situatin. Thrughut Unit 1: label units thrugh multiple steps f a prblem. chse apprpriate units fr real wrld prblems invlving frmulas. use and interpret units when slving frmulas. reprt measured quantities in a way that is reasnable fr the tl used t make the measurement. reprt calculated quantities using the same level f accuracy as used in the prblem statement.

9 Unit 1 Slving and Applying Equatins and Inequalities Remarks Use this pprtunity t review peratins with integers, ratinal numbers, number sense, etc as yu teach this standard. Students may believe that slving an equatin such as 3x + 1 = 7 invlves nly remving the 1, failing t realize that the equatin 1 = 1 is being subtracted t prduce the next step. When using Distributive Prperty, students ften multiply the number (r variable) utside the parentheses by the first term in the parentheses, but neglect t multiply that same number by the ther term(s) in the parentheses. Regarding variables n bth sides, students ften will try t cmbine the terms as if they are n the same side f the equatin rather than eliminating ne f the variables. Students may cnfuse the rule f reversing the inequality when multiplying r dividing by a negative number, with the need t reverse the inequality anytime a negative sign shws up in slving the last step f the inequality. Example: 3x > -15 r x < - 5 (Rather than crrectly using the rule: -3x >15 r x< -5) Students may struggle t slve literal equatins/ frmulas due t nt cntaining any numbers, s reiterating that the same steps (inverse peratins) are used whether dealing with eliminating a variable r number may be helpful. Refer t slpe as rate f change t prepare students fr Unit 2. Resurces = b_exercise_detail.html b_exercise_detail.html b_bnedensity_detail.html

10 Students will learn functin ntatin and develp the cncepts f dmain and range. They explre many examples f functins, including sequencing; they interpret functins given graphically, numerically, symblically, and verbally, translate between representatins, and understand the limitatins f varius representatins. Essential Questin(s): In which ways d yu interpret dmain and range in multiple frmats? Hw can yu represent a functin symblically frm a graph, a verbal descriptin, r a table f values? Hw can yu use peratins t cmbine? What is the inverse f a functin, and hw can yu find the inverse f a linear functin? Learning Gals Standard The students will: MAFS. 912.F-IF.1.1 Understand that a functin frm ne set (called the dmain) t anther set (called the range) assigns t each element f the dmain exactly ne element f the range. If f is a functin and x is an element f its dmain, the f(x) dentes the utput f f crrespnding t the input x. The graph f f is the graph f the equatin y=f(x). MAFS. 912.F-IF.1.2 Use functin ntatin, evaluate functins fr inputs in their dmains, and interpret statements that use functin ntatin in terms f a cntext. MAFS. 912.F-IF.2.4 Fr a functin that mdels a relatinship between tw quantities, interpret key features f graphs and tables in terms f the quantities, and sketch graphs shwing key features given a verbal descriptin f the relatinship. MAFS. 912.F-IF.2.5 Relate the dmain f a functin t its graph and, where applicable, t the quantitative relatinship it describes. MAFS. 912.F-IF.2.6 Calculate and interpret the average rate f change f a functin (presented symblically r as a table) ver a specified interval. Estimate the rate f change frm a graph. MAFS.912.F-IF.3.9 Cmpare prperties f tw functins each represented in a different way (algebraically, graphically, numerically in tables, r by verbal descriptins). Fr example, given a graph f ne quadratic functin and an algebraic expressin fr anther, say which has the larger maximum. MAFS.912.F-LE.1.2 Cnstruct linear and expnential functins, including arithmetic and gemetric sequences, given a graph, a descriptin f a relatinship, r tw input-utput pairs (include reading these frm a table). I can: define and understand functins. define relatin, dmain and range. define a functin as a relatin in which each input (dmain) has exactly ne utput (range). determine if a graph, table r set f rdered pairs represent a functin. determine if stated rules (bth numeric and nn-numeric) prduce rdered pairs that frm a functin. explain that when x is an element f the input f a functin f(x) represents the crrespnding utput. explain that functin ntatin is nt limited t f(x); ther latters can als be used t we can tell different functins apart. explain that the graph f f is the graph f the equatin y=f(x). state the apprpriate dmain f a functin that represents a prblem situatin, defend my chice, and explain why ther numbers might be excluded frm the dmain. identify and eliminate the part(s) f a graph that cause it t fail the vertical line test. When a functin is presented using a table r graph: lcate the infrmatin that explains what each quantity represents. interpret the meaning f an rdered pair. determine if negative inputs make sense in the prblem situatin. determine if negative utputs make sense in the prblem situatin. identify the y-intercept. use the definitin f functin t explain why there can nly be ne y-intercept. use the prblem situatin t explain what an x-intercept

11 MAFS.912.F-BF.2.4 Find inverse functins. a. Slve an equatin f the frm f(x) = c fr a simple functin f that has an inverse and write an expressin fr the inverse. Fr example, f(x) = 2 x 3 r f(x) = (x+1) fr x 1. (x 1) b. Verify by cmpsitin that ne functin is the inverse f anther. c. Read values f an inverse functin frm a graph r a table, given that the functin has an inverse. d. Prduce an invertible functin frm a nn-invertible functin by restricting the dmain. means. identify the x-intercept. use the definitin f functin t explain why sme functins have mre than ne x-intercept. use the prblem situatin t explain what an x-intercept means. explain hw the dmain f a functin is represented in its graph. state, defend and explain the apprpriate dmain f a functin that represents a prblem situatin. When a functin is presented using a verbal r written descriptin: lcate the infrmatin that explains what each quantity represents. decide which quantity shuld be used as the input. identify which parts f the descriptin indicate, if applicable, the functin s y-intercept, x-intercept(s). create a graph that matches the descriptin and indicates all f the key features f the functin. Write and evaluate functins using functin ntatin. decde functin ntatin and explain hw the utput f a functin is matched t its input. cnvert a table, graph, set f rdered pairs r descriptin int functin ntatin by identifying the rule used t turn inputs int utputs and writing the rule. use rder f peratins t evaluate a functin fr a given dmain value. identify the numbers that are nt in the dmain f a functin. chse and analyze inputs (and utputs) that make sense based n the prblem. calculate and interpret the rate f change f a functin. define and explain interval, rate f change and average rate f change. calculate the average rate f change f a functin, represented either by functin ntatin, a graph r a table ver a specific interval. cmpare the rates f change f tw r mre functins. interpret the meaning f the average rate f change in the cntext f the prblem. write the inverse f a functin. define inverse f a functin

12 PART A write the inverse f a functin by slving f(x) = c; fr x explain that after f(x) = c; fr x, c can be cnsidered the input and x the utput write the inverse f a functin in standard ntatin by replacing the x in my inverse equatin with y and replacing the c in my inverse equatin with x. PART B use the cmpsitin f functins t verify that g(x) and f(x) are inverses by shwing that g(f(x))=f(g(x)) = 1. PART C decide if a functin has an inverse using the hrizntal line test use the definitins f functins, inverse functins, and 1:1 functins t explain why the hrizntal line test wrks. list values f an inverse given a table r graph f a functin that has an inverse. PART D identify and eliminate the part f the graph that caused it t fail the vertical line test state the dmain f a relatin that has been altered in rder t pass the hrizntal line test write the inverse f the invertible functin in functin ntatin

13 Unit 3 - Linear and Expnential Relatinships Students build n and infrmally extend their understanding f integer expnents t cnsider expnential functins. They cmpare and cntrast linear and expnential functins, distinguishing between additive and multiplicative change. They interpret arithmetic sequences as linear functins and gemetric sequences as linear functins and gemetric sequences as expnential functins. Essential Questin(s): In what ways can yu recgnize, describe, input and cmpare linear and expnential functins? What are discrete expnential functins and hw can yu represent them? Hw d yu write, graph, and interpret an expnential grwth and decay functin? Hw can yu slve prblems mdeled by equatins invlving variable expnents? Hw can yu recgnize, describe, and cmpare linear and expnential functins? Hw can yu mdel real wrld prblems using an expnential functin? Learning Gals Standard The students will: MAFS.912.F-BF.1.1a Determine an explicit expressin, a recursive prcess, r steps fr calculatin frm a cntext. MAFS.912.F-BF.1.1b Cmbine standard functin types using arithmetic peratins. Fr example, build a functin that mdels the temperature f a cling bdy by adding a cnstant functin t a decaying expnential, and relate these functins t the mdel. MAFS.912.F-BF.2.3 Identify the effect n the graph f replacing f(x) by f(x) + k, k (f(x), f(kx),and f(x+k) fr specific values f k. ( psitive and negative) ; find the value f k given the graphs. Experiment with cases and illustrate an explanatin f the effects n the graph using technlgy MAFS.912.F-IF.3.7 Graph functin expressed symblically and shw key features f the graph, by hand in simple cases and using technlgy fr mre cmplicated functins. a. Graph linear and quadratic functins ; shw intercepts, max and min. e. Graph expnential and lgarithmic functins, shwing intercepts and end behavir, and trignmetric functins shwing perid, midline and amplitude. MAFS.912.F-IF.3.8b Write a functin defined by an expressin in different but equivalent frms t reveal and explain different prperties f the functin. a. Use prperties f expnents t interpret expressins fr expnential functin. I can: Simplify expressins invlving expnents define a linear functin and expnential functin. evaluate and simplify expressins cntaining zer and integer expnents. multiply mnmials. apply multiplicatin prperties f expnents t evaluate and simplify expressins. divide mnmials. apply divisin prperties f expnents t evaluate and simplify expressins. apply prperties f ratinal expnents t simplify expressins. cnvert between radicals and ratinal expnents. recgnize and write expnential functins. classify expnential functins in functin ntatin as grwth r decay explain hw can simple gemetric transfrmatins changes a grwth graph t a decay graph. demnstrate that an expnential functin has a cnstant multiplier r equal intervals. identify situatins that display equal ratis f change ver equal intervals and can be mdeled with expnential functins. distinguish between situatins mdeled with linear functins and withexpnential functins when presented with a realwrld prblem. understand expnential functins and hw they are used. recgnize differences between graphs f expnential functins with different bases. apply expnential functins t mdel applicatins that include grwth and decay in different cntexts.

14 MAFS.912.F-LE.1.1 Distinguish between situatins that can be mdeled with linear functins and with expnential functins. a. Prve that linear functins grw by equal differences ver equal intervals, and that expnential functins grw by equal factrs ver equal intervals. b. Recgnize situatins in which ne quantity changes at a cnstant rate per unit interval relative t anther. c. Recgnize situatins in which a quantity grws r decays by a cnstant percent rate per unit interval relative t anther. MAFS.912.F-LE.1.3 Observe using graphs and tables that a quantity increasing expnentially eventually exceeds a quantity increasing linearly, quadratically, r (mre generally) as a plynmial functin. MAFS.912.A-SSE.2.3c: Chse and prduce an equivalent frm f an expressin t reveal and explain prperties f the quantity represented by the expressin. c. Use the prperties f expnents t transfrm expressins fr expnential functins. MAFS.912.F-LE.2.5: Interpret the parameters in an expnential functin in terms f a cntext. MAFS.912.N-RN.1.1: Explain hw the definitin f the meaning f ratinal expnents fllws frm extending the prperties f integer expnents t thse values, allwing fr a ntatin fr radicals in terms f ratinal expnents. Fr example, we define t be the cube rt f 5 because we want = t hld, s must equal 5. MAFS.912.N-RN.1.2: Rewrite expressins invlving radicals and ratinal expnents using the prperties f expnents. MAFS.912.F-BF.1.2: Write arithmetic and gemetric sequences bth recursively and with an explicit frmula, use them t mdel situatins, and translate between the tw frms. define an expnential functin f(x)=ab x. rewrite expnential functins using the prperties f expnents identify the names and definitins f the parameters a, b and c in the expnential functin f(x)=a(b x )+c. explain the meaning (using apprpriate units) f the cnstant a, b, c and ther pints f an expnential functin when the expnential functin mdels a real-wrld relatinship. cmpse an riginal prblem situatin and cnstruct an expnential functin t mdel it. graph expnential functins. explain the parent functin fr expnentials determine dmain, range, and end behavir ( hrizntal asymptte) by inspectin f the graph use technlgy t graph a plynmial and t find piece values fr the x-intercept(s) and the maximums and minimum (turning pints) substitute cnvenient values fr x t generate a table and graph f an expnential functin. explain hw a simple gemetric transfrmatin changes a grwth graph t a decay graph. interpret the cmpnents f an expnential functin in the cntext f a prblem. describe changes t the parent functin given a functin. explain why f(x) + k translates the riginal graph f f(x) up k units and why f(x) k translates the riginal graph f F(x) dwn k units explain why f(x + k) translates the riginal graph f f(x) left k units and why f(x-k) translates the riginal graph pf f(x) right k units explain why kf(x) vertically stretches r shrinks the graphs f f(x) by a factr f k and predict whether a given value f k will cause a stretch r a shrink explain why f(kx) hrizntally stretches r shrinks the graph f f(x) by a factr f 1 and predict whether a given value f k will k cause a stretch r a shrink describe the transfrmatin that change a graph f f(x) int a different graph when given pictures f the pre-image and image. determine the value f k given the graph f a transfrmed functin.

15 MAFS.912.F-LE.1.2: Cnstruct linear and expnential functins, including arithmetic and gemetric sequences, given a graph, a descriptin f a relatinship, r tw input-utput pairs (include reading these frm a table). MAFS.912.A-SSE.2.4: Derive the frmula fr the sum f a finite gemetric series (when the cmmn rati is nt 1), and use the frmula t slve prblems. MAFS.912.F-IF.1.3 Recgnize that sequences are functins, smetimes defined recursively, whse dmain is a subset f the integers. Fr example, the Fibnacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n 1) fr n 1. graph the listed transfrmatins when given a graph f f(x) and a value f k ( f(x) ± k, f(x±k), kf(x) and f(kx). use a graphing calculatr t generate examples f functins with different k values analyze the similarities and differences between functin with different k values. recgnize frm a graph if the functin is even r dd. explain that a functin is even when f(-x) = f(x) and its graph has y axis symmetry explain that a functin is dd when f(-x) = -f(x) and its graph has 180 rtatinal symmetry. relate linear and expnential functins t arithmetic and gemetric sequences. identify the quantities being cmpared in a real-wrld prblem. write a functin t describe a real-wrld prblem. cmpse tw r mre functins. determine if a functin is linear r expnential given a sequence, a graph, a verbal descriptin r a table. describe the algebraic prcess used t cnstruct the linear functin and expnential functin that passes thrugh tw pints. cnvert a list f numbers (a sequence) int a functin by making the whle numbers (0, 1, 2, etc.) the inputs and the elements f the sequence the utputs. explain that a recursive frmula tells me hw a sequence starts and tells me hw t use the previus value(s) t generate the next element f the sequence. explain that an explicit frmula allws me t find any element f a sequence withut knwing the element befre it. distinguish between explicit and recursive frmulas fr sequences. explain why the recursive frmula fr an arithmetic sequence uses additin and why the explicit frmula uses multiplicatin define a gemetric sequence as a sequence f numbers that is frmed s that the rati f cnsecutive terms is always the same knwn as the cmmn rati. distinguish between arithmetic and gemetric sequences. determine the cmmn rati between tw terms in a gemetric sequence. explain hw t change a term f a gemetric sequence int the next term and write a recursive frmula fr the sequence, a n = r a n 1. write an explicit frmula fr a gemetric sequence, a n =

16 a 1 r n 1. explain why the recursive frmula fr a gemetric sequence uses multiplicatin and why the explicit frmula uses expnentiatin. translate between the recursive and explicit frms f gemetric sequences. decide when a real wrld prblem mdels a gemetric sequence and write an equatin t mdel the situatin. translate between the recursive and explicit frms f arithmetic sequences. evaluate finite gemetric series. define a finite gemetric series and cmmn rati. derive the frmula fr the sum f a finite gemetric series, S n = a 1 ((1 r n)/(1 r)). express the sum f a finite gemetric series. calculate the sum f a finite gemetric series. recgnize real-wrld scenaris that are mdeled by gemetric sequences. use the frmula fr the sum f a finite gemetric series t slve real-wrld prblems.

17 Unit 4 Systems f Equatins and Inequalities Standard The students will: Students explre systems f equatins and inequalities, and they find and interpret their slutins. Essential Questin(s): Hw can we mdel and find slutins in real wrld situatins using systems f equatins and inequalities? Hw can yu slve a system f equatins r inequalities? Can systems f equatins mdel real wrld situatins? MAFS.912A.CED.1.3 Represent cnstraints by equatins r inequalities, and by systems f equatins and/r inequalities, and interpret slutins as viable r nn-viable ptins in a mdeling cntext. MAFS.912A-REI.3.5 Prve that given a system f tw equatins in tw variables, replacing ne equatin by the sum f that equatin and a multiple f the ther prduces a system with the same slutins. MAFS.912.A-REI.3.6 Slve systems f linear equatins exactly and apprximately (with graphs), fcusing n pairs f linear equatins in tw variables. MAFS.912.A-REI.4.12 Graph the slutins t a linear inequality in tw variables as a half-plane (excluding the bundary in the case f a strict inequality), and graph the slutin set t a system f linear inequalities in tw variables as the intersectin f crrespnding half-planes. MAFS.912.A-REI.4.11: Explain why the x-crdinates f the pints where the graphs f the equatins y=f(x) and y=g(x) intersect are the slutins f the equatin f(x)=g(x); find the slutins apprximately, e.g., using technlgy t graph the functins, make tables f values, r find successive apprximatins. Include cases where f(x) and/r g(x) are linear, plynmial, ratinal, abslute value, expnential, and lgarithmic functins. Learning Gals I can: slve systems f equatins by graphing. graph a system f linear equatins and determine the apprximate slutin t the system f linear equatins by estimating the pint f intersectin. graph the system n crdinate axes with apprpriate labels and scales. slve a system f tw linear equatins by graphing and determining the pint f intersectin. explain that a pint f intersectin n the graph f a system f equatins, y=f(x) and y=g(x), represents a slutin t bth equatins use a graphing calculatr t determine the apprximate slutins t a system f equatins, f(x) and g(x). explain that a pint f intersectin n the graph f a system f equatins represents a slutin t bth equatins. use a graphing calculatr t determine the apprximate slutins t a system f equatins. slve systems f equatins algebraically. write the system f equatins and/r inequalities that best mdels the prblem. slve a system f tw linear equatins algebraically using substitutin. slve a system f tw linear equatins algebraically using eliminatin. explain why sme linear systems have n slutins r infinitely many slutins. slve a system f linear equatins algebraically t find an exact slutin. infer that the x-crdinate f the pints f intersectin fr y=f(x) and y=g(x) are als slutins fr f(x)=g(x). interpret slutins in the cntext f the situatin mdeled and decide if they are reasnable. infer that since y=f(x) and y=g(x), f(x)=g(x) by the

18 substitutin prperty. infer that the x-crdinate f the pints f intersectins are slutins fr f(x) = g(x). slve systems f inequalities. slve and graph linear inequalities with tw variables. slve and graph system f linear inequalities. explain that the slutin set fr a system f linear inequalities is the intersectin f the shaded regins f bth inequalities and check pints in the shaded regin t verify slutin.

19 Unit 5 Analyzing Univariate Data Part 1: Students will use multiple representatins and knwledge f the cntext t make judgments abut the apprpriateness f mdels. Essential Questins: Hw can yu represent data? Of the multiple methds available hw d yu chse the best methd? What statistics can yu use t characterize and cmpare the center and spread f data sets and which are mst affected by utliers? Hw can yu cmpare, estimate, and categrize data sets by histgrams, bx plts, and frequency tables? Learning Gals Standard The students will: MAFS.912.S-ID.1.1 Represent data with plts n the real number line (dt plts, histgrams, and bx plts). MAFS.912.S-ID.1.2 Use statistics apprpriate t the shape f the data distributin t cmpare center (median, mean) and spread (interquartile range, standard deviatin) f tw r mre different data sets. I can: display data using the best representatin. chse the best representatin (dt plt, histgram, bx plt) fr a set f data. decide if a representatin preserves all the data values r presents nly the general characteristics f a data set. cnstruct a histgram fr a set f data cnstruct a dt plt fr a set f data and chse the apprpriate scale t represent data n a number line. MAFS.912.S-ID.1.3 Interpret differences in shape, center, and spread in the cntext f the data sets, accunting fr pssible effects f extreme data pints (utliers). MAFS.912.S-ID.1.4 Use the mean and standard deviatin data set t fit it t a nrmal distributin and t estimate ppulatin percentages. Recgnize that they are data sets fr which such a prcedure is nt apprpriate. Use calculatrs, spreadsheets, and tables t estimate area under the nrmal curve, analyze and interpret sets f data given a display. describe the center f the data distributin (mean r median). chse the histgram with the largest mean when shwn several histgrams. describe the spread f the data distributin (interquartile range r standard deviatin). chse the histgram with the greatest standard deviatin when shwn several histgrams. chse the bx-and-whisker plt with the greatest interquartile range when shwn several bx-and-whisker plts. cmpare the distributins f tw r mre data sets by examining their shapes, centers, and spreads when drawn n the same scale. interpret the differences in the shape, center, and spread f a data set in the cntext f a prblem. identify the utliers fr the data set. predict the effect an utlier will have n the shape, center, and spread f a data set. decide whether t include the utliers as part f the data set r t remve them. recgnize and interpret a data set with a nrmal distributin. use mean and standard deviatin f a set f data t fit the data t a nrmal curve

20 use the Rule t estimate the percent f a nrmal ppulatin that falls within 1, 2, r 3 standard deviatins f the mean. recgnize that nrmal distributins are nly apprpriate fr unimdal and symmetric shapes estimate the area under a nrmal curve using a calculatr, table, r spreadsheet

21 Unit 6 Analyzing Bivariate Data Part 2: Students will use regressin techniques t describe and apprximate linear relatinships between quantities. Students will use the linear mdels t lk at residuals t analyze the gdness f fit. Essential Questins: Hw d yu write an equatin t mdel trends and data? Hw d yu write an equatin t shw trends in data? Hw can yu decide whether a crrelatin exists between paired numerical data? Hw can yu find a linear mdel fr a set paired numerical data, and hw d yu evaluate the gdness f fit? Learning Gals Standard The students will: MAFS.912-S-ID.2.6 Represent data n tw quantitative variables n a scatter plt, and describe hw the variables are related. Fit a functin t the data; use functins fitted t data t slve prblems in the cntext f the data. Use given functins r chse a functin suggested by the cntext. Emphasize linear, quadratic, and expnential mdels. Infrmally assess the fit f a functin by pltting and analyzing residuals. Fit a linear functin fr a scatter plt that suggests a linear assciatin. MAFS.912.S-ID.3.7 Interpret the slpe (rate f change) and the intercept (cnstant term) f a linear mdel in the cntext f the data. MAFS.912.S-ID.3.8 Cmpute (using technlgy) and interpret the crrelatin cefficient f a linear fit. MAFS.912.S-ID.3.9 Distinguish between crrelatin and causatin. MAFS.912.S-ID.2.5 Summarize categrical data fr tw categries in tw-way frequency tables. Interpret relative frequencies in the cntext f the data (including jint, marginal, and cnditinal relative frequencies). Recgnize pssible assciatins and trends in the data. I can: mdel bivariate data using linear and expnential functins. identify the independent and dependent variable and describe the relatinship f the variables. cnstruct a scatter plt with an apprpriate scale. identify any utliers n the scatter plt. determine when linear, quadratic, and expnential mdels shuld be used t represent a data set. determine whether linear and expnential mdels are increasing and decreasing. use technlgy t find the functin f best fit fr a scatter plt. use the functin f best fit t make predictins. cmpute the residuals (bserved value minus predicted value) fr the set f data and the functin f best fit. cnstruct a scatter plt f the residuals. analyze the residual plt t determine whether the functin is an apprpriate fit. sketch the line f best fit n a scatter plt that appears linear. write the equatin f the line f best fit (y=mx+b) using technlgy r by using tw pints n the best fit line. interpret the meaning f the slpe in terms f the units stated in the data. interpret the meaning f the y-intercept in terms f the units stated in the data. determine if a linear mdel is the line f best fit. explain the crrelatin cefficient applies nly t quantitative variables and linear mdels f best fit. explain that the crrelatin cefficient must be between -1 and 1 inclusive and explain what each f these values means. explain the crrelatin cefficient as a measure f the gdness f a linear fit. cmpute the crrelatin cefficient (r) using a graphing calculatr

22 r ther apprpriate technlgy. use the crrelatin cefficient t interpret the linear mdel in terms f its sign (i.e., directin) and its magnitude (i.e., strength). use the crrelatin cefficient t determine if a linear mdel is a gd fit fr the data (significance). recgnize that crrelatin des nt imply causatin and that causatin is nt illustrated n a scatter plt. chse tw variables that culd be crrelated because ne is the cause f the ther and defend my selectin. chse tw variables that culd be crrelated even thugh neither variable culd reasnably be cnsidered t be the cause f the ther and defend my selectin. determine if statements f causatin seem reasnable r unreasnable and defend my pinin. display and interpret bivariate categrical data. read and interpret the data displayed in a tw-way frequency table. write clear summaries f data displayed in a tw-way frequency table. calculate percentages using the ratis in a tw-way frequency table t yield relative frequencies. calculate jint, marginal, and cnditinal relative frequencies. interpret and explain the meaning f relative frequencies in the cntext f a prblem. make apprpriate displays f jint, marginal, and cnditinal distributins. describe patterns bserved in the data.

23 Unit 7 Plynmials Part 1: Students will see structure in and will quadratic and plynmials expressins. Essential Questin: In what ways can yu recgnize and create quadratic and plynmial expressins? Hw are mnmials and plynmials related? Can tw algebraic expressins that appear t be different be equivalent? Hw are the prperties f real numbers related t plynmials? Standard Learning Gals The students will: MAFS.912.A-APR.1.1 Understand that plynmials frm a system analgus t the integers, namely, they are clsed under the peratins f additin, subtractin, and multiplicatin; add, subtract, and multiply plynmials. I can: identify and describe a plynmial. define expressin, term, factr, and cefficient. interpret the real-wrld meaning f the terms, factrs, and cefficients f an expressin in terms f their units. MAFS. 912.A-SSE.1.1 Interpret expressins that represent a quantity in terms f its cntext. a. Interpret parts f an expressin, such as terms, factrs, and cefficients. b. Interpret cmplicated expressins by viewing ne r mre f their parts as a single entity. Fr example, interpret P(1+r) n as the prduct f P and a factr nt depending n P. MAFS. 912.A-SSE.1.2 Use the structure f an expressin t identify ways t rewrite it. Fr example, see x 4 - y 4 as (x 2 ) 2 -(y 2 ) 2, thus recgnizing it as a difference f squares that can be factred as (x 2 -y 2 )(x 2 +y 2 ). perfrm peratins n plynmials. apply the definitin f a plynmial t explain why adding, subtracting, r multiplying tw plynmials always prduces a plynmial. add, subtract, and multiply plynmials. explain why equivalent expressins are equivalent. factr plynmials. lk fr and identify clues in the structure f expressins (e.g., like terms, cmmn factrs, difference f squares, perfect squares) in rder t rewrite it anther way. apply mdels fr factring and multiplying plynmials t rewrite expressins.

24 Unit 8 Slving Quadratic Equatins Part 2: Students will create and slve equatins, inequalities, and systems f equatins invlving quadratic expressins. Essential Questins: In what ways can yu represent quadratic equatins using real wrld data? What are the characteristics f quadratic equatin? Hw can yu use quadratic equatins in real-wrld situatins? Learning Gals Standard The students will: MAFS.912.A-CED.1.1: Create equatins and inequalities in ne variable and use them t slve prblems. Include equatins arising frm linear and quadratic functins, and simple ratinal and expnential functins. MAFS.912.A-CED.1.2: Create equatins in tw r mre variables t represent relatinships between quantities. MAFS.912.A-APR.2.2: Knw and apply the Remainder Therem: Fr a plynmial p(x) and a number a, the remainder n divisin by x-a is p(a), s p(a)=0 if and nly if (x-a) is a factr f p(x). MAFS.912.A-APR.2.3: Identify zeres f plynmials when suitable factrizatins are available, and use the zeres t cnstruct a rugh graph f the functin defined by the plynmial. MAFS.912.A-APR.3.4: Prve plynmial identities and use them t describe numerical relatinships. Fr example, the plynmial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used t generate Pythagrean Triples. MAFS.912.A-APR.4.6: Rewrite simple ratinal expressins in different frms; write a(x) b(x) in the frm q(x) + r(x) b(x), where a(x), b(x), q(x), and r(x) are plynmials with the degree f r(x) less than the degree f b(x), using inspectin, lng divisin, r, fr the mre cmplicated examples, a graphing calculatr. MAFS.912.A-REI.2.4: Slve quadratic equatins in ne variable. I can: divide plynmials. divide plynmials using lng divisin and synthetic divisin and apply the Remainder Therem (when apprpriate) t check the answer. apply the Remainder Therem t determine if a divisr (x-a) is a factr f the plynmials p(x). slve quadratic equatins by factring, cmpleting the square, quadratic frmula, and square rt methd. factr a quadratic expressin t find the zers f the functin it represents. identify zers f factred quadratics. identify the multiplicity f the zeres f a factred quadratic. identify and factr a perfect square trinmial. cmplete the square f ax 2 +bx+c t write the quadratic in the frm (x-p) 2 =q. derive the quadratic frmula by cmpleting the square f ax 2 +bx+c. determine the best methd t slve a quadratic equatin in ne variable. slve quadratic equatins by inspectin, finding square rts, cmpleting the square, the quadratic frmula, and factring. classify real numbers as ratinal r irratinal accrding t their definitins. explain that cmplex slutins result when the radicand is negative in the quadratic frmula (b 2-4ac<0). predict whether a quadratic will have a minimum r a maximum based n the value f a. identify the maximum r minimum f a quadratic written in vertex frm.

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