UNIT 2 FUNCTIONS IN THIS MODULE, STUDENTS SYNTHESIZE AND GENERALIZE WHAT THEY HAVE LEARNED ABOUT A VARIETY OF FUNCTION FAMILIES. Established Gals: N-RN.A.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, N-RN.A.2 A-SSE.B.3 we define 5 1 3 t be the cube rt f 5 because we want (5 1 3 3 ) = 5 (1 3 )3 t hld, s (5 1 3 ) 3must equal 5. Rewrite expressins invlving radicals and ratinal expnents using the prperties f expnents. 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. Fr example the expressin 1.15 t can be rewritten as (1.15 1 12 ) 12t 1.012 12t t reveal the apprximate equivalent mnthly interest rate if the annual rate is 15%. Transfer Students will be able t: Write expressins in equivalent frms t slve prblems Create equatins that describe numbers r relatinships Understand the cncept f functin and use functin ntatin Interpret functins that arise in applicatins in terms f a cntext Analyze functins using different representatins. Build a functin that mdels a relatinship between tw quantities Build new functins frm existing functins Cnstruct and cmpare linear, quadratic, and expnential mdels and slve prblems Summarize, represent, and interpret data n tw categrical and quantitative variables ENDURING UNDERSTANDING Students will understand that: Meaning ESSENTIAL QUESTIONS Students Will Keep Cnsidering A-CED.A.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. A-REI.D.11 Explain why the t-crdinates f the pints where the graphs f the equatins t = t(t) and t = t(t) intersect are the slutins f the equatin t(t) = t(t); find the slutins apprximately, e.g., using technlgy t graph the functins, make tables f values, r find successive apprximatins. Include cases where t(t) and/r t(t) are linear, plynmial, ratinal, abslute value, expnential, and lgarithmic functins. F-IF.A.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 t(0) = t(1) = 1, t(t + 1) = t(t) + t(t 1) fr t 1. F-IF.B.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 Page 1 f 7 KNOWLEDGE Students will knw The key features f linear, expnential, plynmial, rt, abslute value, piecewise, simple ratinal, lgarithmic and trignmetric functins Hw parameters intrduced int a functin alter the shape f the graph f the parent functin That lgarithmic functins are inverses f expnential functins The prperties f lgarithms and expnents and their cnnectin t ne anther The cnnectin f the dmain and range f a functin t its inverse Ability t cnnect experience with: Acquisitin SKILLS Students will be skilled at Distinguishing between linear, quadratic, expnential, rt and simple ratinal relatinships given the verbal, numeric and/r graphic representatins Ability t distinguish between linear, quadratic expnential and simple ratinal relatinships given multiple representatins Determining unknwn parameters needed t create an equatin that accurately mdels a given situatin Prducing the graph f a functin that is being used t mdel relatinships, which wuld include apprpriate scales and labels as t accurate display all aspects f the relatin
F-IF.B.5 F-IF.B.6 F-IF.C.7 F-IF.C.8 F-IF.C.9 F-BF.A.1 features given a verbal descriptin f the relatinship. Key features include: intercepts; intervals where the functin is increasing, decreasing, psitive, r negative; relative maximums and minimums; symmetries; end behavir; and peridicity. Relate the dmain f a functin t its graph and, where applicable, t the quantitative relatinship it describes. Fr example, if the functin t(t) gives the number f persn-hurs it takes t assemble t engines in a factry, then the psitive integers wuld be an apprpriate dmain fr the functin. 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. Graph functins expressed symblically and shw key features f the graph, by hand in simple cases and using technlgy fr mre cmplicated cases. e. Graph expnential and lgarithmic functins, shwing intercepts and end behavir, and trignmetric functins, shwing perid, midline, and amplitude. Write a functin defined by an expressin in different but equivalent frms t reveal and explain different prperties f the functin. b. Use the prperties f expnents t interpret expressins fr expnential functins. Fr example, identify percent rate f change in functins such as t = (1.02) t, t = (0.97) t, t = (1.01) 12t, t = (1.2) 10, t and classify them as representing expnential grwth r decay. 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. Write a functin that describes a relatinship between graphing linear and quadratic functins frm Algebra I t graphing expnential and lgarithmic functins this standard as it relates t linear, quadratic and expnential functins frm Algebra I t all functins studied finding the inverse f a linear functin frm Algebra I t finding the inverse f simple expnential, rt and ratinal functins adding, subtracting, multiplying and dividing linear, quadratic and expnential functins frm Algebra I t adding, subtracting, multiplying and dividing any functins writing linear, quadratic and expnential functins in varius frms frm Algebra I t writing all functins in varius frms rearranging frmulas that were linear in nature frm Algebra I t manipulating frmulas that are nt linear in nature frm Algebra I with creating linear, expnential and quadratic equatins in ne variable t include creating simple ratinal functins graphing linear and quadratic functins frm Algebra I t graphing square rt, cube rt and a variety f piece wise defined functins Distinguishing between a mathematical slutin and a cntextual slutin Relating the cncept f dmain t each functin studied Describing the restrictins n the dmain f all functins studied based n real wrld cntext Applying this skill t linear, quadratic, plynmial, rt and simple ratinal functins Prducing a rugh graph f the parent functin fr each type f functin Recgnizing functins in varius frms Recgnizing cmmn attributes f functins frm varius representatins Creating a new functin that is the cmpsitin f tw r mre functins Generalizing the changes that will take place in the graph f any functin as a result f making a particular change t the algebraic representatin f the functin Determining if a functin has an inverse tw quantities. a. Determine an explicit expressin, a recursive prcess, r steps fr calculatin frm a cntext. b. Cmbine standard functin types using arithmetic peratins. Fr example, build a functin that mdels the temperature f a cling bdy by adding a Page 2 f 7
F-BF.A.2 F-BF.B.3 F-BF.B.4 F-LE.A.2 F-LE.A.4 F-LE.B.5 cnstant functin t a decaying expnential, and relate these functins t the mdel. Write arithmetic and gemetric sequences bth recursively and with an explicit frmula, use them t mdel situatins, and translate between the tw frms. Identify the effect n the graph f replacing t(t) by t(t) + t, t t(t), t(tt), and t(t + t) fr specific values f t (bth psitive and negative); find the value f t given the graphs. Experiment with cases and illustrate an explanatin f the effects n the graph using technlgy. Include recgnizing even and dd functins frm their graphs and algebraic expressins fr them. Find inverse functins. a. Slve an equatin f the frm t(t) = t fr a simple functin t that has an inverse and write an expressin fr the inverse. Fr example, t(t) = 2 t 3 r t(t) = (t + 1)/(t 1) fr t 1. 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). Fr expnential mdels, express as a lgarithm the slutin t tt tt = t where t, t, and t are numbers and the base t is 2, 10, r t; evaluate the lgarithm using technlgy. Interpret the parameters in a linear r expnential functin in terms f a cntext. Page 3 f 7
Vcabulary Cmpund Interest Expnential Decay Expnential Expressin Expnential Grwth Scientific Ntatin Arithmetic Sequence Gemetric Sequence Ratinal functin Recursive Expnents nth rt Transfrmatin Dmain Range Inverse Cmpsitin f functins Explicit Rate f change Substitute Evaluate Vertical/Hrizntal Translatin Slide Rtate Flip Reflectin Vertical/Hrizntal Stretch/Shrink Scale Quarterly Semiannually Cntinuusly Prperties f Expnents Lgarithm Instructin and Pacing (suggested rder t teach) TOPIC # f Days Alg2-U3-L1 t L6 5 Remediatin/Enrichment 1 Alg2-U3-L7 t L15 5 Quizzes after every 1 r 2 standards Remediatin/Enrichment 2 Alg2-U3-L16 t L22 6 Quizzes/Tests after every 1 r 2 standards Remediatin/Enrichment 1 Alg2-U3-L23 t L28 5 Review 1 Remediatin/Enrichment 1 Alg2-U3-L29 t L33 3 Remediatin/Enrichment 1 Perfrmance Tasks, Prjects, ixl, swing etc. 1 32 days Page 4 f 7
Base Natural lg Cmmn Lg Expnential Frm Extraneus slutin Intersectin graph Leading cefficient Standard frm Rearrange Piecewise functins Mdel Accurate Apprpriate Odd/Even Functins Parameters Page 5 f 7
Accmmdatins and Resurces 21 st Century Skills Critical Thinking, Creative Thinking, Cllabrating, Cmmunicating, and Technlgy Literacy IXL.cm Desms.cm Edpuzzle- Teacher made vides as well as vide lessns in spanish/ther Online graphing tl Khan Academy www.engageny.rg Resurces Graphing calculatr r Desms nline calculatr simulatin Wlfram Alpha Sftware GeGebra r Gemeter s Sketchpad Sftware Excel r ther spreadsheet sftware, such as Calc (part f the OpenOffice suite) Minimizing the Metal in a Can Lessn http://www.abveandbeyndthecre.cm/#!algebra-ii/c18p6 Ggle Translate Instructinal Strategies Interdisciplinary Cnnectins Differentiatin and Accmmdatins Pleasantville Public Schls recgnizes the imprtance f the varying methdlgies that may be successfully emplyed by teachers within the classrm and, as a result, identifies a wide variety f pssible instructinal strategies that may be used effectively t supprt student achievement. These may include, but nt be limited t, strategies that fall int categries identified by the Framewrk fr Teaching by Charltte Danielsn: Cmmunicating with students Using questining and discussin techniques Engaging students in learning Using assessment in instructin Demnstrating Flexibility and Respnsiveness ELA, Science, and Technlgy Extra time, ELL charts/wrksheets fr vcabulary, mdified quizzes, translatin wrk sheet, step by step instructins, Wrd wall Prvide graphic rganizers Prvide additinal examples and pprtunities fr additinal prblems fr repetitin Prvide tutring pprtunities Prvide retesting pprtunities after remediatin (up t teacher and district discretin) Teach fr mastery nt test Teaching cncepts in different mdalities Adjust pace and hmewrk assignments Offer perfrmance tasks f varied levels Include mre scafflding questins and tasks Page 6 f 7
Cmmn Miscnceptins Students d nt put in the graphing windws crrectly Prper Cnceptins Students d nt knw hw t use lg keys Students d nt knw t use nth rt Students d nt knw use f e Students need mre help where t use parentheses crrectly Minimizing the Metal in a Can Task Average Cst Buying a Car Checking a Calculatin f a Decimal Expnent Curse f Antibitics Extending the Definitins f Expnents, Variatin 2 Lgistics Grwth Mdel, Abstract Versin Lgistics Grwth Mdel, Explicit Mdel Medieval Archer Rumrs Seeing Dts The High Schl Gym Thrwing Baseballs Triangle Series Tw Squares are Equal US Husehlds Perfrmance Task ASSESSMENTS WITH RUBRICS Standard based quizzes/tests after every set f 1 r 2 standards Perfrmance tasks Page 7 f 7