Unit 2 Trigonometric Functions, Identities, and Equations

Transcription

1 Number f : 43 10/30/17 1/19/18 Unit Gals Stage 1 Unit Descriptin: In this unit, students extend their knwledge f angles t rtatinal angles in the plane and radian measure. The six trignmetric functins are defined in terms f a right triangle. The circular functins related t the unit circle and their relatinship t the trignmetric functins are explred, as well as the relatinship between functins f angles and functins f real numbers. Students will analyze the prperties and graphs f trignmetric and circular functins and apply thse functins t real-wrld prblems. Inverse trignmetric and circular functins are studied. The trignmetric identities are derived and used t simplify expressins and prve ther identities. Real wrld prblems are mdeled and slved using trignmetric equatins. Students derive and apply the laws f sines and csines t nn-right triangles. Materials: Graphing calculatrs, Desms Standards fr Mathematical Practice SMP 1 SMP 2 SMP 3 SMP 4 SMP 5 SMP 6 SMP 7 SMP 8 Make sense f prblems and persevere in slving them. Reasn abstractly and quantitatively. Cnstruct viable arguments and critique the reasning f thers. 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. Standards fr Mathematical Cntent Clusters Addressed A-SSE.A Interpret the structure f expressins. A-CED.A Create equatins that describe numbers r relatinships. F-IF.B Interpret functins that arise in applicatins in terms f the cntext. Transfer Gals Students will be able t independently use their learning t Make sense f never-befre-seen prblems and persevere in slving them. Cnstruct viable arguments and critique the reasning f thers. Making Meaning UNDERSTANDINGS Students will understand that A radian is anther way t measure the size f an angle. In the functin f(x) = a f(x b) + c, a, b, and c have the same effect n the shape f the graph in every graph family. Peridic phenmena can be mdeled by trignmetric functins. Functins can be classified int different families, each with its wn characteristics. Functins smetimes have inverses. Inverse functins und each ther. Asympttes represent cnstraints n functins with pssible real-wrld relevance. Algebraic and trignmetric reasning can be used t justify slutins. Acquisitin KNOWLEDGE Students will knw The trignmetric values fr 30, 45, and π radians = 360 The difference between linear and angular velcity. The shapes f the basic trignmetric functins, their reciprcal functins, and inverse functins. ESSENTIAL QUESTIONS Students will keep cnsidering In the functin f(x) =a f(x b) +c, hw d a, b, and c effect the shape f the graph? Is this the same fr all graph families? What is the cnnectin between the characteristics f the graph f a functin and its equatin? Hw can graphing data help t analyze a real-wrld situatin and help in decisin making? SKILLS Students will be skilled at and/r be able t State the dmain and range f a functin. Identify even and dd functins. Use limits t determine the cntinuity f a functin and t describe a functin s end behavir Psted 10/17/17

2 F-BF.A F-BF.B F-TF.A F-TF.B F-TF.C Build a functin that mdels a relatinship between tw quantities. Build new functins frm existing functins. Extend the dmain f trignmetric functins using the unit circle. Mdel peridic phenmena with trignmetric functins. Prve and apply trignmetric identities. G-SRT.D Apply trignmetry t general triangles. Unit Gals Stage 1 Hw the parameters a, b, and c, affect the graph f f(x) =a f(x b) + c. The dmain and range f a functin can be determined algebraically and/r graphically. The best way t express a functin t highlight key features f a graph. Inverse functins und each ther. Trignmetric Identities. Identify, graph, analyze, and describe functin families. Identify and graph transfrmed functins. Justify slutins and verificatins using algebraic and trignmetric reasning. Determine whether a functin has an inverse functin. Find inverse functins algebraically and graphically. Verify algebraically and graphically that ne functin is the inverse f anther Psted 10/17/17

3 Assessed Grade Level Standards Standards fr Mathematical Practice SMP 1 Make sense f prblems and persevere in slving them. SMP 2 Reasn abstractly and quantitatively. SMP 3 Cnstruct viable arguments and critique the reasning f thers. SMP 4 Mdel with mathematics. SMP 5 Use apprpriate tls strategically. SMP 6 Attend t precisin. SMP 7 Lk fr and make use f structure. SMP 8 Lk fr and express regularity in repeated reasning. Standards fr Mathematical Cntent A-SSE.A Interpret the structure f expressins. A-SSE.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. A-SSE.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 ). A-CED.A Create equatins that describe numbers r relatinships. A-CED.2 Create equatins in tw r mre variables t represent relatinships between quantities; graph equatins n crdinate axes with labels and scales. F-IF.B F-IF.4 F-IF.5 F-BF.A F-BF.1 F-BF.B F-BF.3 F-BF.4 Interpret functins that arise in applicatins in terms f the cntext. 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. 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 h(n) gives the number f persn-hurs it takes t assemble n engines in a factry, then the psitive integers wuld be an apprpriate dmain fr the functin. Build a functin that mdels a relatinship between tw quantities. Write a functin that describes a relatinship between tw quantities. a. Determine an explicit expressin, a recursive prcess, r steps fr calculatin frm a cntext. c. (+) Cmpse functins. Fr example, if T(y) is the temperature in the atmsphere as a functin f height, and h(t) is the height f a weather balln as a functin f time, then T(h(t)) is the temperature at the lcatin f the weather balln as a functin f time. Build new functins frm existing functins. 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 (bth 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. Include recgnizing even and dd functins frm their graphs and algebraic expressins fr them. Find inverse functins. 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 Psted 10/17/17

4 Assessed Grade Level Standards F-TF.A Extend the dmain f trignmetric functins using the unit circle. F-TF.3 (+) Use special triangles t determine gemetrically the values f sine, csine, tangent fr π/3, π/4 and π/6, and use the unit circle t express the values f sine, csine, and tangent fr π x, π+x, and 2π x in terms f their values fr x, where x is any real number. F-TF.4 (+) Use the unit circle t explain symmetry (dd and even) and peridicity f trignmetric functins. F-TF.B Mdel peridic phenmena with trignmetric functins. F-TF.6 (+) Understand that restricting a trignmetric functin t a dmain n which it is always increasing r always decreasing allws its inverse t be cnstructed. F-TF.7 (+) Use inverse functins t slve trignmetric equatins that arise in mdeling cntexts; evaluate the slutins using technlgy, and interpret them in terms f the cntext. F-TF.C Prve and apply trignmetric identities. F-TF.9 (+) Prve the additin and subtractin frmulas fr sine, csine, and tangent and use them t slve prblems. F-TF.10 (+) Prve the half angle and duble angle identities fr sine and csine and use them t slve prblems. CA G-SRT.D Apply trignmetry t general triangles. G-SRT.9 (+) Derive the frmula A = 1/2 ab sin(c) fr the area f a triangle by drawing an auxiliary line frm a vertex perpendicular t the ppsite side. G-SRT.10 (+) Prve the Laws f Sines and Csines and use them t slve prblems. G-SRT.11 (+) Understand and apply the Law f Sines and the Law f Csines t find unknwn measurements in right and nn-right triangles (e.g., surveying prblems, resultant frces). Key: [m] = majr clusters; [s] = supprting clusters; [a] = additinal clusters Indicates a mdeling standard linking mathematics t everyday life, wrk, and decisin-making (+) Indicates standards included in curses intended t lead int a furth year f mathematics. CA Indicates a Califrnia-nly standard Psted 10/17/17

5 Assessment Evidence Evidence f Stage 2 Unit Assessment Claim 1: Students can explain and apply mathematical cncepts and carry ut mathematical prcedures with precisin and fluency. Cncepts and skills that may be assessed in Claim1: A-SSE.A Interpret the structure f expressins. Students will use the structure f an expressin t identify ways t rewrite the expressin. Students will interpret parts f an expressin in terms f a cntext. A-CED.A Create equatins that describe numbers r relatinships. Students will create equatins and inequalities in ne r mre variables t represent relatinships in cntext. Students will graph equatins n crdinate axes with labels and scales. Students will represent cnstraints by equatins r inequalities and interpret slutins as viable r nnviable ptins in a real-wrld cntext. Students will rearrange frmulas t highlight a quantity f interest F-IF.B Interpret functins that arise in applicatins in terms f the cntext. Students will interpret key features f graphs and tables in terms f a cntext. Students will sketch graphs shwing key features given a verbal descriptin f a relatinship between tw quantities. Students will relate the dmain f a functin t its graph and, where applicable, t the quantitative relatinship it describes F-BF.A Build a functin that mdels a relatinship between tw quantities. Students will determine an explicit expressin t describe a relatin between tw quantities given a cntext. Student will cmbine functins using arithmetic peratins. Students will cmpse trignmetric functins. F-BF.B Build new functins frm existing functins. Students will Identify the effect n a graph caused when f(x) is replaced by f(x) + k, k f(x), f(kx), and f(x + k) fr specific values f k (bth psitive and negative. Students will find the value f k which transfrms the graph f a parent functin int the graph f a given functin. Students will find the inverse f a functin if that functin has an inverse. Students will verify that tw functins are inverses graphically, by cmpsitin, r by slving fr an inverse. Students will restrict the dmain f a nn-invertible functin t prduce an inverse functin. Students will use the inverse relatinship between expnents and lgarithms t slve prblems invlving lgarithms and expnents. F-TF.A Extend the dmain f trignmetric functins using the unit circle. Students will use special triangles t determine gemetrically the values f sine, csine, and tangent fr π/3, π/4 and π/6. Students will use the unit circle t express the values f sine, csine, and tangent fr π x, π+x, and 2π x in terms f their values fr x, where x is any real number. Students will use the unit circle t explain symmetry (dd and even) and peridicity f trignmetric functins Psted 10/17/17

6 Evidence f Stage 2 F-TF.B Mdel peridic phenmena with trignmetric functins. Students will understand that restricting a trignmetric functin t a dmain n which it is always increasing r always decreasing allws its inverse t be cnstructed. Students will use inverse functins t slve trignmetric equatins that arise in mdeling cntexts; evaluate the slutins using technlgy, and interpret them in terms f the cntext. F-TF.C Prve and apply trignmetric identities. Students will prve the additin and subtractin frmulas fr sine, csine, and tangent. Students will use the additin and subtractin frmulas fr sine, csine, and tangent t slve prblems. Students will prve the half angle and duble angle identities fr sine and csine. Students will use the half angle and duble angle identities fr sine and csine t slve prblems. G-SRT.D Apply trignmetry t general triangles. Students will derive the frmula A = 1/2 ab sin(c) fr the area f a triangle by drawing an auxiliary line frm a vertex perpendicular t the ppsite side. Students will prve the Laws f Sines and Csines. Students will use the Laws f Sines and Csines t slve prblems. Students will apply the Law f Sines and the Law f Csines t find unknwn measurements in right and nn-right triangles Claim 2: Students can slve a range f wellpsed prblems in pure and applied mathematics, making prductive use f knwledge and prblem-slving strategies. Standard clusters that may be assessed in Claim 2: A-SSE.A A-CED.A F-IF.B F-BF.A G-SRT.D Other Evidence Frmative Assessment Opprtunities Claim 3: The student can clearly and precisely cnstruct viable arguments t supprt their wn reasning and critique the reasning f thers. Standard clusters that may be assessed in Claim 3: F-IF.B F-BF.B F-TF.A F-TF.C G-SRT.D Claim 4: The student can analyze cmplex, real-wrld scenaris and can cnstruct and use mathematical mdels t interpret and slve prblems. Standard clusters that may be assessed in Claim 4: A-CED.A F-IF.B F-BF.A F-TF.B G-SRT.D Infrmal teacher bservatins Checking fr understanding using active participatin strategies Exit slips/summaries Tasks Mdeling Lessns (SMP 4) Frmative Assessment Lessns (FAL) Quizzes/Chapter Tests SBAC Interim Assessment Blcks Access Using Frmative Assessment fr Differentiatin fr suggestins. Lcated n the LBUSD website M Mathematics Curriculum Dcuments Psted 10/17/17

7 2 I will review right triangle trignmetry in the Opening Task. Suggested Sequence f Key Events and Instructin Expectatins OPENING TASK Right Triangle Review This Opening Task is a review f the right triangle trignmetry fund in Gemetry and Algebra 2. The tasks may be used individually r tgether. Are Relatinships Predictable? reinfrces the similarity prperty that allws fr all right triangles with the same angle measure t have the same sine, csine and tangent side length ratis. Relatinships with Meaning reviews cmplementary angles and several f the basic trignmetric identities. Finding the Value f a Relatinship prvides practice in finding a right triangle s missing infrmatin. (Activities and Lessns) Supplemental Resurces Cnceptual Understanding: MathVisinPrject: Are Relatinships Predictable? MathVisinPrject: Relatinships with Meaning Prcedural Skills and Fluency: MathVisinPrject: Finding the Value f a Relatinship This review prepares the student t expand their study f trignmetric values, graphs, functins, identities and slutin techniques. These tasks may be revisited as needed thrughut the unit. 5-7 I will use radian measure t slve fr the value f a trignmetric functin by Finding values f trignmetric functins fr acute angles f right triangles. Slving right triangles. Cnverting between degree and radian measures. Recgnizing and slving fr c-terminal angles. Slving fr the linear r angular velcity f a mving pint. Using angle measures t slve real-wrld prblems. Finding the values f trignmetric functins fr any angle. Using the unit circle t find the values f trignmetric functins. Answering questins such as Hw d yu describe angles and angular mvement? What is the difference between ne radian and ne degree? Given that a pint is n the edge f a circle, what effect des dubling the radius f that circle have n the pint s linear speed? Angular speed? Lessn 4.1 Lessn 4.2 Lessn 4.3 Cnceptual Understanding: MathVisinPrject: Diggin It MathVisinPrject: Staking It MathVisinPrject: Water Wheels and the Unit Circle IllustrativeMathematics: What Exactly Is a Radian? IllustrativeMathematics: Bicycle Wheel Prcedural Skills and Fluency: MathVisinPrject: Cnverting Angles Between Radians and Degrees MathVisinPrject: Cterminal Angles Psted 10/17/17

8 Suggested Sequence f Key Events and Instructin Expectatins (Activities and Lessns) Hw can yu re-generate the values fr the standard trignmetric functins? Culd attendance at a theme park be mdeled by a peridic functin? Hw d yu use trignmetry t find unknwn side lengths and angles in right triangles? What patterns d yu see in the table f values fr the trignmetric functins? Starting frm the standard psitin, hw d the magnitudes f sinθ, csθ, and tanθ change as θ increases cntinually frm 0 t 90? Supplemental Resurces Quizlet: Radians t Degrees Quizlet: Unit Circle High Res Unit Circle Khan Academy: Intrductin t the Trignmetric Ratis Khan Academy: Slving fr a Side in a Right Triangle Using the Trignmetric Ratis Khan Academy: Slving fr an Angle in a Right Triangle Using the Trignmetric Ratis Khan Academy: Trignmetric Ratis f Special Triangles Khan Academy: The Reciprcal Trignmetric Ratis Khan Academy: Mdeling with Right Triangles Khan Academy: Intrductin t Radians Khan Academy: Arc Measure Khan Academy: Arc Length (Degrees) Khan Academy: Arc Length (Radians) Khan Academy: Sectrs Khan Academy: The Unit Circle Definitin f Sine, Csine, and Tangent Psted 10/17/17

9 3-4 I will graph trignmetric functins by Using transfrmatins t graph sine and csine functins. Evaluating tangent and reciprcal trignmetric functins. Evaluating inverse functins. Cmpsing trignmetric functins. Answering questins such as If yu re riding a Ferris wheel, hw des yur height change as a functin f time? Hw des the graph f a pint n a Ferris wheel change if the wheel ges faster? Has a smaller diameter? What are the basic characteristics f each parent functin? What d the perids f the functins represent? Hw d the graphs f f(x) + k, k f(x), f(kx), and f(x + k) cmpare t the graph f the parent functin f(x)? Are the transfrmatins f the sine and csine functins similar t the transfrmatins f ther functins yu have studied? Is there an rder in which ne shuld d the steps f a transfrmatin? Hw is the tangent functin different frm the sine and csine functins? Why? With trignmetric functins, what is the difference between reciprcal functins and inverse functins? Suggested Sequence f Key Events and Instructin Expectatins (Activities and Lessns) Lessn 4.4 Lessn 4.5 Lessn 4.6 What d inverse functins allw us t d? What are sme f the representatins we can use t explre the graphs f trignmetric functins? Supplemental Resurces Cnceptual Understanding: Illuminatins: Graphs frm the Unit Circle Desms: Trignmetric Graphing Desms: Sine Transfrmatins Hw D Yu Trignmetric Graphs Which One Desn t Belng: Graphs Which One Desn t Belng: Incmplete Sets Prcedural Skills and Fluency: MathVisinPrject: High Tide MathVisinPrject: Transfrmatins f Trignmetric Graphs MathVisinPrject: Mdel with Trignmetric Functin Quizlet: Basic Trig Graphs Desms: Marbleslides - Peridics Khan Academy: The Graphs f Sine, Csine, and Tangent Khan Academy: Intrductin t Amplitude, Midline, and Extrema f Sinusidal Functins Khan Academy: Finding Amplitude and Midline f Sinusidal Functins frm their Frmulas Psted 10/17/17

10 Suggested Sequence f Key Events and Instructin Expectatins (Activities and Lessns) Supplemental Resurces Khan Academy: Perid f Sinusidal Functins Khan Academy: Graphing Sinusidal Functins Khan Academy: Cnstructing Sinusidal Functins Khan Academy: Inverse Trignmetric Functins Applicatin: Desms: Burning Daylight Graphing Stries: Distance frm Camera (Graph resurce) Graphing Stries: Distance frm Bench (Graph resurce) Psted 10/17/17

11 I will check my understanding f trignmetric functins by participating in the FAL. I will slve blique triangles by Suggested Sequence f Key Events and Instructin Expectatins (Activities and Lessns) FORMATIVE ASSESSMENT LESSON Investigating and prving the Laws f Sine and Csine. Using the Laws f Sine and Csine t find the areas f blique triangles. Answering questins such as When we re slving fr the side lengths f a triangle, why d we need smething ther than the Pythagrean Therem? When given tw sides and a nn-included angle, hw d yu knw when yu have exactly ne triangle? N triangle? Tw triangles? Hw is the triangle area frmula that we learned in elementary schl the same as the frmula fr the area f a triangle given SAS? When d we use Hern s Frmula? Why is it imprtant t knw hw t slve fr the sides f a triangle r its area? When yu re slving a triangle, hw d yu knw when t use trignmetric ratis, the Pythagrean Therem, the Law f Sines r the Law f Csines? Supplemental Resurces Cnceptual Understanding: FAL: Representing Trignmetric Functins Lessn 4.7 Cnceptual Understanding: Illuminatins: Law f Sines Illuminatins: Law f Csines Prcedural Skills and Fluency: Quizlet: Law f Sines and Law f Csines Khan Academy: Law f Sines Khan Academy: Law f Csines Khan Academy: Slving General Triangles Extensin: Cnnect t AP Calculus Related Rates P Psted 10/17/17

12 3-5 I will verify trignmetric identities by Suggested Sequence f Key Events and Instructin Expectatins (Activities and Lessns) Identifying trignmetric identities and using them t find Lessn 5.1 basic trignmetric values. Lessn 5.2 Using basic identities and algebraic prperties t simplify and rewrite trignmetric expressins. Knwing whether equatins are identities. Recgnizing equivalent expressins. Justifying the steps used t transfrm ne side f an equatin int the expressin n the ther side f the equatin. Answering questins such as What is the difference between an equatin and an identity? Why d yu nt need t memrize all three f the Pythagrean Identities? What are sme f the tls and strategies yu have access t when yu are simplifying r verifying trignmetric identities? Hw can a graphing calculatr help t verify whether r nt an equatin is an identity? Supplemental Resurces Cnceptual Understanding: IllustrativeMathematics: Calculatins with Sine and Csine Prcedural Skills and Fluency: IllustrativeMathematics: Finding Trig Values Quizlet: Trignmetry Identities Khan Academy: The Pythagrean Identity Psted 10/17/17

13 1-2 I will slve trignmetric equatins by Suggested Sequence f Key Events and Instructin Expectatins (Activities and Lessns) Using basic identities and algebraic techniques. Lessn 5.3 Investigating and prving sum and difference identities. Lessn 5.4 Using sum and difference identities. Lessn 5.5 Investigating and prving duble-angle and half-angle identities. Using duble-angle and half-angle identities. Answering questins such as What are sme algebraic techniques which can be used t slve trignmetric equatins? When slving a trignmetric equatins, hw d yu knw hw many slutins yu are lking fr? Hw d yu knw whether r nt there are infinite slutins t the trignmetric equatin? Hw can a graphing calculatr be used t slve a trignmetric equatin? Des f(x+y) = f(x) + f(y)? Hw d yu knw? Hw d yu derive the Sum and Difference Identities fr bth sine and csine? Tangent? Given an angle with a terminal side in Quadrant II, in what quadrant wuld a duble angle lie? Half angle? Hw d yu knw? Supplemental Resurces Cnceptual Understanding: Inverse Trig Investigatin Hw D Yu Trig Equatins Illuminatins: Trignmetry fr Slving Prblems Prcedural Skills and Fluency: Inverse Trig War Khan Academy: Slving Basic Sinusidal Equatins Khan Academy: Slving Advanced Sinusidal Equatins Khan Academy: Intrductin t Trignmetric Angle Additin Identities Khan Academy: Using Trignmetric Identities t Slve Prblems Extensin: Cnnect t AP CalculusRates f Change fr Sine and Csine P Psted 10/17/17

14 1-2 I will prepare fr the unit assessment n trignmetric functins, identities and equatins by... Suggested Sequence f Key Events and Instructin Expectatins Incrprating the Standards fr Mathematical Practice (SMPs) alng with the cntent standards t review the unit. (Activities and Lessns) Supplemental Resurces Prcedural Skills and Fluency: MathVisinPrject: Trignmetric Functins Review MathVisinPrject: Graphs f Trignmetric Functins 1-2 Unit Assessment and Key Psted 10/17/17