DEKALB COUNTY SCHOOL DISTRICT RIGOROUS CURRICULUM DESIGN UNIT PLANNING ORGANIZER

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1 DEKALB COUNTY SCHOOL DISTRICT RIGOROUS CURRICULUM DESIGN UNIT PLANNING ORGANIZER Subject(s) Grade and Curse Unit f Study Pacing Mathematics Grade 12 - Pre-Calculus Unit 2- Trignmetric Functins Traditinal: 4 weeks Blck: 2 weeks Unwrapped Pririty Gergia Standards f Excellence Skills and Cncepts Build new functins frm existing functins MGSE9-12.F.BF.4 FIND inverse functins. MGSE9-12.F.BF.4d PRODUCE an invertible functin frm a nn-invertible functin by RESTRICTING the dmain. MGSE9-12.F.TF.4 USE the unit circle t EXPLAIN symmetry (dd and even) and peridicity f trignmetric functins. MGSE9-12.F.TF.7 USE inverse functins t SOLVE trignmetric equatins that arise in mdeling cntexts; EVALUATE the slutins using technlgy, and INTERPRET them in terms f the cntext. 1

2 Unwrapped Skills (Students Need t Be Able t D) Unwrapped Pririty Standards Unwrapped Cncepts (Students Need t Knw) Blm s Taxnmy Levels DOK (Fr Overall Standard) MGSE9-12.F.BF.4 FIND Inverse functins 2 (Understand) 1 (Recall & Reprductin) MGSE9-12.F.BF.4d PRODUCE Invertible functin frm nn-invertible functin 5 (Evaluate) 2 (Skills & Cncepts) RESTRICT Dmain 6 (Create) MGSE9-12.F.TF.4 USE EXPLAIN Unit Circle Symmetry (dd and even) f trignmetric functins Peridicity f trignmetric functins 3 (Apply) 2 (Understand) 2 (Skills & Cncepts) MGSE9-12.F.TF.7 USE SOLVE Inverse functins Trignmetric equatins that arise in mdeling cntexts 3 (Apply) 3 (Apply) 3 (Strategic Thinking/Reasning) EVALUATE Slutins using technlgy 5 (Evaluate) INTERPRET Slutins in terms f the cntext 4 (Analyze) Supprting Standards Build new functins frm existing functins MGSE9-12.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. MGSE9-12.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. 2

3 Standards fr Mathematical Practice 1. Make sense f prblems and persevere in slving them. (Daily) 2. Reasn abstractly and quantitatively. (Daily) 3. Cnstruct viable arguments and critique the reasning f thers. 4. Mdel with mathematics. 5. Use apprpriate tls strategically. 6. Attend t precisin. (Daily) 7. Lk fr and make use f structure. 8. Lk fr and express regularity in repeated reasning. Essential Questins Hw can special right triangles help us find the crdinates f certain angles n the unit circle? Hw des symmetry help us extend ur knwledge f the unit circle t an infinite number f angles? Why des the calculatr nly give ne answer fr an inverse trig functin? Aren t there infinite answers? Hw d inverse trignmetric functins help us slve equatins? Crrespnding Big Ideas Understand the relatinship between right triangle trignmetry and unit circle trignmetry. Use unit circle t explain symmetry (dd and even) and peridicity f trignmetric functins. Use inverse functin t slve trignmetric functins and interpret the slutin accrding t the situatin using technlgy. The inverses f sine, csine and tangent functins are nt functins unless the dmains are limited. 3

4 Unit Assessments Pre-Assessment Student Versin: Precalculus_Unit2_Preassessment_Student.dcx Teacher Versin: Pst-Assessment Student Versin: Dekalb Benchmark Assessment Teacher Versin: Precalculus_Unit2_Preassessment_Teacher.dcx 4

5 Engaging Scenari Current Situatin: Yur family went t an amusement park while they were at the beach. Yu decide t ride the Ferris wheel s yu can lk ut at the cean. Yu are disappinted t find ut that a 100 feet building blcked yur view f the cean fr part f the ride. Yur height frm the grund as yu travel arund the Ferris wheel can be fund using the fllwing equatin, where t = time in secnds frm the beginning f yur ride. Yu are trying t find ut hw lng it will take fr yu t see ver the building and t reach the tp f the ride. Student Challenge: Cnstructing and analyzing the inverse f a sinusidal functin. Predict result using the mdel. Student Rle: Cmplete the fllwing tasks: Create a height versus time table and graph; create the inverse, draw inferences; cmpare dmains and ranges; and create a presentatin f findings. Intended Audience: Class Prduct f Perfrmance: Presentatin f the findings n a pster bard, Pwer Pint r Prezi Perfrmance Task Synpses Task 1: MGSE9-12.F.TF.4 Students will create a height versus time table then graph the results. Students will analyze the symmetry bth graphically and algebraically. Task 2: MGSE9-12.F.BF.4, MGSE9-12.F.BF.4d, MGSE9-12.F.TF.4, MGSE9-12.F.TF.6 Students will create a new graph and equatin by switching the variables in the equatin frm Task 1. Students will cmpare the tw equatins including their dmains, ranges, and symmetries. Lastly, the students will make inferences. Task 3: MGSE9-12.F.TF.7 Students will calculate the desired time fr seeing ver the building and getting t the tp f the ride. Students will als create and present their findings t their peers and teacher utilizing a pster bard r technlgy such as a PwerPint r Prezi. 5

6 Perfrmance Task 1 In Detail Build new functins frm existing functins Task 1: MGSE9-12.F.TF.4 USE the unit circle t EXPLAIN symmetry (dd and even) and peridicity f trignmetric functins. Circumstances f Perfrmance: In-Class (Apprximately minutes) Task 1 Student Directins: Wrk in pairs. Yu will need a ruler, graph paper, and a pster bard (ptinal). Yur height frm the grund as yu travel arund the Ferris wheel can be fund using the fllwing equatin where t = time in secnds frm the beginning f yur ride. a) Make a height vs time table fr yur ride using the equatin. b) Plt the functin. c) What can yu say abut the symmetry f the functin? Justify frm the graph as well as algebraically. Task 1 Teacher Ntes: This task intrduces students t the inverse f a sinusidal functin. They generate data using the given peridic functin and plt them. Then the students will cnclude abut the functin symmetry graphically and algebraically. Remind the students t label the X and Y axis crrectly. Perfrmance Task 1 Scring Guide 4 Advanced 3 Prficient 2 Basic 1 Belw Basic All Prficient criteria plus: Student finds symmetry algebraically and prvides the ratinale. Chse the apprpriate units t plt the data Calculated the data crrectly frm the given functin Explained the symmetry frm the graph accurately. Calculated mst f the data crrectly frm the given functin Explanatin f the symmetry was incmplete r nt clear Calculated mst f the data incrrectly r nt at all Explanatin f the symmetry was nt given r culd nt be understd 6

7 Build new functins frm existing functins MGSE9-12.F.BF.4 FIND inverse functins. Perfrmance Task 2 In Detail MGSE9-12.F.BF.4d PRODUCE an invertible functin frm a nn-invertible functin by RESTRICTING the dmain. MGSE9-12.F.TF.4 USE the unit circle t EXPLAIN symmetry (dd and even) and peridicity f trignmetric functins. MGSE9-12.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. Circumstance f Perfrmance: In-Class (Apprximate Time Task 2 Student Directins: Using the infrmatin frm Task 1, plt a graph by switching the variables (t, h). 1) Hw des the equatin lk after switching the variables? Cnstruct the equatin. 2) What is the relatin between the given equatin and the equatin yu have cnstructed? 3) Des yur newly cnstructed equatin represent a functin? Justify yur answer. 4) If it represents a functin, write the dmain. If nt, then hw can yu make it a functin? Explain. 5) What can yu say abut the symmetry f yur newly cnstructed equatin? 6) Cmpare the dmain and range f the relatins frm bth equatins. 7) What can yu infer? Task 2 Teacher Ntes: Remind the students t label the X and Y axis crrectly. Fr questin 1, Explain t the students that Cnstruct the equatin means writing an equatin. Perfrmance Task 2 Scring Guide 4 Advanced 3 Prficient 2 Basic 1 Belw Basic All Prficient criteria plus: Prvided additinal characteristics f a relatin Clearly explained the relatin between the 2 equatins Crrectly determined if the equatin was a functin Explained the symmetry frm the graph accurately Accurately cmpared the dmain and range fr bth equatins Inferences were crrect and easy t understand Explanatin f the relatin f the 2 equatins were nt cmpletely clear r incmplete Explanatin f the symmetry was incmplete r nt clear Accurately cmpared the dmain r the range fr bth equatins Inferences were partially crrect Explanatin f the relatin f the 2 equatins were nt given r cannt be understd Explanatin f the symmetry was nt given r culd nt be understd Did nt cmpare the dmain r range fr bth equatins r the cmparisn was incrrect Inferences were missing r incrrect 7

8 Perfrmance Task 3 In Detail Build new functins frm existing functins MGSE9-12.F.TF.7 USE inverse functins t SOLVE trignmetric equatins that arise in mdeling cntexts; EVALUATE the slutins using technlgy, and INTERPRET them in terms f the cntext. Circumstances f Perfrmance: In-Class (Apprximate Time minutes) Task 3 Student Directins: Answer the fllwing questins using the functin given in Task 1. 1) Using what yu knw abut inverse trig functins, find ut hw lng it will take until yu can see ver the building? 2) Hw lng will it take yu t reach the tp f the ride? 3) Des the graph represent a peridic functin? Justify yur answer. Present yur findings n a pster bard, Pwer Pint r Prezi t the class. Make sure yu include the graphs. Task 3 Teacher Ntes: May encurage students t use technlgy fr the presentatin. Perfrmance Task 3 Scring Guide 4 Advanced 3 Prficient 2 Basic 1 Belw Basic All Prficient criteria plus: Determined hw lng it wuld take t reach the grund Determined hw lng it wuld take fr the building t blck the view again. Perfrmed the calculatins accurately Determined if the graph was a peridic functin with prper justificatin Perfrmed 1 f the 2 calculatins crrectly Determined if the graph was a peridic functin but withut a prper justificatin Perfrmed nne f the calculatins crrectly Did nt determine if the graph was a peridic functin nr prvided a justificatin 8

9 Instructinal Strategies Research-Based Effective Teaching Strategies Learning Objectives (psted and referenced) Identifying Similarities and Differences Summarizing and Nte Taking Reinfrcing Effrt, Prviding Recgnitin Hmewrk and Practice Nnlinguistic Representatins Cperative Learning Purpseful small grup instructin Increased think time Setting Objectives, Prviding Feedback Check fr Understanding Generating and Testing Hyptheses Cues, Questins, and Advance Organizers Interdisciplinary Nn-Fictin Writing Instructinal Strategies 21st Century Learning Skills Teamwrk and Cllabratin Initiative and Leadership Curisity and Imaginatin Innvatin and Creativity Critical thinking and Prblem Slving Flexibility and Adaptability Effective Oral and Written Cmmunicatin Accessing and Analyzing Infrmatin Other 9

10 Interventin Strategies Interventin Strategies (Tiers 1, 2, 3) Additinal Supprts in Classrm Re-vicing Explaining Prmpting fr participatin Challenging r cuntering Asking Why? Hw? Reread Practice new academic vcab. Assistive technlgy Pre-teach & re-teach in a different way Repetitin Use f manipulatives Cllabrative wrk Direct/explicit instructin Chunking Accmmdating different learning styles Create differentiated text sets Prviding additinal guided practice Specially Designed Instructin fr Special Educatin Students Cnferencing Additinal time Small grup cllabratin Mdify quantity f wrk Take student s dictatin Scaffld infrmatin Differentiated cntent prcess r prduct Cnsistent reward system Refer t students IEP r 504 plan Assistive technlgy Strategies fr English Language Learners Visuals/Realia Frnt-lading Eching/Chral respnse Clr-cding Multiple expsures in different media Pair-share Mdeling Language scafflds: eg, sentence frames Decnstruct cmplex sentences and texts L1 supprt increased pprtunities fr student-student talk Strategic vcabulary instructin Additinal think time 10

11 Tier 1: Lw Preparatin Flexible-Learning Grups by Readiness, Interest, Learning Prfiles Chice f Bks Hmewrk Optins Use f Reading Buddies Varius Jurnal Prmpts Student/Teacher Gal Setting Varied Pacing with Anchr Optins Wrk Alne r Tgether Flexible Seating Varied Scafflding Varied Cmputer Prgrams Design-A-DAY Varied Supplemental Materials Cmputer Mentrs Think-Pair-Share by Readiness, Interest, Learning Prfiles Open-ended Activities Explratins by Interest Optins fr Cmpetitin Tier 3: High Advanced Cntent English/language arts, mathematics, science and/r scial studies curses Resurce Classes Independent/Directed Study Scratic Seminars Strategies fr Gifted Learners Tier 2: Medium Preparatin Gifted Educatin Cluster Classes Gifted Educatin Cllabratin Classes Tiered Activities and Prducts Use f Literature Clubs Multiple Testing Optins Multiple Texts Alternative Assessments Subject Advancement within class Curriculum Cmpacting Tiered Centers Spelling by Readiness Varying Organizers Cmmunity Mentrships Statins Grup Investigatins Students are Assessed in Multiple Ways Student chice in selecting learning activities. Simulatins Tier 4: Advanced/Autnmus Abve grade level accelerated English/language arts, mathematics, science and/r scial studies curses Advanced Placement Classes Internatinal Baccalaureate Classes Internship/Mentrships Whle Grade Acceleratin 11

12 Instructinal Resurces and Materials Suggested Resurces Math Tls: Graph Paper Unit Circle Template Clred pencils Prtractr Mira r flding paper Graphing calculatr Suggested Technlgy Resurces Technlgy Resurces: GADOE Grade Level Curriculum Overview Pre-Calculus GADOE Pre-Calculus Framewrk Unit 2 Math Instructinal Sites: Explanatin f Standards GADOE Grade Level Curriculum Overview Pre-Calculus GADOE Pre-Calculus Framewrk Unit 2 Unit 2 Resurces Pwer Pint Prezi 12

13 Unit Vcabulary Unit Vcabulary Terms Academic / Tier 2 Restrict Cnstruct Interpret Evaluate Mdeling Infer Ferris Wheel Unit-Specific / Dmain / Tier 3 Even Functin Odd Functin Symmetry nn invertible functin Invertible functin Unit Circle Inverse functin Hrizntal Line Test Vertical Line Test Peridicity Sinusids Dmain Range Special Right Triangle C-terminal Angle Reference Angle Terminal side f an angle Interdisciplinary Cnnectins Physical Science SPS8: Obtain, evaluate, and cmmunicate infrmatin t explain the relatinships amng frce, mass, and mtin. a. Plan and carry ut an investigatin t analyze the mtin f an bject using mathematical and graphical mdels. Physics SP1: Obtain, evaluate, and cmmunicate infrmatin abut the relatinship between distance, displacement, speed, velcity, and acceleratin as functins f time. SP1.d: Predict and describe hw changes t initial cnditins affect the resulting mtin. 13

14 Curse: Pre-Calculus Weekly Planner Unit: 2 Trignmetry Functins Pacing Pririty (in bld) and Supprting Standards Engaging Learning Experiences Cre Instructin (in bld) and additinal resurces Frmative Assessments Mathematical Literacy: The ability t read, listen, think creatively, and cmmunicate abut prblem situatins, mathematical representatins, and the validatin f slutins will help students t develp and deepen their understanding f mathematics. (Natinal Cuncil f Teachers f Mathematics, 2000) 4 weeks Traditinal 2 weeks - Blck Make mdificatins t the pacing based upn student pre-assessment results and the daily analysis f student wrk See CAAG fr a listing f all standards. Instructin shuld be supprted with the implementatin f the Standards fr Mathematical Practice (SMP). Students shuld cmplete the listed experiences t supprt understanding and mastery f the math standards. Students are expected t exhibit the SMP. Teachers shuld implement the listed experiences t supprt cre instructin n the math standards. Teachers shuld mdel the SMP during the 3-Part Lessn Structure. Teachers shuld use the assessments t supprt student learning at all levels. The unit pre-assessment shuld be administered prir t unit instructin fr diagnstic purpses. Differentiate instructin based upn the pre-assessment results. Make mdificatins t the pacing based upn student pre-assessment results and the daily analysis f student wrk. Implement the Frmative Assessment Lessn (FAL) t mnitr the prgress f students during the instructinal unit. Intrduce the learning targets n a daily basis in kid friendly language using I Can statements. Prvide instructin using the Cncrete-Representatinal-Abstract (CRA) sequence f instructin. Utilize the Gradual Release Mdel f Respnsibility (I d, We d, Yu d) during instructin t supprt student learning f cncepts/skills. Use UPS Check and the 3 Read Prtcl fr the Prblem Slving Prcess. Use Newman s Prmpt t diagnse the hurdles students face while prblem slving. Incrprate Mathematical Visualizatin t mdel math cncepts, skills, and prblem slving. Remind students that mistakes are respected, expected, and inspected in the mathematics classrm. Weekly cre instructin shuld include the implementatin f Unit Expectatins instructinal tasks frm the Gergia Department f Educatin (GADOE) Perfrmance Assessment (4 Tasks and Scring Guide) lcated at alng with ther district and Frmative Assessment Lessns supplemental instructinal resurces. Included in the planner are GADOE Instructinal Tasks pssible suggestins fr GADOE tasks. Cmpany. Hughtn Mifflin Harcurt Publishing Cmpany. All rights reserved. 14

15 Pacing Pririty (in bld) and Supprting Standards Engaging Learning Experiences Cre Instructin (in bld) and additinal resurces Frmative Assessments Week 1 Week 2 MGSE9-12.F.TF.4 MGSE9-12.F.TF.3 MGSE9-12.F.BF.4, MGSE9-12.F.BF.4d MGSE9-12.F.TF.6 Intr Engaging Scenari and GADOE Pre-Calculus Unit 2: Right Triangles and the Unit Circle Task 1: Graph, find height and symmetry f the Ferris Wheel Task 2: Plt inverse f the functin frm task 1. Use special triangles t determine gemetrically the values f sine, csine, and tangent. Use unit circle t express the values f sine, csine, and tangent. Use unit circle t find the symmetry and peridicity f trig functins. Write a ratinal functin in different frms Find inverse functins Prduce an invertible functin frm a nninvertible functin by restricting the dmain. Pre-Assessment (Unit Pre- Assessment) Discussin f the Essential Questins (thrughut unit) Suggested infrmal prgress mnitring checks (thrughut unit): Questining Hmewrk check Student-led class discussin Exit slip Discussin f the Essential Questins (thrughut unit) Task 1 scring guide Task 2 scring guide Infrmal checks Discussin f the Essential Questins (thrughut unit) Week 3 MGSE9-12.F.BF.4, MGSE9-12.F.BF.4d MGSE9-12.F.TF.6 GADOE Pre-Calculus Unit 2: Inverse Trignmetric Functins Add, subtract, multiply, and divide ratinal expressins Task 2 scring guide Suggested infrmal prgress mnitring checks: Questining Hmewrk check Student-led class discussin Exit slip Cmpany. Hughtn Mifflin Harcurt Publishing Cmpany. All rights reserved. 15

16 Pacing Pririty (in bld) and Supprting Standards Engaging Learning Experiences Cre Instructin (in bld) and additinal resurces Frmative Assessments Discussin f the Essential Questins (thrughut unit) Week 4 MGSE9-12.F.TF.7 Task 3: Slving equatin using inverse trig functin Presentatins Slve trig equatins using inverse functins Evaluate the slutins using technlgy Interpret the slutins Task 3 scring guide Suggested infrmal prgress mnitring checks: Questining Hmewrk check Student-led class discussin Exit slip Pre-Assessment (Unit Pre- Assessment) Cmpany. Hughtn Mifflin Harcurt Publishing Cmpany. All rights reserved. 16