Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
UNIT 1 Prerequisites Total Number of Days: __4.50_ days__ Grade/Course: _Calculus__ ESSENTIAL QUESTIONS How can we utilize equations to solve problems? Why do we want to compare rather than get an exact answer? How can solutions to linear inequalities be graphically represented on a number line Why are linear inequalities useful? What are some types of relationships that can be modeled by graphs? What types of relationships can be modeled by linear graphs What can we do with a system of equations/inequalities that we cannot do with a single equation/inequality? How can linear equations and inequalities be applied to the solution of word problems? What are the characteristics of Quadratic Functions? How do you sketch graphs and write equations for parabolas? How do you sketch the graphs of polynomial functions? How do you divide a polynomial by another polynomial and interpret the result? How are real, imaginary, and complex numbers related? How do you perform operations with complex number? How do you find all zeros of a polynomial ENDURING UNDERSTANDINGS Quadratic equations are necessary for an understanding of acceleration. Systems of linear equations and/or inequalities are used to model and solve real-world problems involving 2 variables. The laws of integer exponents can be extended to rational exponents. To obtain a solution to an equation, no matter how complex, always involves the process of undoing operations. Proportionality involves a relationship in which the ratio of two quantities remains constant as the corresponding values of the quantities change. Real world situations can be modeled and solved by using equations and inequalities. Graphs of lines are functions used to represent changes in data Graphs of equations of parabola of y=x2. Patterns, functions and relationships can be represented graphically, symbolically or verbally. Shape can be preserved during mathematical transformations. Complex numbers guarantee and supports the Fundamental Theorem of Algebra function? How do you sketch the graph of a rational function? How are the solutions to polynomial and rational inequalities found? PACING CONTENT Solving Equations SKILLS Identify different types A.SSE.1.b A.SSE.3 of equations. A.CED.1 Solve linear Equation A.CED.4 A.APR.1 F.IF.6 Ex.1 A.APR.3 3x – 6 = 0 A.APR.7 Ex2. 1 3 26 x x2 .25 STANDARDS (CCCS/MP) x2 x 4 Solve Quadratic equations by factoring, MP.2 completing the square, MP.5 and using the MP.7 quadratic formula. AX2 + BX +C Ex 1. 2x2 +9x+9 = 3 Solve equations involving radicals. Ex.1 2x 7 x 2 Solve equations with RESOURCES OTHER Precalculus (e.g., tech) www.kutasoftware. Pre com/Review of basic Calculus Algebra. with Calculus I with PreLimits 6th Edition Calculus 3rd Pg. 496 Edition Larson, Ronald & Robert P Average: 24 – 40 – odd Hostetler Smarthinking.com Advanced: 89 – 98 even Algebra 2, volume1&2, Common Core Edition. Pearson LEARNING ACTIVITIES/ASSESSME NTS http://www.purplemath .com/modules/solvquad .htm Hands on exercise- TI 84 Graphing pg. 500 Example: 3x – 6 = 0 Reasoning Problem.(enrichment) Standardized Test Prep: SAT/ACT page 243 Problems of the Lesson: pg 504 #31, 49 www.khanacadem Exploration: pg. 497 & y.org/math/algebr 499 a/quadratics/factor Model It: Data Analysising_quadratics Renewable Energy. Pg. 505 # 71 Journal Writing: Interpreting Points of Intersection Others Written tests and absolute values. Ex. Solving Inequalities | x2 3x | 4 x 6 Intervals and Inequalities quizzes Daily Home work Same As Above Pre Calculus with Limits 6th Edition Pg. 541 Average 27 – 33 odd Linear Inequality in one variable Ex: 5x-7>3x+9 .25 Inequalities involving Absolute Values. Ex: |x-5|<2 Polynomial Inequality Ex: X2 x - 6 <0 Rational Inequality 2x 7 Ex: 3 x 5 Word problem from: http://www.sheloves math.com/algebra/b eginningalgebra/wordproblems-in-algebra/ Advanced: 76 – 84 odd http://www.quickma th.com/webMathema tica3/quickmath/gra phs/inequalities/bas ic.jsp Internet examples: http://www.coolmath.co m/algebra/07-solvinginequalities/03-intervalnotation-01.htm TI-84 Graphing. Pg. 542 & 549 # 49 - 51 Standardized Test Prep: SAT/ACT page 243 Problems of the Lesson: pg pg # 71 , 74 Model It: Data AnalysisPrescription Drugs Pg. 550 # 77 Journal Writing: Creating a System Inequalities. Others Written tests and quizzes Daily Home work Rectangular Coordinates To learn how to plot points in the Cartesian plane. use the Pythagorean Theorem a2 + b2 = c2 and the distance formula d ( x2 x1 )2 ( y2 y1 )2 to find distance between two points. How far is it from (4, 3) to (15, 8)? (provide a right triangle figure with coordinates for the vertices) .25 Use the midpoint formula to find the midpoint of a line segment Find the value of p so that (–2, 2.5) is the midpoint between (p, 2) and (–1, 3). Use a coordinate plane and geometric formulas to model and solve real-life problems At 8 AM one day, Amir decides to walk in a straight line on the beach. Pre Calculus with Limits 6th Edition Pg. 2 Average: 53 – 58 odd Advance: 66 – 70 even www.kutasoftwar e.com/functions www.Classzone. co www.khanacademy. org/math/algebra/p ythagorean.../dista nce_formula www.kutasoftware.c om/FreeWorksheet s/.../Midpoint%20F ormula.pdf Internet example: www.purplemath.com/ modules/midpoint.html Hands-on exercise – graphical solution: pg. 3 Problems of the Lesson: #31, 56 Model it: Labor Force pg. 12 # 60 h ttp://lessonplans. Journal Reports Class Discussion and fundingfactory.co m/plan_details.asp questions x?id=271 Others Written tests and quizzes Daily Home work After two hours of making no turns and traveling at a steady rate, Amir was two mile east and four miles north of his starting point. How far did Amir walk and what was his walking speed? Graphs of Equations To sketch graphs of equations. f(x) = ax2 F.BF.1 F.IF.7 Pre Calculus with Limits 6th Edition Pg. 14 Average: 71 – 75 odd Advanced: 99 – 104 even .25 Larger values of a squash the curve Smaller values of a expand it And negative values of a flip it upside down y = (x-4)2 + 6 To find x and y intercepts of graphs of equations http://www.rege ntsprep.org/rege nts/math/algebra /ac1/pracline.ht m Internet example: http://www.mathsisfun. com/sets/graphequation.html Hands-on exercise – graphical solution: pg. 3 Activity-Modeling Linear Data P. 119 – 124 , do all application problems. Including Data Analysis Problems of the Lesson: #31, 49 Model it ; Population Statistics: pg. 24 # 75 Journal Written tests and quizzes Daily Home work Find the x- and yintercepts of 25x2 + 4y2 = 9 To find equations of Circles Find the equation of the circle with the end points of the diameter with (3,-4) and (-5,12). Linear Equations in Two Variables To use slope to graph linear equations in two variables 2y = 3x -12 Find slopes of lines -( ) y + 7x = .25 Write linear equations in two variables 2y = 13x -17 Use slope to identify parallel and perpendicular lines. One line passes through the points (–4, 2) and (0, 3); another line passes through the points (–3, –2) and (3, 2). Are these lines parallel, perpendicular, or neither Same As Above Pre Calculus with Limits page. 25 http://www.purp lemath.com/mod ules/systlin1.htm Internet example: http://www.mathsisfun. com/algebra/lineparallelperpendicular.html Exploration exercise: page 30 Average: 71 – 75 odd Advanced: 99 – 104 even Graphical approach: page 37 # 101 http://www.mat hsisfun.com/alge bra/line-parallelperpendicular.ht ml Problems of the Lesson: #31, 49 Model it ; Data Analysis Pg. 38 # 119 Journal Written tests and quizzes Daily Home work Quadratic Functions and Models Analyze graphs of quadratic functions f(x)= ax2 + bx + c F.IF.7a, 7b, 7e, 8a, 8b, 9 Average: 37 - 49 odd Advanced: 67 - 75 even Write quadratic functions in standard form and use the results to sketch graphs of functions f(x)= a(x-h)2 + k a not equal to zero .25 Pre Calculus with Limits https://www.google.co m/search?q=quadratic +functions+in+real+life &client=firefoxa&rls=org.mozilla:enUS:official&channel=n p&tbm=isch&tbo=u&s ource=univ&sa=X&ei= D4rwUfrOHbS84AOh7 4CQBw&ved=0CD8Qs AQ&biw=1024&bih=62 9 http://math.about.com/ od/algebra1help/a/Qua dratic_Formula.htm Internet example: http://education.ti.com/ en/us/activity/detail?id =26347906DEF24F8F97 F089F6EFC061A1 Group ActivityExploration on P.204 P. 208 – 212 Exploration excercise: page :129 Kangaroo Conundrum: A Study Of A Quadratic Function. Graphical approach: page 137 # 85 Problems of the Lesson: #37 - 42 Problems of the Lesson: #37 – 42 Model it ; Data Analysis Page: 137 # 86 Journal Written tests and quizzes Daily Home work .5 Polynomial Functions of Higher Degree To use transformation to sketch graphs of polynomial functions. A.APR.6. F.BF.1, F.IF.7 Algebra 2, volume1&2, Common Core Edition. kutasoftware.co m Classzone.com http://hotmath.co Internet example: https://www.khanacade my.org/math/algebra2/ polynomial_and_rational The first thing to do here is graph the function without the constant which by this point should be fairly simple for you. Then shift accordingly. Here is the sketch MP4 MP5 MP7 MP8 Pearson Pre Calculus with Limits Average: 37 - 41 odd Advanced: 61 - 74 even m/hotmath_help/t opics/leadingcoefficienttest.html http://tutorial.mat h.lamar.edu/Classe s/Alg/GraphingPol ynomials.aspx /factoring-higher-degpolynomials/v/factoring -5th-degree-polynomialto-find-real-zeros Exploration excercise: page :142 Technology Graphical approach: page: 149 # 23 - 26 Writing about mathematics P. 221, 222 – 225, Point of diminishing returns and graphical reasoning Use leading coefficient test to determine the end behavior of graphs of polynomial functions Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function f(x) = -x3 + 5x. Because the degree is odd and the leading coefficient is negative, the graph rises to the Problems of the Lesson: #57 - 66 Model it ; Tree Growth Pg.: 151, # 97 Journal Written tests and quizzes Daily Home work left and falls to the right as shown in the figure. Evaluate and use zeros of a polynomial functions as sketching aids. Sketch the graph of . Here is a list of the zeroes and their multiplicities. Here is a sketch of the graph. Use the Intermediate Value Theorem to help locate zeros of polynomial functions. A.APR.6. Polynomial To learn how to use F.BF.1, and Synthetic long division to divide F.IF.7 Division polynomials by other polynomials MP1 Divide MP2 3 2 ( 3x – 5x + 10x – 3) MP5 by ( 3x + 1) Pre Calculus with Limits Average: 51 – 61 odd kutasoftware.co m Classzone.com http://www.purpl emath.com/modul es/polydiv3.htm Smarthinking.co m. Advanced: 60 - 68 even Internet example: http://www.wtamu.edu /academic/anns/mps/m ath/mathlab/col_algebra /col_alg_tut37_syndiv.ht m Exploration excercise: model it: page :160 Technology Graphical approach: page: 160 # 65 Problems of the Lesson: #65 .5 Use synthetic division to divide polynomials by binomial of the form (x-k) . Use Remainder Theorem and the Factor Theorem. http://www.mathpo rtal.org/calculators/ polynomialssolvers/syntheticdivisioncalculator.php Group Activity Analyzing a Slant Asymptote P. 160 – 161 , Graphical Analysis, Data Analysis, and Power of an Engine. Model it ; Data Analysis: Military Personnel. Pg.: 160 # 73 Journal Written tests and quizzes Daily Home work Given, Use Remainder Theorem to evaluate f(-2) Complex Numbers To use imaginary unit N.CN.1 N.CN.2 to write complex N.CN.3 numbers. N.CN.7 MP1 MP2 MP5 MP8 .25 Example 1 |3+4i| Resultant value=5 kutasoftware.co m Smarthinking.co m. Average: http://www.math 39 – 43 odd sisfun.com/numb Advanced: ers/complex93– 104 numbers.html even Same as Above Internet example: http://www.themathpag e.com/alg/complexnumbers.htm www.College.hmco.com www.amscopub.com Exploration excercise: page :164 Technology Graphical approach: page: 160 # 65 Problems of the Lesson: #65 Problem model It: Impedance how to find the impedance of an electric circuit Page: 168 Example 2 Daily Home work Write: in complex form Add, subtract, and multiply complex numbers Multiply and simplify in the form of a+bi Use complex conjugates to write the quotient of two complex numbers in standard form Find complex solutions of quadratic equations 16x2 -4x+3 =0 To use the fundamental theorems of Algebra to determine the number of zeros of polynomial functions Zeros of Polynomial Functions Examples Find rational zeros of f(x) = 3x3 -19x2+ 33x-9 .25 P(x) = x 5 + x 3 - 1 is a 5th degree polynomial function, P(x)has exactly 5 complex zeros. Find the conjugate pairs of complex zeros. MP1 MP2 MP7 MP8 Same as Above Average: 48 - 54 odd Advanced: 99 - 105 even kutasoftware.co m Classzone.com Smarthinking.co m. Teacher-made Power Points Graphing calculator Quiz and Test generators Worksheets Internet example: http://www.chesapeake. edu/khennayake/MAT1 13/3.2Solution.htm Exploration excercise: page :183 # 121 Technology Graphical approach: page: 183 # 128 Problems of the Lesson: #103 Problem model It: http://www.sparkno Athletics. tes.com/math/algebr Page: 182 # 112 a2/polynomials/sect Writing Activity ion5.rhtml Comparing Real Zeros and Rational Zeros, pg. 183 # 125 http://hotmath.com/ hotmath_help/topics Daily Home work /descartes-rule-ofsigns.html http://academics.utep. edu/Portals/1788/CALC ULUS%20MATERIAL/2 _5%20ZEROS%20OF% 20POLY%20FN.pdf Example: https://www.khanac ademy.org/math/alg ebra/complexnumbers/complex_n umbers/v/complexconjugates-example Find zeros of polynomials by factoring Solve x2 + 5x + 6 = 0. Learn how to use the Descartes’ Rule of signs and the Upper and Lower Bound Rules to find zeros of polynomials Find the possible number of real roots of the polynomial and verify. f(x)= x3– x2– 14 x + 24 We can verify that there are 2 positive roots and 1 negative root of the given polynomial. Nonlinear Inequalities Solve polynomial inequalities A.APR.3 A.APR.7 Solve and write the solution interval of Solve rational inequalities Solve and sketch the graph of f(x) = .25 ≥1 Use inequalities to model and solve reallife problems An 18-wheel truck stops at a weigh station before passing over a bridge. The weight limit on the bridge is 65,000 pounds. The cab (front) of the truck weighs 20,000 pounds, and the trailer (back) of the truck weighs 12,000 pounds when empty. In pounds, how much cargo can the truck carry and still be allowed to cross the bridge? Same As Above Pg.197 Average: 1 - 11 odd MP1 MP2 MP4 MP7 MP8 Advanced: 30 0 37 even kutasoftware.co m Classzone.com Smarthinking.co m. Teacher-made Power Points Graphing calculator Quiz and Test generators Worksheets Internet example: http://www.intmath.co m/inequalities/3solving-non-linearinequalities.php Exploration excercise: page :200 Technology Graphical approach: page: 204 # 33 Problems of the Lesson: http://www.montere 13, 21, 25 yinstitute.org/course s/Algebra1/COURSE_ Model It: Cable TEXT_RESOURCE/U0 Television. 5_L1_T3_text_final.ht Page: 205 # 73 ml Written tests and quizzes Daily Home work Chapter Review Chapter Wrap Up Assessment To review the chapter Summative Assessment of Polynomial and Rational Functions Same as Above kutasoftware.com Classzone.com MP1 MP2 MP3 MP4 MP5 MP6 MP7 MP8 Same As Above 1day INSTRUCTIONAL FOCUS and VOCABULARIES OF UNIT Chapter Summary with Exercises pg. 207-211 Chapter Practice Test. Pg. 212 Multiple Choice Open Ended Questions from the Instructor’s Resource Book. Formative/Summative: . Written tests and quizzes. . Unit Test with multiple choice and openended. . Notebook assessments. . Journal Reports . Class Discussion and Questions Quizzess, midchapter test and chapter test versions A,B,C,D for basic, Average and Advanced group from the Test Bank of the Larson 6th edition Analyze Quadratic Functions to help students develop how to maximize area or volume (if one dimension is a constant value), projectile motions, or building archways. Analyze of Polynomial Functions students will help students solve polynomial equations by graphing (using technology) and by factoring and applying the polynomial function theorems). Polynomial and Synthetic Division The addition of complex numbers is connected to the addition of vectors. Zeros of Polynomial Functions Rational Functions Nonlinear Inequalities We commonly use quadratic equations in designing the shape of suspension bridge. We use complex numbers in electric circuits and electromagnetism We commonly use zeros of polynomial function in manufacturing of supplies. Rational numbers are used extensively in mathematics, engineering, and science. Marzano’s Six Steps for Teaching Vocabulary: 1. YOU provide a description, explanation or example. (Story, sketch, power point) 2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 3. Ask students to construct a picture, graphic or symbol for each word. 4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 5. Ask students to discuss vocabulary words with one another (Collaborate) 6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.) VOCABULARIES: Quadratic function Parabola Axis Leading Coefficient Vertex X-interc Equation Linear Equation Quadratic Equation Intervals Polynomial Transformation Vertex Intercepts Symmetry X-intercepts Y-intercepts Intermediate Zeroes of a polynomial function Synthetic division Factor theorem Real number Imaginary number Complex number Conjugate Rational Zero Lower and upper bound Asymptote Vertical and horizontal Asymptote Slant Connect 3, 4 Each group should draw picture(s) of a suspension bridge. Examine the shape and determine if quadratic functions is applicable. Write a short journal on your observations of the shape. List all applicable vocabularies in your writing. PARCC/SAT/TIMSS FRAMEWORK/ASSESSMENT http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test http://www.majortests.com/sat/problem-solving-test01 www.khanacademy.org/test-prep/sat 4 Cs Creativity: projects Critical Thinking: Math Journal Collaboration: Teams/Groups/Stations Communication – PowerPoint’s/Presentations 9.4.D Business, Management & Administration Career Cluster 9.4.E Education & Training Career Cluster 21ST CENTURY SKILLS (4Cs & CTE Standards) Three Part Objective Behavior Condition Demonstration of Learning (DOL) 9.4.F Finance Career Cluster 9.4.N Marketing Career Cluster 9.4.O Science, Technology, Engineering & Mathematics Career Cluster 9.4.P Transportation, Distribution & Logistics Career Cluster 9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities. 9.4.12.D.4, 9.4.12.E.4 Solve mathematical problems and use the information to make business decisions and enhance business management duties. 9.4.12.F.4 Solve mathematical problems to obtain information for decision-making in financial settings. 9.4.12.N.4 Solve mathematical problems to obtain information for marketing decision-making. 9.4.12.O.15 Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to fulfill the specific communication needs of that audience. 9.4.H(5) Biotechnology Research and Development 9.4.12.H.(5).2 Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective biotechnology research and development. 9.4.O(1) Engineering and Technology 9.4.12.O.(1).1 Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems. 9.4.12.O.(1).7 Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction). 9.4.O(2) Science and Mathematics 9.4.12.O.(2).2 Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems. Shot-Put A shot-put throw can be modeled using the equation (also in feet). How long was the throw?” , where x is distance traveled (in feet) and y is the height 9.4.12.O.(2).3 Assess the impact that science and mathematics have on society when used to develop projects or products. 9.4.12.O.(2).4 Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and mathematics impact problem-solving in modern society. Coin- Drop A man drops a coin from the top of a cliff that is 200 feet tall. What is the height of the coin 2 seconds after it was dropped? Use the vertical motion model h = -16t2 + 200 where t is the time in seconds, and h is the height in feet 9.4.12.O.(2).6 Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field. MODIFICATIONS/ACCOMMODATIONS Individual student learning styles would be accommodated by: adjusting assessment standards, one-to-one teacher support extra time testing time additional use of visual, auditory and other teaching methods. A wide range of assessments and strategies that complement the individual learning experience would be encouraged. Teacher directed instruction by providing students with more necessary steps in order to solve the problems Small Group Activities - when students are given group guided practice IEP/504 Modifications: Students will be allowed to use the graphing calculator Students will be provided guided notes/graphic organizers to help with organization and to build their note-taking skills in math Modified assessments and assignments (classwork, homework, quizzes/tests) as needed Math Centers (Differentiation) – Review/Revisit topics missed by absentee students APPENDIX (Teacher resource extensions) E-Text, Interactive Digital Resourses, Teacher Resourses. http://.Larsonsuccessnet.com/snapp/login/login.jsp?showLoginPage=true CCSS Mathematical Practices: MP1: MP2: MP3: MP4: MP5: MP6: MP7: MP8: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Abbreviations o Number & Quantity N-RN-The Real Number System N-Q-Quantities N-CN-The Complex Number System N-VM-Vector and Matrix Quantities o Algebra A-SSE-Seeing Structure in Equations A-APR-Arithmetic with Polynomials and Rational Expressions A-CED-Creating Equations A-REI-Reasoning with Equations and Inequalities o Functions F-IF-Interpreting Functions F-BF-Building Functions F-LE-Linear, Quadratic and Exponential Models F-TF-Trigonometric Functions o Geometry G-CO-Congruence G-SRT-Similarity, Right Triangles, & Trigonometry G-C-Circles G-GPE-Expressing Geometric Properties with Equations G-MG-Modeling with Geometry o Statistics and Probability S-ID-Interpreting Categorical & Quantitative Data S-IC-Making Inferences & Justifying Conclusions S-CP-Conditional Probability and Rules of Probability S-MD-Using Probability to Make Decisions Kuta A1: Kuta Software – Infinite Algebra 1 (Free Worksheets) Kuta PA1: Kuta Software – Infinite Pre-Algebra 1 (Free Worksheets) MU: Measuring Up Workbook UNIT 2 Total Number of Days: 5 days Grade/Course: Calculus ESSENTIAL QUESTIONS What is a function ? How is the equation of a line with partial information sketched? How is the domain and range of a function ENDURING UNDERSTANDINGS Functions relationships are fundamental ideas in mathematics. Functions can be represented in several ways, such as a graph generated from a table of x and y or input and output values. Functions can be added, multiplied or subtracted to produce complex determined from the graph of the function? How are the graphs of inverse function related? How can functions be transformed? How does a function’s symmetry affect the behavior of the graph of a function? PACING CONTENT Functions SKILLS To determine whether relations between two variables are functions. functions. Function combinations can impact the range, domain and the graph of the function. Functions model real life problems. STANDARDS (CCCS/MP) F.IF.7, F.BF.1 MP1 MP2 MP5 MP8 Is it a function or relation .25 {(1,2),(2,4),(3,5),(2,6),(1,-3)} Use function notation and evaluate functions A function is represented by f(x) = 2x + 5,Find f(-3) Find the domain of functions. Determine the domain and range of the given function f(x) = Find the difference RESOURCES OTHER calculus (e.g., tech) Algebra 2, kutasoftware.c volume1&2, om Common Core http://www.reg Edition. Pearson entsprep.org/Re Pg. 60 - 65 gents/math/AL GEBRA/AP3/LF Pre Calculus unction.htm with Limits http://www.pur Pg. 40 - 47 plemath.com/m odules/fcns.htm http://www.pur plemath.com/m odules/fcns2.ht m f https://learnzill ion.com/lesso ns/3545-showtherelationshipbetweenvariablesusing-a-graph LEARNING ACTIVITIES/ASSESSME NTS Internet example: http://math.kennesaw.e du/~sellerme/sfehtml/c lasses/math1190/limits. pd Technology Graphical approach: page: 49 # 11 - 12 Problems of the Lesson: 13, 21, 25 Model It: Wildlife . pg: 53 # 102 Average: 72 – 79 odd Advanced: 80 -102 even Journal Teacher made Quizzes Quizzess, midchapter test and chapter test www.Smarthin versions A,B,C,D for basic,Average and king.com Advanced group from the Test Bank of the . Larson 6th edition quotient Evaluate difference of quotients for f(x) = x2 + 2 Quiz and Test generators Worksheets Use functions to model and solve real-life problems Students will examine the relationship between the number of faces, edges, and vertices of various polyhedral using function (Question will be provided) Analyzing Graphs of Functions To use the vertical line test for functions F.BF.1, F.IF.7 MP2 MP4 MP5 .5 Use vertical line test to verify if y = 2x-2 a function? =1 Find the zeros of kutasoftware.c om Classzone.com http://www.co olmath.com/al gebra/15functions/03vertical-linetest-01.htm http://www.yo utube.com/wat ch?v=-xvDn4FOJQ(video) http://calculato r.maconstate.ed u/findingzeroes/(calculat or) http://www.yo utube.com/watc h?v=6YM3TrudI zQ Internet example: https://www.math.ucd avis.edu/~kouba/CalcO ne Technology Graphical approach: page: 49 # 11 - 12 Group Exploration: pg. 59 and 60 Problems of the Lesson: 39, 69, 90 Model It: Data Analysis: temperature Page: 64 # 86 Average: 35 - 45 odd Advanced: 72 - 90 even functions. Example: Teacher made Quizzes Determine the zeros of y = x3 -5x + 1 Determine intervals on which functions are increasing or decreasing. Determine the decrease and increase intervals: f(x) = x3- 6 x2 Determine the relative max and relative min. values of functions. An open rectangular box http://www.bright storm.com/math Writing Journal Quiz and Test generators Worksheets with square base is to be made from 48 ft.2 of material. What dimensions will result in a box with the largest possible volume ? Determine average rate of change of a function. If an object is dropped from a tall building, then the distance it has fallen after t seconds is given be 2 the function d (t ) 16t . Find the average speed over the following intervals: a) Between 1 and 5 seconds b) Between t a and t ah Identify even and odd functions. Example: Is f(x) = A Library of Recognize graphs of Parent Functions parent functions. F.BF.1, F.IF.7 MP4 MP5 MP6 MP7 Identify the parent function f(x) of .25 Identify and graph linear and squaring functions. Identify and graph step and other piecewise-defined functions Internet example: https://myportal.bsd40 5.org/personal/kreiling www.Classzone. k/gprec/Shared%20Doc com uments/Polynomial%20 Review%20Problems.pd https://www.google f https://www.kuta software.com/ .com/search?q=libra ry+of+parent+functi ons&client=firefoxa&rls=org.mozilla:e nUS:official&channel =np&tbm=isch&tbo =u&source=univ&sa =X&ei=LZTwUezpC5 bd4APh1YHIDA&ve d=0CDkQsAQ&biw= 1024&bih=629 Technology Graphical approach: page: 72 # 51, 52 Group Exploration: pg. 69 Problems of the Lesson: 61, 63 Model It: Revenue. pg: 73 # 67 Average: 17 - 27 odd Advanced: 18 - 28 even Writing Journal Teacher made Quizzes Quiz and Test generators Worksheets Example: Transformation of Functions To learn how to use vertical and horizontal shifts in sketching graphs of functions. F.BF.1, F.IF.7 MP2 MP3 MP4 MP5 MP6 .5 kutasoftware.c om Classzone.com http://www.ui owa.edu/~exa mserv/mathm atters/tutorial_ You tube http://www.yo utube.com/wat ch?v=3Q5Sy03 4fok Smarthinking. com. Internet example: http://illuminations. nctm.org/LessonDet ail.aspx?ID=L470 Technology Graphical approach: page: 82 # 65 - 66 Group Exploration: pg. 75 , 76 Problems of the Lesson: 61, 63 Model It: Fuel Use. pg: 82 # 67 Average: 9 - 12 odd Advanced: 9 - 24 even Writing Journal Teacher made Quizzes Quiz and Test generators Worksheets : y =x2 y = x2 +4 y = (x-4)2 +7 Use reflections to sketch graphs of functions Reflections Across the yAxis Consider the following base functions, y = √x, y= 1 x 2 Reflections Across the xAxis Consider the following base functions, f (x) = x2 1 2 y= X 2 Use of non-rigid transformations to sketch graphs of functions Compare the functions and the graph of f(x)= X2 -3 g(x)= X2 +3 Combinations of Functions: Composite Functions Learn how to add, subtract, multiply and divide functions F.BF.1, F.IF.7 MP1 MP2 MP4 MP8 .5 Given f(x) = 2x g(x) = x + 4, and h(x) = 5 – x3, find (f + g)(2), (h – g)(2), (f × h)(2), and (h / g)(2). Find the compositions of one function with another function Worksheets kutasoftware.c om/functions Classzone.com http://www.pur plemath.com/m odules/fcnops.h tm Smarthinking. com. Teacher-made Power Points Graphing calculator Quiz and Test generators Worksheets Internet example: http://www.purplema th.com/modules/fcnc omp5.htm Technology Graphical approach: page: 87 Problems of the Lesson: 47, 59 Model It: Health Care Costs: page. 91 # 61 Average: 5 - 27 odd Advanced: 6 - 28 even Writing Journal: pg. 88 Teacher made Quizzes Quiz and Test generators Worksheets : Given the functions p(x) = x+2 and h(x) = x2 evaluate (h ◦ p)(3) and (h ◦ p)(x) Use combinations and compositions of functions to model and solve real-life problems. You work forty hours a week at a furniture store. You receive a $220 weekly salary, plus a 3% commission on sales over $5000. Assume that you sell enough this week to get the commission. Given the function: f (x) = 0.03x and g(x) = x – 5000 Which of ( f o g)(x)and (g o represents your f )(x) commission? Inverse Functions Find inverse function informally and verify that two functions are inverse functions of each other F.BF.1 F.IF.7 MP1 MP5 MP6 .5 Determine algebraically whether f(x) = 3x-2 and g(x) = are inverses of each other Show that f(g(x)) and g(f(x)) are equal to x Use graphs of functions to determine whether functions have inverse functions Let f(x) = 9 - x2 for x > 0 kutasoftware.c om Classzone.com You tube http://www.yo utube.com/watc h?v=fZxkYZw1h 48 http://www.pur plemath.com/m odules/invrsfcn 7.htm http://home.wi ndstream.net/o krebs/page45.ht ml Smarthinking. com. Teacher-made Power Points Graphing calculator Quiz and Test generators Worksheets http://www.yo utube.com/watc h?v=1NPvkYLjj Internet example: http://home.windstre am.net/okrebs/page4 5.html Technology Graphical approach: page: 100 Group Exploration: pg. 94 Amd 97 Problems of the Lesson: 24, 39, 75 Model It: U.S. Households. page. 101 # 79 Average: 13 - 23 odd Advanced: 29 - 45 even Writing Journal: The existence of an Inverse Function Teacher made Quizzes Quiz and Test generators Worksheets Find the equation for f -1(x) Sketch the graph of f, f -1, and y = x. Use the horizontal line test to determine if functions are oneto-one ws http://www.math warehouse.com/al gebra/relation/on e-to-onefunction.php Find inverse functions algebraically Find the inverse of f(x) = Mathematical Modeling and variation .5.5 F.BF.1 Use mathematical models to approximate F.IF.7 sets of data points A linear model that approximates the data MP2 is y = 1767.0 t + MP4 123916 where y kutasoftware.c Internet example: http://www.regentspre om Classzone.com p.org/Regents/math/alg trig/ATE7/Inverse%20V Smarthinking. ariation.htm com. Teacher-made Technology Graphical approach: page: 110 # 9 represents the number of employees and t = 2 represents 1992. Plot the actual data and model on the same set of plane .How closely does the model represent the data? Write mathematical models for direct variation If y varies directly as x , and the constant of variation is k = , what is y when x = 9 ? Write mathematical models for direct variation as an nth power Example using the ppt: http://www.google.com /#q=model+for++direct +variation+for+nth+po wer Write mathematical model for inverse variation In a formula, Z varies inversely as p. If Z is 200 when p = 4, find Z when p = 10. Write mathematical models for joint variation. If y varies directly as x and z, and y = 5 when x = 3 and z = 4, MP5 MP6 MP7 MP8 Power Points Graphing calculator Quiz and Test generators Worksheets https://www.kh anacademy.org/ math/algebra/a lgebrafunctions/direct _inverse_variati on/v/directvariationmodels Problems of the Lesson: Hooke’s Law pg. 112 # 35 Model It: DataAnalysis: Ocean Temperature. page. 113 # 71 Average: 13 - 27 odd Advanced: 82 - 86 even Teacher made Quizzes Quizzes, mid-chapter test and chapter test versions A,B,C,D for basic, Average and Advanced group from the Test Bank of the Larson 6th edition Quiz and Test generators Worksheets then find y when x = 2 and z = 3. Chapter Review General review of the chapter. MP1 MP3 MP4 Mp5 Chapter Wrap Up Assessment Summative Assessment of Functions and their graphs MP1 MP3 MP4 Mp5 Chapter Summary with Exercises pg. 115 Chapter Practice Test. Pg. 123 Graphing Calculator 1 INSTRUCTIONAL FOCUS and VOCABULARIES OF UNIT Formative/Summative: . Written tests and quizzes. . Daily Home work. . Worksheets. . Project Assessment. . Article Summaries. . Guided practice. . Quizzes, mid-chapter test and chapter test versions A,B,C,D for basic, Average and Advanced group from the Test Bank of the Larson 6th edition . . Notebook assessments. . Journal Reports Functions Analysis of function helps students in finding that the derivative and the integral are operations on functions: Shifting, reflecting, stretching graphs library of parent functions will help students understand what different graphs look like and then how they are transformed based on changes to the equation. combinations of function will help students understand that two functions can be combined to create new functions just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers, Marzano’s Six Steps for Teaching Vocabulary: 1. YOU provide a description, explanation or example. (Story, sketch, power point) 2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 3. Ask students to construct a picture, graphic or symbol for each word. 4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 5. Ask students to discuss vocabulary words with one another (Collaborate) 6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.) VOCABULARIES: Range Domain Independent Dependent Ordered pairs Vertical line test Horizontal line test Vertical Stretch Horizontal stretch Vertical shrink Horizontal shrink Non-rigid Rigid Composite function Inverse function Regression Variation Connect 2, 3, 4 Each group will draw a telephone key pad for the numbers 2 through 9. Create two relations : one mapping numbers onto letters and the other mapping letters onto numbers. Are both relations functions? Explain. PARCC/SAT/TIMSS FRAMEWORK/ASSESSMENT http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test http://www.majortests.com/sat/problem-solving-test01 www.khanacademy.org/test-prep/sat 4 Cs Creativity: projects Critical Thinking: Math Journal Collaboration: Teams/Groups/Stations Communication – PowerPoint’s/Presentations 21ST CENTURY SKILLS (4Cs & CTE Standards) Three Part Objective Behavior Condition Demonstration of Learning (DOL) 9.4.D Business, Management & Administration Career Cluster 9.4.E Education & Training Career Cluster 9.4.F Finance Career Cluster 9.4.N Marketing Career Cluster 9.4.O Science, Technology, Engineering & Mathematics Career Cluster 9.4.P Transportation, Distribution & Logistics Career Cluster 9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities. 9.4.12.D.4, 9.4.12.E.4 Solve mathematical problems and use the information to make business decisions and enhance business management duties. 9.4.12.F.4 Solve mathematical problems to obtain information for decision-making in financial settings. 9.4.12.N.4 Solve mathematical problems to obtain information for marketing decision-making. 9.4.12.O.15 Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to fulfill the specific communication needs of that audience. 9.4.H(5) Biotechnology Research and Development 9.4.12.H.(5).2 Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective biotechnology research and development. 9.4.O(1) Engineering and Technology 9.4.12.O.(1).1 Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems. 9.4.12.O.(1).7 Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction). Measurement: When buying gasoline you notice that 14 gallons is approximately the same amount of gasoline as 53 liters. Use this information to find a linear model that relates gallons to liters. Then use the model to find the numbers of liters in 5 gallons and 25 gallons 9.4.O(2) Science and Mathematics 9.4.12.O.(2).1 9.4.12.O.(2).2 Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems. 9.4.12.O.(2).3 Assess the impact that science and mathematics have on society when used to develop projects or products. 9.4.12.O.(2).4 Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and mathematics impact problem-solving in modern society. 9.4.12.O.(2).6 Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field. MODIFICATIONS/ACCOMMODATIONS 1. E-Text, Interactive Digital Resources, Teacher Resources Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true Individual student learning styles would be accommodated by: adjusting assessment standards, one-to-one teacher support extra time testing time additional use of visual, auditory and other teaching methods. A wide range of assessments and strategies that complement the individual learning experience would be encouraged. Teacher directed instruction by providing students with more necessary steps in order to solve the problems Small Group Activities - when students are given group guided practice IEP/504 Modifications: Students will be allowed to use the graphing calculator Students will be provided guided notes/graphic organizers to help with organization and to build their note-taking skills in math Modified assessments and assignments (classwork, homework, quizzes/tests) as needed Math Centers (Differentiation) – Review/Revisit topics missed by absentee students APPENDIX (Teacher resource extensions) CCSS Mathematical Practices: MP1: MP2: MP3: MP4: MP5: MP6: MP7: MP8: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Abbreviations o Number & Quantity N-RN-The Real Number System N-Q-Quantities N-CN-The Complex Number System N-VM-Vector and Matrix Quantities o Algebra A-SSE-Seeing Structure in Equations A-APR-Arithmetic with Polynomials and Rational Expressions A-CED-Creating Equations A-REI-Reasoning with Equations and Inequalities o Functions F-IF-Interpreting Functions F-BF-Building Functions F-LE-Linear, Quadratic and Exponential Models F-TF-Trigonometric Functions o Geometry G-CO-Congruence G-SRT-Similarity, Right Triangles, & Trigonometry G-C-Circles G-GPE-Expressing Geometric Properties with Equations G-MG-Modeling with Geometry o Statistics and Probability S-ID-Interpreting Categorical & Quantitative Data S-IC-Making Inferences & Justifying Conclusions S-CP-Conditional Probability and Rules of Probability S-MD-Using Probability to Make Decisions Kuta A1: Kuta Software – Infinite Algebra 1 (Free Worksheets) Kuta PA1: Kuta Software – Infinite Pre-Algebra 1 (Free Worksheets) MU: Measuring Up Workbook UNIT 3 Total Number of Days: _4.5days__ Grade/Course: _Calculus__ ESSENTIAL QUESTIONS How do you write and graph exponential functions? How do you evaluate exponential functions for a given value? How do you use transformations to sketch graphs of exponential functions? How do you evaluate logarithms with base a? How do you use transformations to sketch graphs of logarithmic functions? How are patterns of change related to the behavior of functions? How do you recognize, evaluate and graph logarithmic functions? ENDURING UNDERSTANDINGS Exponential functions can be used to model real-life situations. Exponential and logarithmic functions are inverses of each other. Student can solve e, log and natural log. Exponents are used to represent complex expressions. Linear functions have a constant difference, whereas exponential functions have a constant ratio. Real world situations can be represented symbolically and graphically How do you change bases in logarithmic expressions? How do you use properties of logarithms to evaluate or rewrite expressions? PACING CONTENT Exponential Functions and their Graphs SKILLS Learn how to recognize and evaluate exponential functions with base a STANDARDS (CCCS/MP) RESOURCES OTHER Pearson (e.g., tech) F.BF.5 MP1, MP2 MP5 Algebra 2, volume1&2, Common Core Edition. Pearson. Pg. 442 -446 www.kutasoftware. com/exponenial functions www.Classzone.com Pre Calculus with Limits pg. 217 .5 Examples of graphs of Exponential Functions: LEARNING ACTIVITIES/ASSESSME NTS Internet example: http://math.arizona.edu/~ calc/Text/1.2.pdf Standardized Test Prep: SAT/ACT Technology Graphical approach: page: 227 # 33-37 http://www.purpl emath.com/modul es/expofcns5.htm Problems of the Lesson: 17, 53, 65 Model It: Data Analysis: Biology. pg: 228 # 69 Group Exploration: pg. 223, 219 Average: 15 - 30 odd Advanced: 18 - 30 even Graph exponential functions and use the one-to-one property. Writing Journal Teacher made Quizzes Quiz and Test generators Worksheets Suppose Q = f (t) is an exponential function of t. If: f (20) = 88.2 and f (23) = 91.4 Find: (a) the base. (b) the growth rate Evaluate f (25). Use Q= Recognize, evaluate and graph exponential functions with base e. Graph y = e2x. Use exponential functions to model and solve real-life problems. Certain bacteria, given favorable growth conditions, grow continuously at a rate of 4.6% a day. Find the bacterial population after thirty-six hours, if the initial population was 250 bacteria. Logarithmic Functions and their Graphs. Evaluate logarithmic functions with base a =x Graph logarithmic functions. Same as Above MP1 MP4 MP5 .5 kutasoftware.c om Classzone.com http://www.regent sprep.org/Regents /math/algtrig/ATP 8b/logFunction.ht m Internet example: http://www.purplemath .com/modules/graphlog 3.htm Technology Graphical approach: page: 237 # 61 Problems of the Lesson: 4, 15, 79 Model It: Monthly Payment. Page: 237 # 87 Average: 13 - 27 odd Advanced: 82 - 86 even Evaluate and graph natural logarithmic functions. Group Exploration: pg. 230 Teacher made Quizzes Writing Journal: Analyzing a Human Memory Model Quiz and Test generators Worksheets y= Use logarithmic functions to model real-life problems In chemistry, a solution’s pH is defined by the logarithmic equation , where t is the hydronium ion concentration in moles per liter. We usually round pH values to the nearest tenth. http://www.uiowa. edu/~examserv/m athmatters/tutorial _quiz/log_exp/real worldappslogarith m.html a. Find the pH of a solution with hydronium ion concentration 4.5 x 10-5 b. Find the hydronium ion concentration of pure water, which has a pH of 7. Properties of Logarithms. .5 Learn how to use the change-of –base formula to write and evaluate logarithmic expressions. Same as Above kutasoftware.c om/logarithm MP1 MP2 MP5 Classzone.com http://www.pur plemath.com/mo http://www.purplemath .com/modules/graphlog 3.htm Standardized Test Prep: SAT/ACT dules/logrules5. htm Technology Graphical approach: page: Problems of the Lesson: 4, 15, 79 Use properties of logarithms to evaluate or rewrite logarithmic expressions. Model It: Human Memory Model. Pg.244 # 84 Group Exploration: pg. 241 4[lnz+ln(z+5)- 2l(z-5) ] =2 Use properties of logarithms to expand or condense logarithmic expressions Expand the expression Condense the expression 3 log x + 2 log y log z. http://dl.uncw.edu /digilib/mathemati cs/algebra/mat111 hb/eandl/logprop/ logprop.html Average: 19 - 27 odd Advanced: 62 - 72 even Writing Journal Quiz and Test generators Worksheets , kutasoftware.c om Smarthinking.c om. Exponential and Logarithmic Equations Solve complicated exponential equations Technology Graphical approach: page: 254#67 Problems of the Lesson: 4, 15, 79 Solve for x: 23x+1= 5x+6 .5 http://banach.millersvill e.edu/~BobBuchanan/ math160/ExpLogModels /main.pdf Standardized Test Prep: SAT/ACT Solve more complicated logarithmic equations +1 =3 Use exponential and logarithmic equations to model and solve real-life problems. The population of Pittsburgh from2000 to 2007 is modeled by http://www.chilim ath.com/algebra/a dvanced/log/logcondensing.html Model It: Automobiles. Page: 255# 117 Group Exploration: pg. 251 Average: 15 - 23 odd Advanced: 28 - 36 even Writing Journal: Comparing Mathematical Models Teacher made Quizzes Quiz and Test generators Worksheets . P= with t = 0 corresponding to the year 2000. 1. Find the population of Pittsburgh in 2005. 2 . Graph the population as a function of t. 3. From the graph estimate the year in which the population will reach 2:2 million. 4. Confirm this estimates algebraically. Exponential and Logarithmic Models .5 Learn how to recognize the five most common types of models involving exponential and logarithmic functions. Exponential Growth Model Same As Above kutasoftware.c om/exponential function Classzone.com Smarthinking.c om. http://academics.u tep.edu/Portals/17 88/CALCULUS% 20MATERIAL/3_ 5%20EXPO%20A ND%20LOG%20 Internet example: http://www.sosmath.co m/algebra/logs/log5/lo g54/log54.html Technology Graphical approach: page: 258 Problems of the Lesson: 4, 15, 35 Model It: Population Page: 265 #36 MODELS.pdf Exponential Decay Model Gaussian Model Logistic Growth Model Group Exploration: pg. 230 Average: 51 - 59 odd Advanced: 54 - 64 even Writing Journal: Comparing Population Models Teacher made Quizzes Quiz and Test generators Worksheets Natural Logarithmic model Chapter Review .5 To review the chapter Same as Above MP1 MP2 MP3 MP4 MP5 MP6 MP7 MP8 kutasoftware.c om Classzone.com Chapter Summary with Exercises Chapter Practice Test. Quizzes, mid chapter test and chapter test versions A,B,C,D for basic, Average and Advanced group from the Test Bank of the Larson 6th edition Chapter Summative Assessment Summative Assessment of Polynomial and Rational Functions Same As Above Chapter Test MP 1day . Written tests and quizzes. . Daily Home work. . Worksheets. . Project Assessment. . Quizzess, midchapter test and chapter test versions A,B,C,D for basic, Average and Advanced group from the Test Bank of the Larson 6th edition . Notebook assessments. . Journal Reports . Class Discussion and questions INSTRUCTIONAL FOCUS and VOCABULARIES OF UNIT Exponential and logarithmic Functions helps student develop understanding of population growth of virus, bacteri and cell phones. Marzano’s Six Steps for Teaching Vocabulary: 1. YOU provide a description, explanation or example. (Story, sketch, power point) 2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 3. Ask students to construct a picture, graphic or symbol for each word. 4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 5. Ask students to discuss vocabulary words with one another (Collaborate) 6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.) VOCABULARIES: Transformation of graphs Natural exponent Natural logarithmic Exponential growth Average Gaussian Models Connect1, 2 and 3 Find a partner and study the pictures below and explain which of the graph you would apply to the picture. . Connect 6 PARCC/SAT/TIMSS FRAMEWORK/ASSESSMENT http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test http://www.majortests.com/sat/problem-solving-test01 www.khanacademy.org/test-prep/sat 4 Cs Creativity: projects 21ST CENTURY SKILLS (4Cs & CTE Standards) Three Part Objective Behavior Critical Thinking: Math Journal Collaboration: Teams/Groups/Stations Communication – PowerPoint’s/Presentations Condition Demonstration of Learning (DOL) 9.4.D Business, Management & Administration Career Cluster 9.4.E Education & Training Career Cluster 9.4.F Finance Career Cluster 9.4.N Marketing Career Cluster 9.4.O Science, Technology, Engineering & Mathematics Career Cluster 9.4.P Transportation, Distribution & Logistics Career Cluster 9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities. 9.4.12.D.4, 9.4.12.E.4 Solve mathematical problems and use the information to make business decisions and enhance business management duties. 9.4.12.F.4 Solve mathematical problems to obtain information for decision-making in financial settings. 9.4.12.N.4 Solve mathematical problems to obtain information for marketing decision-making. 9.4.12.O.15 Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to fulfill the specific communication needs of that audience. 9.4.H(5) Biotechnology Research and Development 9.4.12.H.(5).2 Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective biotechnology research and development. Biological Cell If you start a biology experiment with 5,000,000 cells and 45% of the cells are dying every minute, how long will it take to have less than 1,000 cells? 9.4.O(1) Engineering and Technology 9.4.12.O.(1).1 Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems. 9.4.12.O.(1).7 Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction). Radioactive. Hospitals utilize the radioactive substance iodine-131 in the diagnosis of conditions of the thyroid gland. The half-life of iodine131 is eight days. a: Determine the decay constant b. (b) If a hospital acquires 2 g of iodine-131, how much of this sample will remain after 20 days ? c: How long will it be until only 0.01 g remains? 9.4.O(2) Science and Mathematics 9.4.12.O.(2).1 9.4.12.O.(2).2 Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems. 9.4.12.O.(2).3 Assess the impact that science and mathematics have on society when used to develop projects or products. 9.4.12.O.(2).4 Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and mathematics impact problem-solving in modern society. 9.4.12.O.(2).6 Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field MODIFICATIONS/ACCOMMODATIONS E-Text, Interactive Digital Resources, Teacher Resources Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true Individual student learning styles would be accommodated by: adjusting assessment standards, one-to-one teacher support extra time testing time additional use of visual, auditory and other teaching methods. A wide range of assessments and strategies that complement the individual learning experience would be encouraged. Teacher directed instruction by providing students with more necessary steps in order to solve the problems Small Group Activities - when students are given group guided practice IEP/504 Modifications: Students will be allowed to use the graphing calculator Students will be provided guided notes/graphic organizers to help with organization and to build their note-taking skills in math Modified assessments and assignments (classwork, homework, quizzes/tests) as needed Math Centers (Differentiation) – Review/Revisit topics missed by absentee students APPENDIX (Teacher resource extensions) CCSS Mathematical Practices: MP1: MP2: MP3: MP4: MP5: MP6: MP7: MP8: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Abbreviations o Number & Quantity N-RN-The Real Number System N-Q-Quantities N-CN-The Complex Number System N-VM-Vector and Matrix Quantities o Algebra A-SSE-Seeing Structure in Equations A-APR-Arithmetic with Polynomials and Rational Expressions A-CED-Creating Equations A-REI-Reasoning with Equations and Inequalities o Functions F-IF-Interpreting Functions F-BF-Building Functions F-LE-Linear, Quadratic and Exponential Models F-TF-Trigonometric Functions o Geometry G-CO-Congruence G-SRT-Similarity, Right Triangles, & Trigonometry G-C-Circles G-GPE-Expressing Geometric Properties with Equations G-MG-Modeling with Geometry o Statistics and Probability S-ID-Interpreting Categorical & Quantitative Data S-IC-Making Inferences & Justifying Conclusions S-CP-Conditional Probability and Rules of Probability S-MD-Using Probability to Make Decisions Kuta A1: Kuta Software – Infinite Algebra 1 (Free Worksheets) Kuta PA1: Kuta Software – Infinite Pre-Algebra 1 (Free Worksheets) MU: Measuring Up Workbook UNIT 4 Total Number of Days: _5 days__ Grade/Course: Calculus__ ESSENTIAL QUESTIONS What is the definition of a radian?. How can you convert from radians to degrees and vice versa? How do you select Radian Mode or Degree Mode on your calculator? The ratios of the side lengths of a triangle can be defined with trigonometric functions. What is the fundamental difference between a degree and a radian? Using radian measure, trigonometric functions can be defined on all real numbers. Trigonometric functions are periodic. What is angular velocity? The unit circle is a means of finding trig functions for any given angle. How does it differ from linear velocity? Any cyclic occurrence can be represented by a trig function. How does the arc length formula allow us to convert between angular and linear velocity? Trig functions can be translated and transformed. Fundamental identities can be used to verify more complicated trig identities. How can the six basic trig functions be used ENDURING UNDERSTANDINGS The characteristics of trigonometric and circular functions and their representations are useful in solving real-world problems. to solve right triangles? What is the main difference between a trig function and its inverse? We can use formulas to find exact value of angles that are combinations of unit circle angles. Any cyclic occurrence can be represented by a trig function. Trig functions can be translated and transformed. How can inverse trig functions be used to calculate unknown angles in a right triangle? What are the two special triangles…45-4590 and 30-60-90 When discussing angles in the Cartesian Plane, which axis is always the initial side? Which direction of rotation is positive and which is negative? Why are 2π and 360o important numbers when discussing coterminal angles? How does the unit circle and the concept of conterminal angles help us to generate graphs of trig How can you evaluate inverse trig functions if a point not on the unit circle is included? functions relate to the parent graphs of the trig functions? How can one compose a function that is periodic but not sinusoidal. How would one be able to compose a function that represents damped harmonic motion. PACING CONTENT Radian and Degree Measure SKILLS Learn how to describe angles. Use radian measure .5 STANDARDS (CCCS/MP) F.TF.1 T.FT. 2, 5. F.BF.2, 3, MP1, MP2 MP5 RESOURCES OTHER Pearson (e.g., tech) www.kutasoftwar Algebra 2, e.com volume1&2, www.Classzone.c Common Core om Edition. Pearson. Pg. 844 - 845 http://www.pur plemath.com/m Pre Calculus odules/radians. with Limits. htm Pg. 282 LEARNING ACTIVITIES/ASSESSME NTS Internet example: http://www.opusmath. com/common-coreclusters/7.g.b-anglearea-surface-areavolume Technology Graphical approach: page: 286 Example: Convert /6 radians to degrees. (/6)*(180/) = (1/1)*(30/1) = 30 Problems of the Lesson: 18, 95 Use degree measure. Average: 18 – 31 odd Advanced: 18 – 47 even Example: Convert 270° to radians. Since 180° equates to π, then: (270/1)*( /180) = (3/1)*( /2) = (3)/2 Use angles to model and solve Model It: Speed of a Bicycle. Pg: 293 #108 Group Exploration: pg. 284 Journal Writing: Teacher made Quizzes real-life problems. Trigonometric Functions: Unit circle. Identify a unit circle and describe its relationship to real world. Same as Above Worksheets kutasoftware.c om Classzone.com Mathgraphs.co m .5 MP1 MP4 MP5 Evaluate trigonometric functions using the unit circle. Example: Evaluate the six trig functions of; t= , t= https://www.kha nacademy.org/m ath/trigonometry /basictrigonometry/unit _circle_tut/v/unit -circle-definitionof-trig-functions1 Internet example: http://coolmath.com/ precalculus-reviewcalculusintro/precalculustrigonometry/28-theunit-circle-01.htm HSPA /SAT Prep Technology Graphical approach: page: 298 Problems of the Lesson: 53 Model It: Harmonic Motion. pg: 300 #57 Group Exploration: pg. 297 Average: 43 - 53 odd Advanced: 59 - 61 even Writing Journal: How do Use the domain and period to evaluate sine and cosine functions. you evaluate trigonometry by using the unit circle Teacher made Quizzes Worksheets t(t+c)=f(t) Use a calculator to evaluate trigonometric functions. Right Triangles Trigonometry Example: Use calculator to evaluate a). sin b) cot 1.5 To evaluate trigonometric function of acute angles Same as Above kutasoftware.c om MP1 MP2 MP5 Classzone.com Internet example: http://www.regentspr ep.org/regents/math/ algebra/AT2/Ltrig.htm HSPA /SAT Prep Technology Graphical approach: page: 303 .5 Problems of the Lesson: 66, 68 Model It: Height. pg: 311# 71 Use the fundamental trigonometric identities Group Exploration: pg. 305 Average: 18 – 31 odd Use a calculator to evaluate trigonometric functions. Advanced: 30 - 42 even Writing Journal: Prove that Tan=sin/cos Teacher made Quizzes https://www.googl e.com/search?q=a pplications+invol ving+right+triang les&client=firefox a&rls=org.mozilla :enUS:official&chan nel=np&tbm=isch &tbo=u&source=u niv&sa=X&ei=_fn xUf75Eqv64AOd4 YGgDA&sqi=2&v ed=0CFEQsAQ&b iw=1024&bih=62 9 Example: Use calculator to evaluate: Cos 28o Sec 28o Use trigonometric functions to model and solve real-life problems. Trigonometric Functions of any Angle. .5 Evaluate trigonometric functions of any angle. Example: Let be any angle in standard position, let (x,y) = P be a point on the terminal side of Same as Above kutasoftware.c om Smarthinking.c om Classzone.com Internet example: http://www.themathp age.com/atrig/functio ns-angle.htm HSPA /SAT Prep Technology Graphical approach: page: 319# 65 - 68 http://www.thema thpage.com/atrig/f Problems of the Lesson: unctions- (other than the origin). Let r be the distance from P to the origin. Then we define: angle.htm 59, 89 Model It: Data Analysis: Meteorology. pg: 319# 87 Group Exploration: pg. 315 Average: 37 - 49 odd Advanced: 71 - 85 even Writing Journal Teacher made Quizzes Worksheets Use reference angles to evaluate trigonometric functions. Example: Find the reference angle: a). θ=300o b). θ= -1350 Evaluate trigonometric functions of reallife numbers. Graphs of Sine and Cosine Functions. Sketch the graphs of basic sine and cosine functions. Same As Above Classzone.com MP .5 Use amplitude and periods to help sketch the graphs of sine and cosine functions. y=d+asin(bx-c) y-d+acos(bx-c) kutasoftware.c om Smarthinking.c om. Internet example: http://patrickjmt.com/gra phing-sine-and-cosinewith-differentcoefficients-amplitudeand-period-ex-1/ HSPA /SAT Prep Interactive Technology Graphical Website: http://www.purple approach: page: 329 # math.com/module 63 s/triggrph.htm Problems of the Lesson: 59, 89 Problem model It: Data Analysis: Astronomy. Page: 330 # 78 Group Exploration: pg. 323 and 324 Average: 18 – 31 odd Advanced: 18 – 47 even Sketch translations of the graphs of sine and cosine functions. Writing Journal: How do you sketch the graphs of sine and cosine functions Teacher made Quizzes Worksheets www.youtube.co m/watch?v=dHU M_ZgZ9Hg Use sine and cosine functions to model real-life data. .5 Graphs of other Trigonometric Functions To sketch the graphs of tangent functions. Same as Above MP kutasoftware.c om Classzone.com http://www.pur Internet example: http://www.intmath. com/trigonometricgraphs/4-graphstangent-cotangentsecant- Example: Sketch the graph of y= tan plemath.com/m odules/triggrph 2.htm Technology Graphical approach: page: 340 # 53, 57 Sketch the graphs of cotangent functions Problems of the Lesson: 59, 867, 89 Example: y=2cot Sketch the graphs of Secant and Cosecant functions Model It: Predator-Prey Video Model. Page: 341# 77 http://www.yout ube.com/watch Group Exploration: pg. ?v=CMvUs338 923O8 Average: 65 - 75 odd Advanced: 65 - 75 even Example: y=sec2x Sketch the graphs of damped trigonometric functions. Writing Journal: Combining Trig Functions. Teacher made Quizzes Worksheets Example: f(x)=e-xsin3x Inverse Trigonometric Functions .5 Evaluate and graph the inverse sine function. cosecant.php HSPA /SAT Prep Same As Above kutasoftware.com Classzone.com MP Internet example: http://www.mathopenr ef.com/triginverse.html https://www.khanaca HSPA /SAT Prep demy.org/math/trigo nometry/unit-circletrigTechnology Graphical func/inverse_trig_func approach: page: 350 # tions/e/inverse_trig_f 69 unctions Problems of the Lesson: 59, 92 Evaluate and graph the other inverse trigonometric functions. Problem model It: Photography. Page: 351#93 Group Exploration: pg. 344 Average: 13 - 23 odd Advanced: 116 - 120 even Example: Sketch a graph of y=arcsinx Evaluate and graph the composition of trigonometric functions. Writing Journal Teacher made Quizzes Example: Find the exact value tan(arccos2/3 Applications and Solve real-life Models problems involving right triangles. .5 Example: Same as above. kutasoftware.co Classzone.com MP http://www.intmath. com/trigonometricfunctions/4-righttriangleapplications.php Solve real-life Internet example: http://academics.utep.e du/Portals/1788/CALC ULUS%20MATERIAL/4_ 8%20APPLICATIONS%2 0AND%20MODELS.pdfl Technology Graphical approach: page: 358 Problems of the Lesson: 27, 37 problems involving directional bearings. Problem model It: Numerical and Graphical Analysis. Page: 363 # 62 Group Exploration: pg. 355 Example: Average: 17 - 31 odd Advanced: 45 - 65 even Writing Journal Teacher made Quizzes Solve real-life problems involving harmonic motion. Example: D=asinωt Chapter Review To review the chapter http://www.docstoc. com/docs/10064014 8/006-PeriodicMotion-Unedited Same as Above MP1 Pg. 365-368 kutasoftware.c om Classzone.com Chapter Summary with Exercises pg. 364 Chapter Practice Test. Pg. 369 Quizzess, midchapter test and chapter test versions A,B,C,D for basic,Average and Advanced group from the Test Bank of the Larson 6th edition Test Summative Assessment of Trigonometry Same As Above Chapter Test MP 1day . Written tests and quizzes. . Daily Home work. . Worksheets. . Project Assessment. . Article Summaries. . Guided practice. . Review Games. . Summative Test . Unit Test with multiple choice and openended. . Notebook assessments. . Journal Reports INSTRUCTIONAL FOCUS and VOCABULARIES OF UNIT Radian and Degree Measure-analysis of the two measures helps students in developing the conversions between them and why radian is preferred in the real world applications like in game library Trigonometric Functions: Analyze the Unit circle is an easy way for students to learn trigonometry, which has an incredible amount of real life applications Right triangle Trigonometry - students can apply Pythagorean Theorem and the six trigonometric functions Graphs of Sine and Cosine functions will help students in developing and understanding of Fourier Analysis and electrical circuit anaysis Marzano’s Six Steps for Teaching Vocabulary: 1. YOU provide a description, explanation or example. (Story, sketch, power point) 2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 3. Ask students to construct a picture, graphic or symbol for each word. 4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 5. Ask students to discuss vocabulary words with one another (Collaborate) 6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.) VOCABULARIES: VOCABULARIES: angular velocity sine cosine tangent cotangent secant cosecant radian degree radian amplitude period vertical translation phase shift horizontal asymptote vertical asymptote sinusoidal function Identity Sum and Difference formulas Double angle formulas Half angle formula amplitude period vertical translation phase shift horizontal asymptote vertical asymptote sinusoidal function Connect 2, 4 & 5 Students work in pairs to discover the properties of trigonometric ratios using an applet. Their task is to complete tables of trig ratios for different triangles (the applet allows students to change the angle measure and base length of a right triangle). They start with sine of varying angles for a triangles with a constant base, then change the base and find the sine of the same angles. In pairs students discuss their observations and particularly focus their attention on comparing the two tables (ideally recognizing that the values are the same regardless of the length of the base). They repeat this exercise for tangent and cosine. PARCC/SAT/TIMMS FRAMEWORK/ASSESSMENT http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test http://www.majortests.com/sat/problem-solving-test01 www.khanacademy.org/test-prep/sat 4 Cs Creativity: projects Critical Thinking: Math Journal Collaboration: Teams/Groups/Stations Communication – PowerPoints/Presentations 21ST CENTURY SKILLS (4Cs & CTE Standards) Three Part Objective Behavior Condition Demonstration of Learning (DOL) 9.4.D Business, Management & Administration Career Cluster 9.4.E Education & Training Career Cluster 9.4.F Finance Career Cluster 9.4.N Marketing Career Cluster 9.4.O Science, Technology, Engineering & Mathematics Career Cluster 9.4.P Transportation, Distribution & Logistics Career Cluster 9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities. 9.4.12.D.4, 9.4.12.E.4 Solve mathematical problems and use the information to make business decisions and enhance business management duties. 9.4.12.F.4 Solve mathematical problems to obtain information for decision-making in financial settings. 9.4.12.N.4 Solve mathematical problems to obtain information for marketing decision-making. 9.4.12.O.15 Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to fulfill the specific communication needs of that audience. 9.4.H(5) Biotechnology Research and Development 9.4.12.H.(5).2 Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective biotechnology research and development. 9.4.O(1) Engineering and Technology 9.4.12.O.(1).1 Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems. 9.4.12.O.(1).7 Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction). Transportation: An airplane takes off 200 yards in front of a 60 foot building. At what angle of elevation must the plane take off in order to avoid crashing into the building? Assume that the airplane flies in a straight line and the angle of elevation remains constant until the airplane flies over the building. x = arctan( ) 5.72 o . The plane must take off at an angle of elevation of about5.72 o in order to avoid hitting the building. 9.4.O(2) Science and Mathematics 9.4.12.O.(2).1 9.4.12.O.(2).2 Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems. 9.4.12.O.(2).3 Assess the impact that science and mathematics have on society when used to develop projects or products. 9.4.12.O.(2).4 Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and mathematics impact problem-solving in modern society. 9.4.12.O.(2).6 Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field. MODIFICATIONS/ACCOMMODATIONS 2. E-Text, Interactive Digital Resources, Teacher Resources Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true Individual student learning styles would be accommodated by: adjusting assessment standards, one-to-one teacher support extra time testing time additional use of visual, auditory and other teaching methods A wide range of assessments and strategies that complement the individual learning experience would be encouraged. Teacher directed instruction by providing students with more necessary steps in order to solve the problems Small Group Activities - when students are given group guided practice IEP/504 Modifications: Students will be allowed to use the graphing calculator Students will be provided guided notes/graphic organizers to help with organization and to build their note-taking skills in math Modified assessments and assignments (classwork, homework, quizzes/tests) as needed Math Centers (Differentiation) – Review/Revisit topics missed by absentee students APPENDIX (Teacher resource extensions) CCSS Mathematical Practices: MP1: MP2: MP3: MP4: MP5: MP6: MP7: MP8: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Abbreviations o Number & Quantity N-RN-The Real Number System N-Q-Quantities N-CN-The Complex Number System N-VM-Vector and Matrix Quantities o Algebra A-SSE-Seeing Structure in Equations A-APR-Arithmetic with Polynomials and Rational Expressions A-CED-Creating Equations A-REI-Reasoning with Equations and Inequalities o Functions F-IF-Interpreting Functions F-BF-Building Functions F-LE-Linear, Quadratic and Exponential Models F-TF-Trigonometric Functions o Geometry G-CO-Congruence G-SRT-Similarity, Right Triangles, & Trigonometry G-C-Circles G-GPE-Expressing Geometric Properties with Equations G-MG-Modeling with Geometry o Statistics and Probability S-ID-Interpreting Categorical & Quantitative Data S-IC-Making Inferences & Justifying Conclusions S-CP-Conditional Probability and Rules of Probability S-MD-Using Probability to Make Decisions Kuta A1: Kuta Software – Infinite Algebra 1 (Free Worksheets) Kuta PA1: Kuta Software – Infinite Pre-Algebra 1 (Free Worksheets) MU: Measuring Up Workbook UNIT 5 Total Number of Days: _2__ days__ Grade/Course: _Calculus__ ESENTIAL QUESTIONS How do we use algebra and basic trig identities to verify complicated trig identities. How do we find trig functions of angles that are combinations of unit circle angles How do you rewrite trigonometric expressions in order to simplify and evaluate functions? How do you verify a trigonometric identity? How do you solve trigonometric equations written in quadratic form or containing more than one angle? How do you simplify expressions and solve equations that contain sums or differences of angles? ENDURING UNDERSTANDINGS Identities are used to evaluate, simplify, and solve trigonometric expressions and equations. Fundamental trigonometric identities, sum and difference formulas together with algebraic techniques are used to simplify expressions, prove identities and solve fundamental identities and can be used to verify more complicated trig identities. We can use formulas to find exact value of angles that are combinations of unit circle angles equations. When trigonometric equations are solved an angle measure is the result. How do you rewrite trigonometric expressions that contain functions of multiple or half-angles, or functions that involve square or products trigonometric expressions? How do you verify a trigonometric identity graphically? How do you use sum and difference identities to evaluate trigonometric expressions? How do you use double-angle and half-angle identities to evaluate trigonometric expressions? PACING CONTENT Using Fundamental Identities SKILLS Recognize and write the fundamental trigonometric identities. Use the fundamental trigonometric identities to STANDARDS (CCCS/MP) F.TF.8, 9 MP1, MP2 MP5 RESOURCES OTHER Pearson (e.g., tech) Algebra 2, www.kutasoftwar volume1&2, e.com Common Core www.Classzone.c Edition. Pearson. om Pre Calculus http://coffman.du with Limits. blin.k12.oh.us/te Pg. 373 achers/teacherp ages/oconnor/se ction5.1precalc. cwk(WP).pdf LEARNING ACTIVITIES/ASSESSME NTS https://teacher.ocps.net /jean.adams/media/51p cf.pdf Technology Graphical approach: page: 375 and 380 # 73 Problems of the Lesson: 6, 14 Problem model It: Analysis. Page: 363 # 62 Group Exploration: pg. evaluate trigonometric functions, simplify trigonometric expressions and rewrite trigonometric expressions. Example: Evaluate: a). sec u=-2/3 Verifying Trigonometric Identities. Simplify: sinxcos2-sinx Learn how verify trigonometric Identities. Example: Verify: (tan2x + 1)(cos2x -1) =-tan2x 374 – Fundamental Trig Identities Average: 15 - 34odd Advanced: 45 - 65 even Writing Journal Worksheets Teacher made Quizzes Same as Above MP1 MP4 MP5 kutasoftware.c om Classzone.com Mathgraphs.co m http://www.purp lemath.com/mo dules/proving.h tm http://www.analyzemat h.com/Trigonometry_2/ Verify_identities.html Warm-up exercise Technology Graphical approach: page: 388 # 39- 44 Problems of the Lesson: 20, 34 Model It: Shadow Length. Pg. 388 # 56 Group Exploration Average: 21 - 27 odd Advanced: 51 - 56 even Writing Journal: Error Analysis. Worksheets Teacher made Quizzes Solving Trigonometric Equations. Learn how to use standard algebraic techniques to solve trigonometric equations. Example: Solve: 3tan2x-1=0 To solve trigonometric equations of quadratic type. Example: find all solutions of; 2sin2x – sinx – 1 =0 To solve trigonometric equations Same as Above MP1 MP2 MP5 kutasoftware.c om Classzone.com mathwarehouse. com/ http://www.rege ntsprep.org/Re gents/math/algt rig/ATT10/trige quations2.htm http://www.purplemath .com/modules/solvtrig.h tm Warm-up exercise HSPA /SAT Prep Technology Graphical approach: page: 397 # 55 Problems of the Lesson: 35, 63 Model It: Data AnalysisUnemployment Rate. Pg. 398 # 76 Group Exploration: 393 Average: 57 - 63 odd Advanced: 75 - 86 even involving multiple angles. Example: Solve: 2cot3t-1=0 Sum and difference formulas Use inverse trigonometric functions to solve trigonometric equations. Solve: sec2x-2tanx=4 Use sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations. Example: Find exact value of (sin42cos12 – cos42sin12) Writing Journal: Worksheets Teacher made Quizzes Same as Above MP kutasoftware.c om Smarthinking.c om Classzone.com http://www.regent sprep.org/Regents /math/algtrig/ATT 14/formulalesson. htm http://www.analyzemat h.com/Trigonometry_2/ Sum_diff_form_trig.html Warm-up exercise HSPA /SAT Prep Technology Graphical approach: page: 405 # 65 Problems of the Lesson: 35, 63 Model It: Harmonic Motion. Pg. 405 # 75 Group Exploration: 400 Average: 69 - 77 odd Advanced: 56 - 75 even Writing Journal: Equations with no Solutions. Worksheets Teacher made Quizzes Multiple Angle and Product-toSum Formulas Use multipleangle formulas to rewrite and evaluate trigonometric functions Same As Above MP Example: Solve: 2cosx+sin2x=0 Use powerreducing formulas to rewrite and evaluate functions kutasoftware.co m Classzone.com Smarthinking.co m. http://www.rege ntsprep.org/Reg ents/math/algtri g/ATT14/formul alesson.htm http://academics.utep.e du/Portals/1788/CALC ULUS%20MATERIAL/5_ 5%20MULTIPLE%20AN GLE%20AND%20PROD UCT%20TO%20SUM%2 0FORMULAS.pdf Problems of the Lesson: 35, 63 Warm-up exercise HSPA /SAT Prep Technology Graphical approach: page: 416# 59 Model It: Mach Number. Pg417 # 121 Group Exploration: Average: 69 - 77 odd Advanced: 56 - 75 even Writing Journal: Deriving an Area Formula Use product-tosum and sum-toproduct formulas to rewrite and evaluate trigonometric functions. Chapter Review Use trigonometric formulas to rewrite real-life models Example: Projectile MotionR= To review the chapter Worksheets Teacher made Quizzes kutasoftware.c om Classzone.com http://www.rege ntsprep.org/Reg ents/math/algtri g/ATT14/formul alesson.htm Quizzess, midchapter test and chapter test versions A,B,C,D for basic,Average and Advanced group from the Test Bank of the Larson 6th edition Test 1day INSTRUCTIONAL FOCUS and VOCABULARIES OF UNIT Using fundamental Identities, solving and Verifying Trigonometric Identities helps students apply their sense of algebraic equations that is proving that the equation is true by showing that both sides equal one another Marzano’s Six Steps for Teaching Vocabulary: 1. YOU provide a description, explanation or example. (Story, sketch, power point) 2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 3. Ask students to construct a picture, graphic or symbol for each word. 4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 5. Ask students to discuss vocabulary words with one another (Collaborate) 6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.) VOCABULARIES: Pythagorean identities • Reciprocal identities • Half angle formulas • Double angle formulas • Sum and difference formulas • Quotient identities • Cofunction identities • Even/odd identities • Power reducing formulas • Product to sum and sum to product formula Connect 4 Each group explore would study the pad below, make a table of frequency combination for each number. Use the sum and difference formulas to write the sound produced by keys 3 and 7. On a Touch-Tone phone, each button produces a unique sound. The sound produced is the sum of two tones, given by y = sin (2lt) and y = sin (2ht) where l and h are the low and high frequencies (cycles per second) shown on the illustration. The sound produced is thus given by y = sin (2lt) + sin (2ht) PARCC FRAMEWORK/ASSESSMENT 4 Cs Creativity: projects Critical Thinking: Math Journal Collaboration: Teams/Groups/Stations Communication – Powerpoints/Presentations 21ST CENTURY SKILLS (4Cs & CTE Standards) Three Part Objective Behavior Condition Demonstration of Learning (DOL) 9.4.D Business, Management & Administration Career Cluster 9.4.E Education & Training Career Cluster 9.4.F Finance Career Cluster 9.4.N Marketing Career Cluster 9.4.O Science, Technology, Engineering & Mathematics Career Cluster 9.4.P Transportation, Distribution & Logistics Career Cluster 9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities. 9.4.12.D.4, 9.4.12.E.4 Solve mathematical problems and use the information to make business decisions and enhance business management duties. 9.4.12.F.4 Solve mathematical problems to obtain information for decision-making in financial settings. 9.4.12.N.4 Solve mathematical problems to obtain information for marketing decision-making. 9.4.12.O.15 Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to fulfill the specific communication needs of that audience. MODIFICATIONS/ACCOMMODATIONS Individual student learning styles would be accommodated by: adjusting assessment standards, one-to-one teacher support extra time testing time additional use of visual, auditory and other teaching methods. A wide range of assessments and strategies that complement the individual learning experience would be encouraged. Teacher directed instruction by providing students with more necessary steps in order to solve the problems Small Group Activities - when students are given group guided practice IEP/504 Modifications: Students will be allowed to use the graphing calculator Students will be provided guided notes/graphic organizers to help with organization and to build their note-taking skills in math Modified assessments and assignments (classwork, homework, quizzes/tests) as needed Math Centers (Differentiation) – Review/Revisit topics missed by absentee students APPENDIX (Teacher resource extensions) CCSS.Mathematical Practices: MP1: MP2: MP3: MP4: MP5: MP6: MP7: MP8: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Standards Abbreviations o Number & Quantity N-RN-The Real Number System N-Q-Quantities N-CN-The Complex Number System N-VM-Vector and Matrix Quantities o Algebra A-SSE-Seeing Structure in Equations A-APR-Arithmetic with Polynomials and Rational Expressions A-CED-Creating Equations A-REI-Reasoning with Equations and Inequalities o Functions F-IF-Interpreting Functions F-BF-Building Functions F-LE-Linear, Quadratic and Exponential Models F-TF-Trigonometric Functions o Geometry G-CO-Congruence G-SRT-Similarity, Right Triangles, & Trigonometry G-C-Circles G-GPE-Expressing Geometric Properties with Equations G-MG-Modeling with Geometry o Statistics and Probability S-ID-Interpreting Categorical & Quantitative Data S-IC-Making Inferences & Justifying Conclusions S-CP-Conditional Probability and Rules of Probability S-MD-Using Probability to Make Decisions Kuta A1: Kuta Software – Infinite Algebra 1 (Free Worksheets) Kuta PA1: Kuta Software – Infinite Pre-Algebra 1 (Free Worksheets) MU: Measuring Up Workbook UNIT 6 Limits and Their Properties Total Number of Days: 21 days Grade/Course: 11-12/calculus ESENTIAL QUESTIONS ENDURING UNDERSTANDINGS Students will understand that • There are different ways of calculating limits. • There are different types of discontinuities. The classic example is the graph of distance versus time for a moving vehicle. In most practical applications, the slope of the graph - which represents speed - will not be constant. •Limits asymptotes and continuity are interconnected calculus topics. • What is a limit? • How is a limit calculated? • What does continuity mean and what are the different types of discontinuities? • What are vertical asymptotes and how are they related to limits? PACING days I day CONTENT Finding Limits Numerically and Graphically SKILLS STANDARDS (CCCS/MP) The limit can be estimated numerically by creating a F.IF.6 Calculate table and graphically by and interpret the drawing a graph. Examples: 1. Evaluate the function f(x) = at several points near x=0 2. Evaluate the limit graphically: Evaluating Limits Analytically average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. F(x) = 3. Evaluate = F.IF.4, 6,7, 7a, 8a, 9 F.LE.3 RESOURCES LEARNING ACTIVITIES/ ASSESSMENTS Larson OTHER Larson (e.g., tech) Basic: Pg 54-55#1, 3, 6. 9-12 I. Average: p54-55 #6, 8, 15, 16, 18 Advanced: P 55 #14-18, 2124 Paul’s online notes page 24 even problems II. III. IV. Thomas (12th Ed.), Pg 74 #1250 John Rogawski V. https://www.khan academy.org/math /differentialcalculus/limits_top ic/limits_tutorial/ v/numericallyestimating-limit https://www.nr.ed u/chalmeta/271D E/Section%203.2. pdf http://pblpathway s.com/calc/C10_2_ 2.pdf http://www.math bootcamps.com/fi nding-limits-usinga-graph/ www.kutasoftware Teacher made Quizzes Quizzess, midchapter test and chapter test versions A,B,C,D for basic,Average and Advanced group from the Test Bank of the Larson 8e edition Hand on activities A. http://w ww.jame srahn.co m/Calcu (2nd Ed.) p 80 #30-48 Evaluate 4. Evaluate Limits using properties of limits _________________________________ _____ Evaluate a limit using properties of limits is a polynomial, then 1. 2. Examples: 1. Apply the property of substitution as long as it do not become undefined 3. Describe and use a strategy for finding limits Evaluate a limit using dividing out and rationalizing techniques Examples: 2. Assume f,g,h are functions ES/limit s.htm F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. FIF.8a,b Larson Basic: P 67 #1-21 odd, 49-55 odd, 67, 71 Average: P 67 #10-22 even, 28, 34, 40, 50-56 even. 74, 76 Evaluate a limit using Squeeze Theorem (Theorem 1.8) Apply two special trigonometric limits (Theorem 1.9) VI. http://calculus201 0.wikidot.com/sol ving-limitsanalytically VII. http://tutorial.mat h.lamar.edu/probl ems/calci/LimitsP roperties.aspx Advanced: P 67 #28, 34, 78=84 even, 105, 111, 123 Thomas P 74 #12-50 selected numbers John Rogawski Pgs. 75 #6-34 Larson Basic: Pgs 68 #50-60 even 3. Squeeze Theorem lusI/PAG F.IF.1, 2 ,4, 7d, 8 1day If .com/limits VIII. IX. https://www.khan academy.org/math /differentialcalculus/limits_top ic/squeeze_theore m/v/squeezetheorem X. http://www.sosma th.com/calculus/li mcon/limcon03/li mcon03.html Average: Pgs 68 #58-70 even Advanced: Pgs 68 #68-84 even, 101, 123 http://tutorial.mat h.lamar.edu/probl ems/calci/OneSide dLimits.aspx Understanding the Definition of a Limit; Investigating Limits as x approaches Infinity Project B. http://w ww.proj ectmaths .ie/works hops/WS 8_NR/po werpoint s/Session %202.pd f C. http://re alteachi ngmeans reallear ning.blo gspot.co m/2011/ 09/dawithfirstprinciple s-andlimits.ht ml Examples: 4. Show that 5. Continuity and One-Sided Limits 1. Determine continuity at a point and continuity on an open interval 2 block periods 2.Apply properties of continuity 2. Interpret and apply Intermediate Value Theorem If 2- x2 ≤ g(x)≤ 2cosx for all x, find the limit as x approaches zero in the function g(x). Examples: 1. a. b. 2. John Rogawski Pgs. 76 #46-74; Pgs 99 #30-46 Calculate the limit of as x approaches -2 from the right (bring in the discussion of STEP FUNCTION from the ONE- Page 57#53,56 XI. Thomas Pgs 74 #44-56 even; Pgs 75 #57-65 Interpret the definition of continuity: Discuss the removable and non-removable discontinuity in Example 1. Journal Writing http://patrickjmt.c om/calculating-alimit I. www.kutasoftware .com/limits at essential continuity II. http://www.calcul us.org/ Average: Pgs 79 #13-19 odd, 32-42 even, 57, 58, 83, 84, 96 III. www.kutasoftware .com/at jump discontinuitie and kinks Advanced: Pgs 79 #23, 26, 41-53 odd, 75-81 odd, 95, 97, 100, IV. http://www.rootm ath.org/calculus/c ontinuity-and-onesided-limits Larson Basic: Pgs 79 #3, 6, 10, 12- 18, 29-32 Page 81#88 Page116#67 Teacher made Quizzes Quizzess, midchapter test and chapter test versions A,B,C,D for basic,Average and Advanced group from the Test Bank of the Larson 8e edition Calcuator Activities 3. SIDED LIMIT. 106 Describe the intervals Thomas Pgs 101 #1, 4, 11-26 [ V. http://tutorial.mat h.lamar.edu/Class es/CalcI/OneSided Limits.aspx I. http://tutorial.mat h.lamar.edu/Class es/CalcI/InfiniteLi mits.aspx II. https://www.khan academy.org/math Rogawski Pgs 90 #1, 4, 5 Discuss property of continuity (Theory 1.11) to solve the above problem. 4. Apply the Intermediate Value Theorem to show that the polynomial function has a zero in the interval [0,1]. 2 block periods Infinite Limits 1.Determine the infinite Limits from Determine infinite limits from left to right and Sketch the vertical asymptotes of the graph of a function given below Kuta worksheet(creat e for three different levels such as easy,medium and hard) Section Project: - Graphs and limits of trigonometric functions as given on pg 90 in Larson the left and the right by creating a table 2.Sketch the vertical asymptotes of the graph of a function Examples 1 For Evaluate A. B. C. Larson Basic: Pgs 89 #2-12 even, 24-32 even, 54 /differentialcalculus/limits_top ic III. www.kutasoftware .com/infinitelimits IV. https://www.math .ucdavis.edu/~kou ba/ProblemsList.h tml V. https://www.khan academy.org/math /differentialcalculus/limits_top ic/limitsinfinity/v/limitsand-infinity VI. http://patrickjmt.c om/calculating-alimit-at-infinity VII. http://www.purpl Average: Pgs 88 #22-46 even, 53-57 Advanced: Pgs 88 # 59, 62, 63 Thomas Pg 114 #1-19 0dd, Pg 114- 115 # 34- 52 even, 80-86 2 block periods 2. Compute the following: a. Rogawski Pgs 85- 86 #1-30 selected problems b. 3. Determine the domain and vertical asymptotes(s), if any, of the following function: emath.com/modul es/asymtote.htm 4. Determine all vertical asymptotes of the graph of VIII. F(x) = Sketch the graph to identify the vertical asymptotes. 2block period Chapter Review& Chapter Test TIMSS/ SAT Prep .5 block periods Content Category: Numbers Equations and Functions Calculus Probability and Statistics Geometry TIMSS Example1. If xy =1 and x is greater than 0, which of the following statements is true? A. When x is greater than 1, y is negative. B. When x is greater than 1, y is greater than 1. C. When x is less than 1, y is less than 1. D. As x increases, y A-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Page 91 of Larson and Resource work sheet Test Form B and C of the Resource Book College Board SAT Guide Page 227 selected problems TIMSS Sample problems I. http://www.edinfor matics.com/timss/p op3/mpop3.htm?su bmit32=Gr.+12+Adv .+Math+Test II. http://www.majort ests.com/sat/proble m-solving-test01 III. S-CPA 5 Understand independence and conditional probability and use them to interpret data http://patrickjmt.c om/findingverticalasymptotes-ofrationalfunctions/ www.khanacadem y.org/test-prep/sat Other Resource Texts Warm up Math minute Teacher created hand outs SAT Book Second Edition Test 1 section3 7 and and 8 page 397 – http://www.edinfor matics.com/timss/ti mss_intro.htm Sample Tests online increases. Gruber’s Complete Preparation for the New SAT E. As x increases, y decreases. SAT Example 1) In a class of 78 students 41 are taking French, 22 are taking German. Of the students taking French or German, 9 are taking both courses. How many students are not enrolled in either course? 4.3.C.1 Solve equations involving absolute value. 2) Iff(x)=│(x²– 50)│, what is the value of f(-5) ? The Princeton Review’s Cracking the New SAT, 2007 Barron’s SAT 2400: Aiming for the Perfect Score Maximum SAT Kaplan SAT 2400, 2006 Edition Up Your Score: The Underground Guide to the SAT, 2007-2008 INSTRUCTIONAL FOCUS OF UNIT The limit process is a fundamental concept of Calculus. One technique to estimate limit is to graph the function and then determine the behavior of the graph as the independent variable approaches a specific value. In this unit, limits are evaluated analytically, graphically, and numerically. Also, identifying where a function is continuous and not continuous; identifying the different types of discontinuity that a function may have; identify the vertical asymptotes of a graph and solving limit problems associated with those asymptotes. Limits are applied in real life to read the Speedometer for unlimited data Also see the statement given below If I keep tossing a coin as long as it takes, how likely am I to never toss a head? Rephrased as a limit problem, we might say If I toss a coin N times, what is the probability p(N) that I have not yet tossed a head? Now what is the limit as N→∞ of p(N)? Academic Vocabularies By Dr Robert Marzano limit continuity sandwich theorem fundamental trig limit infinity tangent normal exist, Asymptote Intermediate value Theorem Marzano’s Six Steps for Teaching Vocabulary: 1. 2. 3. 4. 5. 6. YOU provide a description, explanation or example. (Story, sketch, power point) Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) Ask students to construct a picture, graphic or symbol for each word. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) Ask students to discuss vocabulary words with one another (Collaborate) Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.) 1.Connected Graph(Step 1) When we talk about a continuous function we need to make sure there are no holes or gapes in the graph throughout its domain. So, the graph needs to be connected throughout the domain. A connected graph on its domain is a graph you can draw without lifting the pencil/pen off the paper. (No cheating! Don’t fold part of the paper over and draw on the fold and then unfold it and say – hey that is not connected but I didn’t lift the pencil off the paper!) Simple Test: Did you NEED to lift the pencil/pen off the paper to draw the graph each part of the domain ? Yes: It is not a connected graph No: It is a connected graph For example the domain of y=1/x is (-infinity, 0) and (0, infinity). From (-infinity, 0) it is connected and from (0, infinity) it is connected. 2.Discuss the continuity of F(x) = with the help of a diagram in a journal form (Step2) SAT/TIMSS FRAMEWORK/ASSESSMENT http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test http://www.majortests.com/sat/problem-solving-test01 www.khanacademy.org/test-prep/sat 21ST CENTURY SKILLS (4Cs & CTE Standards) 9.4.D Business, Management & Administration Career Cluster 9.4.E Education & Training Career Cluster Parking Lot cost A student parking lot at a university charges $ 2.00 for the first hour(or part of any hour) and $ 1.00 for each subsequent half hour (or any part) up to a daily maximum of $10.00 a.Sketch a graph of cost as a function of time parked? b. Discuss the significance of discontinuities in the graph to a student who parks there? 9.4.F Finance Career Cluster 9.4.N Marketing Career Cluster 9.4.O Science, Technology, Engineering & Mathematics Career Cluster 9.4.P Transportation, Distribution & Logistics Career Cluster 9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities. 9.4.12.D.4, 9.4.12.E.4 Solve mathematical problems and use the information to make business decisions and enhance business management duties. 9.4.12.F.4 Solve mathematical problems to obtain information for decision-making in financial settings. 9.4.12.N.4 Solve mathematical problems to obtain information for marketing decision-making. 9.4.12.O.15 Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to fulfill the specific communication needs of that audience. 9.4.H(5) Biotechnology Research and Development 9.4.12.H.(5).2 Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective biotechnology research and development. 9.4.O(1) Engineering and Technology 9.4.12.O.(1).1 Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems. 9.4.12.O.(1).7 Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction). 9.4.O(2) Science and Mathematics 9.4.12.O.(2).1 Generator Pollution Develop an understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world. A utility company burns coal to generate electricity.The cost C in dollars of removing p% of the air pollutants in the stack emissions is C= 0≤P≤ 100 Find the cost of removing 15% of the pollutants? Find the limit of C as P→100 9.4.12.O.(2).2 Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems. 9.4.12.O.(2).3 Assess the impact that science and mathematics have on society when used to develop projects or products. 9.4.12.O.(2).4 Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and mathematics impact problem-solving in modern society. 9.4.12.O.(2).6 Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field. MODIFICATIONS/ACCOMMODATIONS Students can create games (bingo, flashcards, monopoly, etc.) to demonstrate their understanding of classifying functions as continuous or discontinuous. APPENDIX (Teacher resource extensions) 3. E-Text, Interactive Digital Resources, Teacher Resources Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Notes to teacher (not to be included in your final draft): 4 Cs Creativity: projects Critical Thinking: Math Journal Collaboration: Teams/Groups/Stations Communication – Powerpoints/Presentations Three Part Objective Behavior Condition Demonstration of Learning (DOL) UNIT 7 Differentiation Total Number of Days: 27 Grade/Course: 11/12 Calculus ESENTIAL QUESTIONS ENDURING UNDERSTANDINGS W Students will understand that hat is a derivative? W ifferentiation is the study of rates of change, which will explored by learning new hat role do derivatives and limits play as a foundation for methods and rules for finding derivatives of functions and then apply to find such the calculus and in practical applications? things as velocity, acceleration, and the rates ofHchange of related variables. ow does the derivative represent an instantaneous rate of change? ow are derivatives used in optimization problems? ow do differential equations describe rates of change? ow do you find the derivatives of the sine function and of the cosine functions? How do you use derivatives to find rates of change? • The tangent line problem leads to the formal definition of a derivative. H he derivative tells us the instantaneous rate of change for a function which means H the same as the slope of the tangent line. H ifferentiation and definite integration are inverse operations. PACI NG 4days CONTENT Derivatives and the tangent line problem SKILLS Examples: 1. Evaluate the slope of the currve y = x2 at the point (2,4), using a numerical method 2. Estimate the derivative of the following function using the definition of the derivative. F(x) = 2 Limit process to find the derivative of a function Examples 1.Let F(x) = 4x2 +5x+6 Determine an equation of the tangent line to the curve Y = f(x) at (1,1.5). Compute f’ by the definition 2. Numerical Derivative of a function Describe that f is differentiable at x=1, i.e., use the limit definition of the derivative to compute f'(1) . he slope of a line in algebra is the average rate of change while the slope of the tangent to a curve at a point in calculus is the instantaneous rate of change (the derivative of a function). RESOURCES LEARNING STANDARDS Pearson ACTIVITIES/ASSES Larson (CCCS/MP) OTHER SMENTS (e.g., tech) CMS IF-4 Demonstrate an understanding of the definition of the derivative of a function at a point, and the notion of differentiability. a. Demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function. b. Demonstrate an understanding of the interpretation of the derivative as instantaneous rate of change. c. Use derivatives to solve a variety of problems coming from physics, chemistry, economics, etc. that involve the rate of change of a function. d. Demonstrate an understanding of the relationship between differentiability and continuity. e. Use derivative formulas to find the Basic Page 103# 1-21 odd Average Page 103 # 12 -32 even Thomas pg 132 13-21 odd Advanced Page 104 28- 38 even Thomas page 133 #32,34 and 35 Basic: Pgs 105 #41, 53 , 55, Kuta Worksheet (simple problems) Average: Pgs 105 # 41-57 odd, Kuta Worksheet (medium problems) Advanced: Examples http://classroom.synonym. com/equations-tangentlines-2838.html http://www.wikihow.com/ Find-the-Equation-of-aTangent-Line Videos: https://www.khanacademy .org/math/differentialcalculus/takingderivatives/derivative_intr o/v/calculus--derivatives1--new-hd-version work sheets file:///C:/Users/DUKE/Do wnloads/CalcI_Complete_Pr actice.pdf http://www.sosmath.com/c alculus/diff/der01/der01.h tml You tube http://www.youtube.com/ watch?v=vzDYOHETFlo https://www.khanacademy .org/math/differentialcalculus/limits_topic Work sheet https://www.math.ucdavis. edu/~kouba/CalcOneDIREC TORY/defderdirectory/Def Using graphing calculators, DI according to student readiness Use data and situations relevant to student interests Whole group and small group cooperative learning Trig function derivative graphic organizer Unit test (multiple choice & free response questions) Summative Assessment: Quizzes Tests Quarterly Assessments Projects Formative Assessments: Demonstration Class discussion Homework Quizzes Larson 8e Test Bank 2.2 Mid chapter test .Constant Rule: Theorem 2.2 Choose various Notation of Derivatives derivatives of algebraic and trigonometric functions. xample 1 The intermediate value theorem. F(x) =5 Power Rule: Theorem 2.3 xample 1 4days xample 2 Find the equation for the tangent line for the graph of F(x) = x2 when x=-2. Constant Multiple Rule: Theorem 2.4 xample 1 Y= Hands on project E Day Light Research project Paul A Foerster Page 117 Larson Worksheet 2.1 Work sheet Larson Resources 2.3 Kuta worksheets IF-5 Apply the rules of differentiation to functions. IF-3Demonstrate knowledge of differentiation using algebraic functions. a. Use the Chain Rule and applications to the calculation of the derivative of a variety of composite functions. b. Find the derivatives of relations and use implicit differentiation in a wide variety of problems from physics, chemistry, economics, Der.html https://www.kuta .com/limits Thomas: Pgs 125 #12-30 xample 2 Y= kπ2 Basic Differentiation Rules http://www.cms.k1 2.nm.us/instruction /secondary/math/ Math Pgs 105-106 #5868 even, Kuta Worksheet (Difficult problems) Average rate change E project based on the example #1 on page 129 of Jon Rogawski E Basic Page 115 # 1- 15 odd 331,33,53,63 Kuta Worksheet (easy problems) Average Page 115#10 -24 even,34- 44 even,62,66 Real life example #81 E Quiz E Larson 8e Test Bank 2.3Teachermade Kuta Worksheet (medium problems) Sum and Difference Rules: Theorem 2.5 Constant rule, Basic Rule,product,qu otient,chain and implicit rule 1. Recognize the constant Rule as (C) = 0 2.Translate the Power Rule n = nxn-1 3. Distinguish the Product and Quotient Rule (fg)’ = f g’ + f’ g . xample 1 xample 2 Find the derivative of Kuta Worksheet (difficult problems) Rates of changes Velocity and acceleration of any moving objects . assessmets E Writing Journals Page 128#109,110 Page 138 #97 E Page 147#70 http://www.math.brown.ed u/utra/derivrules.html http://www.mathsisfun.co m/calculus/derivativesrules.html Page 154#11 E xample 1 A rectangular water tank (see figure below) is being filled at the constant rate of 20 liters / second. The base of the tank has dimensions w = 1 meter and L = 2 meters. What is the rate of change of the height of water in the tank?(express the answer in cm / sec). Chain Rule: Explain d/dx ( (f(x)^r)) = r * f(x)^(r-1) * f '(x) Advanced Page 115-1173848 even,55,62,67,81 Real life example (Thomas Pg 145 #77, 78) Video http://www.youtube.com/ watch?v=DOClHx25P4o http://www.youtube.com/ watch?v=Yyag_L07iMg https://www.khanacademy .org/math/differentialcalculus xample 2 An object is sent through the air. Its height is modelled by the function h(x)=5x^2+3x+65 where h(x) is the height of the object in meters and the x is the time in seconds... estimate the instantaneous rate of change in the object's height at 3s AREI.10. Represent and solve equations and inequalities graphically Basic: Pgs 117 # 83, 85, 90, 92 Average: Pgs 117-118 # 8494 even, 102 Advanced: Pgs 117-118 #9199 odd, 103, 104, Real Life Example #108 E http://www.analyzemath.c om/calculus/Problems/rat e_change.html Thomas Page 133-134 selcted problems Howard Anton page 197 #37-49 0dd 3days Product and Quotient Rule: Product Rule Theorem 2.7 xample 1 Y = (x³ + 5x² -6x + 9) • (7x³ x² -8x + 1) xample 2 Y= product Rule Understand the concept of a function and use function notation FIF.2. https://www.math.hmc.edu /calculus/tutorials/prodrul e/ Basic: Pgs 126- 127 #117 odd, 38-49 odd, 73 Average: Pgs #126-127 # 34-54 even, 6268, 77 Advanced: Pgs #126-127 #48-62 even, 78, 83, 84 Real Life Problem pg 127 #85, 87 Thomas: Pgs 143 #29-49 odd Angton: video https://www.khanacademy .org/math/differentialcalculus/takingderivatives/product_rule/v /quotient-rule E E Pgs 203 #5-16 all . Quotient Rule 2.3: Quotient Rule http://tutorial.math.lamar. edu/Classes/CalcI/quotient rulef.aspx Theorem2.8 xample 1 Differentiate Quizzes Larson 86 Test E Bank 2.3 Teacher made Test Test made with Kuta Software Mid chapter test 3days xample 2 Find all points (x, y) on the graph Where tangent lines are perpendicular to the line Understand the concept of a function and use function notation F.IF.2 E 8x+2y = 1. 3days Section 2.4:Chain Rule Chain Rule Theorem 2.10 d/dx ( (f(x)^r)) = r * f(x)^(r-1) * f '(x) xample 1 Choose the method to differentiate . xample 2 Find an equation of the line tangent to the graph of x= IF-5 Apply the rules of differentiation to functions. a. Use the Chain Rule and applications to the calculation of the derivative of a variety of composite functions. at 4days Recognize when to apply the concept of Guide lines for Implicit Differentiation 1. We assume that the equation we are given has one independent variable, usually x or t, and the dependent variable, usually y, i.e. yis a function http://tutorial.math.lamar. edu/Classes/CalcI/chain rulef.aspx Quizzes Larson 86 Test Bank 2.4 E Kuta Teacher made work sheets Average: Pg 137 #22-28 even, 42-64 even, 74-88 Quiz- 2.4 Advanced Pgs 137 #49- 73 odd, 81, 99, Real Life Problem Pg 139 #101, 103, 106 E Thomas Pgs 167 #12-50 even Anton: Pgs 207 #30, 31, 32 (Word Problem) . Implicit Differentiation Basic: Pg 137 #8-28 even, 44, 46, 68, 74 Basic IF-5 Apply the rules of differentiation to functions. b. Find the derivatives of Page 146 #2-18 even,21 Kuta worksheet easy Average Kuta worksheet Work sheets http://tutorial.math.lamar. edu/Classes/CalcI/Implicit DIff.aspx http://calculator.tutorvista. com/math/584/implicitdifferentiationcalculator.html Home Work Resource section 2.5 Teacher made Kuta work sheets f(g(x)) = f'(g(x))g'(x) of x or t. 2. The derivative we are asked to determine is dy/dx or dy/dt. 3. Since the derivative does not automatically fall out at the end, we usually have extra steps where we need to solve for it. 4. The chain rule is used extensively and is a required technique. 5. Implicit differentiation expands your idea of derivatives by requiring you to take the derivative of both sides of an equation, not just one side. xample 1 Assume that y is a function of x . Find y' = dy/dx for relations and use implicit differentiation in a wide variety of problems from physics, chemistry, economics, medium Quiz – 2.5 Page 146,147 # http://17calculus.com/deri vatives/implicit13-25 differentiation/ odd,33,41,57,67 Advanced www.kutasoftware.com/impli Page 146- 147 # 15-27 multiple 0f 3,42,46,48,56,68 Kuta worksheet difficult Thomas Page 174 # 12,20,29,34,47 Anton Page 207 #30,31,32 citdifferentiation Videos https://www.khanacademy .org/math/differentialcalculus/takingderivatives/implicit_differe ntiation/v/implicitderivative-of-y---cos-5x--3y#! E http://www.youtube.com/ watch?v=jv4gTxWqeBE x 3 + y3 = 4 . xample 2 Determine the slope and concavity of the graph of x2y + y4 = 4 + 2x at the point (-1, 1) . Differentiate the following E equations explicitly, finding y as a function of x. Solve for y´=dy/dx. 4days Related Rates Example 2 Water is being poured into a conical reservoir at the rate of pi cubic feet per second. The reservoir has a radius of 6 feet across the top and a height of 12 feet. At what rate is the depth of the water increasing when the depth is 6 feet? Review for chapter Test Chapter 2 test 2days TIMSS/ SAT Basic Page 154 # 13,15,17 Kuta worksheet Easy Example 1. A ladder 10 feet long is standing straight up against the side of a house. The base of the ladder is pulled away from the side of the house at the rate of 2 feet per second. How high up the side of the house will the top of the ladder be 1 second after the base begins being pulled away from the house? How high up the side of the house will the top of the ladder be after 2 seconds, 3 seconds, 4 seconds, 5 seconds? Work sheets http://www2.seminolest ate.edu/lvosbury/Calcul usI_Folder/RelatedRateP roblems.htm Average Page 154-155 # 13,18,21,22 Kuta worksheet Medium Advanced Kuta worksheet Difficult Teacher made Kuta work sheets Quiz – 2.6 http://tutorial.math.lam ar.edu/problems/calci/r elatedrates.aspx www.kutasoftware.com/ relatedrates Page 154 – 155# 18,19,22,27 A-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Home Work Resource section 2.6 Video https://www.khanacade my.org/math/differentia lcalculus/derivative_appli cations/rates_of_change/ v/rates-of-changebetween-radius-andarea-of-circle Test Bank-Chapter Test-versions A,and C Prep Content Category: umbers Equations and Functions alculus robability and Statistics Geometry Construct a viable argument to justify a solution method. N IV. Sample problems from TIMSS Example1: http://www.ed informatics.co m/timss Stu wants to wrap some ribbon around a box as shown and have 25 cm left to tie a bow. V. ttp://www.edinfor matics.com/timss/ pop3/mpop3.htm? submit32=Gr.+12+ Adv.+Math+Test SAT study guide C Practice Test#2 Section 2 and P section 5(Math sections) ttp://www.majorte sts.com/sat/proble m-solving-test02 4.2.B.1 VI. How long a piece of ribbon does he need? A) 6 cm. B) 2 cm C) Understand and visualize geometric transformations (translations, rotations, and reflections). (4.2.B.1) ww.khanacademy .org/test-prep/sat 4 http://www.edinformati cs.com/timss 5 6 5 cm. D) 7 1 cm. E) 7 7 cm. SAT Sample problems Example 1: Clean soap powder is packed in cube-shaped cartons. A carton measures 10 cm on each side. The company decides to increase the length of each edge of the carton by 10 per cent. How much does the volume increase? A) 0 cu.cm. B) 1 cu. cm. C) 00 cu. cm. D) 31 cu. cm. 2. A triangular prism consists of rectangular and triangular faces. Each rectangular face has area r and each triangular face has area t .What is the total surface area of the figure, in terms of r and t ? A) 2r + t Practice Test Page 452-457 and `page 463 - 468 College Board SAT Study Guide Page 305 - 313 Other Resource Texts Gruber’s Complete Preparation for the New SAT The Princeton Review’s Cracking the New SAT, 2007 Barron’s SAT 2400: Aiming for the Perfect Score Maximum SAT Kaplan SAT 2400, 2006 Edition Up Your Score: The Underground Guide to the SAT, 2007-2008 1 2 1 3 B) 3r + 2t C) 4r + 3t D) 6rt E) 32rt INSTRUCTIONAL FOCUS OF UNIT This unit focuses on the following: the tangent line problem largely applied to find the velocity and acceleration of any moving objects.; using the formal definition of a derivative; finding derivatives using the power rule product rule, quotient rule, chain rule, product rule, quotient rule, chain rule. Students will investigate the definition of the derivative and find rates of change using the limit of the difference quotient. Students will interpret the derivative as the instantaneous rate of change or as the limit of the average rate of change. Students will find related rates and use related rates to solve real-life problems. Tangent LineApplication : 1. If we are traveling in a car around a corner and we hit something slippery on the road (like oil, ice, water or loose gravel) and our car starts to skid, it will continue in a direction tangent to the curve. Academic Vocabularies by Dr. Robert Marzano The constant rule: The power rule The constant multiple rule: Product Rule: Quotient Rule Higher order derivatives Second derivative Third derivative The Chain Rule The General Power Rule: The average rate of change The instantaneous rate of change The position function Velocity Average velocity Instantaneous velocity, An implicit function Related rates Speed, velocity, acceleration , instantaneous, limit, derivative, differentiable, concavity Marzano’s Six Steps of Teaching Academic Vocabularies 7. 8. 9. 10. 11. explanation or example. (Story, sketch, power point) explain meaning in their own words. (Journal, community circle, turn to your neighbor) graphic or symbol for each word. expand their word knowledge. (Add to their notes, use graphic organizer format) YOU provide a description, Ask students to restate or reAsk students to construct a picture, Engage students in activities to Ask students to discuss vocabulary words with one another (Collaborate) 12. words. (Bingo with definitions, Pictionary, Charades, etc.) Example 1-Step 3 Have students play games with the Calvin and Phoebe elope in a hot air balloon, which rises at a constant rate of 3 meters per second. Five seconds after they cast off, Phoebe's jilted suitor Bonzo McTavish races up in his Porsche. He parks 50 meters from the launch pad, and runs toward the pad at 2 meters per second. At what rate is the distance between Bonzo and the balloon changing when the balloon is 30 meters above the ground? Example 2. Step5 Assume that h(x) = f( g(x) ) , where both f and g are differentiable functions. If g(-1)=2, g'(-1)=3, and f'(2)=-4 , what is the value of h'(-1) ? . PARCC /SAT FRAMEWORK/ASSESSMENT 3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test solving-test01 http://www.edinformatics.com/timss/pop http://www.majortests.com/sat/problem- www.khanacademy.org/test-prep/sat 21ST CENTURY SKILLS (4Cs & CTE Standards) 21st Century Skills: Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation 9.4.D Business, Management & Administration Career Cluster 9.4.E Education & Training Career Cluster 9.4.F Finance Career Cluster 9.4.N Marketing Career Cluster 9.4.O Science, Technology, Engineering & Mathematics Career Cluster 9.4.P Transportation, Distribution & Logistics Career Cluster 9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities. 9.4.12.D.4, 9.4.12.E.4 Solve mathematical problems and use the information to make business decisions and enhance business management duties. 9.4.12.F.4 Solve mathematical problems to obtain information for decision-making in financial settings. 9.4.12.N.4 Solve mathematical problems to obtain information for marketing decision-making. 9.4.12.O.15 Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to fulfill the specific communication needs of that audience. 9.4.H(5) Biotechnology Research and Development 9.4.12.H.(5).2 Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective biotechnology research and development. 9.4.O(1) Engineering and Technology 9.4.12.O.(1).1 Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems. Find the velocity and acceleration of a particle with the given position of s(t) = t3 - 2t2 - 4t + 5 at t = 2 where t is measured in seconds and s is measured in feet. Velocity is found by taking the derivative of the position. At 2 seconds, the velocity is 0 feet per second. The acceleration is found by taking the derivative of the velocity function, or the second derivative of the position. 9.4.12.O.(1).7 Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction). 9.4.O(2) Science and Mathematics 9.4.12.O.(2).1 Develop an understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world. 9.4.12.O.(2).2 Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems. Aircraft Applications: Suppose one airplane flies over BWI Airport at a given rate, flying east. Twelve minutes later a second plane flies over BWI at its rate, flying north. A typical related rate application would calculate the rate at which they were separating at a later point in time. This is essentially the problem illustrated with boats by the applet in this section. 9.4.12.O.(2).3 Assess the impact that science and mathematics have on society when used to develop projects or products. 9.4.12.O.(2).4 Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and mathematics impact problem-solving in modern society. 9.4.12.O.(2).6 Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field. MODIFICATIONS/ACCOMMODATIONS Students can create flashcards of the derivative graphs of the function graphs provided by the teacher. Students can then get into groups where they can match function graphs to their derivative graphs. APPENDIX (Teacher resource extensions) 4. Resources, Teacher Resources Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true E-Text, Interactive Digital Mathematical Practices 9. persevere in solving them. 10. Make sense of problems and Reason abstractly and quantitatively. 11. critique the reasoning of others. Construct viable arguments and 12. Model with mathematics. 13. Use appropriate tools strategically. 14. Attend to precision. 15. Look for and make use of structure. 16. in repeated reasoning. Look for and express regularity Notes to teacher (not to be included in your final draft): 4 Cs Creativity: projects Critical Thinking: Math Journal Collaboration: Teams/Groups/Stations Communication – Powerpoints/Presentations Three Part Objective Behavior Condition Demonstration of Learning (DOL) UNIT 8 Application of Differentiation . Total Number of Days: 25 days Grade/Course: 12/Calculus ESENTIAL QUESTIONS What do the derivatives of a function tell us about that function? Why the Mean Value Theorem is considered such an important theorem in calculus. What is the physical interpretation of the MVT? How do we use the first and second derivative? What is an optimization problem and how do we solve it? How Can we solve real world problems using the calculus? PACI NG CONTENT SKILLS ENDURING UNDERSTANDINGS Solve extrema on an interval problem applied in Highway design. Solve problems that apply the Rolle’s Theorem and mean value theorem. Use the first and second derivatives to identify when a function is increasing, decreasing, concave up and/or concave down applied in Beam Deflection,Sales growth etc. . Combine together calculus skills with skills from previous math courses to accurately graph a wide variety of functions. Given a car’s velocity at two different check points, how can the police officer apply the MVT to determine if the car was exceeding the speed limit over the time interval? STANDARDS (CCCS/MP) RESOURCES Larson Pearson OTHER (e.g., tech) Extrema on an Interval Determine the relative maximum and minimum using “Extreme Value Theorem” Examples: 1. Find the value of the relative maximum and minimum for F(x) = IF-3D Demonstrate knowledge of differentiation using algebraic functions. a. Use differentiation and algebraic manipulations to sketch, by hand, graphs of functions. LEARNING ACTIVITIES/ASSESS MENTS Using graphing calculators, DI according to student readiness Use data and situations relevant to student interests Whole group and small group cooperative learning Trig function derivative graphic organizer Unit test (multiple choice 2. Prove graphically that f(x)=cos(x) has relative maxima at all even multiples of pi and relative minima at all odd multiples of pi Definition of an extrema of a function on an interval 2 days Lesson for Goal #2: To determine the critical numbers on a closed interval Examples: 1. Find the extrema of f(x)= x2+2x-4 at interval [-1,1] 2. Find the extrema of f(x)= x2-2-cos(x) at interval [-1,3] b. Identify maxima, minima, inflection points, and intervals where the function is increasing and decreasing. c. Use differentiation and algebraic manipulations to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts. CMS IF-4f Use formulas to find derivatives of inverse trigonometric functions, exponential functions and logarithmic functions. IF-5 Apply the rules of differentiation to functions. c. Demonstrate an understanding of and apply Rolle’s Theorem, the Mean Value Theorem. & free response questions) Summative Assessment: Basic: Page 169#4,8,13 and 15 Average: Page 169 # 5,6,8,11,17 and 18 Advanced: Page 169 #8,10,12, 18, 20 Kuta Worksheet Easy, Medium, or Difficult To locate the absolute extrema using critical numbers Examples: 1. Determine the Quizzes Tests Quarterly Assessments Projects Formative Assessments: http://www.spar knotes.com/math /calcab/applicati onsofthederivativ e/section1.rhtml Demonstration Class discussion Homework Home Work Assignments Larson Resource worksheet 3.1 Teacher made Kuta worksheet for the section Quizzes Larson 8e Test Bank 3.1 Lesson for Goal #3: The definition of relative extrema of a function on an open interval https://www.kutas oftware.com/extre ma Thomas: Page 228 #6-18 even Anton: Pgs 277 #4-16 even absolute extrema for the following function and interval: g(t)=2t3+ 3t2-12t+4 4 days Kuta Worksheet Easy, Medium, or Difficult Extrema on a closed interval 2. Determine the maximum and minimum of the function on the interval . First start by finding the roots of the function derivative: Basic: Page 169#19-31 odd, 40 Average: Page 169 #25-39 odd, 42 Advanced: Page 169 #14, 26, 34, 35, 39, 47, 48 Real Life Problems: Pg 171 # 59. 65, 69 Thomas: Pgs 229 #30-40 even Now evaluate the function at all critical points and endpoints to find the extreme values. Kuta Worksheet Easy, Medium, or Difficult https://www.kha nacademy.org/m ath/differentialcalculus/derivati ve_applications/c ritical_points_gra phing/v/testingcritical-pointsfor-local-extrema Rolle’s Theorem and the Mean Value Theorem To determine if Rolle’s Theorem applies to a function using a created graph 3 days Examples: 1. Determine whether Rolle’s Theorem’s hypotheses are satisfied &, if so, find a number c for which f’(c) = 0. 2. f(x) = x³ - 2x² - x + 2 on [-1,2] Basic: Page 176 #4-18 odd Average: Page 176 # 6-30 odd Advanced: Page 176 #10-32 even Thomas: Page 236 #3-30 (multiples of 3) Kuta Worksheet Easy, Medium, or Difficult https://chaffeem ath.wikispaces.co m/file/view/Outl ine+3.2+The+Mea n+Value+Theore m.pdf http://www.cliffs notes.com/math/ calculus/calculus /applications-ofthederivative/meanvalue-theorem 3. f(x) = sin x on [0,π] Instantaneous rate of change 2 days Mean Value Theorem The Mean Value Theorem: If f) is continuous on and differentiable on (a,b ), then there is at least one point in (a,b ) at which: Example: 1. Verify the conclusion of the Mean Value Theorem Basic: Pg 177 #37, 40, 42, 44, 52 Average: Pgs 177 #40-50 even, 55 Advanced: Pgs 177 #46-50 even, 58, 59 Anton: Pg 324 #6-14 even Video https://www.you tube.com/watch? v=fI6w2kL295Y https://www.kha nacademy.org/m ath/differentialcalculus/takingderivatives/secan t-line-slope- Home Work Assignments Larson Resource worksheet 3.2 Teacher made Kuta worksheet for the section Resource Quiz A,B,or D version for f(x)= x 2−3 x−2 on [−2,3]. Real Life Problem: Pgs 237 (Thomas) #51, 56 Kuta Worksheet Easy, Medium, or Difficult 2. A balloon is inflated by an electric pump. Determine the rate of change of volume with respect to radius when the radius measures exactly 6 cm. tangent/v/appro ximatinginstantaneousrate-of-changeword-problem-1 Problems http://tutorial.m ath.lamar.edu/Cl asses/CalcI/Tang ents_Rates.aspx 3. A football is punted into the air. Model the football’s height using the polynomial function f (t) = -4.9t2+ 16t + 1, where f (t) represents the height in meters at t seconds. Determine the instantaneous rate of change of height at 1 s, 2 s, and 3 s. 2 days Increasing and Decreasing Functions and the First Derivative Test IF- 3Demonstrate Home Work Assignments Larson Resource worksheet 3.3 Teacher made Kuta worksheet for the Definiton of Inc/Dec function and Theorem3.5 Examples: 1. Calculate the Intervals of Increase and Decrease for the Following Function 2. For f(x) = sin x + cos x on [0,2π], determine all intervals where f is increasing or decreasing. The First Derivative Test 4 days Theorem 3.6 explaining the First derivative Test. Explain the relative minimum and maximum Examples: knowledge of differentiation using algebraic functions. a. Use differentiation and algebraic manipulations to sketch, by hand, graphs of functions. b. Identify maxima, minima, inflection points, and intervals where the function is increasing and decreasing. c. Use differentiation and algebraic manipulations to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts. IF-4f Use formulas to find derivatives of inverse trigonometric functions, exponential functions and logarithmic functions. IF-5 Apply the rules of differentiation to functions. c. Demonstrate an understanding of and apply Rolle’s Theorem, the Mean Value Theorem. Basic: Pgs 186- 187 #4-20 even Average: Pgs 186- 187 10-32 even Advanced: Pgs 186- 187 #14-40 even section Resource Quiz A,B,or D version https://www.kutas oftware.com/incre asing/decreasing Anton: Pgs 276 #9-21 odd Thomas: Pgs 228 #24, 28, 31 Kuta Worksheet Easy, Medium, or Difficult Video https://www.you tube.com/watch? v=vQYQxpHLfoE Basic: Pg 186 #4-20 even, 43, 45, 80 Average: Pg 186 #24-46 even, Writing Journal Page 177#56 Page 195#48 1. Find the local maximum and minimum values of the function using the first derivative test f(x) = x4 – 2x2 + 3 82, 83, 86 Advanced: Pg 186 #31-51 odd, 60, 74, 80, 83 Kuta Worksheet (Easy, Medium, Difficult) 21. Determine intervals on which a function is concave upward or concave downward Thomas: Pg 242 #39, 50, 62, 64 Anton: Pg 287 #4, 11, 35, 60 .Find the relative extrema for the function . Using the First Derivative Test Find f '(x). Work sheets http://archives.m ath.utk.edu/visua l.calculus/3/grap hing.5/index.html https://www.kutas oftware.com/First video http://www.online mathlearning.com/ derivativetest.html Find all critical numbers of f. critical numbers are –1, 0 and 1. Step 3: Determine intervals. Concavity The intervals are(–∞, –1), (– 1, 0), (0, 1), and (1,∞). 4 days Theorem 3.7 IF- 3Demonstrate knowledge of differentiation using algebraic functions. a. Use differentiation http://www.educ ation.com/studyhelp/article/deri vative-testderivative-testrelative/ Steps for testing concavity The First derivative of a function gives the slope. When the slope continually increases, the function is concave upward. When the slope continually decreases, the function is concave downward. With the second derivative. When the second derivative is positive, the function is concave upward. When the second derivative is negative, the function is concave downward. Example:1. Find the concavity of f(x) = 5x3 + 2x2 − 3x and algebraic manipulations to sketch, by hand, graphs of functions. b. Identify maxima, minima, inflection points, and intervals where the function is increasing and decreasing. Basic: Pg 195 #1-15 odd, 27, 58 Average: Pg 195 #12-34 even, 57, 64 Advanced: Pg 196 #24-36 even, 63, 64, 68 Real Life Problem: Pg 196 #68, 69 Kuta Worksheet (Easy, Medium, Difficult) Video https://www.kha nacademy.org/m ath/differentialcalculus/derivati ve_applications/c oncavityinflectionpoints/v/concavit y--concaveupwards-andconcavedownwardsintervals Home Work Assignments Larson Resource worksheet 3.4 Teacher made Kuta worksheet for the section Resource Quiz A,B,or D version http://www.yout ube.com/watch?v =dfyRWMEFSEU Thomas: Pg 242 #57, 63, 75 Finney: Pg 215 #10-28 odd The derivative is f'(x) = 15x2 + 4x − 3 The second derivative is f''(x) = 30x + 4 And 30x + 4 is negative up to x = −4/30 = −2/15, and positive from there onwards. So: f(x) is concave Notes http://www.sosm ath.com/calculus /diff/der15/der1 5.html Hands on activity/Project http://www.math sisfun.com/calcul us/concave-updownconvex.html Graphical,Numerical,Analy tical Analysis of maximum Volume of a box Page 223 of Larson Text Book Points of inflection downward up to x = −2/15 f(x) is concave upward from x = −2/15 on Determine where the function is concave up and concave down: Group Activity Paper Folding Page 230 of Finney #48 GO FISH FOR DERIVATIVES http://www.fcps.edu/Lak eBraddockSS/high_school /pdfs/hands_document.pd f Theorem 3.8 Inflection points are where the function changes concavity Example 2. 2days The Second Derivative Test Determine the points of inflection for .y = x3− 3x2 + 3x − 1. Since there is a change of concavity at x = 1, and so there is a point of inflection at x = 1. Second derivative test for extrema Theorem 3.9 https://www.kutas oftware.com/point of inflection Teacher Made Kuta Worksheet (Easy, Medium, Difficult) Example 1: Find any local extrema of f(x) = x 4 − 8 x 2 using the Second Derivative Test. f′(x) = 0 at x = −2, 0, and 2. Because f″(x) = 12 x 2 −16, you find that f″(−2) = 32 > 0, and f has a local minimum at (−2,−16); f″(2) = 32 > 0, and f has local maximum at (0,0); and f″(2) = 32 > 0, and f has a local minimum (2,−16). Example 2: Find any local extrema of f(x) = sin x + cos x on [0,2π] using the Second Derivative Test. f′(x) = 0 at x = π/4 and 5π/4. Because f″(x) = −sin x −cos x, you find that as a local maximum at Basic: Pg 195 #27-39 odd Average: Pg 195 #28-32 even https://www.kha nacademy.org/m ath/differentialcalculus/derivati ve_applications/c oncavityinflectionpoints/v/inflectio n-points Advanced: Pg 195 #36-48 even Problems . Also, local minimum at and f h video . and f has a . http://www.cliffs notes.com/math/ calculus/calculus /applications-ofthederivative/secon d-derivative-testfor-local-extrema video http://www.yout ube.com/watch?v =QtXCIxB6kW8 TIMSS Example: 1. 2 days Chapter Review Chapter Test 4) The value of Page 242 of Larson Resource Test b and D forms A) 0 B) Chapter review End of unit test C) D) TIMSS/SAT 2. ) Which of the following graphs has these features:? f'(0) >0, f'(1) <0 and f''(x) is always negative. Sample problems from http://www.edinfor matics.com/timss VII. SAT practice problem How many positive integers less than or equal to 100 are multiples of 3 or multiples of 5 or multiples of both 3 and 5? A) 41 B) 47 C)50 D)53 SAT practice-Student produced response question Page 359-361 Recap concepts Page 358 VIII. http://ww w.edinfor matics.co m/timss/p op3/mpop 3.htm?sub mit32=Gr. +12+Adv.+ Math+Test http://ww w.majorte sts.com/sa t/problem -solvingtest02 www.collegeboard. com SAT Practice test 3(page 513) section 2,5 and 8 I. http://www.majo rtests.com/sat/pr oblem-solvingtest03 E)59 • www.takesat.com SAT 2. There are n students in a biology class, and only 6 of them are seniors. If 7 juniors are added to the class, how many students in the class will not be seniors? IX. A)n-2 B)n-3 www.kha nacadem y.org/test -prep/sat C)n-1 D)n+2 E)n+1 .5.E. Translate words into a mathematical expression or equation. Other Resource Texts Gruber’s Complete Preparation for the New SAT The Princeton Review’s Cracking the New SAT, 2007 Barron’s SAT 2400: Aiming for the Perfect Score Maximum SAT Kaplan SAT 2400, 2006 Edition Up Your Score: The Underground Guide to the SAT, 2007-2008 INSTRUCTIONAL FOCUS OF UNIT This unit focuses on the following topics related rates, extrema on an interval, intermediate value related rates, extrema on an interval, intermediate value summarizing all skills from calculus and previous math courses in curve sketching, and optimization problems.The increasing ,decreasing functional concept can be applied to any data that can be represented as a graph. Concavity is also well explained on graphs with bends or smooth turns Academic Vocabularies by Dr. Marzano Maximum minimum point of inflection, tangent line optimization Critical numbers Absolute Extrema Relative Extrema Critical Point Increasing/decreasing functions Anti-derivative Point of Inflection Marzano’s Six Steps for Teaching Academic Vocabularies 13. YOU provide a description, explanation or example. (Story, sketch, power point) 14. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 15. Ask students to construct a picture, graphic or symbol for each word. 16. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 17. Ask students to discuss vocabulary words with one another (Collaborate) 18. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.) Example 1. Step 1 The graph of f ', the derivative of a function f, is shown in Figure 7.2-19. Find the points of inflection of f and determine where the function f is concave upward and where it is concave downward on [–3, 5]. Example 2—Step 3 Use the First derivative Test to determine the extrema from the table PARCC FRAMEWORK/ASSESSMENT http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test http://www.majortests.com/sat/problem-solving-test01 www.khanacademy.org/test-prep/sat 21ST CENTURY SKILLS (4Cs & CTE Standards) st 21 Century Skills: Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation 9.4.D Business, Management & Administration Career Cluster 9.4.E Education & Training Career Cluster 9.4.F Finance Career Cluster 9.4.N Marketing Career Cluster 9.4.O Science, Technology, Engineering & Mathematics Career Cluster 9.4.P Transportation, Distribution & Logistics Career Cluster 9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities. 9.4.12.D.4, 9.4.12.E.4 Solve mathematical problems and use the information to make business decisions and enhance business management duties. 9.4.12.F.4 Solve mathematical problems to obtain information for decision-making in financial settings. 9.4.12.N.4 Solve mathematical problems to obtain information for marketing decision-making. 9.4.12.O.15 Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to fulfill the specific communication needs of that audience. 9.4.H(5) Biotechnology Research and Development 9.4.12.H.(5).2 Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective biotechnology research and development. Using data from the UN World Population Prospects, graph the population data for three different countries from the year 1950-2010. Then, describe how the population has changed over the past 60 years and will change for the next 90 using the Calculus concepts of increasing/decreasing, concave up/concave down, relative extrema and inflection points. If inclined, try to connect some of the trends that you see to history. 9.4.O(1) Engineering and Technology 9.4.12.O.(1).1 Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems. 9.4.12.O.(1).7 Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction). 9.4.O(2) Science and Mathematics 9.4.12.O.(2).1 Develop an understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world. 9.4.12.O.(2).2 Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems. 9.4.12.O.(2).3 Assess the impact that science and mathematics have on society when used to develop projects or products. 9.4.12.O.(2).4 Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and mathematics impact problem-solving in modern society. 9.4.12.O.(2).6 Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field. Highway Design In order to build a highway, it is necessary to fill a section of a valley where the grades (slopes) of the sides are 9% and 6%. The top of the filled region will have the shape of a parabolic arc that is tangent to the two slopes at the points and The horizontal distance between the points and is 1000 feet. (a) Find a quadratic function Y = a x2 + bx + c, - 500≤ x ≤ 500,that describes the top of the filled region. (b) Construct a table giving the depths d of the fill to X = -500,-400,-300,-200, -100, 0, 100, 200, 300, 400, and 500. (c) What will be the lowest point on the completed highway? Will it be directly over the point where the two hillsides come together? MODIFICATIONS/ACCOMMODATIONS Students can create flashcards of the derivative graphs of the function graphs provided by the teacher. Students can then get into groups where they can match function graphs to their derivative graphs. APPENDIX (Teacher resource extensions) 5. E-Text, Interactive Digital Resources, Teacher Resources Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true Mathematical Practices 17. Make sense of problems and persevere in solving them. 18. Reason abstractly and quantitatively. 19. Construct viable arguments and critique the reasoning of others. 20. Model with mathematics. 21. Use appropriate tools strategically. 22. Attend to precision. 23. Look for and make use of structure. 24. Look for and express regularity in repeated reasoning. 6. Notes to teacher (not to be included in your final draft): 4 Cs Creativity: projects Critical Thinking: Math Journal Collaboration: Teams/Groups/Stations Communication – Powerpoints/Presentations Three Part Objective Behavior Condition Demonstration of Learning (DOL) UNIT 9 Integration Total Number of Days: 26 days Grade/Course: 12/Calculus ESENTIAL QUESTIONS ENDURING UNDERSTANDINGS How do you use indefinite integral notation for antiderivatives? How do you find a particular solution of a differential equation? How do you use sigma notation to write and evaluate a sum? How do you find the area of a plane region using limits? How do you use Riemann Sums to approximate area? How do you evaluate a definite integral using limits? How do you evaluate a definite integral using properties of definite integrals? How do you evaluate a definite integral using the Fundamental Theorem of Calculus? How do you use the Mean Value Theorem for Integrals? How do you find the average value of a function over a closed interval? How do you use the Second Fundamental Theorem of Calculus? How are integrals used to measure changing quantities? Integration is a process that is closely related to differentiation, which will be explored by learning new methods and rules for solving definite and indefinite integrals, including the Fundamental Theorem of Calculus, and then applying these rules. Integration is applied to calculate the Escape velocity, Tree and population growth etc. Mean value Theorem is applied in medicine for treatment of tumors for approximating the area where as average value is applied in Blood flow and Respiratory cycle etc. PACING CONTENT SKILLS STANDARDS (CCCS/MP) RESOURCES Larson Pearson OTHER (e.g., tech) Anti-derivatives and indefinite integration Example 1. 1. Determine the general solution to the differential equation using the notation of AntDerivative 3 days Write the general solution of a differential equation using the variable and constant of Integration = IF-6 Students will apply the rules of integration to functions. a. Apply the definition of the integral to model problems in physics, economics, etc., obtaining results in terms of integrals. Basic: Pg 255 #1-13 odd, 21, 28, 30 Kuta easy worksheet Average: Pg 255 #12-28 even, 36, 38 Kuta mediumworkshee t Advanced: Pg 255 #26-42 even, 68, 70 Kuta difficultworkshee t Thomas: Pg 285-286 #32, 42, 50, 66, 70 Anton: Pg 363 #21, 29, 36, 39 Work sheet www.kuta/different ialequation LEARNING ACTIVITIES/ASSE SSMENTS Using graphing calculators, DI according to student readiness Use data and situations relevant to student interests Whole group and small group cooperative learning Trig function derivative graphic organizer Unit test (multiple choice & free response questions) Summative Assessment: Quizzes Tests 3 days Indefinite integral notation for anti derivative Problems 2. The marginal cost function for producing x units is C = 23+16x - 3x and the total cost for producing 1 unit is Rs.40. Explain the method to determine the total cost function and the average cost function. http://www.textboo ksonline.tn.nic.in/b ooks/12/std12-bmem-2.pdf http://tutorial.math .lamar.edu/Classes/ DE/UndeterminedC oefficients.aspx A-CED. For Goals 2 and 3: Notes http://www.sosmat h.com/tables/diffeq /diffeq.html A-REI F--‐BF.5. The antiderivative of a function ƒ(x) if for all x in the domain of ƒ, F'(x) = ƒ(x) ∫ƒ(x) dx = F(x) + C, where C is a constant. Refer to page 250 for basic integration rules Basic: Pg 256-257 #50, 59, 67 Kuta easy worksheet video https://www.khana cademy.org/math/d ifferentialequations Average: Pg 256-257 #60, 63, 70 Kuta medium worksheet Advanced: Pg 256-257 #62, 70, 82, 84 Kuta difficult worksheet Quarterly Assessments Projects Formative Assessments: Demonstration Class discussion Homework Journals: 4.1 Page 257 #65 4.2 Page 269#73 4.3 Page280#53 4.4 Page292#54 4.5 Page 306#109 4.6 Page 314#22 Hands on activities /projects https://www.youtu http://people.wallawa lla.edu/~ken.wiggins/ examplesactivities/co ntents.html Examples: Integrate be.com/watch?v=lx x_i0zCxrU 1. Thomas: Pg 288 #119, 89, 90, 92 2. 3days Basic integration rules to find antiderivatives of indefinite integrals Identify the Initial condition Problems http://www.mathmagic.com/calculus /integral.htm Anton: Pg 364 #41, 42, 55 Initial condition and particular solutions y=F(x) is called an initial condition Quiz Test Bank 4.1 Teacher made assessment using Kuta software Examples: 1. A ball is thrown upward with an initial velocity of 64 ft/sec from an initial height of 80 feet. Worksheet www.kuta/basicint egration A. Illustrate a method to find the position function, giving the height, s, as a function of the time, t B. When does the ball hit the ground. 2. F’’(x)=sin(x); F’(0)=1; F(0)=6; Solve for the differential equation. Home Work Larson Resource Book 4.1-first two pages IF-6 Students will apply the rules of integration to functions. a. Apply the definition of the integral to model problems in physics, economics, etc., obtaining results in terms of integrals. Home Work Larson Resource Book 4.1 pages 3 and 4 Practice problems from Paul’s online notes Area Apply sigma notation to write and evaluate a sum 3days Sigma Notation Theorem 4.2- Summation formula The starting point for the summation or the lower limit of the summation is 1 The stopping point for the summation or the upper limit of summation 5 Example 1: For Goals 1-3: You tube Basic: Pg 267-269 #1-11 odd, 21, 26, 27, 31, 47, 49, 53, 57 Kuta easy worksheet Average: Pg 267-269 #1222 even, 26, 29, 32, 50, 54, 63 Kuta mediumworkshee t https://www.khana cademy.org/math/p recalculus/seq_indu ction/geometricsequenceseries/v/sigmanotation-sum Advanced: Pg 267-269 #20, 26, 29, 30, 51-56, 61, 66, 75 Kuta difficult Notes http://tutorial.math .lamar.edu/Classes/ CalcI/SummationNo tation.aspx Home Work Larson Resource Book 4.2 Resource Quiz 4.2 Teacher made assessment using Kuta software Evaluate worksheet Example 2. Evaluate Approximate the area of a plane region Example 1. Compare the upper and lower sums to approximate the area of the region defined by the function Y= on the interval [0.2]. Use four subintervals with equal ,2 width. (Hint: Draw a picture) There are four subintervals n = 4 so the width of each subinterval is equal to 2 = Example 2. Estimate the area between and the x-axis using n=5 subintervals IF-6 Students will apply the rules of integration to functions On the calculator, interpret a summation as follows: You Tube http://www.youtub e.com/watch?v=VT WRy1VQbPI Extra Practice http://tutorial.math .lamar.edu/classes/ calcI/areaproblem. aspx 4 days Area of a plane region using limits When finding the area under a curve for a region, it is often easiest to approximate area using a summation series. This approximation is a summation of areas of rectangles. The rectangles can be either left-handed or right-handed and, depending on the concavity, will either overestimate or underestimate the true area. Example 1: Use the limit definition of Area to estimate the area of f(x) = x3 between x=0 and x=1 Example 2 Apply the limit process to find the area of the region between the graph of the http://college.cenga ge.com/mathematic s/larson/calculus_a nalytic/7e/students /downloads/summ aries/ex4_2_4.pdf http://sites.csn.edu /gcohen/181/21_ar ea_riemann.pdf function and the x-axis over the given interval. y=16-x2 [-4,4] 3days Riemann Sum and Definite Integral Basic: Pg 278-279 #3-11 odd, 18, 23, 29,41 Kuta easy worksheet Note that the Riemann sum when each xi is the righthand endpoint of the subinterval [ai-1, ai] is when each xi is the lefthand endpoint of the subinterval [ai-1, ai] is and when each xi is the lefthand midpoint of the subinterval [ai-1, ai] is Definition of Riemann Sum Example: 1. WORD PROBLEM A rectangular canal, 5m wide and 100m long has an uneven bottom. Depth measurements are taken Average: Pg 278-279 #1329 odd, 35, 47 IF-6 Students will apply the rules of integration to functions. a. Apply the definition of the integral to model problems in physics, economics, etc., obtaining results in terms of integrals. b. Demonstrate knowledge of the Fundamental Theorem of Calculus, and use it to interpret integrals as anti-derivatives. c. Use definite Kuta mediumworkshee t Advanced: Pg 278-279 #2432 even, 36, 40, 47 Kuta difficultworkshee t https://www.khana cademy.org/math/i ntegralcalculus/indefinitedefiniteintegrals/definite_i ntegrals/v/riemann -sums-and-integrals http://archives.mat h.utk.edu/visual.cal culus/4/riemann_s ums.4/ Home work Resource work sheet 4.3 Quiz 4.3 from Larson Test Bank Teacher made Assessments using Kuta software MID CHAPTER ASSESSMENT http://www.lakelan dschools.org/webpa ges/dcox/files/Rie mann%20sum%20 practice.pdf Hands on activities /projects http://people.wallawa lla.edu/~ken.wiggins/ examplesactivities/co ntents.html at every 20m along the length of the canal. Use these depth measurements to construct a Riemann sum using right endpoints to estimate the volume of water Dista Depth in the nce (m) canal. (m) See table 0 2.0 to the 20 1.6 left. 40 1.8 60 2.1 80 2.1 100 1.9 2. Approximate the area under the curve with a Riemann sum, using six sub-intervals and right endpoints. 3: Evaluate the Riemann sum for f( x) = x 2 on [1,3] using the four subintervals of equal length, where x i is the right endpoint in the ith integrals in problems involving area, velocity, acceleration, and the volume of a solid. Section project Demonstrating the Fundamental Theorem Page 294 subinterval (see Figure ) . Examples http://www.sosmat h.com/calculus/inte g/integ02/integ02. html http://tutorial.math .lamar.edu/Classes/ CalcI/DefnofDefinit eIntegral.aspx Limits and properties of definite integrals Theorems 4.5, 4.6, 4.7, 4.8 explaining the properties of integral limits Example: 1 Evaluate the following definite integral Example 2. S IF-6 Students will apply the rules of integration to functions. a. Apply the definition of the integral to model problems in physics, economics, etc., obtaining results in terms of integrals. b. Basic: Pg 280-281 #49, 55, 58 Kuta easy worksheet Average: Pg 280-281 #56, 58, 62 Kuta mediumworkshee t Problems (Note: x must be in radians ) 4days The Fundamental Theorem of Calculus Theorem 4.9 Let f (x) be continuous on [a, b]. If F(x) is any anti derivative of f (x), then, Demonstrate knowledge of the Fundamental Theorem of Calculus, and use it to interpret integrals as antiderivatives. c. Use definite integrals in problems involving area, velocity, acceleration, and the volume of a solid. d. Compute, by hand, the integrals of a wide variety of functions using substitution. e. Apply the rules of integration to functions. Use definite integrals in problems involving Advanced: Pg 280-281 #57, 62, 70 Kuta difficultworkshee t Thomas: Pgs 322 #10-40 multiples of 5 http://www.mathsi sfun.com/calculus/i ntegrationdefinite.html You tube http://www.youtub e.com/watch?v=wyc adSRDID4 Anton: Pgs 394 #12-22 even https://www.khana cademy.org/math/i ntegralcalculus/indefinitedefiniteintegrals/fundamen tal-theorem-ofcalculus/v/fundame ntal-theorem-ofcalculus Basic: Pg 291#6-18, 29 Average: Pg 291 #11-27 odd, 35 Home work http://apcentral.col legeboard.com/apc Larson Resource work sheet 4.4 )dx = F(b) - F(a) Apply FTC to evaluate Fundamental Theorem of Calculus (FTC) Example 1. Example 2. area between two f. Apply the rules of integration to functions. Use advanced techniques to evaluate integrals, including integration by parts, trigonometri c integrals, trigonometri c substitution Advanced: Pg 291 #14-38 even /public/repository/ AP_CurricModCalcul usFundTheorem.pd f Thomas: Pg 322 #29-41 odd, 50 Anton: Pg 363 #5-13 odd Kuta Software Worksheets crated by Teacher You tube https://www.khana cademy.org/math/d ifferentialcalculus/derivative_ applications/mean_ value_theorem/v/m ean-value-theorem Mean Value Theorem(MVT) Theorem 4.10 The Mean Value Theorem is one of the most important theoretical tools in Calculus. It states that Problems http://tutorial.math .lamar.edu/Classes/ CalcI/MeanValueTh eorem.aspx Test Bank Quiz 4.4 Teacher made assessment using Kuta soft ware if f(x) is defined and continuous on the interval [a,b] and differentiable on (a,b), then there is at least one number c in the interval (a,b) (that is a < c < b) such that Basic: Pg 291#40, 44, 46, 48 Average: Pg 291 #40-48 even 2days Mean Value Theorem Example 1. Advanced: Pg 291 #43-51 odd, 62 Find a value of c such that the conclusion of the mean value theorem is satisfied for Real Life Problem: Pg 292 #62, 64 f(x) = -2x 3 + 6x – 2 Example2. Determine a and b for the function: If it satisfies the hypothesis of mean value theorem on the interval [2, 6]. Definition for Average Value: Let f be a function which is continuous on the closed interval [a, b]. The average value of f from x = a Thomas: Pg 333 #10-30 even Anton: Pg 363 #31, 36, 39 http://www.vitutor. com/calculus/lhopi tal/mean_problems. html http://archives.mat h.utk.edu/visual.cal culus/5/average.1/ Kuta Software Worksheets crated by Teacher You tube https://www.khana cademy.org/averag evalue to x = b is the integral Example 1. Assume that in a certain city the temperature (in ◦F) t hours after 9 A.M. is represented by the function T(t) = 50 + 14 sin Find the average temperature in that city during the period from 9 A.M. to 9 P.M. Second Fundamental Theorem(4.11) 2days Average value of a Function Example 1. Calculate Worksheet www.kuta/average value Basic: Pg 293-294 #6773 odd, 79, 83, 87 Average: Pg 293-294 #7080 even, 84, 90 Advanced: Pg 293-294 #7494 even Thomas: Pg 334 #57-65 Anton: Pg 364 #53-57 odd http://www.youtub e.com/watch?v=Cz_ GWNdf_68 www.kuta/secondfu ndamentaltheorem Kuta Software Worksheets crated by Teacher Practice Problems Example 2. Calculate http://wwwmath.mit.edu/~djk/ 18_01/chapter14/e xample04.html Second Fundamental Theorem Lesson Goal #1 Examples Pattern recognition and change of variables involve a U- substitution. http://college.cenga ge.com/mathematic s/larson/calculus_a nalytic/7e/students /downloads/summ aries/ex4_5_1.pdf Theorem 4.12 Example 1: You tube by trying to identify the pattern https://www.khana cademy.org/pattern s [ f (g(x))g'(x) ]dx Let u=g(x)=5x2+1 and du = g’(x) = 10 xdx = ( +1) + C Example2: FOR GOALS 1 and 2: Basic: Pg 304-306 #1626 even,44-54 even, 60, 64, 68, 74, 99 Theorems 4.13 and 4.14 Average: Pg 304-306 #1628 even, 42, 4951 odd, 60, 62, 70, 74, 82, 96, 100 Example: Advanced: Lesson for goal #2: 1. Find 2. 4days Integrate Pg 304-306 #2636 even, 41, 42, 49-61 odd, 70, 71, 106 Real Life Problem: Pg 304-306 #111, 113 Home work Larson Resource work sheet 4.5 Test Bank Quiz 4.5 Teacher made assessment using Kuta soft ware Kuta Software Worksheets crated by Teacher Integration by Substitution Basic: Pg 314 #2, 8, 18, 40, 45 Average: Pg 314 #3, 14, 32, 39, 40, 45, 51 Lesson for Goal #1: Theorem 4.16 and 4.18 Trapezoidal Rule: 1day Use Pattern recognition to find an indefinite integral Simpsons Rule: Advanced: Pg 314 #10, 18, 34, 38, 40, 44, 51, 52 Anton: Pg 567 #39, 40, 41(REAL LIFE WORD PROBLEMS) http://tutorial.math .lamar.edu/Classes/ CalcII/Approximati ngDefIntegrals.aspx Chapter Review 2days Chapter Test https://www.math. ucdavis.edu/~koub a/CalcTwoDIRECTO RY/usubdirectory/ USubstitution.html Page 316-317 Resource test Band C versions You tube https://www.youtu be.com/watch?v=R TX-ik_8i-k Content Category: Numbers Equations and Functions Calculus Probability and Statistics Geometry http://www.mathw ords.com/t/trapezo id_rule.htm Larson Resource work sheet 4.6 TIMSS Example1: Determine all complex number z that satisfy the equation https://www.khana cademy.org/math/i ntegralcalculus/indefinitedefiniteintegrals/riemannsums/v/trapezoidal -approximation-ofarea-under-curve where denotes the conjugate of z. Change of variable for definite integrals A)( -1/3 + 5/3i) B)( -3 -i) C)(1 -i) D)answer not given SAT Example. Amy has to visit towns B and C in any order. The roads connecting these towns with her home are shown on the diagram. How many different Home work X. http://www.edinf ormatics.com/tim ss/pop3/mpop3. htm?submit32=Gr .+12+Adv.+Math+ Test5,6 http://www.erikt hered.com/tutor/ Sample problems from XI. http://www.edinfor matics.com/timss XII. Test Bank Quiz 4.6 Teacher made assessment using Kuta soft ware Chapter Review Chapter Test http://www.edinform atics.com/timss/pop3 /mpop3.htm?submit3 2=Gr.+12+Adv.+Math +Test http://www.majortest s.com/sat/problem- Numerical Integration Approximation of definite integral using Trapezoidal and Simpsons Rule routes can she take starting from A and returning to A, going through both B and C (but not more than once through each) and not travelling any road twice on the same trip? A. . 8 B. 6 C. 10 D. 4 E. 2 TIMSS/SAT A-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 4.4.3 C.2: Represen t all possibilities for a simple counting situation in an organized way and draw conclusions from this representatio n. solving-test02 Other Resource Texts Gruber’s Complete Preparation for the New SAT The Princeton XIII. Review’s Cracking the New SAT, 2007 www.khanacademy. org/test-prep/sat Barron’s SAT 2400: Aiming for the Perfect Score Maximum SAT Kaplan SAT 2400, 2006 Edition Up Your Score: The Underground Guide to the SAT, 20072008 http://www.edinfor matics.com/timss INSTRUCTIONAL FOCUS OF UNIT calculating an anti-derivative, solving initial value problems, finding area under a curve by summing rectangles, using integrals to find the exact area under a curve, total area vs. net area, mean value theorem for integrals, and u-substitution. Academic Vocabularies By Dr. Robert Marzano Anti-derivative Approximating sum Area between Area under a curve Constant of Integration Definite Integral Differential Equation Indefinite Integral Initial Condition: Integrand Upper and Lower limit Variable of Integration The integral sign ∫ Substitution Trapezoidal Rule Simpson Rule Marzano’s Six Steps for Teaching Academic Vocabularies 19. 20. 21. 22. explanation or example. (Story, sketch, power point) explain meaning in their own words. (Journal, community circle, turn to your neighbor) picture, graphic or symbol for each word. to expand their word knowledge. (Add to their notes, use graphic organizer format) YOU provide a description, Ask students to restate or reAsk students to construct a Engage students in activities 23. vocabulary words with one another (Collaborate) 24. with the words. (Bingo with definitions, Pictionary, Charades, etc.) Example 1 -Step 2 Ask students to discuss Have students play games A new piece of industrial equipment will depreciate in value, rapidly at first and then less rapidly as time goes by. Suppose that the rate in dollars per year of depreciation of a new milling machine is given by V’(t) = 500(t − 12) where V (t) is the value of the machine after t years. What is the total loss in value of the machine in the first five years? In the second five years? Example 2. – Step 6 http://www.intmath.com/integration/millionaire-calculus-game.php PARCC FRAMEWORK/ASSESSMENT http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test http://www.majortests.com/sat/problem-solving-test01 www.khanacademy.org/test-prep/sat 21ST CENTURY SKILLS (4Cs & CTE Standards) 21st Century Skills: Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation 9.4.D Business, Management & Administration Career Cluster 9.4.E Education & Training Career Cluster 9.4.F Finance Career Cluster 9.4.N Marketing Career Cluster 9.4.O Science, Technology, Engineering & Mathematics Career Cluster 9.4.P Transportation, Distribution & Logistics Career Cluster 9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities. 9.4.12.D.4, 9.4.12.E.4 Solve mathematical problems and use the information to make business decisions and enhance business management duties. 9.4.12.F.4 Solve mathematical problems to obtain information for decision-making in financial settings. 9.4.12.N.4 Solve mathematical problems to obtain information for marketing decision-making. 9.4.12.O.15 Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to fulfill the specific communication needs of that audience. 9.4.H(5) Biotechnology Research and Development 9.4.12.H.(5).2 Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective biotechnology research and development. Finding a Population Function Since 1990, the rate of increase in the number of divorced adults (in millions) in the United States from 1990 through 2005 can be modeled by D= -4.9t2+ 12t+770 Where t is the year, with corresponding to 1990. The number of divorced adults in 2005 was 22.1 million. (Source: U.S. Census Bureau) a. Find the model for the number of divorced adults in the United States. b. Use the model to predict the number of divorced adults in 2012. 9.4.O(1) Engineering and Technology 9.4.12.O.(1).1 Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems. 9.4.12.O.(1).7 Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction). 9.4.O(2) Science and Mathematics http://www.youtube.com/watch?v=EBfxiKQLnJ4 Example2 9.4.12.O.(2).1 Develop an understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world. 9.4.12.O.(2).2 Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems. 9.4.12.O.(2).3 Assess the impact that science and mathematics have on society when used to develop projects or products. 9.4.12.O.(2).4 Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and mathematics impact problem-solving in modern society. 9.4.12.O.(2).6 Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field. Volume of a pyramid using Integration Find the volume of a pyramid with a square base that is 20 meters tall and 20 meters on a side at the base. As with most of our applications of integration, we begin by asking how we might approximate the volume. Since we can easily compute the volume of a rectangular prism (that is, a “box”), we will use some boxes to approximate the volume of the pyramid, as shown in figure 9.3.1: on the left is a cross-sectional view, on the right is a 3D view of part of the pyramid with some of the boxes used to approximate the volume MODIFICATIONS/ACCOMMODATIONS Students can create flashcards of the basic rule of integrations provided by the teacher. Students can then get into groups where they can match the solution to their questions APPENDIX (Teacher resource extensions) 7. E-Text, Interactive Digital Resources, Teacher Resources Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true Mathematical Practices 25. Make sense of problems and persevere in solving them. 26. Reason abstractly and quantitatively. 27. Construct viable arguments and critique the reasoning of others. 28. Model with mathematics. 29. Use appropriate tools strategically. 30. Attend to precision. 31. Look for and make use of structure. 32. Look for and express regularity in repeated reasoning. 8. Notes to teacher (not to be included in your final draft): 4 Cs Creativity: projects Critical Thinking: Math Journal Collaboration: Teams/Groups/Stations Communication – Powerpoints/Presentations Three Part Objective Behavior Condition Demonstration of Learning (DOL) UNIT 10 SEQUENCE and SERIES ``` Total Number of Days: 10 days Grade/Course: 12/CALCULUS ESENTIAL QUESTIONS How are sequences and series used to model many mathematical ideas and realistic situations? How are sequences & series related? What is an arithmetic sequence/series? What is a geometric sequence/series? 9. How is the Principle of Mathematical Induction used to prove statements are true for all natural numbers? How do we develop equations to describe sequences and series, then use them to solve other problems? ENDURING UNDERSTANDINGS A finite sequence/series contains a finite number of terms. An infinite sequence/series contains an infinite number of terms. Arithmetic and geometric sequences and series have a common difference or a common ratio, respectively. Sequence and series formulas are used to find a specific term or a total up to a specific term. The summation symbol (sigma) can be used to quickly write sequence and series formulas. Sequences and series can be used as prediction tools. Use the binomial theorem and Pascal’s triangle to generate binomial coefficients for certain types of sequences & series Use the compound interest formula to model finance problems PACI NG CONTENT SKILLS STANDARDS (CCCS/MP) RESOURCES Larson Pearson OTHER (e.g., tech) Vocabulary The nth term of an Arithmetic sequence To find any term of an arithmetic sequence: http://www.regent sprep.org/Regents/ math/algtrig/ATP2 /ArithSeq.htm where a1 is the first term of the sequence, d is the common difference, n is the number of the term to find. Problems http://tutorial.mat h.lamar.edu/Classe s/CalcII/SeriesIntro .aspx To find the sum of a certain number of terms of an arithmetic sequence: 2 Block periods where Sn is the sum of n terms (nth partial sum), a1 is the first term, an is the nth term. Example 1. Evaluate the 25th term of the sequence -7, - LEARNING ACTIVITIES/AS SESSMENTS You tube https://www.khana cademy.org/math/ precalculus/seq_in duction Using graphing calculators, DI according to student readiness Use data and situations relevant to student interests Whole group and small group cooperative learning Trig function derivative graphic organizer Unit test (multiple choice & free response questions) F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use Class work Page 705-707 Summative Assessment: Quizzes Tests Quarterly The sum of an Arithmetic Sequence 4, -1, 2, ... n = 25; a1 = -7, d = 3 them to model situations, and translate between the two forms. Basic #2,4,8,2028even,45,48 Kuta basic worksheet Average Example 2. A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern. If the theater has 20 rows of seats, how many seats are in the theater? We wish to find "the sum" of all of the seats. n = 20, a1 = 60, d = 8 and we need a20 for the sum. Now, use the sum formula: #1638even,53,54,78,8 0 Kuta medium worksheet Assessme nts Projects Formative Assessments: Demonstr ation Class discussion Homewor k Advanced Kuta difficult work sheet # 8,20,35,50-64 even.79,81,86,88 Home work: Resource 9,2 worksheet Teacher made Quiz and Test using Kuta software/Larso n Test Bank Teacher made Mid Chapter Test Hands on activities /projects There are 2720 seats. http://people.wall awalla.edu/~ken.w iggins/examplesact ivities/contents.ht ml Summation notation to write sum Summation notation is used to denote a sum of terms. Usually, the terms follow a pattern or formula. Example 1: Evaluate: F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. https://www.khana cademy.org/math/i ntegralcalculus/sequences _series_approx_calc /calculusseries/v/writingseries-sigmanotation Writing Journals Page 704 Writing about Mathematics Example 2. Page 713 Writing about Mathematics Use sigma notation to represent -3 + 6 - 9 + 12 - 15 + ... A.SSE.4 for 50 terms term position 1 2 3 4 term -3 6 -9 12 When will a geometric series converge and when will it diverge? 2 block periods Geometric Sequence To find any term of a geometric sequence: Class work Basic Kuta Easy work sheet Page 714- 717 #26-34 even,49,64,83 Practice problems http://www.regent sprep.org/Regents/ math/algtrig/ATP2 /GeoSeq.htm Average The sum of a Geometric series where a1 is the first term of the sequence, r is the common ratio, n is the number of the term to find. Example 1.: A ball is dropped from a height of 8 feet. The ball bounces to 80% of its previous height with each bounce. How high (to the nearest tenth of a foot) does the ball bounce on the fifth bounce? Example2: Lesson for goal#2 To find the sum of a certain number of terms of a geometric sequence: where Sn is the sum of n terms (nth partial sum), a1 is the first term, r is the common ration. Example 1. Evaluate using a Kuta medium worksheet # 34.39,50-64 even,84,86 Advanced Kuta difficult worksheet #38,40,50-68 even,78,85,88,90,98 You tube https://www.khana cademy.org/math/ precalculus/seq_in duction/precalcgeometricsequences/e/geom etric_sequences_1 Notes and extra practice problems http://www.purple math.com/modules /series5.htm formula: Example2: Evaluate S10 for 250, 100, 40, 16 ... Mathematical Induction is a special way of Mathematical Induction proving things. It has only 2 steps: Step 1. Show it is You tube true for the first one https://www.khanac ademy.org/math/pre calculus/seq_inducti on/proof_by_inducti on/v/proof-byinduction Step 2. Show that if any one is true then the next one is true Example 1. Prove that the sum of the first n natural numbers is given by Page 726-727 this formula: Basic # 2-16 even,32,36,54,58 Average 10-28even,50,52,63 Advanced examples http://www.themath page.com/aprecalc/ mathematicalinduction.htm 1+2+3+. . . +n= #20-32 even.58,52,67 n(n + 1) 2 Example 2. 1 block Sum of powers of Integers. Prove by mathematical induction: http://www.math.rut gers.edu/~erowland/ sumsofpowers.html 13 + 23 +33+………n3 = n2(n + 1)2 4 The formulas for sums of powers of the first n positive integers can be proved using the Principle of Mathematical Induction and, in Riemann sums during an introduction to definite integration. http://www.maa.org/ publications/periodi cals/convergence/su ms-of-powers-ofpositive-integersintroduction = n(n + 1) . 2 Examples http://www.mathcen tre.ac.uk/resources/u ploaded/mc-tysigma-2009-1.pdf Example 1: Evaluate using the sum of powers You tube http://www.youtube. com/watch?v=iR9B Tx4tPN8 = TIMSS/SAT + Example 2: -REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Evaluate Chapter Test http://www.educatio n.gov.za/LinkClick.a spx?fileticket=gWrc oxvqUMI%3D&tabi d=666&mid=1861 Example 3. Evaluate TIMSS Practice Problems Other SAT Content Category: Numbers Equations and Functions Calculus Probability and Statistics Geometry Resource Texts Gruber’s Complete Preparation for the http://www.edinform New SAT atics.com/timss/pop3 The Princeton /mpop3.htm?submit3 2=Gr.+12+Adv.+Math Review’s Cracking the New SAT, 2007 +Test Barron’s SAT 2400: Aiming for the Perfect Score Maximum SAT Kaplan SAT 2400, 2006 Edition Up Your Score: The Underground SAT practice test#6 from SAT study guide Guide to the SAT, 2007-2008 Page689-(second 100 SAT Math Tips Edition) and How to Master Section2,4,and 8) Them Now! Charles Gulotta TIMSS Example1: Example :1 Describe a method for estimating the perimeter of figure C. Example 2: Which of the following graphs has these features:? f'(0) >0, f'(1) <0 and f''(x) is always negative. 4.5.A.1 Practice algebra skills and problem solving strategies 4.3.B.4 Solve various other algebraic equations 8 Practice SAT Tests http://www.edinf ormatics.com/tim ss/pop3/mpop3.h tm?submit32=Gr. +12+Adv.+Math+ Test (two variables, simultaneous equations, quadratic equations 4.2.C.1 8 SAT Practice Tests Review concepts of geometry and apply strategies SAT http://www.major tests.com/sat/prob lem-solving-test 4.4.B.2 SAT Example:1. 3x + y = 19 , and x + 3y = 1. Find the value of 2x + 2y? Understand geometric notation, lines and angles, geometric probability. Practice problems A.20 B. 18 C.11 D.10 E.5 Example :2. What is the slope of the line l? http://www.majortes ts.com/sat/problemsolving-test08 A.-3 B. C.0 D.10 E. INSTRUCTIONAL FOCUS OF UNIT Students will connect their prior study of algebraic patterns with the concepts in this unit and expands their understandings and skills related to sequences and series. Students explore the basics Characteristics of arithmetic and geometric sequences and series, connecting these ideas to functions whose domain are a subset of the integers. Academic Vocabularies By Dr. Robert Marzano sequence, series, finite,infinite terms, factorial, recursive, sigma notation and summation, partial sums, common difference & common ratio, compound interest, arithmetic & geometric sequences & series, Binomial theorem, binomial coefficients, Pascal’s triangle, Marzano’s Six Steps for Teaching Academic Vocabularies 25. 26. 27. 28. explanation or example. (Story, sketch, power point) explain meaning in their own words. (Journal, community circle, turn to your neighbor) picture, graphic or symbol for each word. YOU provide a description, Ask students to restate or reAsk students to construct a Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 29. 30. vocabulary words with one another (Collaborate) with the words. (Bingo with definitions, Pictionary, Charades, etc.) Ask students to discuss Have students play games Example 1 Sum of a Geometric Series Step 1 A ball is dropped from a table that is twenty‐four inches high. The ball always rebounds three fourths of the distance fallen. Approximately how far will the ball have traveled when it finally comes to rest? Figure 1. Follow the bouncing ball. Notice that this problem actually involves two infinite geometric series. One series involves the ball falling, while the other series involves the ball rebounding. Falling, Rebounding, Use the formula for an infinite geometric series with –1 < r < 1. The ball will travel approximately 168 inches before it finally comes to rest. Example 2 -Step 5 http://quizlet.com/6257649/sequences-and-series-unit-vocabulary-flash-cards/ PARCC FRAMEWORK/ASSESSMENT http://www.edinformatics.com/timss/pop3/mpop3.htm?submit32=Gr.+12+Adv.+Math+Test http://www.majortests.com/sat/problem-solving-test01 www.khanacademy.org/test-prep/sat 21ST CENTURY SKILLS (4Cs & CTE Standards) 21st Century Skills: Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation 9.4.D Business, Management & Administration Career Cluster 9.4.E Education & Training Career Cluster 9.4.F Finance Career Cluster 9.4.N Marketing Career Cluster 9.4.O Science, Technology, Engineering & Mathematics Career Cluster 9.4.P Transportation, Distribution & Logistics Career Cluster 9.4.12.D.2, 9.4.12.E.2, 9.4.12.F.2, 9.4.12.N.2, 9.4.12.O.2, 9.4.12.P.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities. 9.4.12.D.4, 9.4.12.E.4 Solve mathematical problems and use the information to make business decisions and enhance business management duties. You visit the Grand Canyon and drop a penny off the edge of a cliff. The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic sequence. What is the total distance the object will fall in 6 seconds? Arithmetic sequence: 16, 48, 80, ... The 6th term is 176. Now, we are ready to find the sum: 9.4.12.F.4 Solve mathematical problems to obtain information for decision-making in financial settings. 9.4.12.N.4 Solve mathematical problems to obtain information for marketing decision-making. 9.4.12.O.15 Prepare science, technology, engineering, and mathematics material in oral, written, or visual formats to provide information to an intended audience and to fulfill the specific communication needs of that audience. 9.4.H(5) Biotechnology Research and Development A mine worker discovers an ore sample containing 500 mg of radioactive material. It is discovered that the radioactive material has a half life of 1 day. Find the amount of radioactive material in the sample at the beginning of the 7th day. 500 mg of ore. Half life of one day means that half of the amount remains after 1 day. Begin of day 1 Begin of day 2 Begin of day 3 500 mg 250 mg 125 mg End of day 1 End of day 2 End of day 3 250 mg 125 mg 62.5 mg ... ... Decide to either work with the "beginning" of each day, or the "end" of each day, as each can yield the answer. Only the starting value and number of terms will differ. We will use "beginning": 9.4.12.H.(5).2 Apply biochemistry, cell biology, genetics, mathematics, microbiology, molecular biology, organic chemistry, and statistics concepts to conduct effective biotechnology research and development. 9.4.O(1) Engineering and Technology 9.4.12.O.(1).1 Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems. 9.4.12.O.(1).7 Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction). 9.4.O(2) Science and Mathematics 9.4.12.O.(2).1 Develop an understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world. 9.4.12.O.(2).2 Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems. 9.4.12.O.(2).3 Assess the impact that science and mathematics have on society when used to develop projects or products. 9.4.12.O.(2).4 Use scientific and mathematical problem-solving skills and abilities to develop realistic solutions to assigned projects, and illustrate how science and mathematics impact problem-solving in modern society. 9.4.12.O.(2).6 Demonstrate the knowledge and technical skills needed to obtain and succeed in a chosen scientific and mathematical field. MODIFICATIONS/ACCOMMODATIONS Students can create flashcards of the basic rule of integrations provided by the teacher. Students can then get into groups where they can match the solution to their questions APPENDIX (Teacher resource extensions) 9. E-Text, Interactive Digital Resources, Teacher Resources Login at https://www.Larsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true Mathematical Practices 33. Make sense of problems and persevere in solving them. 34. Reason abstractly and quantitatively. 35. Construct viable arguments and critique the reasoning of others. 36. Model with mathematics. 37. Use appropriate tools strategically. 38. Attend to precision. 39. Look for and make use of structure. 40. Look for and express regularity in repeated reasoning. 10. Notes to teacher (not to be included in your final draft): 4 Cs Creativity: projects Critical Thinking: Math Journal Collaboration: Teams/Groups/Stations Communication – Powerpoints/Presentations Three Part Objective Behavior Condition Demonstration of Learning (DOL)