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UNIT 1 - POLYNOMIALS Total Number of Days: 15 days (A and B days meet alternate days) Grade/Course: 11/Algebra 2 ESSENTIAL QUESTIONS How are the real solutions of a quadratic equation related to the graph of the related quadratic function? What does the degree of a polynomial tell you about its related polynomial function? For a polynomial function, how are factors, zeros and x-intercepts related? For a polynomial function, how are factors and roots related? PACING 1 1 day Q1 CONTENT Unit 1 PreAssessment 1. Complex number system SKILLS Complex number system Quadratic Functions with Complex Solutions Functions Polynomials Find the value of i raised to a power. Example 1: Evaluate i17. ENDURING UNDERSTANDINGS A basis for the complex numbers is a number whose square is -1. Every quadratic equation has complex number solutions. Knowing the zeros of polynomial functions can help you understand the behavior of its graph. The degree of a polynomial equation tells you how many roots the equation has. You can use much of what you know about multiplying and dividing fractions to multiply and divide rational expressions. When solving an equation involving rational expressions multiplying by the common denominator can result in extraneous solutions. RESOURCES LEARNING STANDARDS Pearson ACTIVITIES/ASSESSME Pearson (CCCS/MP) OTHER NTS (e.g., tech) N.CN.1, 2, 7, 9 Rewrite questions A.SSE.1, similar to the Unit 1 test. A.APR.A.1,2,3 15 MC, 5 SR, 5 OE MP 1-8 N.CN.1 Know there is a complex number i such that i2 = −1, and every complex number has the form a + bi with a and b real. Algebra 2 Student Textbook Section 4-8 Basic: Problems 1-7 EXS: 8-46, 48, 50, 56, http://www.suitc aseofdreams.net/ Powers_i.htm#A1 Algebra Lab Algebra 2 Student Textbook Section 4-8 Solve It http://www.regen Page 248 or PowerAlgebra tsprep.org/Regent s/math/algtrig/A TO6/powerlesson. htm (textbook website) Algebra 2 Interactive Digital Path Chapter 4 MP 2, 7 57, 73-89 Average: Problems 1-7 EXS: 9-43 odd, 45-69, 73-89 BASIC Using the Distributive SKILLS Property for problem REVIEW solving PARCC/HSPA PREP Last 1/3 of the period. A –CED.1 Create equations and inequalities in one variable and use them to solve problems. MP 4, 2 Vocabulary Support PowerAlgebra (textbook website) Algebra 2 Other Resources Teacher Resources Advanced: Chapter 4 Problems 1-7 Additional EXS: 9-43, Vocabulary 45-72, 73-89 Support 4-8 Review of the WORKSHEET: http://tullyschools. distributive org/hsteachers/dne property uman/IntegratedAlg Algebra 1 ebra/IntegratedAlge textbook pg braNotes/01Septem 96 Problem ber/008ChapterWo rdProblems/Chapte 3 or PowerAlgebr rWordProblems.pdf a (textbook website) Algebra 1 Interactive Digital Path Chapter 2 Chapter 2-3 View Instruction Problem 3 Chapter 4-8 View Solve it Assessments Quiz – Teacher created Students will create the pattern of powers of i. Questions on finding the value of i raised to a power. Example: Half the sum of six times a number and eighteen is the same as twice the number less twenty. 2. Complex number system Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Example 1: Evaluate (8 + 3i) + (–1 – 5i). 2 days Q 2,3,4 Example 2: Evaluate (8 + 3i) – (–1 – 5i). Example 3: Evaluate (8 + 3i)(–1 – 5i). N.CN.2 Use Properties of operations to add, subtract, and multiply complex numbers MP 2, 7 Algebra 2 Student Textbook Section 4-8 Basic: Problems 1-7 EXS: 8-46, 48, 50, 56, 57, 73-89 Average: Problems 1-7 EXS: 9-43 odd, 45-69, 73-89 Advanced: Problems 1-7 EXS: 9-43, 45-72, 73-89 http://www.regents prep.org/Regents/ math/algtrig/ATO6 /lessonadd.htm Algebra Lab PowerAlgebra (textbook website) Algebra 2 Textbook Interactive Digital Path Chapter 4 Chapter 4-8 View Instruction Problems 3, http://www.regents 3 Alternate, 4 prep.org/Regents/ http://www.regents prep.org/Regents/ math/algtrig/ATO6 /multlesson.htm math/algtrig/ATO6 /practicepageadd.ht m http://www.regents prep.org/Regents/ math/algtrig/ATO6 /multprac.htm Section A pg 34, Student Activity 3, Powers of i http://www.projectmaths.ie/ documents/teachers/tl_comp lex_numbers.pdf Assessment Quiz - Teacher created One of each to add, subtract, and multiply complex numbers. BASIC SKILLS REVIEW PARCC/HSPA PREP Last 1/3 of the period. Factoring Trinomials where a = 1 Example 1: p2 + 13p + 40 Example 2: n2 – 16n + 60 Example 3: x2 – 14x + 48 Example 4: x2 – 2x – 35 A.SSE.A.1a Interpret parts of an expression, such as terms, factors, and coefficients. MP 7 Extra practice Algebra 1 review section 8-5 PowerAlgebra (textbook website) Algebra 2 Interactive Digital Path Chapter 8 Chapter 8-5 View Solve it Instruction Problems 1 to 5 Classwork http://www.regents prep.org/Regents/ math/ALGEBRA/AV 6/Ltri1.htm http://www.youtub e.com/watch?v=aYI UQ-6IJu8 http://www.regents prep.org/Regents/ math/ALGEBRA/AV 6/PracFact2.htm Algebra Lab TI-84 Activity http://mathbits.com/MathBit s/TISection/Algebra1/Factori ng.htm Homework http://www.kutasoftware.co m/FreeWorksheets/Alg1Wor ksheets/Factoring%201.pdf 3. Quadratic Functions with Complex Solutions Solve quadratic equations using the quadratic formula. Example 1: x2 – 2x + 10 = 0 Example 2: x2 – 6x = –9 2 days Q 5,6,7 N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. MP 2, 7 Algebra 2 Textbook TE Lesson 47 pg 240-244 Basic: Problems 1-4 EXS: 11-42, 57-59, 67, 78-90 Average: Problems 1-4 EXS: 11-37 odd, 39-69, 78-90 Advanced: Problems 1-4 EXS: 11-37 odd, 39-77, 78-90 http://www.virtual nerd.com/algebra2/quadratics/formu ladiscriminant/quadr aticformula/complexsolutions-quadraticformula-Example Algebra Lab Algebra 2 Solve It - Text pg 240 or PowerAlgebra (textbook website) Algebra 2 Interactive Digital Path Chapter 4 http://www.regents Chapter 4-7 View prep.org/Regents/ Solve it math/ALGEBRA/AE 5/indexAE5.htm http://terzicmath.w eebly.com/uploads/ 5/7/3/3/5733011/ real-and-complexsolutions.pdf Algebra Lab Dynamic Activity at same website as above Assessment Quiz - Teacher created quiz – three questions – all with complex solutions including one that can be solved by the square root method. BASIC SKILLS REVIEW PARCC/HSPA PREP Applications of Percents Create equations and inequalities in one variable and use them to solve problems. Last 1/3 of the period. Algebra 2 text Basic Review Student Text pg 972 Worksheet: Application problems http://teachers.s with sales discount, duhsd.net/mlew markup, and sales tax. is1/syllabus_file s/PA7/Chapter %209/PA7%20 Worksheetdiscounts_markups%203.pdf 5-6 ALL MP 4,2 4. Functions 1 day Q8,9,10 A –CED.1 Identify what parts of equations represent. Example 1: C = M – x/100 * M The equation above represents the final cost of an item after a discount. Which part of the formula is the discount? Example 2: The expression P(1.05)2 gives the number of dollars in an investment account over years after the initial amount is invested. The account earns a simple annual interest. a. What does 2 represent N.CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. MP 2 1. C = P + x *P 100 The equation above represents the final cost of an item after sales tax. Which part of the formula is the tax? 2. The expression 7000(1.02)t gives the number of dollars in an investment account over years after the initial amount is invested. The account earns a simple annual interest. a. What does 7000 represent in the context of the problem? b. What does 1.02 Algebra Lab Take some word problems from the web that are similar to the ones on the test and write questions about the word problems. Enrichment Have students create their own problems and solutions. in the context of the problem? b. What does 1.05 represent in the context of the problem? BASIC SKILLS REVIEW PARCC/HSPA PREP Example 3: A 30-ounce solution that is 25 percent acid has x ounces of pure acid added to it. The following expression is used to answer some questions about the mixture. (0.25(30) + x)/(30 + x) a. What does 30 + x represent? b. What does 0.25(30) + x represent? Applications of adding fractions Last 1/3 of the period. Part of a MID UNIT period TEST Mid unit test to determine areas of weakness that need to be addressed before the represent in the context of the problem? 3. A 25-ounce solution that is 55 percent acid has x ounces of pure acid added to it. The following expression is used to answer some questions about the mixture. 0.55(25) + x 25 + x a. What does 25 + x represent? b. What does 0.55(25) + x represent? 7.EE.3 Solve multi-step reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. N.CN.1, 2, 7, 9 MP 7, 5 Algebra 2 Basic Review Student Text pg 973 Example 1 and Q 1, 2, 5 Teacher created material. Teacher created material. Example problem: Paul practices a tenth of an hour of basketball on Friday, two and a quarter hours of basketball on Saturday, and one and five-sixths hours of basketball on Sunday. How many hours of basketball did he practice altogether? Test to determine skills that need reteaching. Use problems similar to state unit test. 5. Polynomi als Write polynomial expressions. Example 1: Rewrite the expression 6x + [2x/(x+5)] as one rational expression that is equivalent to the expression for all x values, where x ≠ –5. 2 days Q11, 12 Example 2: A box in the shape of a rectangular prism has a width that is 5 inches greater than the height and a length that is 2 inches greater than the width. Write a polynomial expression in standard form for the volume of the box. Explain the meaning of any variables used. the pre-assessment that have been covered. A.SSE.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For Example, interpret P(1+r)n as the product of P and a factor not depending on P. MP 3, 6 EXAMPLE 1 Basic/ELLS Rewrite the expression 3x + [2x/5] as one rational expression that is equivalent to the expression for all x values. EXAMPLE 2 Basic/ELLS Regular Rewrite the expression 7x + [5x/(x+8)] as one rational expression that is equivalent to the expression for all x values, where x ≠ –8. Enrichment Example 2 Give questions Two questions to similar to answer similar to the Example 2 but basic and regular. some have sides that are less than A third question where students are the height. required to explain These questions how the process is should also similar to finding the include some common with fractions denominator of a for the change. regular fraction. Enrichment Give questions similar to Example 2. Regular Give questions similar to Example 2 but some have sides that are less than the height. Assessment Example 1 Two questions to answer similar to the basic and regular. A third question where students are required to explain how the process is similar to finding the common denominator of a regular fraction. Rewrite the expression 7x + [5x/(x+8)] as one rational expression that is equivalent to the expression for all x values, where x ≠ –8. Last 1/3 of the period. BASIC SKILLS REVIEW PARCC/HSPA PREP Polynomials Solving equations using the distributive property and combining like terms. 1 day Q 13 Example 2: (8x3 - 8x - 4) - (-16x6 + 9x3 - 6x2) Example 3: (-3x + 8) (-4x3 + 8x2 - Teacher created material. Create equations and inequalities in one variable and use them to solve problems. MP 4,2 A.APR.A.1 Arithmetic operations with polynomials. Example 1: (-14x7 + 19x6 - 17) + (5x5 - 10x4 + 18) A –CED.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply Basic (2r + 9r4) – (8r - 7r2 + 4r4) Regular (3b3 + 8) – (9b3 + 7 + b4) + (5b4 + 6) Enrichment (4b5 – 3b3 + 8) – (4b4 + http://www.you tube.com/watch ?v=hVCVW2cfcs A Example: The perimeter of a rectangle is 104m. The length is 7m more than twice the width. What are the dimensions of the rectangle? http://www.mat hsisfun.com/alge bra/polynomials -addingsubtracting.html Algebra Lab http://www.regentspre p.org/Regents/math/AL GEBRA/teachres/TRcub es.htm http://www.kut asoftware.com/F reeWorksheets/ Alg1Worksheets /Adding+Subtra cting%20Polyno mials.pdf http://www.regentspre p.org/Regents/math/AL GEBRA/AV2/indexAV2.h tm Assessment 2x + 12) polynomials. MP 3 9b3 + 7 + b) + (5b2 + 6) Verify results http://educa tion.ti.com/e n/us/activity /detail?id=B E17AD49E4E 14CDA81B9B A66A3587E1 0 BASIC SKILLS REVIEW PARCC/HSPA PREP Using the TI-84 to find the roots of quadratics. A.APR.3 Example 1 http://www.gle ncoe.com/sec/m ath/algebra/alge bra1/algebra1_0 4/study_guide/p dfs/alg1_pssg_G 063.pdf Teacher created material. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Last 1/3 of the period. MP 2 Polynomials 1 day Q14 Determine the roots of polynomial functions. A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. MP 7 Algebra 2 Basic, Regular, Advance Page 355 question 4 http://www.ck1 2.org/book/CK12-Algebra-I--Honors/r3/secti on/7.9/ Two questions to answer similar to the basic and regular. At least one should not have like terms in each expression. http://mathbits.com/Ma thBits/TISection/Algebr a2/zerofunctions.htm Students will be given quadratics to find the roots using the calculator. Then students will be given a cubic and quartic to do the same. Algebra Lab Algebra 2 Teachers Edition pg 346 Performance Task 3 http://www.aug ustatech.edu/ma th/molik/Polyno mialFunctions.p df Assessment Given a graph, determine the roots. Given a polynomial expression, determine the roots. Last 1/3 of the period. BASIC SKILLS REVIEW PARCC/HSPA PREP Multiplying and factoring perfect square trinomials and differences of squares. A.APR.3 http://www.the mathpage.com/a lg/differencetwo-squares.htm Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. MP 3 Polynomials Difference of Squares Difference of Cubes Sum of Cubes 2 days Q15 A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. MP 7 Last 1/3 of the period. BASIC SKILLS REVIEW PARCC/HSPA PREP Write the equations of lines. A.CED.2 Create equations in two or more Algebra 2 Student Textbook Section 5-3 Basic: Problems 1-4 EXS: 11-35 odd, 37, 38 Average: Problems 1-4 EXS: 25-35 odd, 37, 38, 39-49 odd, 51-57 which by class) Advanced: Problems 1-4 From Average plus 58-60 Student Textbook Section 2-3 & 2-4 http://www.the mathpage.com/a lg/perfectsquaretrinomial.htm http://www.mat hsisfun.com/alge bra/polynomials -difference-twocubes.html Example: Multiply (x + 9)(x – 9) (x + 6)2 Factor x2 – 100 9x2 – 24x + 16 Algebra Lab Algebra 2 Teachers Edition pg 346 Performance Task 3 http://www.mat hsisfun.com/defi nitions/differenc e-ofsquares.html Assessment Given polynomials, students will determine if they are Difference of Squares, Difference of Cubes, Sum of Cubes. http://www.edu cation.com/stud yhelp/article/pre Example: What is the equation of the line containing the variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. MP 4, 2 Polynomials Linear Factors and End Behavior 1 day A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. MP 7 Q16,17, 18 Last 1/3 of the BASIC SKILLS REVIEW PARCC/HSPA PREP Counting Principle S.CP.9 Use permutations Selected problems should be determined by the skills of the class. -calculus-helpslope-equationline/ http://www.edu cation.com/stud yhelp/article/pre -calculus-helppre-calculuschapter-1/ Algebra 2 http://www.pur Student plemath.com/m Textbook odules/polyends Section 5-1 & .htm 5-2 Basic: Problems 1-4 EXS: pg 285 9-39 odd Average: Problems 1-4 EXS: pg 286 40-54 Advanced: Problems 1-4 From Average plus pg 287 55-57 Student Textbook Section 11-1 points (0, −1) and (5, 1)? Find an equation of the line containing the point (−1, −5) and parallel to the line y = 2 x − 4. Algebra Lab Algebra 2 Teachers Edition pg 280 Solve it or PowerAlgebra (textbook website) Algebra 2 Interactive Digital Path Chapter 5 Chapter 5-1 View Solve it Assessment Given a graph, determine the roots. Given a polynomial expression, determine the roots and describe the end behavior. http://www.regents Example: prep.org/Regents/ A movie theater sells 3 math/ALGEBRA/AP sizes of popcorn (small, R1/PracCnt.htm period. and combinations to compute probabilities of compound events and solve problems. All: Problem 1 EXS: pg 678 9-11 medium, and large) with 3 choices of toppings (no butter, butter, extra butter). How many possible ways can a bag of popcorn be purchased? MP 2 Last 1/3 of the period. Basic Skills Mixed problem solving PARCC/HSPA problems based on unit Prep basic skill review. Teacher created material. Quiz on Basic Skills Teacher created material. Test to determine skills that need reteaching. A.APR.3 S.CP.9 END UNIT TEST Part of a period A –CED.1, 2 A.SSE.A.1a 7.EE.3 End unit test to determine areas of weakness that need to be addressed before the state unit test. A.SSE.1 A.APR.1, 3 Use problems similar to the pre-assessment that have been covered since the mid-unit test. INSTRUCTIONAL FOCUS OF UNIT 1. Use Properties of operations to add, subtract, and multiply complex numbers. 2. Solve quadratic equations with real coefficients that have complex solutions. 3. Show that the fundamental Theorem of Algebra is true for quadratic polynomials 4. Interpret coefficients, terms, degree, powers (positive and negative), leading coefficients and monomials in polynomial and rational expressions in terms of context. 5. Restructure by performing arithmetic operations on polynomial/rational expressions. 6. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For Example, calculate mortgage payments. 7. Use an appropriate factoring technique to factor expressions completely including expressions with complex numbers. 8. Explain the relationship between zeros and factors of polynomials and use zeros to construct a rough graph of the function defined by the polynomial. PARCC FRAMEWORK/ASSESSMENT PARCC EXEMPLARS: www.parcconline.org (copy & paste the url or link into search engine) Complex Numbers: http://www.projectmaths.ie/documents/teachers/tl_complex_numbers.pdf Polynomials: http://www.projectmaths.ie/students/stran4lc/student-activity-polynomial.pdf 1. It is given that: 24 + 10𝑥 − 𝑥 2 = 𝑝 − (𝑥 − 5)2 . Find the value of p. HSPA EXEMPLARS: Number And Numerical Operations: http://www.riverdell.org/cms/lib05/NJ01001380/Centricity/Domain/56/Number%20Sense%20Practice%20Problems.pdf Question 81 - Candra had a gift card for $130. She spent $20.83 on Friday and $56.51 on Saturday using her gift card. Then, on Sunday she returned an item she bought on Saturday, and $13.76 was credited back to her gift card. How much money is left on Candra's gift card on Monday? Candra’s friend, Alex, also received a gift card. The amount on his gift card is 20% less than Candra’s. How much does Alex have on his gift card. If Alex bought three items and returned one to end up spending 75% of the balance of his card, create four transactions to reflect this. Patterns And Algebra: http://mslaplantesmathclass.weebly.com/uploads/1/7/3/7/17373129/4.10.2013.pdf Billie Sue's BBQ had their grand opening this year. The first month they did not turn a profit. However, each month thereafter, they have had steady profits of $1,600 per month. If x represents the number of months they have profited, which of the following equations represents the amount of profits after x months? A. y = 1,600x B. y = 1,600 – x C. y = 1,600x + x D. y = x + 1,600 To make low fat cottage cheese, milk containing 4% butterfat is mixed with 10 gallons of milk containing 1% butterfat to obtain a mixture containing 2% butterfat. How many gallons of the richer milk is used? If the equation is (.02)(10 + x) = (.01)(10) + (.04)(x) explain how each of the following relate to the original question. a. What does (.02)(10 + x) represent? b. What does (.01)(10) represent? C. What does (.04)(x) represent? 21ST CENTURY SKILLS (4Cs & CTE Standards) 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). 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 APPENDIX (Teacher resource extensions) 1. E-Text, Interactive Digital Resources, Teacher Resources Login at https://www.pearsonsuccessnet.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. SLO 6 Communicate the precise answer to a real-world problem. 7. Look for and make use of structure. SLO 5 Identify structural similarities between integers and polynomials. SLO 7 Identify expressions as single entities, e.g. the difference of two squares. 8. Look for and express regularity in repeated reasoning. SLO 6 Arrive at the formula for finite geometric series by reasoning about how to get from one term in the series to the next. All of the content presented in this course has connections to the standards for mathematical practices. * This course includes exponential and logarithmic functions as modeling tools. (PARCC Model Content Frameworks) UNIT 2 – Expressions and Equations (1) Total Number of Days: 16 days (A and B days meet alternate days) Grade/Course: 11/Algebra 2 ESENTIAL QUESTIONS What are the operations that apply to all function? How can Geometric and Analytic representations be used to describe the behavior of the function? How are algebraic, numeric, and graphic representations of functions related? How does representing functions graphically help you solve a system of equations? How does writing equivalent equations help you solve a system of equations? What does the degree of a polynomial tell you about its related polynomial function? For a polynomial function, how are factors, zeros and xintercepts related? For a polynomial function, how are factors and roots related? PACING 1 CONTENT Unit 2 PreAssessment SKILLS Exponents and Radicals Rational expression Equations in one ENDURING UNDERSTANDINGS You can add, subtract, multiply, and divide functions based on how you perform these operations for real numbers. One difference, however, is that you must consider the domain of each function. To solve a system of equations, find a set of values that replace the variables in the equations and make each equation true. You can solve a system of equations by writing equivalent systems until the value on one variable is clear. Then substitute to find the values of the other variables. You can factor many quadratic trinomials into products of two binomials. To find the zeros of a quadratic function, you must set the equation equal to zero.. A polynomial function has distinguishing “behaviors”. You can look at its algebraic form and know something about its graph. You can look at its graph and know something about its algebraic form. Knowing the zeros of polynomial functions can help you understand the behavior of its graph. You can divide polynomials using steps that are similar to the long division steps that you use to divide whole numbers. The degree of a polynomial equation tells you how many roots the equation has. RESOURCES LEARNING STANDARDS ACTIVITIES/ASSESSME OTHER (CCCS/MP) Pearson NTS (e.g., tech) N.RN.1, 2 Rewrite questions similar A.APR.6 to the Unit 2 test. A.REI.1,6 15 MC, 5 SR, 5 OE A.SSE.3 1. Exponents and Radicals. 2 variable F.IF.4 System of Equations Equivalent Expressions Key features of graphs 1. Solve exponential N.RN.1 equations. Explain how the Ex: definition of the 1 𝑥 meaning of rational If (53 ) = 5, what is the exponents follows value of x? Explain your from extending the reasoning. properties of integer exponents 2. Rewrite radical to those values, expression in allowing for a exponential forms: notation for radicals in terms of Ex: rational exponents. Rewrite the 3 expression 5√125 as a N.RN.2 power of 5. Rewrite expressions involving radicals and rational exponents using the properties of exponents. MP 1, 8 Algebra 2 Student Textbook Section 6-1 Basic: Problems 1-4 EXS: 10-30, 33-37, 5667 Average: Problems 1-4 EXS: 11-29 odd, 31-48, 56-67 Advanced: Problems 1-4 11-29 odd, 31-67 Student Textbook Section 6-4 Basic: Problems Interactive Digital Path 61 Roots and 86-1 Radical Expressions PowerAlgebra (textbook website) Algebra 2 Interactive Digital Path Chapter 6 Chapter 6-1 View Solve it Dynamic Activity 6-4 Rational Exponents PowerAlgebra (textbook website) Algebra 2 Interactive Digital Path Chapter 6 Chapter 6-4 View Dynamic Algebra Lab Algebra 2 Student Textbook Section 6-1 Concept Byte Page 360 Algebra Lab Student Textbook Section 6-4 Solve It Page 381 or on web at PowerAlgebra (textbook website) Algebra 2 Interactive Digital Path Chapter 6 Chapter 6-4 View Solve it Assessment Three question quiz: Two calculation questions similar to the Examples shown. One reasoning question requiring a written response that shows a student understands how the exponent power relates to the radical. 1-6 EXS: 10-67, 72-82 E, 56-67, 98119 Activity Average: Problems 1-6 EXS: 11-65 odd, 67-90, 98-119 Advanced: Problems 1-6 11-65 odd, 67-119 Basic Skills PARCC/HSPA Last 1/3 Prep of the period. 2. Rational expression 3 Simplify radical expressions N-RN.A.2 Dividing Polynomials using long or synthetic method. A.APR.6 Ex: Divide 2 A) 4x + 2x + 1 , where x -2 x ¹ 2? B) Simplify Rewrite expressions involving radicals and rational exponents using the properties of exponents. MP 4 Rewrite rational expressions. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + Student Textbook Section 5-4 Basic: Problems 1-5 EXS: 9-41, 43, 48, 50, 67-85 Teacher created material. This will be a review of the process to simplify radicals. This skill will be applied in the next test prep. Interactive Digital Path 54 Dividing Polynomials Algebra Lab Student Textbook Section 5-4 Solve It Page 303 or on the web at http://www.pea rsonsuccessnet.c om/snpapp/lear n/navigateIDP.d o?method=vlo&i nternalId=13111 2100000061&is Html5Sco=false http://www.pearsonsuccessne t.com/snpapp/learn/navigateI DP.do?method=vlo&internalId =131112100000061&isHtml5 Sco=false Algebra Lab x 2 - 3x - 10 x2 - 5x + 6 ¸ 2 2 2 x - 11x + 5 2 x - 7 x + 3 r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated Examples, a computer algebra system. MP 1, 2, 3, 6 Average: Problems 1-5 EXS: 9-39 odd, 40-62, 67-85 Advanced: Problems 1-5 EXS: 9-39 odd, 40-66, 67-85 Student Textbook Section 8-4 Basic: Problems 1-4 EXS: 8-27, 31-33, 37, 50-67 Average: Problems 1-4 EXS: 9-35 odd, 27-44, 50-67 Advanced: Problems 1-4 EXS: 9-25 odd, 27-67 Interactive Digital Path 84 Rational Expressions http://www.pea rsonsuccessnet.c om/snpapp/lear n/navigateIDP.d o?method=vlo&i nternalId=13111 2100000094&is Html5Sco=false Student Textbook Section 8-4 Solve It Page 527 or on web at http://www.pearsonsuccessne t.com/snpapp/learn/navigateI DP.do?method=vlo&internalId =131112100000094&isHtml5 Sco=false Click on Solve It Tab Assessment Three question quiz: Two calculation questions similar to the Examples shown. One reasoning question requiring a written response where students will discuss the processes of long and synthetic division. Basic Skills PARCC/HS PA Prep Problem solving with radicals. 3. Equations in One variable Solving radical Equations Last 1/3 of the period. 2 EX: solve for x 5 - 2 x + 5 = 12 N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. MP 4 A.REI.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. A.REI.2 Understand solving equations as a process of reasoning and explain the Student Textbook Section 65 Basic: Problems 1-5 EXS: 9-47, 57-60, 6365 Average: Problems 1-5 EXS: 9-43 odd 45-67, 73-96 Advanced: Problems 1-5 EXS: 9-43 odd, 45-72 Teacher created material. Three part open ended question where students will find the width, area, and area of a shaded region where the answers must be written in radical form. Interactive Digital Path 65 Solving Square Root and Other Radical Expressions Algebra Lab Student Textbook Section 6-5 Solve It Page 390 or on the web at http://www.pea rsonsuccessnet.c om/snpapp/lear n/navigateIDP.d o?method=vlo&i nternalId=13111 2100000075&is Html5Sco=false http://www.pearsonsuccessne t.com/snpapp/learn/navigateI DP.do?method=vlo&internalId =131112100000075&isHtml5 Sco=false Assessment Three question quiz: Two calculation questions similar to the Example shown. One reasoning question requiring a written response where students will discuss why the solution normally has ± in front of it and in what situations it is not needed. Basic Skills PARCC/HS PA Prep Solving radical word problems. Last 1/3 of the period. Part of a period MID UNIT TEST 4. System of Equations Mid unit test to determine areas of weakness that need to be addressed before the state unit test. 1. Solve system of linear equations by graphing. EX: 3 1 y x 3 2 1 1 y x 6 3 reasoning. Solve simple rational and radical equations in one variable, and give Examples showing how extraneous solutions may arise. N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. MP 3 N.RN.1, 2 A.APR.6 A.REI.1, 2 A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Student Textbook Section 3-1 Basic: Problems 1-4 EXS: 7-28, 30-36 even, 38-43, 53- Teacher created material. Example: A shop needs to make a frame where the height is 1/2 its width. It is to be enlarged to have an area of 60.5 square inches. What will be the dimensions of the enlargement? Teacher created material. Test to determine skills that need reteaching. Interactive Digital Path 31 Solving Systems Using Tables and Graphs http://www.pea rsonsuccessnet.c om/snpapp/lear n/navigateIDP.d o?method=vlo&i Algebra Lab Student Textbook Section 3-1 Solve It Page 134 or on the web at http://www.pearsonsuccessne t.com/snpapp/learn/navigateI DP.do?method=vlo&internalId =131112100000075&isHtml5 Sco=false Algebra Lab 2. Solve the system of equations algebraically. EX: 2 x 3y 8 4 x 2y 10 3. Solve system of equations graphically and algebraically. EX: y = 2x + 5 y = x 2 + 4x - 10 A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For Example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. MP 6 67 Average: Problems 1-4 EXS: 7-27 odd 29-47, 53-67 Advanced: Problems 1-4 EXS: 7-27 odd, 29-52 Student Textbook Section 3-2 Basic: Problems 1-5 EXS: 10-43, 53-57, 6780 Average: Problems 1-5 EXS: 11-41 odd, 43-61, 67-80 Advanced: Problems 1-5 EXS: 11-41 nternalId=13111 2100000036&is Html5Sco=false Interactive Digital Path 32 Solving Systems Algebraically http://www.pea rsonsuccessnet.c om/snpapp/lear n/navigateIDP.d o?method=vlo&i nternalId=13111 2100000037&is Html5Sco=false Section 3-1 Dynamic Activity on the web at http://www.pearsonsuccessne t.com/snpapp/learn/navigateI DP.do?method=vlo&internalId =131112100000075&isHtml5 Sco=false Algebra Lab Student Textbook Section 3-2 Solve It Page 142 or on the web at http://www.pearsonsuccessne t.com/snpapp/learn/navigateI DP.do?method=vlo&internalId =131112100000037&isHtml5 Sco=false Algebra Lab Section 3-2 Dynamic Activity on the web at http://www.pearsonsuccessne t.com/snpapp/learn/navigateI DP.do?method=vlo&internalId =131112100000037&isHtml5 Sco=false odd, 43-66, 67-80 Basic Skills PARCC/HSPA Prep Problem Solving with Systems of Linear Equations Last 1/3 of the period. 5. Equivalent Expression s. Use properties of exponents to simplify exponential functions. Ex: Simplify the following function () f x = 3x × 23x+2 2 A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. MP 4,2 A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions. For Example the expression 1.15t can Student Textbook Section 6-4 Basic: Problems 1-6 EXS: 10-67, 72-82 E, 56-67, 98119 Average: Problems 1-6 EXS: 11-65 odd, 67-90, 98-119 Advanced: Problems Teacher created material. Example: Interactive Digital Path 64 Rational Exponents Algebra Lab Student Textbook Section 6-4 Solve It Page 381 or on web at http://www.pea rsonsuccessnet.c om/snpapp/lear n/navigateIDP.d o?method=vlo&i nternalId=13111 2100000073&is Html5Sco=false http://www.pearsonsuccessne t.com/snpapp/learn/navigateI DP.do?method=vlo&internalId =131112100000073&isHtml5 Sco=false Interactive Digital Path 65 Solving Square Root and Other Radical Expressions http://www.pea rsonsuccessnet.c At Pinho’s, Sam bought 2 donuts and 5 muffins spending $14.25. Amy bought 3 donuts and 2 muffins spending $9. How much do they charge for a donut? How much for a muffin? Click on Dynamic Activity Tab Algebra Lab Student Textbook Section 6-5 Solve It Page 390 or on the web at http://www.pearsonsuccessne t.com/snpapp/learn/navigateI DP.do?method=vlo&internalId =131112100000075&isHtml5 Sco=false be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. MP 4, 3 1-6 11-65 odd, 67-119 Student Textbook Section 6-5 om/snpapp/lear n/navigateIDP.d o?method=vlo&i nternalId=13111 2100000075&is Html5Sco=false Assessment Two question quiz: Two calculation questions similar to the Example shown. Basic: Problems 1-5 EXS: 9-47, 57-60, 6365 Average: Problems 1-5 EXS: 9-43 odd 45-67, 73-96 Advanced: Problems 1-5 EXS: 9-43 odd, 45-72 Basic Skills PARCC/HSPA Prep Last 1/3 of the period. Multiplying with scientific notation. N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Teacher created material. Example: The speed of light is approximately 3 x 10 8 m/s. How far does light travel in 6.0 x 101 seconds? 6. Key features of graphs End behavior of a function. EX: Find the end behavior of the function f(x)=x4 – 4 x3 + 3 x + 25. 2 MP 3, 8 F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. MP 4, 2 Last 1/3 Basic Skills of the PARCC/HSPA period. Prep Mixed problem solving problems based on unit basic skill review. N-RN.A.2 A.REI.6 Student Textbook Section 5-1 Basic: Problems 1-4 EXS: 8-39, 40-50 Even, 58-71 Average: Problems 1-4 EXS: 9-39, 40-54 Even, 58-71 http://hotmath.c om/hotmath_hel p/topics/endbehavior-of-afunction.html Interactive Digital Path 51 Polynomial Functions http://www.pea rsonsuccessnet.c om/snpapp/lear n/navigateIDP.d o?method=vlo&i nternalId=13111 2100000075&is Html5Sco=false Advanced: Problems 1-4 EXS: 9-39 odd, 40-71 67 Page 283 Teacher created material. Algebra Lab Student Textbook Section 5-1 Solve It Page 280 or on web at http://www.pearsonsuccessne t.com/snpapp/learn/navigateI DP.do?method=vlo&internalId =131112100000073&isHtml5 Sco=false Assessments Three question quiz: Two questions similar to the Example shown. One reasoning question requiring a written response where students will discuss how they can sometimes determine the roots visually. Quiz on Basic Skills END UNIT TEST Part of a period End unit test to determine areas of weakness that need to be addressed before the state unit test. A.REI.6, 7 A.SSE.3 F.IF.4 Teacher created material. Test to determine skills that need reteaching. INSTRUCTIONAL FOCUS OF UNIT 1. Use properties of integer exponents to explain and convert between expressions involving radicals and rational exponents, using correct notation. For Example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. 2. Rewrite simple rational expressions in different forms using inspection, long division, or, for the more complicated Examples, a computer algebra system. 3. Solve simple equations in one variable and use them to solve problems, justify each step in the process and the solution and in the case of rational and radical equations show how extraneous solutions may arise. 4. Solve systems of linear equations and simple systems consisting of a linear and a quadratic equation in two variables, algebraically and graphically. Write equivalent expressions for exponential functions using the properties of exponents.Interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. PARCC FRAMEWORK/ASSESSMENT PARCC EXEMPLARS: www.parcconline.org (copy & paste the url or link into search engine) Complex Numbers: http://www.projectmaths.ie/documents/teachers/tl_complex_numbers.pdf Polynomials: http://www.projectmaths.ie/students/stran4lc/student-activity-polynomial.pdf It is given that: 24 + 10𝑥 − 𝑥 2 = 𝑝 − (𝑥 − 5)2 . Find the value of p. HSPA EXEMPLARS: Number And Numerical Operations: http://www.riverdell.org/cms/lib05/NJ01001380/Centricity/Domain/56/Number%20Sense%20Practice%20Problems.pdf Question 81 - Candra had a gift card for $130. She spent $20.83 on Friday and $56.51 on Saturday using her gift card. Then, on Sunday she returned an item she bought on Saturday, and $13.76 was credited back to her gift card. How much money is left on Candra's gift card on Monday? Candra’s friend, Alex, also received a gift card. The amount on his gift card is 20% less than Candra’s. How much does Alex have on his gift card. If Alex bought three items and returned one to end up spending 75% of the balance of his card, create four transactions to reflect this. Patterns And Algebra: http://mslaplantesmathclass.weebly.com/uploads/1/7/3/7/17373129/4.10.2013.pdf Billie Sue's BBQ had their grand opening this year. The first month they did not turn a profit. However, each month thereafter, they have had steady profits of $1,600 per month. If x represents the number of months they have profited, which of the following equations represents the amount of profits after x months? A. y = 1,600x B. y = 1,600 – x C. y = 1,600x + x D. y = x + 1,600 To make low fat cottage cheese, milk containing 4% butterfat is mixed with 10 gallons of milk containing 1% butterfat to obtain a mixture containing 2% butterfat. How many gallons of the richer milk is used? If the equation is (.02)(10 + x) = (.01)(10) + (.04)(x) explain how each of the following relate to the original question. a. What does (.02)(10 + x) represent? b. What does (.01)(10) represent? C. What does (.04)(x) represent? 21ST CENTURY SKILLS (4Cs & CTE Standards) 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). 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 APPENDIX (Teacher resource extensions) E-Text, Interactive Digital Resources, Teacher Resources Login at https://www.pearsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true Mathematical Practices 1. Make sense of problems and persevere in solving them. SLO 3 Use problems that involve may givens of the need to be composed or decomposed before they can be solved 2. Reason abstractly and quantitatively. SLO 5 Using properties of exponents to determine if two expressions are equal. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. * 5. Use appropriate tools strategically. SLO 6 Use graphing technically when available. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. All of the content presented in this course has connections to the standards for mathematical practices. * This course includes exponential and logarithmic functions as modeling tools. (PARCC Model Content Frameworks) UNIT 3 – Expressions and Equations (2) Total Number of Days: 16 days (A and B days meet alternate days) Grade/Course: 11/Algebra 2 ESENTIAL QUESTIONS How does representing functions graphically help you solve a system of equations? For a polynomial function, how are factors, zeros and x-intercepts related? How do you model a quantity that changes regularly over time by the same percentage? How can you model periodic behavior? PACING CONTENT SKILLS ENDURING UNDERSTANDINGS To solve a system of equations, find a set of values that replace the variables in the equations and make each equation true. A polynomial function has distinguishing “behaviors”. You can look at its algebraic form and know something about its graph. You can look at its graph and know something about its algebraic form. Periodic behavior is behavior that repeats over intervals of equal length. An angle with a full circle of rotation measure 2π radians. RESOURCES STANDARDS LEARNING OTHER ACTIVITIES/ASSESSMENTS (CCCS/MP) Pearson (e.g., tech) 1 Unit 3 PreAssessment Systems of equations Arithmetic Sequences Geometric Sequences Inverse of functions. Functions and their graphs. Exponential functions. Radian Measures Trig A.REI.11 F.BF.2 F.BF.4 F.IF.4 F.IF.7 F.LE.5 F.TF.1 F.TF.2 F.TF.8 F.TF.5 1. Systems of equations Determine the solution from the graph of the system. A.REI.11 SECTION 4-9 Find approximate solutions for the intersections of functions and explain why the xcoordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x) involving linear, polynomial, rational, absolute value, and exponential Basic: Problem 1 Pg 262 Q 8 – 13 EXAMPLE 1: What is the solution set? 2 EXAMPLE 2: For the functions a and b defined below, sketch a graph Rewrite questions similar to the Unit 2 test. 15 MC, 5 SR, 5 OE Average: Problem 1 Pg 262 - 263 Q 8 – 13, 41 46 Advanced: Problem 1 Pg 262 - 263 Q 8 – 13, 41 – 46, 53-55 Understand that where the graphs intersect is the solution. Understand that when looking at the table of values in the graphing calculator the solutions are where all the y values are the same. Algebra Lab Concept Byte pg 477 Questions 1 – 3 Algebra Lab Concept Byte pg 484-485 Questions 1 – 12 Algebra Lab Teacher created worksheets with systems of two different types of equations including: Rational with linear. Square root with absolute value. Last 1/3 of the period. Basic Skills PARCC/HSPA Prep without the use of a calculator and use the graph to identify the solution set to a(x) = b(x). a(x) = 4/x b(x) = 1/2x + 1 EXAMPLE 3: For the functions defined above, fill in the tables of values. Then give the solution set to g(x) = h(x). Explain your answer. g(x) = √𝑥 − 4 h(x) = | x – 6 | Patterns in mathematics functions. Then have students make up some of their own. They should graph them and determine the solution set. If the solution set is not easy to find, students should adjust the parameters of the problem until it is. MP 1, 4, 5 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a Algebra Lab http://en.wikib ooks.org/wiki/ SA_NC_Doing_In vestigations/Ch apter_6#Numbe r_Patterns_Activ ity_1 After covering the patterns, have students create their own pattern, then write a question and solution. http://wwwrohan.sdsu.edu /~ituba/math3 03s08/mathide as/mmi10_01_0 2.pdf This should also include a paragraph or two in writing where students can explain they understand the process. rational number. 2. Arithmetic Sequences 2 Basic Skills PARCC/HSPA Prep Last 1/3 of the period. MP Write the explicit and F.BF.2 recursive form of an arithmetic sequence. Write arithmetic and geometric EXAMPLE: sequences both recursively and Julia makes $2.00 an with an explicit hour for first hour of formula, use them work, $4.00 her to model second hour, $6.00 situations, and her third hour and so translate between on. How much the two forms. money will she earn on her 12th hour of MP 3 work? Write a recursive and explicit rule for this problem. Combinations and S.CP.9 Permutations Use permutations and combinations to compute probabilities of compound events and solve problems. Section 9-2 Basic: Problems 1-4 EXS: 7-25, 3135, 49-53, 61, 75-88 http://www.alge bralab.org/lesso ns/lesson.aspx?fi le=algebra_ariths eq.xml Algebra Lab Solve it pg 572 also on http://www.pearsonsuccessnet.c om/snpapp/learn/navigateIDP.d o?method=vlo&internalId=13111 2100000100&isHtml5Sco=false Dynamic Activity also on same web site. Average: Problems 1-4 EXS: 7-25 odd, 26-62, 75-88 Advanced Problems 1-4 EXS: 7-25 odd, 26-88 http://www.mat hsisfun.com/com binatorics/combi nationspermutations.ht ml Example: Permutation: In how many ways can 4 of 7 different kinds of bushes be planted along a http://www.rege walkway? ntsprep.org/Reg ents/math/algtri g/ATS5/Lcomb.h tm Combination: How many ways are there to select 3 bracelets from a box of 20? 3. Geometric Sequences 2 Write the explicit and F.BF.2 recursive form of a geometric sequence. Write arithmetic EXAMPLE: and geometric sequences both A ball is dropped, recursively and and for each bounce with an explicit after the first bounce, formula, use them the ball reaches a to model height that is a situations, and constant percent of translate between the preceding height. the two forms. After the first bounce, it reaches a MP 4, 2 height of 30 feet, and after the third bounce it reaches a height of 10.8 feet. Section 9-3 Basic: Problems 1-4 EXS: 7-31, 3644 even, 4850, 59, 63-81 http://www.alge bralab.org/lesso ns/lesson.aspx?fi le=Algebra_GeoS eq.xml Algebra Lab Solve it pg 580 also on http://www.pearsonsuccessnet.c om/snpapp/learn/navigateIDP.d o?method=vlo&internalId=13111 2100000102&isHtml5Sco=false Dynamic Activity also on same web site. Average: Problems 1-4 EXS: 7-31 odd, 32-59, 63-81 Advanced: Problems 1-4 EXS: 7-31 odd, 32-62, 63-81 Write an explicit rule for the height after the nth bounce, an, where n represents the bounce number. Last 1/3 of the period. Basic Skills PARCC/HSPA Prep Simple probability S.MD.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). http://www.mat hsisfun.com/dat a/probability.ht ml Example: There are five balls in a bag: 2 red, 2 blue, and 1 white. What is the probability of randomly choosing a red ball? 4. Geometric Sequences Write the recursive formula of a geometric sequence in visual form. Example: 2 Write a recursive formula for the pattern shown. Basic Skills PARCC/HSPA Prep Last 1/3 of the period. Dependent and Independent Events 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. S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. http://www.skw irk.com.au/p-c_s12_u-223_t599_c2236/VIC/5/Patt erns-numberandgeometric/Patter ns-andalgebra/Patterns -andalgebra/Maths/ http://www.mat hsisfun.com/dat a/probability.ht ml Example: You throw a die twice. What is the probability of throwing a six and then a second six? Are these independent or dependent events? A bowl contains 4 peaches and 4 apricots. Maxine randomly selects one, puts it back, and then randomly selects another. What is the probability that both selections were apricots? 5. Inverse of functions. Find the inverse of a given function. EXAMPLES: Write an expression for the inverse of f(X) = 4x + 3. 7 What is the inverse function for x 2 , where 3 2 f(x) = x0? F.BF.4 Section 6-7 Determine the inverse function for a simple function that has an inverse and write an expression for it. Basic: Problems 1-6 EXS: 8-43, 4854 even, 65, 75-95 MP 4 Average: Problems 1-6 EXS: 9-41 odd, 42-67, 75-95 Advanced: Problems 1-6 EXS: 9-42-74, 75-95 Section 7-3 2 4 find x–3 f 1 x , where x > 3. If f(x) = Basic: Problems 1-5 EXS:12-47, 58-61, 72-76 even, 85-98 Average: Problems 1-5 EXS:13-43 odd, 44-79, 85-98 Advanced: Problems 1-5 EXS:13-43, odd, 44-84, 85-98 http://www.alge bralab.org/lesso ns/lesson.aspx?fi le=Algebra_Funct ionsRelationsInv erses.xml http://www.mat hworksheetsgo.c om/sheets/algeb ra-2/functionsandrelations/inverse -functionsworksheet.php Basic Skills PARCC/HSPA Prep Mutually exclusive and non-mutually exclusive events. Last 1/3 of the period. 6. Functions and their graphs. Describing key features of functions. EXAMPLE 1: 2 Describe each of the following key features of the graph of f(x) = (x – 4)(x – 3)2(x + 1)3 S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is http://www.nuts hellmath.com/te xtbooks_glossary _demos/demos_c ontent/alg2_com pound_probabilit y.html http://www.mat hsisfun.com/dat a/probabilityevents-mutuallyexclusive.html USE THE FOLLOWING SECTIONS2-3 2-5, 4-1, 4-2, 4-3, 5-1, 5-8, 13-1, 13-4, 13-5 Section 5-1 Basic: Problems 1-4 EXS: 8-39, 4050, even, 51, 58-71 Average: Problems 1-4 EXS: 9-39 odd, 40-54, 58-71 Advanced: Example: A pair of dice is rolled. What is the probability that the sum of the numbers rolled is either 7 or 11? A pair of dice is rolled. What is the probability that the sum of the numbers rolled is either an even number or a multiple of 3? http://www.you tube.com/watch? v=GALfCd-2XRQ Algebra Lab Concept Byte pp 459-460 http://learni.st/ users/S33572/b oards/2366reading-andinterpretinggraphs-commoncore-standard-912-f-if-4 Algebra Lab Concept Byte pp 477 http://olhs.olentangy.k12.oh.us/t eachers/kevin_streib/Algebra%2 0I The above web site has the solution keys to some really good questions on this topic. However, the original worksheets are not available. Example 2: Use the graph of the function f to answer the following questions. For what values of x is f < 0? increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Problems 1-4 9-39 odd, 4071 F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Average: Problems 1-5 13-35 odd, 36-47, 50-76 MP 6, 4 Section 8-3 Basic: Problems 1-5 EXS: 13-42, 50-76 Advanced: Problems 1-5 EXS: 13—35 odd, 36-49, 50-76 USE THE FOLLOWING SECTIONS 7-1, 7-2, 7-3, 13-4, 13-5, 13-6, 13-7, 13-8 Basic Skills PARCC/HSPA Prep Geometric Probability S.MD.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Last 1/3 of the period. 7. Exponenti al functions. Modeling with exponential functions. EXAMPLE: 2 The population, in thousands, of a certain city can be modeled by the function P t 180 0.94 0.25t where t is the number of years since 2000. What was the population of the city in the year 2000? What is the rate of change of the city’s population? http://www.mat hsisfun.com/dat a/probabilityevents-mutuallyexclusive.html F.LE.5 Section 7-2 Interpret the parameters in a linear or exponential function in terms of a context. Basic: Problems 1-5 EXS: 7-33, 35, 38-40 even, 44-62 MP 1, 2, 6, 5 , http://www.nuts hellmath.com/te xtbooks_glossary _demos/demos_c ontent/alg2_com pound_probabilit y.html Average: Problems: 1-5 EXS: 7-29 odd, 31-41, 44-62 Advanced: Problems 1-5 EXS: 7-29 odd, 31-62 http://www.alge bralab.org/lesso ns/lesson.aspx?fi le=Algebra_Expo nentsApps.xml http://www.ope ntextbookstore.c om/precalc/1.3/ Chapter%204.pd f Example: Find the probability that a randomly chosen point in the figure lies in the shaded region. Give all answers in fraction and percent forms. Algebra Lab Solve it pg 442 also found on http://www.pearsonsuccessnet.c om/snpapp/learn/navigateIDP.d o?method=vlo&internalId=13111 2100000082&isHtml5Sco=false Dynamic activity on same site as above. Last 1/3 of the period. Basic Skills PARCC/HSPA Prep Measures of Central Tendency 8. Radian Measures Measuring angles with radians. EXAMPLE: Convert as required. 60° = 3 = 1 5 radian degrees T.TF.1 Section 13-3 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Basic: Problems 1-4 EXS: 6-34, 3549, 54-68 http://www.rege ntsprep.org/Reg ents/math/ALGE BRA/AD2/measu re.htm http://www.keswick.hs.yrdsb.edu .on.ca/DeptResources/Math/MBF 3CWebsite/Resources/Statistics/ MeasureofCentralTendencyPracti ce.pdf http://www.mat hwarehouse.com /trigonometry/r adians/convertdegee-toradians.php Algebra Lab Solve it pg 844 also found on http://www.pearsonsuccessnet.c om/snpapp/learn/navigateIDP.d o?method=vlo&internalId=13111 2100000140&isHtml5Sco=false Average: Problems 1-4 7-33 odd, 3550, 54-68 Advanced: Problems 1-4 7—33 odd, 35-53, 54-68 Last 1/3 of the period. Basic Skills PARCC/HSPA Prep Bar and circle graphs http://www.mat hsisfun.com/dat a/bargraphs.html http://www.mat hsisfun.com/dat a/piecharts.html Algebra Lab Pg 982 http://nces.ed.gov/nceskids/crea teagraph/default.aspx 9. Trig functions Use trig identities to solve problems. EXAMPLE: 1 If sinθ = 5/7 and cosθ < 0, then in which quadrant does the terminal side of θ lie when it is placed in standard position? What are the values of cosθ and tanθ ? Explain your reasoning and show your work. T.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. F.TF.8 Prove the Pythagorean identity sin 2 (θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. USE THE FOLLOWING SECTIONS Section 13-4 Section 13-5 Section 13-6 Section 14-1 http://www.ope ntextbookstore.c om/precalc/1.3/ Chapter%205.pd f Algebra Lab Solve it pg 851 also found on http://www.pearsonsuccessnet.c om/snpapp/learn/navigateIDP.d o?method=vlo&internalId=13111 2100000141&isHtml5Sco=false Dynamic activity on same site as above. Algebra Lab Solve it pg 861 also found on http://www.pearsonsuccessnet.c om/snpapp/learn/navigateIDP.d o?method=vlo&internalId=13111 2100000143&isHtml5Sco=false Dynamic activity on same site as above. Last 1/3 of the period. Basic Skills PARCC/HSPA Prep Descriptive Statistics and Histograms 10. Periodic functions Modeling trig functions. EXAMPLE: The amount of daylight, in hours per day, can be approximated by the function 𝑑(𝑡) = 1 75 25 2𝜋 (𝑡 − cos( 5 7 365 + 9)9)) where t is the number of days since the most recent January 1 (including January 1). Using this approximation, what are the maximum and minimum amounts of daylight throughout the year? Maximum: _________ Minimum: _________ http://www.mat hsisfun.com/dat a/histograms.ht ml T.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. USE THE FOLLOWING SECTIONS 13-4 13-5 13-6 13-7 http://www.ope ntextbookstore.c om/precalc/1.3/ Chapter%206.pd f Algebra Lab Pg 983 http://nces.ed.gov/nceskids/crea teagraph/default.aspx Algebra Lab Solve it pg 868 also found on http://www.pearsonsuccessnet.c om/snpapp/learn/navigateIDP.d o?method=vlo&internalId=13111 2100000144&isHtml5Sco=false Dynamic activity on same site as above. Algebra Lab Solve it pg 875 also found on http://www.pearsonsuccessnet.c om/snpapp/learn/navigateIDP.d o?method=vlo&internalId=13111 2100000145&isHtml5Sco=false Dynamic activity on same site as above. END UNIT TEST Part of a period End unit test to determine areas of weakness that need to be addressed before the state unit test. A.REI.11 F.BF.2 F.BF.4 F.IF.4 F.IF.7 F.LE.5 F.TF.1 F.TF.2 F.TF.8 F.TF.5 Teacher created material. Test to determine skills that need reteaching. Use problems similar to the pre-assessment that have been covered since the midunit test. INSTRUCTIONAL FOCUS OF UNIT Find approximate solutions for the intersections of functions and explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x) involving linear, polynomial, rational, absolute value, and exponential functions. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Determine the inverse function for a simple function that has an inverse and write an expression for it. Graph functions expressed symbolically and show key features of the graph (including intercepts, intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity) by hand in simple cases and using technology for more complicated cases. Interpret the parameters in a linear or exponential function in terms of a context. Uses the radian measure of an angle to find the length of the arc in the unit circle subtended by the angle and find the measure of the angle given the length of the arc. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers (interpreted as radian measures of angles traversed counterclockwise around the unit circle) and use the Pythagorean identity (sin θ )2 + (cos θ )2 = 1 to find sin θ , cos θ , or tan θ , given sin θ , cos θ , or tan θ , and the quadrant of the angle. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. PARCC FRAMEWORK/ASSESSMENT 21ST CENTURY SKILLS (4Cs & CTE Standards) 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). 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 APPENDIX (Teacher resource extensions) E-Text, Interactive Digital Resources, Teacher Resources Login at https://www.pearsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. SLO 7 Make connections between the unit circle and trigonometric functions. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. * 5. Use appropriate tools strategically. 6. Attend to precision. SLO 7 Use precise language to explain why trigonometric functions are radian measures of angles traversed counter-clockwise around the unit circle. 7. Look for and make use of structure. SLO 3 Use the structure of a function to determine if it has an inverse. 8. Look for and express regularity in repeated reasoning. All of the content presented in this course has connections to the standards for mathematical practices. * This course includes the exponential and logarithmic functions as modeling tools. (PARCC Model Content Frameworks) UNIT 4 MODELING WITH FUNCTIONS Total Number of Days: 14 days (A and B days meet every other day) Grade/Course: 11/Algebra 2 ESENTIAL QUESTIONS What are the different types of functions? How can Geometric and Analytic representations be used to describe the behavior of the function? How are exponential functions and logarithmic functions related? PACING 1 1 CONTENT SKILLS Unit 4 Pre- Equation of parabola Assessmen Graphing functions t Average rate of change Writing functions Compare graphs of different functions Combine functions Transformations of graphs Exponential and logarithmic models 1. Write the equation of Equation parabolas using the of distance formula. Parabola EX: Q1 Find the equation of a parabola with focus (0,4) ENDURING UNDERSTANDINGS There are sets of functions, called families, in which each function is a transformation of a special function called the parent. You can use logarithms to solve exponential equations; and conversely, you can use exponents to solve logarithmic properties. You can translate periodic functions in the same way that you translate other functions. RESOURCES LEARNING STANDARDS Pearson ACTIVITIES/ASSESS Pearson (CCCS/MP) OTHER MENTS (e.g., tech) G.PE.2 Rewrite questions N.Q.2 similar to the Unit 4 F.IF.4, 6, 7, 8, 9 test. F.BF.1, 3 F.LE.4 G.PE.2 Section 10-2 Derive the equation of a parabola given a focus and directrix. Basic: Problems 1-5 EXS: 7-33, 38-42 even, 45, 55, 59-69 http://www.m athwarehouse. com/quadratic /parabola/foc us-anddirectrix-ofparabola.php http://swh.spr ALGEBRA LAB Solve it! Pg 622 and on Interactive Digital Path Here’s Why It Works Activity (paper folding) and directrix y = -3 Average: Problems 1-5 EXS: 7-33 odd, 34-55, 59-69 ingbranchisd.c om/LinkClick.a spx?fileticket= WxTPGm3oaY%3D&tabi d=16646 Advanced: Problems 1-5 EXS: 7-33 odd, 34-69 Last 1/3 of the period. 1 Basic Skills PARCC/HS PA Prep Distance Formula G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 1. Equation of Parabola Write the equation of parabolas in vertex form. G.PE.2 Section 10-2 Derive the equation of a parabola given a focus and directrix. Q2 Find the equation of a parabola with focus (0,4) and directrix x = -3 Basic: Problems 1-5 EXS: 7-33, 38-42 even, 45, 55, 59-69 EX: http://www.y outube.com/w atch?v=PuqdjX yBavY Average: Problems 1-5 EXS: 7-33 odd, 34-55, http://www.m athwarehouse. com/geometry /parabola/sta ndard-andvertexform.php Teachers Edition pg 623 ASSESSMENT • Identify the parts of a parabola • Explain how the distance formula relates to this • Find an equation using the distance formula Example: A 50 feet ladder is placed 35 feet away from a wall. The distance from the ground straight up to the top of the wall is 60 feet. Check whether the ladder reaches the top of the wall? ALGEBRA LAB Dynamic Activity on Interactive Digital Path ASSESSMENT • Find the equation of two parabolas in vertex form. • One should open up or down and the other open left or right. 59-69 Advanced: Problems 1-5 EXS: 7-33 odd, 34-69 Last 1/3 of the period. Basic Skills PARCC/HS PA Prep Real Number Systems 2. Graphing Functions Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. N.Q.2 Define appropriate quantities for the purpose of descriptive modeling. EX 1: You are given the graph below; create a word problem that matches the information labeled on the graph. F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description Q 3, 4, 5 2 N.Q.2 Define appropriate quantities for the purpose of descriptive modeling. http://www.m athsisfun.com/ decimalfractionpercentage.ht ml Section CB 24 http://www.m athsisfun.com/ sets/functionfloorceiling.html Page 90 & 91 Examples 1-5 http://www.m EXS: 1 athsisfun.com/ sets/functionspiecewise.html http://www.m athsisfun.com/ sets/functionexponential.ht ml http://a4a.lear nport.org/foru • At least on should open in the negative direction. • Explain how the focus and directrix will give a clue as to what direction the parabola opens in. Example: Which of the following is NOT a correct statement? a) 63% of 63 is less than 63. b) 115% of 63 is more than 63. c) 1/3% of 63 is the same as 1/3 of 63. d) 100% of 63 is equal to 63. ALGEBRA LAB Worksheets for practice available on http://www.ciclt.net/ul/ okresa/MATHEMATICS% 20II%20Unit%205%20St ep%20and%20Piecewise %20Functions.pdf ASSESSMENT Quiz - Graph a step, piecewise, and exponential function OR • Use a project to assess where students do the EX 2: The amount of snow in mm during a major snow storm is given by the function h(x) below, where x is the time in hours, 0 x 10 2 3 10 h x x 10 x 30 10 3 30 x 50 8 Graph the function h(x) on the coordinate plane below. Describe the change in the height of the snow on the ground during the 50-hour period. of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. b. Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. e. Graph exponential and logarithmic functions, showing intercepts and end behavior. MP 6, 4 m/topics/piec ewisefunction-cellphoneactivity?xg_sou rce=activity http://www.re gentsprep.org/ Regents/math /ALGEBRA/AE 7/ExpDecayL.h tm same thing but with more detailed explanation. Last 1/3 of the period. 2 Basic Skills PARCC/HS PA Prep Conversions N.Q.2 Define appropriate quantities for the purpose of descriptive modeling. 3. Average Calculate the average rate rate of of change for a function. change EX 1: Q 6, 7, 8 Find the average rate of change for the function f(x) = 5(3)x on intervals of length1, starting at 0. What do you observe about the rate of change? EX 2: A data set with equally spaced inputs is given. x Last 1/3 of the period. Basic Skills PARCC/HS PA Prep 2 3 5 1 1 3 y 2 6 2 Direct Variation F.IF.6 Calculate and interpret the 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. Textbook: Page 336 Q 20-23 Page 337 Q 35 Page 760 Q 17 http://www.y outube.com/w atch?v=XKCZn 5MLKvk Example: Average%20R ate%20of%20 Change/Avera ge%20Rate%2 0of%20Change .pdf http://earthmath.kennes aw.edu/main_site/revie w_topics/rate_of_change. htm http://www.y outube.com/w atch?v=iJ_0nP UUlOg http://www.y outube.com/w atch?v=iJ_0nP UUlOg&feature =youtu.be MP 1, 4, 5, 7 Change 75 km/hr to m/min. Show your process. ASSESSMENT Quiz – Rate of Change from a table and another from a graph. 7 4 2 A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Lesson 2-2 Pg 71-72 Q 25, 26, 34 http://courses. dcs.its.utexas.e du/speedwayfiles/highscho ol/ASKME/sha red/template.p hp?moviePath =../ALG-1A04321/flash/u nit02/u02tu08 propDire/&mo vieName=u02t u08propDire.s wf Example: The exchange rate from U.S. dollars to British pound sterling (£) was approximately $1.79 to £1 in 2004. Write and solve a direct variation equation to determine how many pounds sterling you would receive in exchange for US$90. 4. Writing functions Q 9, 10, 11 Write and explain a function to represent quantity. F.IF.8 EX 1: Write a function for the volume of the following shape. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. MP 7 r h 2 EX 2: The density, D, of the water in the ocean is related to the pressure, p, underwater and the height, ℎ of the water column in meters by 𝑝 the function 𝐷 = 𝑔ℎ where g is the acceleration due to gravity. (Facts: Density of sea seawater is about 1025 kg/m3, and g is about 10 m/s2) Textbook: Page 414 http://www.so phia.org/volu me-ofcompositefigures-3/volume-ofcompositefigures-tutorial Algebra Lab (for Example 1) For the geometric composite figures, have students find the volume of the composites to start. http://www.so phia.org/volu me-ofcompositefigures-3/volume-ofcompositefigures--5tutorial http://map.ma thshell.org/ma terials/downlo ad.php?fileid= 684 http://www.m athwarehouse. com/geometry /parabola/sta ndard-tovertexform.php Then have students look at the original formulas for each and combine the formulas without the numbers from the problem. Last have students use literal equations to rewrite the combined formulas for other variables. Last 1/3 of the period. Part of a period 2 Basic Skills PARCC/HS PA Prep Find the gravity, g, given the density and the pressure. Indicate the type of proportionality, direct or inverse, that relates height to pressure and density. EX 3: Rewrite the equation of the parabola f (x) = 2x 2 + x - 3 , in vertex form and find the vertex. Proportional Division MID UNIT TEST Mid unit test to determine areas of weakness that need to be addressed before the state unit test. 5. Compare graphs of different functions Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. G.PE.2 N.Q.2 F.IF.4, 6, 7, 8 F.IF.9 Compare properties of two functions each represented in a different http://www .cimt.plymo uth.ac.uk/pr ojects/mepr es/book8/b k8i7/bk8_7i 3.htm Teacher created material. Page 860 Q 1-8 http://a4a.lear nport.org/page /comparingfunctions http://a4a.lear Example: If three students share $180 in the ratio 1 : 2 : 3, how much is the largest share? Test to determine skills that need reteaching. Algebra Lab Give students different graphs to function. Have them Q 12, 13, 14 verbal descriptions) EX 1: Compare the graph of the two parabolas: - The graph of f is a parabola with vertex (0,0) and focus (6,0). - g:y = 1 2 x 4 way (algebraically, graphically, numerically in tables, or by verbal descriptions). For Example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F.BF.3 EX 2: Compare the following graph to the function () g(x) = 2cos x + 1. Include maximum, minimum, amplitudes, and periods. y x Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. MP 1, 3, 5, 6, 8 nport.org/page /comparingfunctions graph them on the graphing calculator and sketch them by hand. Have students describe the characteristics of each graph then compare the characteristics across the different graphs. Assessment Give students different graphs to function and ask them to list some key characteristics. Then have them compare pairs of functions. Last 1/3 of the period. 1 Basic Skills PARCC/HS PA Prep Angles with parallel lines 6. Combine functions Write a function that combines two relations or more. Q 15, 16, 17, 18 EX: Write a function that gives the area A, as a function of x for a square and 2 semicircles: G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. F.BF.1 Write a function that describes a relationship between two quantities. b. Combine standard function types using arithmetic operations. For Example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. http://www.re gentsprep.org/ Regents/math /geometry/GP 8/Lparallel.ht m Example: What is the value of x? http://www.m athwarehouse. com/geometry /angle/parallel -lines-cuttransversal.ph p http://scc.scds b.edu.on.ca/St udents/onlinec ourses/Sacche tto/AFIC%20w eb%20page/p df%20files/78%20Applicati ons%20of%20 Logs%20&%2 0Exp.pdf http://www.purplemath. com/modules/quadprob. htm http://www.nlreg.com/c ooling.htm Assessment Four questions, one of each related to the unit test questions. Last 1/3 of the period. Basic Skills PARCC/HS PA Prep 7. Transform ation of graphs Q 19, 20, 21 2 Triangle congruence. EX: Explain the differences between the two functions: f(X) = x2 g(X) = 3x2+4 G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. MP 3, 5, 8 http://www.re gentsprep.org/ Regents/math /geometry/GP 4/Ltriangles.ht m Example problem: ΔABC ≅ ΔDEF as shown. Find x. http://www.re gentsprep.org/ Regents/math /geometry/GP 4/PracCon2.ht m Section 7-2 Basic: Problems 1-5 EXS: 7-333, 35, 38-40 even, 44-62 Average: Problems 1-5 EXS: 7-29 odd, 31-41, 44-62 Advanced: Problems 1-5 EXS: 7-29 odd, 31-62 https://teache r.ocps.net/theo dore.klenk/ma thwebpage/me dia/calchorizo ntalandvertical stretches.pdfh ow http://www.y outube.com/w atch?v=kFw3X U0wisU Algebra Lab Give students different functions to graph the horizontal and vertical stretch and shrink. Students should come up with a set of rules so they understand what changes in the function create what changes in the graph. Assessment Quiz – three questions similar to the unit test. Last 1/3 of the period. Basic Skills PARCC/HS PA Prep Angles in circles 8. Exponenti al and logarithmi c models Solve exponential equations using logarithms. Q 22, 23, 24 2 EX: What is the value of x that satisfy the equation: 32x = 12 G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. F.LE.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. MP 4 Section 7-5 Basic: Problems 1-6 EXS: 7-45, 46-54 even, 60, 61, 84100 Average: Problems 1-6 EXS: 7-45 odd, 46-78, 84-100 Advanced: Problems 1-6 EXS: 7-45 odd, 46-83, 84-100 http://www.re gentsprep.org/ Regents/math /geometry/GG 2/CylinderPag e.htm Example problem: A cylinder and a cone each have a radius of 3 cm. and a height of 8 cm. What is the ratio of the volume of the cone to the volume of the cylinder? http://www.e du.gov.on.ca/e ng/studentsuc cess/lms/files/ tips4rm/mhf4 u_unit_5.pdf Algebra Lab Instruction problem 1 from Interactive Digial Path Lesson 75. http://www.youtube.co m/watch?v=5R5mKpLsf Yg Last 1/3 of the period. Part of a period Basic Skills PARCC/H SPA Prep Mixed problem solving problems based on unit basic skill review. A –CED.1, 2 A.SSE.A.1a 7.EE.3 A.APR.3 S.CP.9 Teacher created material. Quiz on Basic Skills END UNIT TEST End unit test to determine areas of weakness that need to be addressed before the state unit test. A.REI.6, 7 A.SSE.3 F.IF.4 Teacher created material. Test to determine skills that need reteaching. INSTRUCTIONAL FOCUS OF UNIT Derive the equation of a parabola given a focus and directrix. Graph functions that model relationships between two quantities, expressed symbolically, and show key features of the graph (including intercepts, intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity) by hand in simple cases and using technology for more complicated cases. Estimate, calculate and interpret the average rate of change of a function presented symbolically, in a table, or graphically over a specified interval. Rewrite a function in different but equivalent forms to identify and explain different properties of the function. Analyze and compare properties of two functions when each is represented in a different form (algebraically, graphically, numerically in tables, or by verbal descriptions). Construct a function that combines standard function types using arithmetic operations to model a relationship between two quantities. Identify and illustrate (using technology) an explanation of the effects on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. PARCC FRAMEWORK/ASSESSMENT OVERVIEW http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsAlgII_Nov2012V3_FINAL.pdf http://www.parcconline.org/sites/parcc/files/ESTableAlgebra2EOYforPARCC_FinalV2.pdf EXAMPLES https://sites.google.com/site/jacobsmathdepartment/parcc-assessments http://www.parcconline.org/samples/mathematics/high-school-mathematics 21ST CENTURY SKILLS (4Cs & CTE Standards) 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). 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 TEXTBOOK RELATED MATERIALS Guided Instruction – Virtual Nerd Videos available on the Interactive Digital Path (see Appendix for login site) Algebra 2 Companion – Vocabulary worktext to be used as the lesson is taught Reteaching Worksheet – Simplified explanations of the concepts with additional practice TOOLS Graphing Calculators where applicable Manipulatives where applicable Graphic Organizers Sketching the problem where applicable Interactive web sites You Tube STRATEGIES Highlight important ideas Pair or group activities Visual and graphic depictions of the problem Peer tutoring Formative assessments to determine need Frequent feedback to students Appropriate pacing of the material Allow adequate processing time Monitor student work and responses TEACHER WEB SITES FOR IDEAS http://nichcy.org/research/ee/math http://www.cehd.umn.edu/nceo/presentations/NCTMLEPIEPStrategiesMathGlossaryHandout.pdf http://floridarti.usf.edu/resources/format/pdf/Classroom%20Cognitive%20and%20Metacognitive%20Strategies%20for%20Teachers_Revised_SR_09.08.10.p df http://www2.edc.org/accessmath/resources/strategiestoollist.pdf http://www.glencoe.com/sec/teachingtoday/subject/intervention_strategies.phtml APPENDIX (Teacher resource extensions) E-Text, Interactive Digital Resources, Teacher Resources Login at https://www.pearsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true Mathematical Practices 1. Make sense of problems and persevere in solving them. SLO 6 Use more complex real-world context than those use in Algebra I. 2. Reason abstractly and quantitatively. SLO 3 Interpret the rate of change in context. SLO 8 Convert between exponential and logarithmic models 3. Construct viable arguments and critique the reasoning of others. SLO 4 Justify why two different forms of a function are equivalent. 4. Model with mathematics. * 5. Use appropriate tools strategically. SLO 7 Use technology when available. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. All of the content presented in this course has connections to the standards for mathematical practices. * This course includes the exponential and logarithmic functions as modeling tools. (PARCC Model Content Frameworks) UNIT 5 Inference and Conclusions from Data Total Number of Days: 11 days (A and B days meet every other day) Grade/Course: 11/Algebra 2 ESENTIAL QUESTIONS How can we gather, organize and display data to communicate and justify results in the real world? How can we analyze data to make inferences and/or predictions, based on surveys, experiments, probability and observational studies? PACING 1 CONTENT SKILLS Unit 5 Pre- Sample space, events Assessment outcomes, unions, intersections and complements Independent events Conditional probability and independence The definition of conditional probability. Random sample, statistic, parameter, population and sample The meaning of theoretical and experimental statistics Survey, an experiment, and an observational study ENDURING UNDERSTANDINGS You can describe and compare sets of data using various statistical measures, depending on what characteristics you want to study. Standard deviation is a measure of how far the numbers in a data set deviate from the mean. You can get good statistical information about population by studying a sample of the population. RESOURCES LEARNING STANDARDS Pearson ACTIVITIES/ASSE Pearson (CCCS/MP) OTHER SSMENTS (e.g., tech) S.CP.1, 2, 5, 6 Rewrite questions S.IC.1, 2, 3, 4, 5, 6 similar to the Unit 4 test. Mean, proportion and margin of error Conduct an experiment or simulation and the meaning of significance Describe the union and intersection of events, and the complement of an event. 1 1. Sample space, events outcomes, unions, EX: intersectio ns and complemen ts Q 1,2 Use the calendar above to list the outcomes in the events “a date after April 15 and not on a Monday or Saturday” Last 1/3 of the period. 1 S.CP.1 NONE Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and, ”not”). Find: List the outcomes of multiple events MP 2, 4 Basic Skills PARCC/HS PA Prep 3D Geometry G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. 2. Independe nt events Justify that two events are S.CP.2 independent through the use of Understand that two probability rules. events A and B are ALGEBRA 1 TEXT CB 125 http://www.y outube.com/ watch?v=2Rx W3UWpi2c http://www.v irtualnerd.co m/commoncore/hssstatisticsprobability/H SS-CPconditionalprobabilityrules/A/1/sa mple-spacedefinition http://www.virtualne rd.com/commoncore/hss-statisticsprobability/HSS-CPconditionalprobabilityrules/A/2/dependentindependent-eventsExample http://www. mathopenref. com/cubevol ume.html Example: If a cube has a volume of 64cu.cm, the length of ONE edge would = A. 6 cm. B. 4 cm. C. 8 cm. D. 16 cm. http://www.y outube.com/ watch?v=WD P_O3msUXk http://www.illustrati vemathematics.org/ill ustrations/950 Q 3,4,5 EX: 50 g 100 g 200 g Total independent if the probability of A and B occurring together is Tea Packe Instan Tota the product of their bag t19 tea t4tea l probabilities, and use 3 s 34 0 59 this characterization to 16 determine if they are 24 100 independent. Use the frequency table above to find the probability that a person buys: a) Instant tea b) 100g tea c) 200 g Tea bags c) 50g packet tea EX 2: The table below shows the enrollment in art and biology classes at a small school. Enrolled in Biology classes Did not enroll in biology classes Enrolled in Art classes Did not enroll in art classes x 54 45 81 What is the value of x if the events “selected student is S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table Section 11-4 Problems 14 Basic: EXS: 8-26, 34-49 Average: EXS: 9-19 odd, 20-31, 34-49 Advanced: EXS: 9-19 odd, 20-49 http://www.cpalms.o rg/RESOURCES/URLr esourcebar.aspx?Reso urceID=vRQBCoxzGfo =D enrolled in Art classes” and “selected student is enrolled in Biology classes” are independent? Show your work. as a sample space to decide if events are independent and to approximate conditional probabilities. For Example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. MP 2, 4 Last 1/3 of the period. Basic Skills PARCC/HS PA Prep Maximize Area G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with http://cims.n yu.edu/~kiryl /Calculus/Sec tion_4.5-Optimization %20Problems /Optimization _Problems.pdf Example: John has 100 feet of fencing and wants to fence off the largest possible space. John says a circle would be best, but his cousin Jack says a square typographic grid systems based on ratios). 3. Conditional probability and independe nce. Recognize conditional probability and independence in everyday situations. EX 1: Q 6,7,8 1 A checkerboard has 64 sectors of equal size, with 32 white sectors and 32 black sectors. When throwing a coin on the board, the coin is equally likely to land on a white sector or a black sector. A boy throws a coin four times and lands on black. He guesses that the next time the coin will land on black. Is his guessing accurate? Why or why not? would give you the largest space. Who is correct? Draw diagrams, show work, and explain your solution. S.CP.5 Sect 11-4 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For Example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Problems 14 MP 2, 4 Basic: EXS: 8-26, 34-49 Average: EXS: 9-19 odd, 20-31, 34-49 Advanced: EXS: 9-19 odd, 20-49 http://www.youtube.c om/watch?v=tbBW2V VFgso EX 2: There are 6 black pens and 8 blue pens in a jar. If you take a pen without looking and then take another pen without replacing the first. Explain why the two events are not independent. Describe the change that could make them independent. Last 1/3 of the period. 1 Basic Skills PARCC/HS PA Prep 3D Geometry G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. 4. The definition of conditional probability. Compute the probability of A or B using the Addition Rule for probability. S.CP.6 Section 11-3 Problems 15 Q 9,10,11 A bag contains 4 red, 2 blue, 6 green, and 8 white marbles. What is the probability of selecting a white marble at Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. EX 1: http://www.v irtualnerd.co m/prealgebra/peri meter-areavolume/volu me/volumeExamples/cyli nder-heightfrom-volume Basic: EXS: 9-37, 45-62 Average: Example: Mr. Braunsdorf has a circular above ground swimming pool. If the 20 ft diameter pool holds 1256 ft3 of water, how deep is it? (Use 7r = 3.14) http://www.mathgoo dies.com/lessons/vol 6/addition_rules.html random from this bag, not replacing the marble, and then selecting another white marble? Round your answer to the nearest tenth of a percent if necessary. EX 2: Last 1/3 of the period. 1 S.CP.7 EXS: 9-29 odd, 31-42, 45-62 Apply the Addition Rule, P(A or B) = P(A) + Advanced: P(B) – P(A and B), and EXS: 9-29 interpret the answer in odd, 31-62 terms of the model. You shuffle a standard deck of playing cards and choose a card at random. What is the probability that you choose a face card (jack, queen, king, or a club)? MP 1, 2 Basic Skills PARCC/HS PA Prep Midpoint Formula 5. Random sample, statistic, parameter, Make inferences about a population from a random sample G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). S.IC.1 Section 11-7 Understand statistics as a process for Problems 13 http://www.s ophia.org/ap plying-themidpointformula-withoneendpoint/app lying-themidpointformula-withone-endpointtutorial Example: Given: M (-1,-2) is the midpoint of AB where A is (-4,2). Find the coordinates of the other endpoint, B. http://www.sophia.or g/standard-deviationtutorial population and sample Q 12,13 EX: making inferences about population The points scored in each game parameters based on a played by the boys’ and girls’ random sample from basketball teams last season that population. are given, MP 1, 2 Boys Team: 56, 81, 80, 75, 48, 65, 90, 66, 70, 70 Girls Team: 60, 72, 61, 58, 78, 65, 66, 55, 65, 73 1 Basic Skills PARCC/HS PA Prep Shaded Region 6. The meaning of theoretical and Design a simulation that models a desired event EX: Average: EXS: 7-13 odd, 14-21, 24-37 Advanced: EXS: 7-13 odd, 14-37 Interpret the data as to which team is more consistent in their scoring (use the standard deviation). Last 1/3 of the period. Basic: EXS: 6-13, 15, 17, 20, 21, 24-27 G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). S.IC.2 Section 11-2 Decide if a specified model is consistent Problem 1, 2, 3, 5 http://www.o nlinemathlear ning.com/are a-shadedregion.html Example: Find the area of a given shaded region. http://www.sophia.or g/standard-deviationtutorial experiment al statistics Suppose we throw a coin 10 times, and we only see heads 3 Q 14, 15 times. What can we say about the fairness of this coin? with results from a given data-generating process, e.g., using simulation. For Example, a model says a spinning coin falls heads up with probability 0. 5. Would a result of 5 tails in a row cause you to question the model? MP 2, 4 Last 1/3 of the period. 1 Basic: EXS: 8-28, 31-35, 3751 Average: EXS: 9-27 odd 28-35, 37-51 Advanced: EXS: 9-27 odd, 28-36, 37-51 Basic Skills PARCC/HS PA Prep Scientific Notation N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 7. Survey, an experiment , and an observatio nal study Recognize a survey, an experiment and an observational study S.IC.3 Section 11-8 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how Problem 1-3 Q 16, 17,18 EX: Jason is interested in finding the number of students who would be willing to donate $10 http://www.c hem.tamu.edu /class/fyp/m athrev/mrscnot.html Basic: EXS: 6-12, 15-19, 23, 26-38 Example: The speed of light is approximately 299.8 million meters per second. What is that speed in scientific notation form? http://www.regentsp rep.org/Regents/math /algtrig/ATS1/StatSur veylesson.htm or an hour of time to help a local food bank? Explain how can randomization be applied? randomization relates to each. MP 1, 2 Average: EXS: 7-11 odd, 15-19, 21-23, 2638 Advanced: EXS: 7-11 odd, 16-19, 23, 24, 2638 Last 1/3 of the period. 2 Basic Skills PARCC/HS PA Prep Ordering Rational Numbers 8. Mean, proportion and margin of error EX: The midterm scores for 20 random students (in a class of 100): Q 19,20 82 45 37 98 100 74 87 89 63 76 75 61 43 99 86 75 92 65 80 86 Estimate the mean score of all students and identify the range of scores within 2 standard deviations of the mean. N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. S.IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. MP 1, 2 http://www.e tap.org/demo /Algebra1/les son2/instruct ion4tutor.htm l Section 11-7 Problem 1-3 Basic: EXS: 6-13, 15, 17, 20, 21, 24-27 Average: EXS: 7-13 odd, 14-21, 24-37 Advanced: EXS: 7-13 odd, 14-37 Example: Arrange the following numbers in order from LEAST to GREATEST: 1/3, 2/5, 0.6, 0.125 http://psy2.ucsd.edu/ ~dhuber/ch5_hays.pd f http://www.regentsp rep.org/Regents/math /algtrig/ATS1/Disper sion.htm Last 1/3 of the period. 2 Basic Skills PARCC/HS PA Prep Angles 9. Conduct an experiment or simulation and the meaning of significanc e Decide if the differences between parameters are significant EX: Q 21, 22 G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). S.IC.5 (Use Algebra 1 Text Resources Section 124) Use data from a randomized experiment to compare two treatments; use simulations to decide if Section 11-6 differences between parameters are significant. Problem 1-5 S.IC.6 1. The heights, in millimeters, of 10 seedlings from 2 seed types are shown. Seed A: 56, 61, 48, 51, 59, 65, http://www.f reemathhelp.c om/felizanglestriangle.html Evaluate reports based on data. MP 1, 2 Basic: EXS: 7-23, 25, 30-43 Average: EXS: 7-15 Example: If A is a right angle, and m B = 43o, then m C = http://www.regentsp rep.org/Regents/math /algtrig/ATS1/Central Tendency.htm 49, 71, 69, 64 Seed B: 55,63, 58, 47, 61, 46, 53, 47, 41, 59 Make box-and-whisker plots comparing the samples. Which seed type is, on average, taller? END UNIT Part of a TEST period End unit test to determine areas of weakness that need to be addressed before the state unit test. odd, 16-27, 30-43 Advanced: EXS: 7-15 odd, 16-43 S.CP. 1-7 S.IC.1-6 Teacher created material. Test to determine skills that need reteaching. INSTRUCTIONAL FOCUS OF UNIT Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,””not”). Use two-way frequency tables to determine if events are independent and to calculate/approximate conditional probability. Use everyday language to explain independence and conditional probability in real-world situations. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A and apply the addition [P(A or B) = P(A) + P(B) – P(A and B)] rule of probability in a uniform probability model; interpret the results in terms of the model. Make inferences about population parameters based on a random sample from that population. Determine if the outcomes and properties of a specified model are consistent with results from a given data-generating process using simulation. Identify different methods and purposes for conducting sample surveys, experiments, and observational studies and explain how randomization relates to each. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. Use data from a randomized experiment to compare two treatments and use simulations to decide if differences between parameters are significant; evaluate reports based on data. PARCC FRAMEWORK/ASSESSMENT OVERVIEW http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsAlgII_Nov2012V3_FINAL.pdf http://www.parcconline.org/sites/parcc/files/ESTableAlgebra2EOYforPARCC_FinalV2.pdf EXAMPLES https://sites.google.com/site/jacobsmathdepartment/parcc-assessments http://www.parcconline.org/samples/mathematics/high-school-mathematics 21ST CENTURY SKILLS (4Cs & CTE Standards) 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). 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 APPENDIX (Teacher resource extensions) E-Text, Interactive Digital Resources, Teacher Resources Login at https://www.pearsonsuccessnet.com/snpapp/login/login.jsp?showLoginPage=true Probability concepts: http://www.marlboro.edu/academics/study/mathematics/courses/probability Probability and statistics problems: http://learn.tkschools.org/mwilkinson/Algebra%20II/alg2%20Chapter%2012%20Notes.pdf Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. SLO 6 Compare theoretical and empirical data. 3. Construct viable arguments and critique the reasoning of others. SLO 7 Explain when and why you would use a sample survey, experiment, or an observational study; develop the meaning of statistical significance. 4. Model with mathematics.* 5. Use appropriate tools strategically. 6. Attend to precision. SLO 9 Examine the scope and nature of conclusions drawn in the reports. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. All of the content presented in this course has connections to the standards for mathematical practices. *This course includes the exponential and logarithmic functions as modeling tools. (PARCC Model Content Frameworks)