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Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Algebra 2 Standards Aligned With the Algebra 2 PARCC Assessment Performance Based Assessment (PBA/MYA) and End Of Year Assessment (EOY) Cluster Standard PBA/MYA Unit 1: Inferences and Conclusions from Data HS.S.ID.A Summarize, represent, and interpret data on a single count of measurement variable. HS.S.IC.A Understand and evaluate random processes underlying statistical experiments. HS.S.IC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies. HS.S.MD.B Use probability to evaluate outcomes of decisions. EOY HS.S.ID.A.4 X X HS.S.IC.A.2 X X HS.S.IC.A.1 HS.S.IC.B.3 HS.S.IC.B.4 HS.S.IC.B.5 HS.S.IC.B.6 HS.S.MD.B.6 HS.S.MD.B.7 X X X X X X Page | 1 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Algebra 2 Standards Aligned With the Algebra 2 PARCC Assessment Performance Based Assessment (PBA/MYA) and End Of Year Assessment (EOY) Cluster Standard PBA/MYA Unit 2: Polynomial, Rational, and Radical Relationships HS.A.SSE.A Interpret the structure of expressions. HS.A.SSE.B Write expressions in equivalent forms to solve problems. HS.N.CN.A Perform arithmetic operations with complex numbers. HS.N.CN.C Use complex numbers in polynomial identities and equations. HS.F.IF.C Analyze functions using different representations. HS.A.APR.A Performa arithmetic operations on polynomials. HS.A.APR.B Understand the relationship between zeros and factors of polynomials. HS.A.APR.C Use polynomial identities to solve problems. HS.A.APR.D Review rational expressions. HS.A.SSE.A.1 EOY HS.A.SSE.A.2 X X HS.N.CN.A.1 X X HS.A.SSE.B.4 HS.N.CN.A.2 HS.N.CN.C.7 HS.N.CN.C.8 HS.N.CN.C.9 HS.F.IF.C.7 X X X X X HS.A.APR.A.1 X HS.A.APR.B.3 X HS.A.APR.B.2 HS.A.APR.C.4 HS.A.A.PR.C.5 HS.A.APR.D.6 HS.A.APR.D.7 HS.A.REI.A Understand solving equations as a process of reasoning HS.A.REI.A.2 and explain the reasoning. HS.A.REI.D Represent and solve equations and inequalities HS.A.REI.D.11 graphically. Unit 3: Trigonometric Functions HS.F.TF.A HS.F.TF.A.1 Extend the domain of trigonometric functions using the HS.F.TF.A.2 unit circle. HS.F.TF.B Model period phenomena with trigonometric HS.F.TF.B.5 functions. HS.F.TF.C HS.F.TF.C.8 Prove and apply trigonometric identities. X X X X X X X X X X X X X X X X X X X Page | 2 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Algebra 2 Standards Aligned With the Algebra 2 PARCC Assessment Performance Based Assessment (PBA/MYA) and End Of Year Assessment (EOY) Cluster Standard Unit 4: Modeling with Functions HS.A.CED.A Create equations that describe numbers or relationships. HS.F.BF.A Build a function that models a relationship between two quantities. HS.F.IF.B Interpret functions that arise in applications in terms of a context. HS.F.IF.C Analyze functions using different representations. HS.F.BF.B Build new functions from existing functions. HS.F.LE.A Construct and compare linear, quadratic, and exponential models and solve problems. PBA/MYA HS.A.CED.A.1 X HS.A.CED.A.2 HS.A.CED.A.3 HS.A.CED.A.4 HS.F.BF.A.1 HS.F.IF.B.4 HS.F.IF.B.5 HS.F.IF.B.6 HS.F.IF.C.7 X HS.F.IF.C.8 X HS.F.BF.B.4 X HS.F.IF.C.9 HS.F.BF.B.3 HS.F.LE.A.4 EOY X X X X X X X X Page | 3 Curriculum Guide 2014-2015 High School Algebra 2 Unit Overview Poudre School District Building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include polynomial, rational, and radical functions. Students work closely with the expressions that define the functions, and continue to expand and hone their abilities to model situations and to solve equations, including solving quadratic equations over the set of complex numbers and solving exponential equations using the properties of logarithms. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas for this course, organized into four units are as follows: Unit 1: Inferences and Conclusions from Data In this unit, students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They identify different ways of collecting data – including sample surveys, experiments, and simulations – and the role that randomness and careful design play in the conclusions that can be drawn. Students will be able to… • Summarize, represent, and interpret data on a single count of measurement variable. HS.S.ID.A.4 • Understand and evaluate random processes underlying statistical experiments. HS.S.IC.A.1, HS.S.IC.A.2 • Make inferences and justify conclusions from sample surveys, experiments, and observational studies. HS.S.IC.B.3, HS.S.IC.B.4, HS.S.IC.B.5, HS.S.IC.B.6 • Use probability to evaluate outcomes of decisions. HS.S.MD.B.6, HS.S.MD.B.7 Page | 4 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 2: Polynomial, Rational, and Radical Relationships This unit develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. The unit culminates with the fundamental theorem of algebra. Rational numbers extend the arithmetic of integers by allowing division by all numbers except 0. Similarly, rational expressions extend the arithmetic of polynomials by allowing division by all polynomials except the zero polynomial. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Students will be able to… • Interpret the structure of expressions. HS.A.SSE.A.1, HS.A.SSE.A.2 • Perform arithmetic operations with complex numbers. HS.N.CN.A.1, HS.N.CN.A.2 • Use complex numbers in polynomial identities and equations. HS.N.CN.C.7, HS.N.CN.C.8, HS.N.CN.C.9 • Analyze functions using different representations. HS.F.IF.C.7 • Perform arithmetic operations on polynomials. HS.A.APR.A.1 • Understand the relationship between zeros and factors of polynomials. HS.A.APR.B.2, HS.A.APR.B.3 • Use polynomial identities to solve problems. HS.A.APR.C.4, HS.A.APR.C.5 • Review rational expressions. HS.A.APR.D.6, HS.A.APR.D.7 • Understand solving equations as a process of reasoning and explain the reasoning. HS.A.REI.A.2 • Write expressions in equivalent forms to solve problems. HS.A.SSE.B.4 • Represent and solve equations and inequalities graphically. HS.A.REI.D.11 Unit 3: Trigonometric Functions Building on their previous work with functions, and on their work with trigonometric ratios and circles in Geometry, students now use the coordinate plane to extend trigonometry to model periodic phenomena. Students will be able to… • Extend the domain of trigonometric functions using the unit circle. HS.F.TF.A.1, HS.F.TF.A.2 • Model periodic phenomena with trigonometric functions. HS.F.TF.B.5 • Prove and apply trigonometric identities. HS.F.TF.C.8 Page | 5 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 4: Modeling with Functions In this unit students synthesize and generalize what they have learned about a variety of function families. They extend their work with exponential functions to include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying function. They identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as “the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions” is at the heart of this unit. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling context. Students will be able to… • Create equations that describe numbers or relationships. HS.A.CED.A.1, HS.A.CED.A.2, HS.A.CED.A.3, HS.A.CED.A.4 • Build a function that models a relationship between two quantities. HS.F.BF.A.1 • Interpret functions that arise in applications in terms of a context. HS.F.IF.B.4, HS.F.IF.B.5, HS.F.IF.B.6 • Analyze functions using different representations. HS.F.IF.C.7, HS.F.IF.C.8, HS.F.IF.C.9 • Build new functions from existing functions. HS.F.BF.B.3, HS.F.BF.B.4 • Construct and compare linear, quadratic, and exponential models and solve problems. HS.F.LE.A.4 Page | 6 Curriculum Guide 2014-2015 High School Algebra 2 Unit 1: Inferences and Conclusions from Data Poudre School District In this unit, students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They identify different ways of collecting data – including sample surveys, experiments, and simulations – and the role that randomness and careful design play in the conclusions that can be drawn. Sub-Unit A: The Normal Curve HS.S.ID.A.4 Sub-Unit B: Data and Populations HS.S.IC.A.1 HS.S.IC.A.2 HS.S.IC.B.3 HS.S.IC.B.4 HS.S.IC.B.5 HS.S.IC.B.6 HS.S.MC.B.6 HS.S.MD.B.7 Page | 7 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 1: Inferences and Conclusions from Data Sub-Unit A: The Normal Curve August 25, 2014 – August 29, 2014 Common Core State Standards Explanations/Examples Resources HS.S.ID.A: Summarize, represent, and interpret data on a single count or measurement variable. While students may have heard of the normal distribution, it is unlikely that they will have prior experience using it to make specific estimates. Build on students’ understanding of data distributions to help them see how the normal distribution uses area to make estimates of frequencies (which can be expressed as probabilities). Emphasize that only some data are well described by a normal distribution. HS.S.ID.A.4 HS.S.ID.A.4 Lessons Use the mean and standard deviation of a data Students may use spreadsheets, graphing calculators, HS.S.ID.A set to fit it to a normal distribution and to statistical software and tables to analyze the fit Prentice Hall estimate population percentages. Recognize that between a data set and normal distributions and • PH A2 there are data sets for which such a procedure is estimate areas under the curve. o Ch 12.7 not appropriate. Use calculators, spreadsheets, • The bar graph below gives the birth weight of a and tables to estimate areas under the normal The Practice of Statistics population of 100 chimpanzees. The line shows curve. how the weights are normally distributed about the • Ch 2.2 mean, 3250 grams. Estimate the percent of baby chimps weighing 3000-3999 grams. Tasks Activities Practice Assessments Page | 8 Curriculum Guide 2014-2015 Common Core State Standards HS.S.ID.A.4 (continued) High School Algebra 2 Unit 1: Inferences and Conclusions from Data Sub-Unit A: The Normal Curve August 25, 2014 – August 29, 2014 Explanations/Examples HS.S.ID.A.4 (continued) • Poudre School District Resources Determine which situation(s) is best modeled by a normal distribution. Explain your reasoning. o o Annual income of a household in the U.S. Weight of babies born in one year in the U.S. Page | 9 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 1: Inferences and Conclusions from Data Sub-Unit B: Data and Populations September 2, 2014 – October 3, 2014 Common Core State Standards Explanations/Examples Resources HS.S.IC.A: Understand and evaluate random processes underlying statistical experiments. For S.IC.2, include comparing theoretical and empirical results to evaluate the effectiveness of a treatment. HS.S.IC.A.1 HS.S.IC.A.1 Lessons Understand statistics as a process for making HS.S.IC.A inferences to be made about population The Practice of Statistics parameters based on a random sample from that • Ch 4.1 population. Tasks HS.S.IC.A.2 HS.S.IC.A.2 Decide if a specified model is consistent with Possible data-generating processes include (but are Activities results from a given data-generating process, e.g., not limited to): flipping coins, spinning spinners, using simulation. For example, a model says a rolling a number cube, and simulations using the Practice spinning coin will fall heads up with probability random number generators. Students may use 0.5. Would a result of 5 tails in a row cause you to graphing calculators, spreadsheet programs, or applets Assessments question the model? to conduct simulations and quickly perform large numbers of trials. The law of large numbers states that as the sample size increases, the experimental probability will approach the theoretical probability. Comparison of data from repetitions of the same experiment is part of the model building verification process. • Have multiple groups flip coins. One group flips a coin 5 times, one group flips a coin 20 times, and one group flips a coin 100 times. Which group’s results will most likely approach the theoretical probability? Page | 10 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 1: Inferences and Conclusions from Data Sub-Unit B: Data and Populations September 2, 2014 – October 3, 2014 Common Core State Standards Explanations/Examples Resources HS.S.IC.B: Make inferences and justify conclusions from sample surveys, experiments, and observational studies. In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make comparisons. These ideas are revisited with a focus on how they way in which data is collected determines the scope and nature of the conclusions that can be drawn from the data. The concept of statistical significance is developed is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment. For S.IC.4 and 5, focus on the variability of results from experiments – that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness. HS.S.IC.B.3 HS.S.IC.B.3 Lessons Recognize the purposes of and differences among Students should be able to explain HS.S.IC.B sample surveys, experiments, and observational techniques/applications for randomly selecting study The Practice of Statistics studies; explain how randomization relates to subjects from a population and how those • Ch 4.1 each. techniques/applications differ from those used to randomly assign existing subjects to control groups or HS.S.IC.B.4/HS.S.IC.B.5/HS.S.IC.B.6 experimental groups in a statistical experiment. Prentice Hall • PH A2 In statistics, an observational study draws inferences o Ch 12.3 about the possible effect of a treatment on subjects, Tasks where the assignment of subjects into a treated group HS.S.IC.B.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. versus a control group is outside the control of the investigator (for example, observing data on academic achievement and socio-economic status to see if there is a relationship between them). This is in contrast to controlled experiments, such as randomized controlled trials, where each subject is randomly assigned to a treated group or a control group before the start of the treatment. HS.S.IC.B.4 Students may use computer generated simulation models based upon sample surveys results to estimate population statistics and margins of error. Activities Practice Assessments Page | 11 Curriculum Guide 2014-2015 Common Core State Standards HS.S.IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. HS.S.IC.B.6 Evaluate reports based on data. High School Algebra 2 Unit 1: Inferences and Conclusions from Data Sub-Unit B: Data and Populations September 2, 2014 – October 3, 2014 Explanations/Examples HS.S.IC.B.5 Students may use computer generated simulation models to decide how likely it is that observed differences in a randomized experiment are due to chance. Poudre School District Resources Treatment is a term used in the context of an experimental design to refer to any prescribed combination of values of explanatory variables. For example, one wants to determine the effectiveness of weed killer. Two equal parcels of land in a neighborhood are treated; one with a placebo and one with weed killer to determine whether there is a significant difference in effectiveness in eliminating weeds. HS.S.IC.B.6 Explanations can include but are not limited to sample size, biased survey sample, interval scale, unlabeled scale, uneven scale, and outliers that distort the line-ofbest-fit. In a pictogram the symbol scale used can also be a source of distortion. As a strategy, collect reports published in the media and ask students to consider the source of the data, the design of the study, and the way the data are analyzed and displayed. • A reporter used the two data sets below to calculate the mean housing price in Arizona as $629,000. Why is this calculation not representative of the typical housing price in Arizona? Page | 12 Curriculum Guide 2014-2015 Common Core State Standards HS.S.IC.B.5 (continued) High School Algebra 2 Poudre School District Unit 1: Inferences and Conclusions from Data Sub-Unit B: Data and Populations September 2, 2014 – October 3, 2014 Explanations/Examples HS.S.IC.B.5 (continued) o King River area {1.2 million, 242000, 265500, 140000, 281000, 265000, 211000} o Resources Toby Ranch homes {5 million, 154000, 250000, 250000, 200000, 160000, 190000} HS.S.MD.B: Use probability to evaluate outcomes of decisions. Extend to more complex probability models. Include situations such as those involving quality control, or diagnostic tests that yield both false positive and false negative results. HS.S.MD.B.6 (+) HS.S.MD.B.6 (+) Lessons Use probabilities to make fair decisions Students may use graphing calculators or programs, HS.S.MD.B (e.g., drawing by lots, using a random number spreadsheets, or computer algebra systems to model Prentice Hall generator). and interpret parameters in linear, quadratic or • PH A2 exponential functions. o Ch 12.1 HS.S.MD.B.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). HS.S.MD.B.7 (+) Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to model and interpret parameters in linear, quadratic or exponential functions. Tasks Activities Practice Assessments Page | 13 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 2: Polynomial, Rational, and Radical Relationships This unit develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. The unit culminates with the fundamental theorem of algebra. Rational numbers extend the arithmetic of integers by allowing division by all numbers except 0. Similarly, rational expressions extend the arithmetic of polynomials by allowing division by all polynomials except the zero polynomial. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Sub-Unit A: Complex Numbers HS.A.SSE.A.1 HS.N.CN.A.1 HS.N.CN.A.2 HS.A.SSE.A.2 HS.N.CN.C.8 HS.N.CN.C.7 HS.F.IF.C.7 HS.N.CN.C.9 Page | 14 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 2: Polynomial, Rational, and Radical Relationships Sub-Unit B: Polynomial Functions HS.APR.A.1 HS.A.APR.B.3 HS.A.APR.B.2 HS.A.APR.C.4 HS.A.APR.C.5 Sub-Unit C: Rational Functions HS.A.REI.A.2 HS.A.APR.D.7 HS.A.APR.D.6 HS.A.SSE.B.4 Sub-Unit D: Solving Systems HS.A.REI.D.11 Page | 15 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 2: Polynomials, Rational and Radical Relationships Sub-Unit A: Complex Numbers October 6, 2014 – October 31, 2014 Common Core State Standards Explanations/Examples Resources HS.A.SSE.A: Interpret the structure of expressions. Extend to polynomial and rational expressions. HS.A.SSE.A.1 HS.A.SSE.A.1 Lessons Interpret expressions that represent a quantity in Students should understand the vocabulary for the HS.A.SSE.A terms of its context. parts that make up the whole expression and be able Prentice Hall to identify those parts and interpret their meaning in a. Interpret parts of an expression, such as • PH A2 terms of a context. terms, factors, and coefficients. o Ch 5 b. Interpret complicated expressions by o Ch 6.1-6.5 viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the Tasks product of P and a factor not depending on P. HS.A.SSE.A.2 HS.A.SSE.A.2 Activities Use the structure of an expression to identify Students should extract the greatest common factor ways to rewrite it. For example, (whether a constant, a variable, or a combination of Practice see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a each). If the remaining expression is quadratic, difference of squares that can be factored as students should factor the expression further. Assessments (x2 – y2)(x2 + y2). • Factor HS.N.CN.A: Perform arithmetic operations with complex numbers. HS.N.CN.A.1 HS.N.CN.A.1 Lessons Know there is a complex number i such that HS.N.CN.A i2 = −1, and every complex number has the form Prentice Hall a + bi with a and b real. • PH A2 HS.N.CN.A.2 HS.N.CN.A.2 o Ch 5.6 Use the relation i2 = –1 and the commutative, • Simplify the following expression. Justify each associative, and distributive properties to add, Tasks step using the commutative, associative and subtract, and multiply complex numbers. distributive properties. Activities Practice Page | 16 Curriculum Guide 2014-2015 Common Core State Standards HS.N.CN.A.2 (continued) High School Algebra 2 Unit 2: Polynomials, Rational and Radical Relationships Sub-Unit A: Complex Numbers October 6, 2014 – October 31, 2014 Explanations/Examples HS.N.CN.A.2 (continued) Assessments Solutions may vary; one solution follows: Poudre School District Resources HS.N.CN.C: Use complex numbers in polynomial identities and equations. Limit to polynomials with real coefficients. HS.N.CN.C.7 HS.N.CN.C.7 Lessons Solve quadratic equations with real coefficients HS.N.CN.7/HS.N.CN.8 • Within which number system can x2 = – 2 be that have complex solutions. Prentice Hall solved? Explain how you know. • PH A2 • Solve x2+ 2x + 2 = 0 over the complex numbers. o Ch 5.6-5.8 • Find all solutions of 2x2 + 5 = 2x and express them in the form a + bi. HS.N.CN.8 HS.N.CN.C.8 HS.N.CN.C.8 Prentice Hall Extend polynomial identities to the complex • PH A2 numbers. For example, rewrite x2 + 4 as o Ch 6.4-6.5 (x + 2i)(x – 2i). Page | 17 Curriculum Guide 2014-2015 High School Algebra 2 Unit 2: Polynomials, Rational and Radical Relationships Sub-Unit A: Complex Numbers October 6, 2014 – October 31, 2014 Common Core State Standards Explanations/Examples HS.N.CN.C.9 HS.N.CN.C.9 Know the Fundamental Theorem of Algebra; • How many zeros does have? Find all show that it is true for quadratic polynomials. the zeros and explain, orally or in written format, your answer in terms of the Fundamental Theorem of Algebra. • How many complex zeros does the following polynomial have? How do you know? Poudre School District Resources Lessons (continued) HS.N.CN.9 Prentice Hall • PH A2 o Ch 6.6 Tasks Activities Practice Assessments HS.F.IF.C: Analyze functions using different representations. Relate F.IF.7c to the relationship between zeros of quadratic functions and their factored forms. HS.F.IF.C.7 HS.F.IF.C.7 Lessons Graph functions expressed symbolically and Key characteristics include but are not limited to HS.F.IF.C.7 show key features of the graph, by hand in simple maxima, minima, intercepts, symmetry, end behavior, Prentice Hall cases and using technology for more complicated and asymptotes. Students may use graphing • PH A2 cases. calculators or programs, spreadsheets, or computer o Ch 5.2-5.4 c. Graph polynomial functions, identifying zeros algebra systems to graph functions. o Ch 6.1-6.2 when suitable factorizations are available, o p. 312 • Describe key characteristics of the graph of and showing end behavior. f(x) = │x – 3│ + 5. Tasks • Sketch the graph and identify the key characteristics of the function described below. Activities Practice Assessments Page | 18 Curriculum Guide 2014-2015 Common Core State Standards HS.F.IF.C.7 (continued) High School Algebra 2 Unit 2: Polynomials, Rational and Radical Relationships Sub-Unit A: Complex Numbers October 6, 2014 – October 31, 2014 Explanations/Examples HS.F.IF.C.7 (continued) • • • Poudre School District Resources Graph the function f(x) = 2x by creating a table of values. Identify the key characteristics of the graph. Graph f(x) = 2 tan x – 1. Describe its domain, range, intercepts, and asymptotes. Draw the graph of f(x) = sin x and f(x) = cos x. What are the similarities and differences between the two graphs? Page | 19 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 2: Polynomials, Rational and Radical Relationships Sub-Unit B: Polynomial Functions November 3, 2014 – December 12, 2014 Common Core State Standards Explanations/Examples Resources HS.A.APR.A: Perform arithmetic operations on polynomials. Extend beyond the quadratic polynomials found in Algebra 1. HS.A.APR.A.1 HS.A.APR.A.1 Lessons Understand that polynomials form a system HS.A.APR.A.1 analogous to the integers, namely, they are Prentice Hall closed under the operations of addition, • PH A2 subtraction, and multiplication; add, subtract, o Ch 5.6 and multiply polynomials. o Ch 6.1-6.5 Tasks Activities Practice Assessments HS.A.APR.B: Understand the relationship between zeros and factors of polynomials. HS.A.APR.B.2 HS.A.APR.B.2 Lessons Know and apply the Remainder Theorem: For a The Remainder theorem says that if a polynomial p(x) is HS.A.APR.B.2 polynomial p(x) and a number a, the remainder divided by x – a, then the remainder is the constant p(a). Prentice Hall on division by x – a is p(a), so p(a) = 0 if and only That is, So if p(a) = 0 then p(x) • PH A2 if (x – a) is a factor of p(x). o Ch 6.3 = q(x)(x-a). • Let . Evaluate p(-2). What does your answer tell you about the factors of p(x)? [Answer: p(-2) = 0 so x+2 is a factor.] HS.A.APR.B.3 Prentice Hall • PH A2 o Ch 6.4 Tasks Page | 20 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 2: Polynomials, Rational and Radical Relationships Sub-Unit B: Polynomial Functions November 3, 2014 – December 12, 2014 Common Core State Standards Explanations/Examples Resources HS.A.APR.B.3 HS.A.APR.B.3 Activities Identify zeros of polynomials when suitable Graphing calculators or programs can be used to Practice factorizations are available, and use the zeros to generate graphs of polynomial functions. construct a rough graph of the function defined • Factor the expression and Assessments by the polynomial. explain how your answer can be used to solve the equation . Explain why the solutions to this equation are the same as the xintercepts of the graph of the function . HS.A.APR.C: Use polynomial identities to solve problems. This cluster has many possibilities for optional enrichment, such as relating the example in A.APR.4 to the solution of the system u2+v2=1, v=t(u+1), relating the Pascal triangle property of binomial coefficients to (x+y)n+1=(x+y)(x+y)n, deriving explicit formulas for the coefficients, or proving the binomial theorem by induction. HS.A.APR.C.4 HS.A.APR.C.4 Lessons Prove polynomial identities and use them to HS.A.APR.C.5 • Use the distributive law to explain why describe numerical relationships. For example, Prentice Hall x2 – y2 = (x – y)(x + y) for any two numbers x and y. the polynomial identity • PH A2 Derive the identity (x – y)2 = x2 – 2xy + y2 from (x2+y2)2 = (x2– y2)2 + (2xy)2 can be used to o Ch 6.8 (x + y)2 = x2 + 2xy + y2 by replacing y by –y. generate Pythagorean triples. Use an identity to explain the pattern Tasks 22 – 1 2 = 3 Activities 32 – 22 = 5 Practice 42 – 32 = 7 52 – 42 = 9 Assessments 1)2 [Answer: (n + whole number n.] n2 = 2n + 1 for any Page | 21 Curriculum Guide 2014-2015 High School Algebra 2 Unit 2: Polynomials, Rational and Radical Relationships Sub-Unit B: Polynomial Functions November 3, 2014 – December 12, 2014 Common Core State Standards Explanations/Examples HS.A.APR.C.5 HS.A.APR.C.5 Know and apply the Binomial Theorem for the • Use Pascal’s Triangle to expand the expression expansion of (x + y)n in powers of x and y for a . positive integer n, where x and y are any • Find the middle term in the expansion of . numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) ↑ 4C0 ↑ 4C1 ↑ 4C2 ↑ 4C3 Poudre School District Resources ↑ 4C4 Page | 22 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 2: Polynomials, Rational and Radical Relationships Sub-Unit C: Rational Functions January 6, 2015 – January 23, 2015 Common Core State Standards Explanations/Examples Resources HS.A.APR.D: Review rational expressions. The limitations on rational functions apply to the rational expressions in A.APR.6. A.APR.7 requires the general division algorithm for polynomials. HS.A.APR.D.6 HS.A.APR.D.6 Lessons Rewrite simple rational expressions in different The polynomial q(x) is called the quotient and the HS.A.APR.D.6 polynomial r(x) is called the remainder. Expressing a Prentice Hall forms; write a(x)/b(x) in the form q(x) + rational expression in this form allows one to see • PH A2 r(x)/b(x), where a(x), b(x), q(x), and r(x) are different properties of the graph, such as horizontal o Ch 9.3 polynomials with the degree of r(x) less than the asymptotes. degree of b(x), using inspection, long division, or, HS.A.APR.D.7 for the more complicated examples, a computer • Find the quotient and remainder for the rational Prentice Hall 3 2 algebra system. 𝑥𝑥 −3𝑥𝑥 +𝑥𝑥−6 expression and use them to write the 2 • PH A2 𝑥𝑥 +2 o Ch 9.4-9.5 expression in a different form. • HS.A.APR.D.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. 2𝑥𝑥+1 Express 𝑓𝑓(𝑥𝑥) = in a form that reveals the 𝑥𝑥−1 horizontal asymptote of its graph. [Answer: Error! Digit expected., so the horizontal asymptote is y = 2.] HS.A.APR.D.7 • Use the formula for the sum of two fractions to explain why the sum of two rational expressions is another rational expression. • 1 Tasks Activities Practice Assessments 1 Express 2 +1 − 2 −1 in the form 𝑎𝑎(𝑥𝑥)/𝑏𝑏(𝑥𝑥), where 𝑥𝑥 𝑥𝑥 a(x) and b(x) are polynomials. Page | 23 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 2: Polynomials, Rational and Radical Relationships Sub-Unit C: Rational Functions January 6, 2015 – January 23, 2015 Common Core State Standards Explanations/Examples Resources HS.A.REI.A: Understand solving equations as a process of reasoning and explain the reasoning. Extend to simple rational and radical equations. HS.A.REI.A.2 HS.A.REI.A.2 Lessons Solve simple rational and radical equations in HS.A.REI.A.2 • one variable, and give examples showing how Prentice Hall extraneous solutions may arise. • PH A2 • o Ch 7.5 o Ch 9.6 𝑥𝑥+2 • =2 • 𝑥𝑥+3 Tasks Activities Practice Assessments HS.A.SSE.B: Write expressions in equivalent forms to solve problems. Consider extending A.SSE.4 to infinite geometric series in curricular implementations of this course description. HS.A.SSE.B.4 HS.A.SSE.B.4 Lessons Derive the formula for the sum of a finite • In February, the Bezanson family starts saving for a HS.A.SSE.B.4 geometric series (when the common ratio is not Prentice Hall trip to Australia in September. The Bezanson’s 1), and use the formula to solve problems. For expect their vacation to cost $5375. They start with • PH A2 example, calculate mortgage payments. o Ch 11.5 $525. Each month they plan to deposit 20% more than the previous month. Will they have enough Tasks money for their trip? Activities Practice Page | 24 Curriculum Guide 2014-2015 Common Core State Standards High School Algebra 2 Unit 2: Polynomials, Rational and Radical Relationships Sub-Unit C: Rational Functions January 6, 2015 – January 23, 2015 Explanations/Examples Poudre School District Assessments Resources Page | 25 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 2: Polynomials, Rational and Radical Relationships Sub-Unit D: Solving Systems January 26, 2015 – January 30, 2015 Common Core State Standards Explanations/Examples Resources HS.A.REI.D: Represent and solve equations and inequalities graphically. Include combinations of linear, polynomial, rational, radical, absolute value, and exponential functions. HS.A.REI.D.11 HS.A.REI.D.11 Lessons Explain why the x-coordinates of the points where Students need to understand that numerical solution HS.A.REI.D.11 the graphs of the equations y = f(x) and methods (data in a table used to approximate an Prentice Hall y = g(x) intersect are the solutions of the equation algebraic function) and graphical solution methods • PH A2 f(x) = g(x); find the solutions approximately, e.g., may produce approximate solutions, and algebraic o Ch 3.1-3.3 using technology to graph the functions, make solution methods produce precise solutions that can o p. 296-297 tables of values, or find successive be represented graphically or numerically. Students o p. 589 approximations. Include cases where f(x) and/or may use graphing calculators or programs to generate g(x) are linear, polynomial, rational, absolute tables of values, graph, or solve a variety of functions. Tasks value, exponential, and logarithmic functions. • Given the following equations determine the x value that results in an equal output for both Activities functions. Practice Assessments Page | 26 Curriculum Guide 2014-2015 High School Algebra 2 Unit 3: Trigonometric Functions Poudre School District Building on their previous work with functions, and on their work with trigonometric ratios and circles in Geometry, students now use the coordinate plane to extend trigonometry to model periodic phenomena. HS.F.TF.A.2 HS.F.TF.A.1 HS.F.TF.B.5 HS.F.TF.C.8 Page | 27 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 3: Trigonometric Functions February 2, 2015 – February 20, 2015 Common Core State Standards Explanations/Examples Resources HS.F.TF.A: Extend the domain of a trigonometric function using the unit circle. HS.F.TF.A.1 HS.F.TF.A.1 Lessons Understand radian measure of an angle as the HS.F.TF.A length of the arc on the unit circle subtended by Prentice Hall the angle. • PH A2 o Ch 13.2-13.3 HS.F.TF.A.2 HS.F.TF.A.2 Explain how the unit circle in the coordinate Students may use applets and animations to explore Tasks plane enables the extension of trigonometric the unit circle and trigonometric functions. Students functions to all real numbers, interpreted as may explain (orally or in written format) their Activities radian measures of angles traversed understanding. counterclockwise around the unit circle. Practice Assessments HS.F.TF.B: Model periodic phenomena with trigonometric functions. HS.F.TF.B.5 HS.F.TF.B.5 Lessons Choose trigonometric functions to model Students may use graphing calculators or programs, HS.F.TF.B.5 periodic phenomena with specified amplitude, spreadsheets, or computer algebra systems to model Prentice Hall frequency, and midline. trigonometric functions and periodic phenomena. • PH A2 o Ch 13.1 • The temperature of a chemical reaction oscillates o Ch 13.4-14.6 between a low of C and a high of C. The temperature is at its lowest point when t = 0 and Tasks completes one cycle over a six hour period. a. b. Sketch the temperature, T, against the elapsed time, t, over a 12 hour period. Find the period, amplitude, and the midline of the graph you drew in part a). Activities Practice Assessments Page | 28 Curriculum Guide 2014-2015 Common Core State Standards HS.F.TF.B.5 (continued) High School Algebra 2 Poudre School District Unit 3: Trigonometric Functions February 2, 2015 – February 20, 2015 Explanations/Examples HS.F.TF.B.5 (continued) c. Write a function to represent the relationship between time and temperature. d. e. Resources What will the temperature of the reaction be 14 hours after it began? At what point during a 24 hour day will the reaction have a temperature of C? HS.F.TF.C: Prove and apply trigonometric identities. An Algebra II course with an additional focus on trigonometry could include the (+) standard F.TF.9: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. This could be limited to acute angles in Algebra II. HS.F.TF.C.8 HS.F.TF.C.8 Lessons Prove the Pythagorean identity HS.F.TF.C.8 sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), Prentice Hall cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) • PH A2 and the quadrant of the angle. o Ch 14.1 Tasks Activities Practice Assessments Page | 29 Curriculum Guide 2014-2015 High School Algebra 2 Unit 4: Modeling with Functions Poudre School District In this unit students synthesize and generalize what they have learned about a variety of function families. They extend their work with exponential functions to include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying function. They identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as “the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions” is at the heart of this unit. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling context. Sub-Unit A: Writing Functions HS.A.CED.A.1 HS.A.CED.A.2 HS.A.CED.A.4 HS.F.IF.B.5 HS.F.BF.A.1 Curriculum Guide 2014-2015 High School Algebra 2 Unit 4: Modeling with Functions Poudre School District Sub-Unit B: Interpreting Functions HS.F.IF.B.6 HS.F.IF.C.8 HS.F.IF.B.4 HS.F.IF.C.7 HS.F.IF.C.9 HS.F.BF.B.3 HS.A.CED.A.3 Sub-Unit C: Inverse Functions HS.F.BF.B.4 HS.F.LE.A.4 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 4: Modeling with Functions Sub-Unit A: Writing Functions February 23, 2015 – March 13, 2015 Common Core State Standards Explanations/Examples Resources HS.A.CED.A: Create equations that describe numbers or relationships. For A.CED.1, use all available types of functions to create such equations, including root functions, but constrain to simple cases. While functions used in A.CED.2, 3, and 4, will often be linear, exponential, or quadratic the types of problems should draw from more complex situations than those addressed in Algebra 1. For example, finding the equation of a line through a given point perpendicular to another line allows one to find the distance from a point to a line. Note that the examples given for A.cED.4 applies to earlier instances of this standard, not to the current course. HS.A.CED.A.1 HS.A.CED.A.1 Lessons Create equations and inequalities in one variable Equations can represent real world and mathematical HS.A.CED.A.2 and use them to solve problems. Include problems. Include equations and inequalities that arise Prentice Hall equations arising from linear and quadratic when comparing the values of two different functions, • PH A2 functions, and simple rational and exponential such as one describing linear growth and one o Ch 2.2-2.7 functions. describing exponential growth. o Ch 5.1 o Ch 5.3 • Given that the following trapezoid has area 54 o Ch 5.5 cm2, set up an equation to find the length of the o Ch 6.1 base, and solve the equation. • HS.A.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Lava coming from the eruption of a volcano follows a parabolic path. The height h in feet of a piece of lava t seconds after it is ejected from the volcano is given by ℎ(𝑡𝑡) = −𝑡𝑡 2 + 16𝑡𝑡 + 936. After how many seconds does the lava reach its maximum height of 1000 feet? Tasks Activities Practice Assessments HS.A.CED.A.2 Page | 32 Curriculum Guide 2014-2015 Common Core State Standards HS.A.CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. High School Algebra 2 Unit 4: Modeling with Functions Sub-Unit A: Writing Functions February 23, 2015 – March 13, 2015 Explanations/Examples HS.A.CED.A.4 • The Pythagorean Theorem expresses the relation between the legs a and b of a right triangle and its hypotenuse c with the equation a2 + b2 = c2. o Why might the theorem need to be solved for c? o Solve the equation for c and write a problem situation where this form of the equation might be useful. o • Solve Poudre School District Resources for radius r. Motion can be described by the formula below, where t = time elapsed, u=initial velocity, a = acceleration, and s = distance traveled s = ut+½at2 o Why might the equation need to be rewritten in terms of a? o Rewrite the equation in terms of a. HS.F.BF.A: Build a function that models a relationship between two quantities. Develop models for more complex or sophisticated situations than in previous courses. HS.F.BF.A.1 HS.G.BF.A.1 Lessons Write a function that describes a relationship Students will analyze a given problem to determine the HS.F.BF.A between two quantities. function expressed by identifying patterns in the Prentice Hall function’s rate of change. They will specify intervals of b. Combine standard function types using • PH A2 increase, decrease, constancy, and, if possible, relate arithmetic operations. For example, build a o Ch 2.2-2.7 them to the function’s description in words or function that models the temperature of a o Ch 5.1 graphically. Students may use graphing calculators or cooling body by adding a constant function to o Ch 5.3 programs, spreadsheets, or computer algebra systems a decaying exponential, and relate these o Ch 5.5 to model functions. functions to the model. o Ch 6.1 Page | 33 Curriculum Guide 2014-2015 Common Core State Standards HS.F.BF.A.1 (continued) High School Algebra 2 Unit 4: Modeling with Functions Sub-Unit A: Writing Functions February 23, 2015 – March 13, 2015 Explanations/Examples HS.F.BF.A.1 (continued) • You buy a $10,000 car with an annual interest rate of 6 percent compounded annually and make monthly payments of $250. Express the amount remaining to be paid off as a function of the number of months, using a recursion equation. • • A cup of coffee is initially at a temperature of 93º F. The difference between its temperature and the room temperature of 68º F decreases by 9% each minute. Write a function describing the temperature of the coffee as a function of time. Poudre School District Tasks Resources Activities Practice Assessments The radius of a circular oil slick after t hours is given in feet by 𝑟𝑟 = 10𝑡𝑡 2 − 0.5𝑡𝑡, for 0 ≤ t ≤ 10. Find the area of the oil slick as a function of time. HS.F.IF.B: Build new functions from existing functions. Use transformations of functions to find models as students consider increasingly more complex situations. HS.F.IF.B.5 HS.F.IF.B.5 Lessons Relate the domain of a function to its graph and, Students may explain orally, or in written format, the HS.F.IF.B.5 where applicable, to the quantitative relationship existing relationships. Prentice Hall it describes. For example, if the function h(n) gives • PH A2 the number of person-hours it takes to assemble n o Ch 2.1 engines in a factory, then the positive integers would be an appropriate domain for the function. Tasks Activities Practice Assessments Page | 34 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 4: Modeling with Functions Sub-Unit B: Interpreting Functions March 23, 2014 – April 24, 2015 Common Core State Standards Explanations/Examples Resources HS.F.IF.B: Interpret functions that arise in applications in terms of a context. Emphasize the selection of a model function based on behavior of data and context. HS.F.IF.B.4 HS.F.IF.B.4 Lessons For a function that models a relationship Students may be given graphs to interpret or produce HS.F.IF.B.4 between two quantities, interpret key features of graphs given an expression or table for the function, by Prentice Hall graphs and tables in terms of the quantities, and hand or using technology. • PH A2 sketch graphs showing key features given a o Ch 2.2 • A rocket is launched from 180 feet above the verbal description of the relationship. Key o Ch 2.5 ground at time t = 0. The function that models this features include: intercepts; intervals where the o Ch 5.1 situation is given by h = – 16t2 + 96t + 180, where function is increasing, decreasing, positive, or o Ch 6.1 t is measured in seconds and h is height above the negative; relative maximums and minimums; o Ch 8.1 ground measured in feet. symmetries; end behavior; and periodicity. o Ch 9.3 o What is a reasonable domain restriction for t o Ch 13.1 in this context? o o o o o • Determine the height of the rocket two seconds after it was launched. Determine the maximum height obtained by the rocket. Determine the time when the rocket is 100 feet above the ground. Tasks Activities Practice Assessments Determine the time at which the rocket hits the ground. How would you refine your answer to the first question based on your response to the second and fifth questions? Compare the graphs of y = 3x2 and y = 3x3. Page | 35 Curriculum Guide 2014-2015 Common Core State Standards HS.F.IF.B.4 (continued) High School Algebra 2 Unit 4: Modeling with Functions Sub-Unit B: Interpreting Functions March 23, 2014 – April 24, 2015 Explanations/Examples HS.F.IF.B.4 (continued) • • • HS.F.IF.B.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. Poudre School District Resources . Find the domain of R(x). Also Let find the range, zeros, and asymptotes of R(x). Let . Graph the function and identify end behavior and any intervals of constancy, increase, and decrease. It started raining lightly at 5am, then the rainfall became heavier at 7am. By 10am the storm was over, with a total rainfall of 3 inches. It didn’t rain for the rest of the day. Sketch a possible graph for the number of inches of rain as a function of time, from midnight to midday. HS.F.IF.B.6 The average rate of change of a function y = f(x) over 𝛥𝛥𝛥𝛥 𝑓𝑓(𝑏𝑏)−𝑓𝑓(𝑎𝑎) an interval [a,b] is = 𝑏𝑏−𝑎𝑎 𝛥𝛥𝛥𝛥 In addition to finding average rates of change from functions given symbolically, graphically, or in a table, Students may collect data from experiments or simulations (ex. falling ball, velocity of a car, etc.) and find average rates of change for the function modeling the situation. Page | 36 Curriculum Guide 2014-2015 Common Core State Standards HS.F.IF.B.6 (continued) High School Algebra 2 Unit 4: Modeling with Functions Sub-Unit B: Interpreting Functions March 23, 2014 – April 24, 2015 Explanations/Examples HS.F.IF.B.6 (continued) • Use the following table to find the average rate of change of g over the intervals [-2, -1] and [0,2]: x g(x) -2 2 -1 -1 0 -4 2 -10 • The table below shows the elapsed time when two different cars pass a 10, 20, 30, 40 and 50 meter mark on a test track. o o Poudre School District Resources For car 1, what is the average velocity (change in distance divided by change in time) between the 0 and 10 meter mark? Between the 0 and 50 meter mark? Between the 20 and 30 meter mark? Analyze the data to describe the motion of car 1. How does the velocity of car 1 compare to that of car 2? Car 1 Car 2 d t t 10 4.472 1.742 20 6.325 2.899 30 7.746 3.831 40 8.944 4.633 50 10 5.348 Page | 37 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 4: Modeling with Functions Sub-Unit B: Interpreting Functions March 23, 2014 – April 24, 2015 Common Core State Standards Explanations/Examples Resources HS.F.IF.C: Analyze functions using different representation. Focus on applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate. HS.F.IF.C.7 HS.F.IF.C.7 Lessons Graph functions expressed symbolically and Key characteristics include but are not limited to HS.F.IF.C.7 show key features of the graph, by hand in simple maxima, minima, intercepts, symmetry, end behavior, Prentice Hall cases and using technology for more complicated and asymptotes. Students may use graphing • PH A2 cases. calculators or programs, spreadsheets, or computer o Ch 2.2 b. Graph square root, cube root, and piecewise- algebra systems to graph functions. o Ch 2.5 defined functions, including step functions o p. 71 • Describe key characteristics of the graph of and absolute value functions. o Ch 5.2-5.4 f(x) = │x – 3│ + 5. o Ch 6.1-6.2 e. Graph exponential and logarithmic functions, • Sketch the graph and identify the key o p. 312 showing intercepts and end behavior, and characteristics of the function described below. o Ch 7.8 trigonometric functions, showing period, o Ch 8.1 midline, and amplitude. o Ch 9.3 Tasks Activities Practice Assessments • Graph the function f(x) = 2x by creating a table of values. Identify the key characteristics of the graph. Page | 38 Curriculum Guide 2014-2015 Common Core State Standards HS.F.IF.C.7 (continued) HS.F.IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. HS.F.IF.C.9 Compare properties of two functions each represented in a different 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. High School Algebra 2 Unit 4: Modeling with Functions Sub-Unit B: Interpreting Functions March 23, 2014 – April 24, 2015 Explanations/Examples HS.F.IF.C.7 (continued) • Graph f(x) = 2 tan x – 1. Describe its domain, range, intercepts, and asymptotes. • Draw the graph of f(x) = sin x and f(x) = cos x. What are the similarities and differences between the two graphs? Poudre School District Resources HS.F.IF.C.8 HS.F.IF.C.9 • Examine the functions below. Which function has the larger maximum? How do you know? Page | 39 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 4: Modeling with Functions Sub-Unit B: Interpreting Functions March 23, 2014 – April 24, 2015 Common Core State Standards Explanations/Examples Resources HS.F.BF.B: Build new functions from existing functions. Use transformation of functions to find models as students consider increasingly more complex situations. For F.BF.3, note the effect of multiple transformations on a single graph and the common effect of each transformation across function types. HS.F.BF.B.3 HS.F.BF.B.3 Lessons Identify the effect on the graph of replacing f(x) Students will apply transformations to functions and Prentice Hall by f(x) + k, k f(x), f(kx), and f(x + k) for specific recognize functions as even and odd. Students may use • PH A2 values of k (both positive and negative); find the graphing calculators or programs, spreadsheets, or o Ch 2.6 value of k given the graphs. Experiment with computer algebra systems to graph functions. o Ch 5.3 cases and illustrate an explanation of the effects o Ch 7.8 • Is f(x) = x3 - 3x2 + 2x + 1 even, odd, or neither? on the graph using technology. Include o Ch 8.2 Explain your answer orally or in written format. recognizing even and odd functions from their o Ch 9.3 • Compare the shape and position of the graphs of graphs and algebraic expressions for them. o Ch 13.7 and , and explain the differences in terms of the algebraic expressions Tasks for the functions. Activities Practice Assessments • Describe effect of varying the parameters a, h, and k have on the shape and position of the graph of f(x) = a(x-h)2 + k. Page | 40 Curriculum Guide 2014-2015 Common Core State Standards HS.F.BF.B.3 (continued) High School Algebra 2 Unit 4: Modeling with Functions Sub-Unit B: Interpreting Functions March 23, 2014 – April 24, 2015 Explanations/Examples HS.F.BF.B.3 (continued) • Compare the shape and position of the graphs of to , and explain the differences, orally or in written format, in terms of the algebraic expressions for the functions. • • Poudre School District Resources Describe the effect of varying the parameters a, h, and k on the shape and position of the graph f(x) = ab(x + h) + k., orally or in written format. What effect do values between 0 and 1 have? What effect do negative values have? Compare the shape and position of the graphs of y = sin x to y = 2 sin x. Page | 41 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 4: Modeling with Functions Sub-Unit B: Interpreting Functions March 23, 2014 – April 24, 2015 Common Core State Standards Explanations/Examples Resources HS.A.CED.A: Create equations that describe numbers or relationships. While functions used in A.CED.2, 3, and 4 will often be linear, exponential, or quadratic, the types of problems should draw from more complex situations that those addressed in Algebra 1. For example, finding the equation of a line through a given point perpendicular to another line allows one to find the distance from a point to a line. HS.A.CED.A.3 HS.A.CED.A.3 Lessons Represent constraints by equations or HS.A.CED.A.3 • A club is selling hats and jackets as a fundraiser. inequalities, and by systems of equations and/or Prentice Hall Their budget is $1500 and they want to order at inequalities, and interpret solutions as viable or • PH A2 least 250 items. They must buy at least as many non-viable options in a modeling context. For o Ch 3.4 hats as they buy jackets. Each hat costs $5 and example, represent inequalities describing each jacket costs $8. nutritional and cost constraints on combinations o Write a system of inequalities to represent the Tasks of different foods. situation. Activities o Graph the inequalities. o If the club buys 150 hats and 100 jackets, will Practice the conditions be satisfied? o What is the maximum number of jackets they Assessments can buy and still meet the conditions? Page | 42 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 4: Modeling with Functions Sub-Unit C: Inverse Functions April 27, 2015 – May 15, 2015 Common Core State Standards Explanations/Examples Resources HS.F.BF.B: Build new functions from existing functions. Use transformations of functions to find models as students consider increasingly more complex situations. Extend F.BF.4a to simple rational, simple radical, and simple exponential functions; connect F.BF.4a to F.LE.4. HS.F.BF.B.4 HS.F.BF.B.4 Lessons Find inverse functions. Students may use graphing calculators or programs, HS.F.IF.B spreadsheets, or computer algebra systems to model Prentice Hall a. Solve an equation of the form f(x) = c for a functions. • PH A2 simple function f that has an inverse and o Ch 7.7 write an expression for the inverse. For • For the function h(x) = (x – 2)3, defined on the 3 example, f(x) =2 x or f(x) = (x+1)/(x-1) for domain of all real numbers, find the inverse Tasks x ≠ 1. function if it exists or explain why it doesn’t exist. • • Graph h(x) and h-1(x) and explain how they relate to each other graphically. Find a domain for f(x) = 3x2 + 12x - 8 on which it has an inverse. Explain why it is necessary to restrict the domain of the function. Activities Practice Assessments Page | 43 Curriculum Guide 2014-2015 High School Algebra 2 Poudre School District Unit 4: Modeling with Functions Sub-Unit C: Inverse Functions April 27, 2015 – May 15, 2015 Common Core State Standards Explanations/Examples Resources HS.F.LE.A: Construct and compare linear, quadratic, and exponential models and solve problems. Consider extending this unit to include the relationship between properties of logarithms and properties of exponents, such as the connection between the properties of exponents and the basic logarithm property that 𝐥𝐥𝐥𝐥𝐥𝐥 𝒙𝒙𝒙𝒙 = 𝐥𝐥𝐥𝐥𝐥𝐥 𝒙𝒙 + 𝐥𝐥𝐥𝐥𝐥𝐥 𝒚𝒚. HS.F.LE.A.4 HS.F.LE.A.4 Lessons For exponential models, express as a logarithm Students may use graphing calculators or programs, HS.F.LE.A.4 ct the solution to ab = d where a, c, and d are spreadsheets, or computer algebra systems to analyze Prentice Hall numbers and the base b is 2, 10, or e; evaluate the exponential models and evaluate logarithms. • PH A2 logarithm using technology. o Ch 8.3-8.6 • Solve 200 e0.04t = 450 for t. Solution: We first isolate the exponential part by dividing both sides of the equation by 200. e0.04t = 2.25 Now we take the natural logarithm of both sides. ln e0.04t = ln 2.25 Tasks Activities Practice Assessments The left hand side simplifies to 0.04t, by logarithmic identity 1. 0.04t = ln 2.25 Lastly, divide both sides by 0.04. t = ln (2.25) / 0.04 t 20.3 Page | 44 Performance Level Descriptors – Algebra II Equivalent Expressions HS.N.RN.A.2 A.Int.1 HS.A.REI.A.2 HS.A.SSE.A.2-3 HS.A.SSE.A.2-6 HS.A.SSE.B.3c-2 Interpreting Functions HS.A.APR.A.2 HS.A.REI.D.11-2 HS.F.IF.B.4-2 Algebra II: Sub-Claim A The student solves problems involving the Major Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Uses mathematical properties and structure of polynomial, exponential, rational and radical expressions to create equivalent expressions that aid in solving mathematical and contextual problems with three or more steps required. Rewrites exponential expressions to reveal quantities of interest that may be useful. Uses mathematical properties and relationships to reveal key features of polynomial, exponential, rational, trigonometric and logarithmic functions, using them to sketch graphs and identify characteristics of the relationship between two quantities, and applying the remainder theorem where appropriate. Identifies how changing the Uses mathematical properties and structure of polynomial, exponential, rational and radical expressions to create equivalent expressions that aid in solving mathematical and contextual problems with two steps required. Rewrites exponential expressions to reveal quantities of interest that may be useful. Uses mathematical properties and relationships to reveal key features of polynomial, exponential, rational, trigonometric and logarithmic functions, using them to sketch graphs and identify characteristics of the relationship between two quantities, and applying the remainder theorem where appropriate. Uses mathematical properties and structure of polynomial, exponential and rational expressions to create equivalent expressions. Uses provided mathematical properties and structure of polynomial and exponential expressions to create equivalent expressions. Rewrites exponential expressions to reveal quantities of interest that may be useful. Interprets key features of graphs and tables, and uses mathematical properties and relationships to reveal key features of polynomial, exponential and rational functions, using them to sketch graphs. Uses provided mathematical properties and relationships to reveal key features of polynomial and exponential functions, using them to sketch graphs. Page | 45 Performance Level Descriptors – Algebra II Algebra II: Sub-Claim A The student solves problems involving the Major Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command parameters of functions impacts key features of graphs. Rate of Change HS.FIF.B.6-2 HS.F.IF.B.6-7 Calculates and interprets the average rate of change of polynomial, exponential, logarithmic or trigonometric functions (presented symbolically or as a table) over a specified interval, and estimates the rate of change from a graph. Compares rates of change associated with different intervals. Building Functions HS.A.SSE.B.4-2 F-Int.3 HS.F.BF.A.1b-1 HS.F.BF.A.2 Builds functions that model mathematical and contextual situations, including those requiring multiple trigonometric functions, sequences and combinations of these and other functions, and uses the models to solve, interpret and generalize about problems. Calculates and interprets the average rate of change of polynomial, exponential, logarithmic or trigonometric functions (presented symbolically or as a table) over a specified interval, and estimates the rate of change from a graph. Calculates the average rate of change of polynomial and exponential functions (presented symbolically or as a table) over a specified interval, and estimates the rate of change from a graph. Calculates the average rate of change of polynomial and exponential functions (presented symbolically or as a table) over a specified interval. Builds functions that model mathematical and contextual situations, including those requiring trigonometric functions, sequences and combinations of these and other functions, and uses the models to solve, interpret and generalize about problems. Builds functions that model mathematical and contextual situations, including those requiring trigonometric functions, sequences and combinations of these and other functions, and uses the models to solve and interpret problems. Builds functions that model mathematical and contextual situations, limited to those requiring arithmetic and geometric sequences, and uses the models to solve and interpret problems. Page | 46 Performance Level Descriptors – Algebra II Statistics & Probability HS.S.IC.B.3-1 Algebra II: Sub-Claim A The student solves problems involving the Major Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Determines why a sample survey, experiment or observational study is most appropriate. Determines why a sample survey, experiment or observational study is most appropriate. Given an inappropriate choice of a sample survey, experiment or observational study, determines how to change the scenario to make the choice appropriate. Given an inappropriate choice of a sample survey, experiment or observational study, identifies and supports the appropriate choice. Determines whether a sample survey, experiment or observational study is most appropriate. Identifies whether a given scenario represents a sample survey, experiment or observational study. Page | 47 Performance Level Descriptors – Algebra II Interpreting Functions HS.F.IF.C.7c HS.F.IF.C.7e-1 HS.F.IF.C.7e-2 HS.F.IF.C.8b HS.F.IF.C.9-2 F-Int.1-2 Algebra II: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Given multiple functions in different forms (algebraically, graphically, numerically and by verbal description), writes multiple equivalent versions of the functions, and identifies and compares key features. Given multiple functions in different forms (algebraically, graphically, numerically and by verbal description), writes multiple equivalent versions of the functions, and identifies and compares key features. Graphs polynomial, exponential, trigonometric and logarithmic functions, showing key features. Graphs exponential, polynomial and trigonometric functions, showing key features. Given functions represented algebraically, graphically, numerically and by verbal description, writes multiple equivalent versions of the functions and identifies key features. Given functions represented algebraically, graphically, numerically and by verbal description, writes equivalent versions of the functions, and identifies key features. Graphs exponential and polynomial functions, showing key features. Graphs polynomial functions, showing key features. Uses commutative and associative properties to perform operations with complex numbers. Determines how the changes of a parameter in functions impact their other representations. Equivalent Expressions HS.N.CN.A.1 HS.N.CN.A.2 HS.A.APR.D.6 Uses commutative, associative and distributive properties to perform operations with complex numbers. Uses commutative, associative and distributive properties to perform operations with complex numbers. Uses commutative, associative and distributive properties to perform operations with complex numbers. Rewrites simple rational expressions using inspection or long division, and determines how different Rewrites simple rational expressions using inspection or long division. Rewrites simple rational expressions using inspection. Page | 48 Performance Level Descriptors – Algebra II Algebra II: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command forms of an expression reveal useful information. Function Transformations HS.F.BF.B.3-2 HS.F.BF.B.3-3 HS.F.BF.B.3-5 Trigonometry HS.F.TF.A.1 HS.F.TF.C.8-2 Given a context that infers particular transformations, identifies the effects on graphs of polynomial, exponential, logarithmic and trigonometric functions, and determines if the resulting function is even or odd. Identifies the effects of multiple transformations on graphs of polynomial, exponential, logarithmic and trigonometric functions, and determines if the resulting function is even or odd. Identifies the effects of a single transformation on graphs of polynomial, exponential, logarithmic and trigonometric functions – including f(x)+k, kf(x), f(kx), and f(x+k) – and determines if the resulting function is even or odd. Identifies the effects of a single transformation on graphs of polynomial, exponential, logarithmic and trigonometric functions – limited to f(x)+k and kf(x) – and determines if the resulting function is even or odd. Given a trigonometric value and quadrant for an angle, utilizes the structure and relationships of trigonometry, including relationships in the unit circle, to identify other trigonometric values for that angle, and describes the relationship between the radian measure and the subtended arc in the circle in contextual situations. Given a trigonometric value and quadrant for an angle, utilizes the structure and relationships of trigonometry, including relationships in the unit circle, to identify other trigonometric values for that angle, and describes the relationship between the radian measure and the subtended arc in the circle. Given a trigonometric value and quadrant for an angle, utilizes the structure and relationships of trigonometry, including relationships in the unit circle, to identify other trigonometric values for that angle. Given a trigonometric value and quadrant for an angle, utilizes the structure and relationships of trigonometry to identify other trigonometric values for that angle. Page | 49 Performance Level Descriptors – Algebra II Solving Equations and Systems HS.N.CN.C.7 HS.A.REI.B.4b-2 HS.A.REI.C.6-2 HS.A.REI.C.7 F-Int.3 F-BF.Int.2 HS.F.LE.A.2-3 HS-Int.3-3 Algebra II: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Solves multi-step contextual word problems to find similarities and differences between solution approaches involving linear, exponential, quadratic (with real or complex solutions) and trigonometric equations and systems of equations, using inverses where appropriate. HS.S.ID.A.4 HS.S.ID.B.6a-1 HS.S.ID.B.6a-2 Solves problems involving linear, exponential, quadratic (with real or complex solutions) and trigonometric equations and systems of equations, using inverses where appropriate. Solves problems involving linear, exponential and quadratic (with real solutions) equations and systems of equations, using inverses where appropriate. Constructs linear and exponential function models in multi-step contextual problems with mathematical prompting. Constructs linear and exponential function models in multi-step contextual problems. Constructs linear and exponential function models in multi-step contextual problems with mathematical prompting. Uses the means and standard deviations of data sets to fit them to normal distributions. Uses the means and standard deviations of data sets to fit them to normal distributions. Uses the means and standard deviations of data sets to fit them to normal distributions. Uses the means and standard deviations of data sets to fit them to normal distributions. Fits exponential or trigonometric functions to data in order to solve multistep contextual problems. Fits exponential and trigonometric functions to data in order to solve multistep contextual problems. Fits exponential functions to data in order to solve multistep contextual problems. Uses fitted exponential functions to solve multi-step contextual problems. Constructs linear and exponential function models in multi-step contextual problems. Data – Univariate and Bivariate Solves multi-step contextual word problems involving linear, exponential, quadratic (with real or complex solutions) and trigonometric equations and systems of equations, using inverses where appropriate. Identifies when these procedures are not appropriate. Page | 50 Performance Level Descriptors – Algebra II Inference HS.S.IC.A.2 S-IC.Int.1 Probability S-CP.Int.1 Algebra II: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Uses sample data to make, justify and critique inferences and conclusions about the corresponding population. Uses sample data to make inferences and justify conclusions about the corresponding population. Decides if specified models are consistent with results from given data-generating processes. Decides if specified models are consistent with results from given data-generating processes. Recognizes, determines and uses conditional probability and independence in multistep contextual problems, using appropriate set language and appropriate representations, including two-way frequency tables. Recognizes, determines and uses conditional probability and independence in multistep contextual problems, using appropriate set language and appropriate representations, including two-way frequency tables. Applies the Addition Rule of probability and interprets answers in context. Applies the Addition Rule of probability. Uses sample data to make inferences about the corresponding population. Identifies when sample data can be used to make inferences about the corresponding population. Recognizes, determines and uses conditional probability and independence in contextual problems, using appropriate set language and appropriate representations, including two-way frequency tables. Recognizes and determines conditional probability and independence in contextual problems. Page | 51 Performance Level Descriptors – Algebra II Reasoning HS.C.3.1 HS.C.3.2 HS.C.4.1 HS.C.5.4 HS.C.5.11 HS.C.6.2 HS.C.6.4 HS.C.7.1 HS.C.8.2 HS.C.8.3 HS.C.9.2 HS.C.11.1 HS.C.12.2 HS.C.16.3 HS.C.17.2 HS.C.17.3 HS.C.17.4 HS.C.17.5 HS.C.18.4 HS.C.CCR Algebra II: Sub-Claim C The student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Clearly constructs and communicates a complete response based on: Clearly constructs and communicates a complete response based on: Constructs and communicates a response based on: • • • • • • • • a response to a given equation or system of equations a chain of reasoning to justify or refute algebraic, function or number system propositions or conjectures a response based on data a response based on the graph of an equation in two variables, the principle that a graph is a solution set or the relationship between zeros and factors of polynomials a response based on trigonometric functions and the unit circle a response based on • • • • • a response to a given equation or system of equations a chain of reasoning to justify or refute algebraic, function or number system propositions or conjectures, a response based on data a response based on the graph of an equation in two variables, the principle that a graph is a solution set or the relationship between zeros and factors of polynomials a response based on trigonometric functions and the unit circle a response based on • • • • • Constructs and communicates an incomplete response based on: a response to a given • a response to a given equation or system of equation or system of equations equations a chain of reasoning to • a chain of reasoning to justify or refute justify or refute algebraic, function or algebraic, function or number system number system propositions or propositions or conjectures conjectures • a response based on a response based on data data a response based on the • a response based on the graph of an equation in graph of an equation in two variables, the two variables, the principle that a graph is principle that a graph is a solution set or the a solution set or the relationship between relationship between zeros and factors of zeros and factors of polynomials polynomials • a response based on a response based on trigonometric functions trigonometric functions and the unit circle and the unit circle • a response based on a response based on Page | 52 Performance Level Descriptors – Algebra II Algebra II: Sub-Claim C The student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command • • • • • • transformations of functions OR a response based on properties of exponents by: using a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate) providing an efficient and logical progression of steps or chain of reasoning with appropriate justification performing precise calculations using correct gradelevel vocabulary, symbols and labels providing a justification of a conclusion • • • • • • transformations of functions OR a response based on properties of exponents by: using a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate) providing a logical progression of steps or chain of reasoning with appropriate justification performing precise calculations using correct gradelevel vocabulary, symbols and labels providing a justification of a conclusion • • • • • • transformations of functions OR a response based on properties of exponents by: using a logical approach based on a conjecture and/or stated assumptions providing a logical, but incomplete, progression of steps or chain of reasoning performing minor calculation errors using some grade-level vocabulary, symbols and labels providing a partial justification of a conclusion based on own calculations • • • • • • transformations of functions OR a response based on properties of exponents by : using an approach based on a conjecture and/or stated or faulty assumptions providing an incomplete or illogical progression of steps or chain of reasoning making an intrusive calculation error using limited gradelevel vocabulary, symbols and labels providing a partial justification of a conclusion based on own calculations Page | 53 Performance Level Descriptors – Algebra II Algebra II: Sub-Claim C The student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command • evaluating, interpreting • evaluating the validity • determining whether • an argument or conclusion is generalizable evaluating, interpreting and critiquing the validity and efficiency of others’ responses, approaches and reasoning – utilizing mathematical connections (when appropriate) – and providing a counterexample where applicable and critiquing the validity of others’ responses, approaches and reasoning – utilizing mathematical connections (when appropriate) of others’ approaches and conclusions Page | 54 Performance Level Descriptors – Algebra II Algebra II: Sub-Claim D The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems, knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure and/or looking for and expressing regularity in repeated reasoning. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Modeling HS.D.2-3 HS.D.2-4 HS.D.2-7 HS.D.2-10 HS.D.2-12 HS.D.2-13 HS.D.3-5 HS.D.3-6 HS.D.CCR Devises a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: Devises a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: • • • • • using stated assumptions and approximations to simplify a real-world situation mapping relationships between important quantities selecting appropriate tools to create the appropriate model analyzing relationships mathematically between important quantities (either given or created) to draw conclusions • • • • using stated assumptions and approximations to simplify a real-world situation mapping relationships between important quantities selecting appropriate tools to create the appropriate model analyzing relationships mathematically between important quantities (either given or created) to draw conclusions interpreting Devises a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: • • • • • using stated assumptions and approximations to simplify a real-world situation illustrating relationships between important quantities using provided tools to create appropriate but inaccurate model analyzing relationships mathematically between important given quantities to draw conclusions interpreting mathematical results in Devises a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: • • • • • • using stated assumptions and approximations to simplify a real-world situation identifying important given quantities using provided tools to create inaccurate model analyzing relationships mathematically to draw conclusions writing an expression, equation or function to describe a situation using securely held content incompletely reporting a conclusion, Page | 55 Performance Level Descriptors – Algebra II Algebra II: Sub-Claim D The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems, knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure and/or looking for and expressing regularity in repeated reasoning. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command mathematical results in a simplified context with some inaccuracy • interpreting the context of the within the reporting • reflecting on whether mathematical results in situation the results make sense • indiscriminately using the context of the data from a data source • reflecting on whether situation • modifying the model if the results make sense it has not served its • reflecting on whether • improving the model if purpose the results make sense it has not served its • writing an expression, • improving the model if • • • • • it has not served its purpose writing a complete, clear and correct expression, equation or function to describe a situation analyzing and/or creating constraints, relationships and goals justifying and defending models which lead to a conclusion using geometry to solve design problems using securely held • • • • purpose writing a complete, clear and correct expression, equation or function to describe a situation using geometry to solve design problems using securely held content, briefly, but accurately reporting the conclusion identifying and using relevant data from a data source • • • • equation or function to describe a situation using geometry to solve design problems using securely held content, incompletely reporting a conclusion selecting and using some relevant data from a data source making an evaluation or recommendation Page | 56 Performance Level Descriptors – Algebra II Algebra II: Sub-Claim D The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems, knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure and/or looking for and expressing regularity in repeated reasoning. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command content, accurately • making an appropriate • • reporting and justifying the conclusion identifying and using relevant data from a data source making an appropriate evaluation or recommendation evaluation or recommendation Page | 57