Mathematics II chapter

Mathematics II Chapter of the Mathematics Framework for California Public Schools: Kindergarten Through Grade Twelve Adopted by the California State Board of Education, November 2013 Published by the California Department of Education Sacramento, 2015 Mathematics II T Mathematics III Mathematics II Mathematics I he Mathematics II course focuses on quadratic expressions, equations, and functions and on comparing the characteristics and behavior of these expressions, equations, and functions to those of linear and exponential relationships from Mathematics I The need for extending the set of rational numbers arises, and students are introduced to real and complex numbers Links between probability and data are explored through conditional probability and counting methods and involve the use of probability and data in making and evaluating decisions The study of similarity leads to an understanding of right-triangle trigonometry and connects to quadratics through Pythagorean relationships Circles, with their quadratic algebraic representations, finish out the course The courses in the Integrated Pathway follow the structure introduced in the K–8 grade levels of the California Common Core State Standards for Mathematics (CA CCSSM); they present mathematics as a coherent subject and blend standards from different conceptual categories The standards in the integrated Mathematics II course come from the following conceptual categories: Modeling, Functions, Number and Quantity, Algebra, Geometry, and Statistics and Probability The course content is explained below according to these conceptual categories, but teachers and administrators alike should note that the standards are not listed here in the order in which they should be taught Moreover, the standards are not topics to be checked off after being covered in isolated units of instruction; rather, they provide content to be developed throughout the school year through rich instructional experiences California Mathematics Framework Mathematics II 541 What Students Learn in Mathematics II In Mathematics II, students extend the laws of exponents to rational exponents and explore distinctions between rational and irrational numbers by considering their decimal representations Students learn that when quadratic equations not have real solutions, the number system can be extended so that solutions exist, analogous to the way in which extending whole numbers to negative numbers allows to have a solution Students explore relationships between number systems: whole numbers, integers, rational numbers, real numbers, and complex numbers The guiding principle is that equations with no solutions in one number system may have solutions in a larger number system Students consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions They select from these functions to model phenomena Students learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function Students also learn that when quadratic equations not have real solutions, the graph of the related quadratic function does not cross the horizontal axis Additionally, students expand their experience with functions to include more specialized functions—absolute value, step, and other piecewise-defined functions Students in Mathematics II focus on the structure of expressions, writing equivalent expressions to clarify and reveal aspects of the quantities represented Students create and solve equations, inequalities, and systems of equations involving exponential and quadratic expressions Building on probability concepts introduced in the middle grades, students use the language of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability Students use probability to make informed decisions, and they should make use of geometric probability models whenever possible Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right-triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem In Mathematics II, students develop facility with geometric proof They use what they know about congruence and similarity to prove theorems involving lines, angles, triangles, and other polygons They also explore a variety of formats for writing proofs In Mathematics II, students prove basic theorems about circles, chords, secants, tangents, and angle measures In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center, and the equation of a parabola with 542 Mathematics II California Mathematics Framework a vertical axis when given an equation of its horizontal directrix and the coordinates of its focus Given an equation of a circle, students draw the graph in the coordinate plane and apply techniques for solving quadratic equations to determine intersections between lines and circles, between lines and parabolas, and between two circles Students develop informal arguments to justify common formulas for circumference, area, and volume of geometric objects, especially those related to circles Examples of Key Advances from Mathematics I Students extend their previous work with linear and exponential expressions, equations, and systems of equations and inequalities to quadratic relationships • A parallel extension occurs from linear and exponential functions to quadratic functions: students begin to analyze functions in terms of transformations • Building on their work with transformations, students produce increasingly formal arguments about geometric relationships, particularly around notions of similarity Connecting Mathematical Practices and Content The Standards for Mathematical Practice (MP) apply throughout each course and, together with the Standards for Mathematical Content, prescribe that students experience mathematics as a coherent, relevant, and meaningful subject The Standards for Mathematical Practice represent a picture of what it looks like for students to mathematics and, to the extent possible, content instruction should include attention to appropriate practice standards The CA CCSSM call for an intense focus on the most critical material, allowing depth in learning, which is carried out through the Standards for Mathematical Practice Connecting content and practices happens in the context of working on problems, as is evident in the first MP standard (“Make sense of problems and persevere in solving them”) Table M2-1 offers examples of how students can engage in each mathematical practice in the Mathematics II course California Mathematics Framework Mathematics II 543 Table M2-1 Standards for Mathematical Practice—Explanation and Examples for Mathematics II Standards for Mathematical Practice MP.1 Make sense of problems and persevere in solving them MP.2 Reason abstractly and quantitatively MP.3 Construct viable arguments and critique the reasoning of others Students build proofs by induction and proofs by contradiction CA 3.1 (for higher mathematics only) MP.4 Model with mathematics Explanation and Examples Students persevere when attempting to understand the differences between quadratic functions and linear and exponential functions studied previously They create diagrams of geometric problems to help make sense of the problems Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects Students construct proofs of geometric theorems based on congruence criteria of triangles They understand and explain the definition of radian measure Students apply their mathematical understanding of quadratic functions to real-world problems Students also discover mathematics through experimentation and by examining patterns in data from real-world contexts MP.5 Students develop a general understanding of the graph of an equation Use appropriate tools strategically or function as a representation of that object, and they use tools such as graphing calculators or graphing software to create graphs in more complex examples, understanding how to interpret the result MP.6 Attend to precision MP.7 Look for and make use of structure MP.8 Look for and express regularity in repeated reasoning 544 Mathematics II Students begin to understand that a rational number has a specific definition and that irrational numbers exist When deciding if an equation can describe a function, students make use of the definition of function by asking, “Does every input value have exactly one output value?” Students apply the distributive property to develop formulas such as They see that the expression takes the form of “5 plus ‘something’ squared,” and therefore that expression can be no smaller than Students notice that consecutive numbers in the sequence of squares 1, 4, 9, 16, and 25 always differ by an odd number They use polynomials to represent this interesting finding by expressing it as California Mathematics Framework Standard MP.4 holds a special place throughout the higher mathematics curriculum, as Modeling is considered its own conceptual category Although the Modeling category does not include specific standards, the idea of using mathematics to model the world pervades all higher mathematics courses and should hold a significant place in instruction Some standards are marked with a star () symbol to indicate that they are modeling standards—that is, they may be applied to real-world modeling situations more so than other standards Modeling in higher mathematics centers on problems that arise in everyday life, society, and the workplace Such problems may draw upon mathematical content knowledge and skills articulated in the standards prior to or during the Mathematics II course Examples of places where specific Mathematical Practice standards can be implemented in the Mathematics II standards are noted in parentheses, with the standard(s) also listed Mathematics II Content Standards, by Conceptual Category The Mathematics II course is organized by conceptual category, domains, clusters, and then standards The overall purpose and progression of the standards included in Mathematics II are described below, according to each conceptual category Standards that are considered new for secondary-grades teachers are discussed more thoroughly than other standards Conceptual Category: Modeling Throughout the CA CCSSM, specific standards for higher mathematics are marked with a  symbol to indicate they are modeling standards Modeling at the higher mathematics level goes beyond the simple application of previously constructed mathematics and includes real-world problems True modeling begins with students asking a question about the world around them, and the mathematics is then constructed in the process of attempting to answer the question When students are presented with a real-world situation and challenged to ask a question, all sorts of new issues arise (e.g., Which of the quantities present in this situation are known, and which are unknown? Can a table of data be made? Is there a functional relationship in this situation?) Students need to decide on a solution path that may need to be revised They make use of tools such as calculators, dynamic geometry software, or spreadsheets They try to use previously derived models (e.g., linear functions), but may find that a new formula or function will apply Students may see when trying to answer their question that solving an equation arises as a necessity and that the equation often involves the specific instance of knowing the output value of a function at an unknown input value Modeling problems have an element of being genuine problems, in the sense that students care about answering the question under consideration In modeling, mathematics is used as a tool to answer questions that students really want answered Students examine a problem and formulate a mathematical model (an equation, table, graph, or the like), compute an answer or rewrite their expression to reveal new information, interpret and validate the results, and report out; see figure M2-1 This is a new approach for many teachers and may be challenging to implement, but the effort should show students that mathematics is relevant to their lives From a pedagogical perspective, modeling gives a concrete basis from which to abstract the mathematics and often serves to motivate students to become independent learners Figure M2-1 The Modeling Cycle Problem Formulate Validate Compute Interpret Report The examples in this chapter are framed as much as possible to illustrate the concept of mathematical modeling The important ideas surrounding quadratic functions, graphing, solving equations, and rates of change are explored through this lens Readers are encouraged to consult appendix B (Mathematical Modeling) for further discussion of the modeling cycle and how it is integrated into the higher mathematics curriculum Conceptual Category: Functions The standards of the Functions conceptual category can serve as motivation for the study of standards in the other Mathematics II conceptual categories Students have already worked with equations in which they have to “solve for ” as a search for the input of a function that gives a specified output; solving the equation amounts to undoing the work of the function The types of functions that students encounter in Mathematics II have new properties For example, while linear functions show constant additive change and exponential functions show constant multiplicative change, quadratic functions exhibit a different change and can be used to model new situations New techniques for solving equations need to be constructed carefully, as extraneous solutions may arise or no real-number solutions may exist In general, functions describe how two quantities are related in a precise way and can be used to make predictions and generalizations, keeping true to the emphasis on modeling that occurs in higher mathematics The core question when students investigate functions is, “Does each element of the domain correspond to exactly one element of the range?” (University of Arizona [UA] Progressions Documents for the Common Core Math Standards 2013c, 8) 546 Mathematics II California Mathematics Framework Interpreting Functions F-IF Interpret functions that arise in applications in terms of the context [Quadratic] 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 increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity  Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes  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  Analyze functions using different representations [Linear, exponential, quadratic, absolute value, step, piecewise-defined] Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases  a Graph linear and quadratic functions and show intercepts, maxima, and minima  b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions  Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context b Use the properties of exponents to interpret expressions for exponential functions For example, , , , and identify percent rate of change in functions such as , and classify them as representing exponential growth or decay 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 Standards F-IF.4–9 deal with understanding the concept of a function, interpreting characteristics of functions in context, and representing functions in different ways (MP.6) Standards F-IF.7–9 call for students to represent functions with graphs and identify key features of the graph They represent the same function algebraically in different forms and interpret these differences in terms of the graph or context For instance, students may easily see that the function crosses the -axis at (0,6), since the terms involving are simply when But then they factor the expression defining to obtain , revealing that the function crosses the -axis at (–2,0) and (–1,0) because those points correspond to where (MP.7) In Mathematics II, students work with linear, exponential, and quadratic functions and are expected to develop fluency with these types of functions, including the ability to graph them by hand California Mathematics Framework Mathematics II 547 Students work with functions that model data and with choosing an appropriate model function by considering the context that produced the data Students’ ability to recognize rates of change, growth and decay, end behavior, roots, and other characteristics of functions becomes more sophisticated; they use this expanding repertoire of families of functions to inform their choices for models Standards F-IF.4–9 focus on applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate Example: Population Growth F-IF.4–9 The approximate population of the United States, measured each decade starting in 1790 through 1940, can be modeled with the following function: In this function, represents the number of decades after 1790 Such models are important for planning infrastructure and the expansion of urban areas, and historically accurate long-term models have been difficult to derive y 14 y = Q( t ) 13 U.S Population in Tens of Millions 12 11 10 1 10 11 12 13 14 15 t Number of Decades after 1790 Questions: a According to this model, what was the population of the United States in the year 1790? b According to this model, when did the U.S population first reach 100,000,000? Explain your answer c According to this model, what should the U.S population be in the year 2010? Find the actual U.S population in 2010 and compare with your result d For larger values of , such as findings 548 Mathematics II , what does this model predict for the U.S population? Explain your California Mathematics Framework Example: Population Growth (continued) F-IF.4–9 Solutions: a The population in 1790 is given by , which is easily found to be 3,900,000 because b This question asks students to find such that Dividing the numerator and denominator on the left by 100,000,000 and dividing both sides of the equation by 100,000,000 simplifies this equation to Algebraic manipulation and solving for result in This means that after 1790, it would take approximately 126.4 years for the population to reach 100 million c Twenty-two (22) decades after 1790, the population would be approximately 190,000,000, which is far less (by about 119,000,000) than the estimated U.S population of 309,000,000 in 2010 d The structure of the expression reveals that for very large values of , the denominator is dominated by Thus, for very large values of , Therefore, the model predicts a population that stabilizes at 200,000,000 as increases Adapted from Illustrative Mathematics 2013m Building Functions F-BF Build a function that models a relationship between two quantities [Quadratic and exponential] Write a function that describes a relationship between two quantities  a Determine an explicit expression, a recursive process, or steps for calculation from a context  b Combine standard function types using arithmetic operations  Build new functions from existing functions [Quadratic, absolute value] Identify the effect on the graph of replacing by , , , and for specific values of (both positive and negative); find the value of 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 Find inverse functions for a simple function a Solve an equation of the form expression for the inverse For example, California Mathematics Framework that has an inverse and write an Mathematics II 549 Conceptual Category: Statistics and Probability In grades seven and eight, students learned some basics of probability, including chance processes, probability models, and sample spaces In higher mathematics, the relative frequency approach to probability is extended to conditional probability and independence, rules of probability and their use in finding probabilities of compound events, and the use of probability distributions to solve problems involving expected value (UA Progressions Documents 2012d, 13) Building on probability concepts that were developed in grades six through eight, students use the language of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability Students should make use of geometric probability models wherever possible They use probability to make informed decisions (National Governors Association Center for Best Practices, Council of Chief State School Officers [NGA/CCSSO] 2010a) Conditional Probability and the Rules of Probability S-CP Understand independence and conditional probability and use them to interpret data [Link to data from simulations or experiments.] 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”)  Understand that two events and are independent if the probability of and occurring together is the product of their probabilities, and use this characterization to determine if they are independent  , and interpret independence of Understand the conditional probability of given as and as saying that the conditional probability of given is the same as the probability of , and the conditional probability of given is the same as the probability of  Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified Use the two-way table 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  Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations  Use the rules of probability to compute probabilities of compound events in a uniform probability model Find the conditional probability of given as the fraction of ’s outcomes that also belong to , and interpret the answer in terms of the model  Apply the Addition Rule, model  , and interpret the answer in terms of the (+) Apply the general Multiplication Rule in a uniform probability model, , and interpret the answer in terms of the model  (+) Use permutations and combinations to compute probabilities of compound events and solve problems  566 Mathematics II California Mathematics Framework To develop student understanding of conditional probability, students should experience two types of problems: those in which the uniform probabilities attached to outcomes lead to independence of the outcomes, and those in which they not (S-CP.1–3) The following examples illustrate these two distinct possibilities Example: Guessing on a True–False Quiz S-CP.1–3 If there are four true-or-false questions on a quiz, then the possible outcomes based on guessing on each question may be arranged as in the table below: Possible outcomes: Guessing on four true–false questions Number correct Outcomes Number correct 3 3 CCCC ICCC CICC CCIC CCCI 2 2 2 Out- Number comes correct CCII CICI CIIC ICCI ICIC IICC 1 1 Outcomes CIII ICII IICI IIIC IIII C indicates a correct answer; I indicates an incorrect answer By counting outcomes, one can find various probabilities For example: and Noticing that shows that the two events—getting the first question correct and the second question correct—are independent Adapted from UA Progressions Documents 2012d California Mathematics Framework Mathematics II 567 Example: Work-Group Leaders S-CP.1–3 Suppose a five-person work group consisting of three girls (April, Briana, and Cyndi) and two boys (Daniel and Ernesto) wants to randomly choose two people to lead the group The first person is the discussion leader and the second is the recorder, so order is important in selecting the leadership team In the table below, “A” represents April, “B” represents Briana, “C” represents Cyndi, “D” represents Daniel, and “E” represents Ernesto There are 20 outcomes for this situation: Selecting two students from three girls and two boys Number of girls Outcomes 2 1 1 1 AB AC BC AD AE BD BE CD CE DE BA CA CB DA EA DB EB DC EC ED Notice that the probability of selecting two girls as the leaders is as follows: whereas and But since , the two events are not independent One can also use the conditional-probability perspective to show that these events are not independent Since and , these events are seen to be dependent Adapted from UA Progressions Documents 2012d 568 Mathematics II California Mathematics Framework Students also explore finding probabilities of compound events (S-CP.6–9) by using the Addition Rule and the general Multiplication Rule A simple experiment in which students roll two number cubes and tabulate the possible outcomes can shed light on these formulas before they are extended to application problems Example S-CP.6–9 On April 15, 1912, the RMS Titanic rapidly sank in the Atlantic Ocean after hitting an iceberg Only 710 of the ship’s 2,204 passengers and crew members survived Some believe that the rescue procedures favored the wealthier first-class passengers Data on survival of passengers are summarized in the table at the end of this example, and these data will be used to investigate the validity of such claims Students can use the fact represents the event that a passenger that two events and are independent if survived, and represents the event that the passenger was in first class The conditional probability is compared with the probability For a first-class passenger, the probability of surviving is the fraction of all first-class passengers who survived That is, the sample space is restricted to include only first-class passengers to obtain: The probability that a passenger survived is the number of all passengers who survived divided by the total number of passengers: Since , the two given events are not independent Moreover, it can be said that being a passenger in first class did increase the chances of surviving the accident Students can be challenged to further investigate where similar reasoning would apply today For example, what are similar statistics for Hurricane Katrina, and what would a similar analysis conclude about the distribution of damages? (MP.4) Titanic passengers Survived Did not survive Total First-class 202 123 325 Second-class 118 167 285 Third-class 178 528 706 Total passengers 498 818 1,316 Adapted from Illustrative Mathematics 2013q California Mathematics Framework Mathematics II 569 Using Probability to Make Decisions S-MD Use probability to evaluate outcomes of decisions [Introductory; apply counting rules.] (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator)  (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game)  Standards S-MD.6 and S-MD.7 involve students’ use of probability models and probability experiments to make decisions These standards set the stage for more advanced work in Mathematics III, where the ideas of statistical inference are introduced See the University of Arizona Progressions document titled “High School Statistics and Probability” for further explanation and examples: http://ime.math.arizona edu/progressions/ (UA Progressions Documents 2012d [accessed April 6, 2015]) 570 Mathematics II California Mathematics Framework California Common Core State Standards for Mathematics Mathematics II Overview Number and Quantity The Real Number System  Extend the properties of exponents to rational exponents  Use properties of rational and irrational numbers The Complex Number Systems  Perform arithmetic operations with complex numbers  Use complex numbers in polynomial identities and equations Algebra Seeing Structure in Expressions  Interpret the structure of expressions  Write expressions in equivalent forms to solve problems Arithmetic with Polynomials and Rational Expressions  Perform arithmetic operations on polynomials Mathematical Practices Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Creating Equations  Create equations that describe numbers or relationships Reasoning with Equations and Inequalities  Solve equations and inequalities in one variable  Solve systems of equations Functions Interpreting Functions  Interpret functions that arise in applications in terms of the context  Analyze functions using different representations Building Functions  Build a function that models a relationship between two quantities  Build new functions from existing functions Linear, Quadratic, and Exponential Models  Construct and compare linear, quadratic, and exponential models and solve problems  Interpret expressions for functions in terms of the situation they model Trigonometric Functions  Prove and apply trigonometric identities C alifornia Mathematics Framework Mathematics II 571 Geometry Congruence  Prove geometric theorems Similarity, Right Triangles, and Trigonometry  Understand similarity in terms of similarity transformations  Prove theorems involving similarity  Define trigonometric ratios and solve problems involving right triangles Circles  Understand and apply theorems about circles  Find arc lengths and areas of sectors of circles Expressing Geometric Properties with Equations  Translate between the geometric description and the equation for a conic section  Use coordinates to prove simple geometric theorems algebraically Geometric Measurement and Dimension  Explain volume formulas and use them to solve problems Statistics and Probability Conditional Probability and the Rules of Probability  Understand independence and conditional probability and use them to interpret data  Use the rules of probability to compute probabilities of compound events in a uniform probability model Using Probability to Make Decisions  572 Use probability to evaluate outcomes of decisions Mathematics II California Mathematics Framework Mathematics II M2 Number and Quantity The Real Number System N-RN Extend the properties of exponents to rational exponents Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents For example, we define to be the cube root of because we want to hold, so must equal Rewrite expressions involving radicals and rational exponents using the properties of exponents Use properties of rational and irrational numbers Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a non-zero rational number and an irrational number is irrational The Complex Number System N-CN Perform arithmetic operations with complex numbers [ as highest power of ] Know there is a complex number such that real Use the relation complex numbers , and every complex number has the form with and and the commutative, associative, and distributive properties to add, subtract, and multiply Use complex numbers in polynomial identities and equations [Quadratics with real coefficients] Solve quadratic equations with real coefficients that have complex solutions (+) Extend polynomial identities to the complex numbers For example, rewrite as * (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials Algebra Seeing Structure in Expressions A-SSE Interpret the structure of expressions [Quadratic and exponential] 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 as the product of P and a factor not depending on P  Use the structure of an expression to identify ways to rewrite it For example, see recognizing it as a difference of squares that can be factored as as , thus * (+) Indicates additional mathematics to prepare students for advanced courses  Indicates a modeling standard linking mathematics to everyday life, work, and decision-making California Mathematics Framework Mathematics II 573 M2 Mathematics II Write expressions in equivalent forms to solve problems [Quadratic and exponential] Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression  a Factor a quadratic expression to reveal the zeros of the function it defines  b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines  c Use the properties of exponents to transform expressions for exponential functions For example, the expression can be rewritten as to reveal the approximate equivalent monthly interest rate if the annual rate is 15%  Arithmetic with Polynomials and Rational Expressions A-APR Perform arithmetic operations on polynomials [Polynomials that simplify to quadratics] 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 polynomials Creating Equations A-CED Create equations that describe numbers or relationships Create equations and inequalities in one variable including ones with absolute value and use them to solve problems Include equations arising from linear and quadratic functions, and simple rational and exponential functions CA  Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales  Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations  [Include formulas involving quadratic terms.] Reasoning with Equations and Inequalities A-REI Solve equations and inequalities in one variable [Quadratics with real coefficients] Solve quadratic equations in one variable a Use the method of completing the square to transform any quadratic equation in into an equation of the form that has the same solutions Derive the quadratic formula from this form b Solve quadratic equations by inspection (e.g., for ), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation Recognize when the quadratic for real numbers and formula gives complex solutions and write them as Solve systems of equations [Linear-quadratic systems] 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 and the circle 574 Mathematics II California Mathematics Framework M2 Mathematics II Functions Interpreting Functions F-IF Interpret functions that arise in applications in terms of the context [Quadratic] 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 increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity  Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes  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  Analyze functions using different representations [Linear, exponential, quadratic, absolute value, step, piecewisedefined] Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases  a Graph linear and quadratic functions and show intercepts, maxima, and minima  b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions  Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context b Use the properties of exponents to interpret expressions for exponential functions For example, identify percent , , , and , and classify them as rate of change in functions such as representing exponential growth or decay 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 Building Functions F-BF Build a function that models a relationship between two quantities [Quadratic and exponential] Write a function that describes a relationship between two quantities  a Determine an explicit expression, a recursive process, or steps for calculation from a context  b Combine standard function types using arithmetic operations  California Mathematics Framework Mathematics II 575 M2 Mathematics II Build new functions from existing functions [Quadratic, absolute value] Identify the effect on the graph of replacing by , , , and for specific values of (both positive and negative); find the value of 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 Find inverse functions a Solve an equation of the form the inverse For example, for a simple function that has an inverse and write an expression for Linear, Quadratic, and Exponential Models F-LE Construct and compare linear, quadratic, and exponential models and solve problems [Include quadratic] Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function  Interpret expressions for functions in terms of the situation they model Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity CA  Trigonometric Functions F-TF Prove and apply trigonometric identities Prove the Pythagorean identity , or and the quadrant of the angle and use it to find , , or given , Geometry Congruence G-CO Prove geometric theorems [Focus on validity of underlying reasoning while using variety of ways of writing proofs.] 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 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 11 Prove theorems about parallelograms Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals 576 Mathematics II California Mathematics Framework Mathematics II Similarity, Right Triangles, and Trigonometry M2 G-SRT Understand similarity in terms of similarity transformations Verify experimentally the properties of dilations given by a center and a scale factor: a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged b The dilation of a line segment is longer or shorter in the ratio given by the scale factor Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar Prove theorems involving similarity [Focus on validity of underlying reasoning while using variety of formats.] 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 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures Define trigonometric ratios and solve problems involving right triangles Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles Explain and use the relationship between the sine and cosine of complementary angles Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems  8.1 Derive and use the trigonometric ratios for special right triangles (30°, 60°, 90° and 45°, 45°, 90°) CA Circles G-C Understand and apply theorems about circles Prove that all circles are similar 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 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle (+) Construct a tangent line from a point outside a given circle to the circle Find arc lengths and areas of sectors of circles [Radian introduced only as unit of measure] Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector Convert between degrees and radians CA California Mathematics Framework Mathematics II 577 M2 Mathematics II Expressing Geometric Properties with Equations G-GPE Translate between the geometric description and the equation for a conic section Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation Derive the equation of a parabola given a focus and directrix Use coordinates to prove simple geometric theorems algebraically Use coordinates to prove simple geometric theorems algebraically For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, ) lies on the circle centered at the origin and containing the point (0, 2) [Include simple circle theorems.] Find the point on a directed line segment between two given points that partitions the segment in a given ratio Geometric Measurement and Dimension G-GMD Explain volume formulas and use them to solve problems Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone Use dissection arguments, Cavalieri’s principle, and informal limit arguments Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems  Know that the effect of a scale factor greater than zero on length, area, and volume is to multiply each by , and , respectively; determine length, area, and volume measures using scale factors CA , Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve real-world and mathematical problems CA Statistics and Probability Conditional Probability and the Rules of Probability S-CP Understand independence and conditional probability and use them to interpret data [Link to data from simulations or experiments.] 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”)  Understand that two events and are independent if the probability of and occurring together is the product of their probabilities, and use this characterization to determine if they are independent  , and interpret independence of and Understand the conditional probability of given as as saying that the conditional probability of given is the same as the probability of , and the conditional probability of given is the same as the probability of  578 Mathematics II California Mathematics Framework Mathematics II M2 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified Use the two-way table 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  Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations  Use the rules of probability to compute probabilities of compound events in a uniform probability model Find the conditional probability of answer in terms of the model  given Apply the Addition Rule, as the fraction of ’s outcomes that also belong to , and interpret the , and interpret the answer in terms of the model  (+) Apply the general Multiplication Rule in a uniform probability model, and interpret the answer in terms of the model  , (+) Use permutations and combinations to compute probabilities of compound events and solve problems  Using Probability to Make Decisions S-MD Use probability to evaluate outcomes of decisions [Introductory; apply counting rules.] (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator)  (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game)  California Mathematics Framework Mathematics II 579 This page intentionally blank ... correct 3 3 CCCC ICCC CICC CCIC CCCI 2 2 2 Out- Number comes correct CCII CICI CIIC ICCI ICIC IICC 1 1 Outcomes CIII ICII IICI IIIC IIII C indicates a correct answer; I indicates an incorrect answer.. .Mathematics II T Mathematics III Mathematics II Mathematics I he Mathematics II course focuses on quadratic expressions, equations, and... practice in the Mathematics II course California Mathematics Framework Mathematics II 543 Table M2-1 Standards for Mathematical Practice—Explanation and Examples for Mathematics II Standards for