Algebra II. Introduction

Transcription

1 Introduction The purpose of this course is to extend students understing of functions the real numbers, to increase the tools students have for modeling the real world. They extend their notion of number to include complex numbers see how the introduction of this set of numbers allows the solutions of polynomial equations the Fundamental Theorem of Algebra. Students deepen their understing of the concept of function, apply equation solving function concepts to many different types of functions. The system of polynomial functions, analogous to the integers, is extended to the field of rational functions, analogous to the rational numbers. Students explore the relationship between exponential functions their inverses, the logarithmic functions. Trigonometric functions are extended to all real numbers, their graphs properties are studied. Finally, students statistics knowledge is extended to understing the normal distribution, probability concepts, they are challenged to make inferences based on sampling, experiments, observational studies. The stards in the traditional Algebra II course come from the following conceptual categories:,, Number Quantity, Algebra, Statistics Probability. The content of the course will be exped upon according to these conceptual categories, but teachers administrators alike should note that the stards are not topics to be checked off a list during isolated units of instruction, but rather content that should be present throughout the school year through rich instructional experiences. In addition, the stards should not necessarily be taught in the order in which they appear within each unit, but rather in a coherent manner. This Algebra II Scope Sequence reflects current thinking related to the intent of the Common Core State Stards for Mathematics (CCSSM) is designed to be used for the planning of mathematics instruction for the school year. The document includes a one page year at a glance, an overview of the scope sequence of the mathematical themes stards to be taught through the course of the year. A scope sequence refers to the breadth depth of a specific curriculum. The scope is how much to teach of a subject over the course of a semester or year. The sequence is the order you are going to teach the lessons in. The second part of the document details the content focus stards to be addressed in each unit. It provides essential questions, academic vocabulary how the mathematical practice stards support the content stards, resources for differentiation assessment.

2 An Overview: What Students Learn in Algebra II Building on their work with linear, quadratic, exponential functions, in Algebra II students extend their repertoire of functions to include polynomial, rational, radical functions. Students work closely with the expressions that define the functions continue to exp hone their abilities to model situations to solve equations, including solving quadratic equations over the set of complex numbers solving exponential equations using the properties of logarithms. Based on their previous work with functions, on their work with trigonometric ratios circles in Geometry, students now use the coordinate plane to extend trigonometry to model periodic phenomena. 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. Students see how the visual displays summary statistics they learned in earlier grades relate to different types of data to probability distributions. They compute different types of probabilities to simulate events. They identify different ways of collecting data including sample surveys, experiments, simulations the role that romness careful design play in the conclusions that can be drawn. Finally, in Algebra II, students extend their understing of modeling: they identify appropriate types of functions to model a situation, they adjust parameters to improve the model, they compare models by analyzing appropriateness of fit making judgments about the domain over which a model is a good fit. The description of modeling as the process of choosing using mathematics statistics to analyze empirical situations, to underst them better, to make decisions is one of the main themes of this course. The narrative discussion diagram of the modeling cycle should be considered when knowledge of functions, statistics, algebraic reasoning is applied in a modeling context.

3 Examples of Key Advances from Previous Courses In Algebra I, students added, subtracted multiplied polynomials. In Algebra II, students divide polynomials with remainders, leading to the factor remainder theorems. This is the underpinning for much of advanced algebra, including the algebra of rational expressions. Themes from middle school algebra continue deepen during high school. As early as grade 6, students began thinking about solving equations as a process of reasoning (6.EE.5). This perspective continues throughout Algebra I Algebra II (A REI.4). Reasoned solving plays a role in Algebra II because the equations students encounter can have extraneous solutions (A REI.2) inverse functions play a role in solving equations in addition to inverse operations. In Algebra I, students worked with quadratic equations with no real roots. In Algebra II, they extend the real numbers to complex numbers, one effect is that they now have a complete theory of quadratic equations: Every quadratic equation with complex coefficients has (counting multiplicities) two roots in the complex numbers. In grade 8, students learned the Pythagorean Theorem used it to determine distances in a coordinate system (8.G.6 8). In Geometry, students proved theorems using coordinates (G GPE.4 7). In Algebra II, students will build on their understing of distance in coordinate systems draw on their growing comm of algebra to connect equations graphs of conic sections (e.g., G 67 GPE.1). In Geometry, students began trigonometry through a study of right triangles. In Algebra II, they extend the three basic functions to the entire unit circle apply trigonometry to model periodic phenomena. As students acquire mathematical tools from their study of algebra functions, they apply these tools in statistical contexts (e.g., S ID.6). In a modeling context, they might informally fit an exponential function to a set of data, graphing the data the model function on the same coordinate axes.

4 **Algebra II extends functions from linear, exponential, quadratic, absolute value, piecewise to include polynomial, simple rational, logarithmic, trigonometric (sinusoid only), root/radical step functions. Building functions limited to linear, exponential, e.g. 2 or 3 polynomial, simple rational sinusoidal. Onevariable equations include linear, quadratic, exponential (irrational solutions included), polynomial, rational logarithmic. Systems limited to linear quadratic. Red stards + stards are specific to honors sections subject to the limitations described here. Intro Unit 1 week Unit 1 Exponential Extend the properties of exponents to rational exponents N RN 1,2 Write expressions in equivalent forms to solve problems (exponential logarithmic) A SSE 3c, 4 Scope Sequence Year at a Glance SEMESTER 1 WEMESTER 2 & Equations Statistics Probability Equations Unit 2 Trigonometric Extend the domain of the trigonometric functions using the unit circle F TF 1, 2, 2.1(CA) Model periodic phenomena 5, 8 Analyze functions using different representations (trig functions) F IF 7e Unit 3 Extend the properties of exponents to rational exponents N RN 1,2 Interpret the structure of expressions A SSE 1, 1a, 1b Solve equations inequalities graphically A REI 11 Interpret functions in context F IF 4, 5, 6 Unit 4 Polynomial Rational Expressions 4weeks Translate between the geometric description equation for a conic (parabola circle only) G GPE 3.1(CA) Use properties of rational irrational numbers N RN 3 Perform arithmetic operations on complex numbers N CN 1, 2 Unit 5 Probability Underst independence conditional probability use them to interpret data S CP 1, 2, 3, 4, 5 Use the rules of probability to compute probabilities of compound events in a uniform probability model 6, 7, 8+, 9+ Use probability to evaluate outcomes of decisions S MD 6+, 7+ Unit 6 Statistics Rom Processes 5 week Underst evaluate rom processes underlying statistical experiments S IC 1, 2 Make inferences justify conclusions from sample surveys, experiments observational studies S IC 3, 4, 5, 6 Summarize, represent interpret data (normal distribution) S ID 4 2 weeks

5 Intro Unit 1 week Unit 1 Exponential Represent solve equations inequalities graphically (exponential logarithmic) A REI 11 Interpret functions that arise in context (linear exponential) F IF 4, 5 Underst the concept of a function use function notation F IF 3 Analyze function using different representations F IF 7e, 8b, 9 Build a function that models a relationship between two quantities F BF 2 Equations (continued) Unit 2 Trigonometric Unit 3 3 Weeks Analyze functions using different representations F IF 7b, 7c Build functions that model relationships between two quantities F BF 1 Build new functions from existing functions F BF 3 Construct compare linear, quadratic, exponential models (include more general polynomials) F LE 3 Equations continued) Unit 4 Polynomial Rational Expressions Use complex numbers in polynomial (polynomial rational) N CN 7, 8+, 9+ Interpret the structure of expressions (polynomial rational) A SSE 1, 2 Perform arithmetic operations on polynomials (beyond quadratic) A APR 1 Underst the relationship between zeros factors of polynomials A APR 2, 3 Use polynomial identities to solve problems A APR 4, 5+ Rewrite rational expressions A APR 6, 7+ Statistics Probability (continued) Unit 5 Probability Unit 6 Statistics Rom Processes Use probability to evaluate outcomes of decisions S MD 6+, 7+ 2 weeks

6 Intro Unit 1 week Unit 1 Exponential Build new functions from existing functions F BF 4a Construct compare linear, quadratic exponential models (focus on exponential logarithmic) F LE 1, 2, 3, 4, 5 Rearrange formulas to highlight quantities of interest A CED 4 Equations (continued) Unit 2 Trigonometric Unit 3 3 Weeks Create equations that describe numbers or relationships A CED 1, 2, 3 Equations continued) Unit 4 Polynomial Rational Expressions Underst solving equations as a process of reasoning explain the reasoning (simple radical rational) A REI 2 Represent solve equations inequalities graphically (combined types) A REI 11 Statistics Probability (continued) Unit 5 Probability Unit 6 Statistics Rom Processes 2 weeks

7 1. Make sense of problems persevere in solving them. 2. Reason abstractly quantitatively. 3. Construct viable arguments critique the reasoning of others. 4. Model with mathematics. Mathematical Practices 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for make use of structure. 8. Look for express regularity in repeated reasoning. in High School Mathematics Students develop increasing levels of proficiency as they progress from grade 8 mathematics through Algebra II in terms of modeling. Each of the high school mathematics courses has specific modeling project based units built into the schedule. problems are 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. This will be a new approach for many teachers will be challenging to implement, but the effort will produce students who can appreciate that mathematics is relevant to their lives. From a pedagogical perspective, modeling gives a concrete basis from which to abstract the mathematics often serves to motivate students to become independent learners. Figure 1: The modeling cycle. Students examine a problem formulate a mathematical model (an equation, table, graph, etc.), compute an answer or rewrite their expression to reveal new information, interpret their results, validate them, report out.

8 EMPHASES BY CLUSTER (PARCC/SBAC Algebra II Major Clusters (70%) Areas of intensive focus, where students need fluent understing application of the core Supporting Clusters (20%) Rethinking linking; areas where some material is being covered, but in a way that applies core understings ALGEBRA Seeing the Structure in Expressioins o Interpret the structure of expressions. o Write expressions in equivalent forms to solve problems. Arithmetic with Polynomials Rational Exprressions o Perform arthimetic operations on polynomials. o Underst the relationship between zeros factors of polynomials. Creating Equations o Create equations that descrbe numbers or relationships. Reasoning with Eations Inequalities o Underst solving equations as a process of reasoning explain the reasoning. o Solve equations inequalities in one variable. o Solve systems of equations. Arithmetic with Polynomials Rational Exprressions o Rewrite rational expressions. o Use polynomial identities to solve problems.

9 Major Clusters (70%) Areas of intensive focus, where students need fluent understing application of the core Algebra II Supporting Clusters (20%) Rethinking linking; areas where some material is being covered, but in a way that applies core understings FUNCTIONS STATISTICS & PROBABILTY NUMBER & QUANTITY Interpreting o Interpret functions that arise in applications in terms of the context. Building o Buils a function that models a relationship between two quantities.. Making Inferences Justifying Conclusions o Make inferences justify conclusions from sample surveys, experiments, observational studies. Interpreting o Analyze functions using different representaions. Linear, Quadratics Exponential Models o Construct compare linear, quadratic, exponential models solve problems. Building o Build new functions from exisiting functions. Trigonometric o Extend the domain of trogonometric functions using the unit circle.. o Model periodic phenomena with trigonometric fuctions. o Prove apply trigonometric identies. Making Inferences Justifying Conclusions o Underst evaluate rom processes underlying statistical experiemnts. Interpreting Categorical Quantatative Data o Summarize, represent, interpret data on a single count or measureable variable. Using Probability to Make Decision o Use probability to evaluate outcomes of decisions. The Complex Number Systems o Perform arithmetic operations with complex numbers.