Download Algebra 2 Scope and Sequence

Algebra II
Introduction The purpose of this course is to extend students’ understanding of functions and the real numbers, and to increase the tools students have for modeling the real world. They extend their notion of number to include complex numbers and see how the introduction of this set of numbers allows the solutions of polynomial equations and the Fundamental Theorem of Algebra. Students deepen their understanding of the concept of function, and apply equation solving and 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 and their inverses, the logarithmic functions. Trigonometric functions are extended to all real numbers, and their graphs and properties are studied. Finally, students’ statistics knowledge is extended to understanding the normal distribution, probability concepts, and they are challenged to make inferences based on sampling, experiments, and observational studies. The standards in the traditional Algebra II course come from the following conceptual categories: Modeling, Functions, Number and Quantity, Algebra, and Statistics and Probability. The content of the course will be expanded upon according to these conceptual categories, but teachers and administrators alike should note that the standards 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 standards 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 and Sequence reflects current thinking related to the intent of the Common Core State Standards for Mathematics (CCSSM) and is designed to be used for the planning of mathematics instruction for the 2013 – 2014 school year. The document includes a one page year at a glance, an overview of the scope and sequence of the mathematical themes and standards to be taught through the course of the year. A scope and sequence refers to the breadth and 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 and standards to be addressed in each unit. It provides essential questions, academic vocabulary and how the mathematical practice standards support the content standards, resources for differentiation and assessment. Fresno Unified Scope & Sequence Final Draft 2013-2014
Algebra II
Algebra II
An Overview: What Students Learn in Algebra II Building on their work with linear, quadratic, and exponential functions, in Algebra II 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. Based 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. 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 and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They compute different types of probabilities to simulate events. 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. Finally, in Algebra II, students extend their understanding of modeling: 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 one of the main themes of this course. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and algebraic reasoning is applied in a modeling context. Fresno Unified Scope & Sequence Final Draft 2013-2014
Algebra II
Algebra II
Examples of Key Advances from Previous Courses 
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In Algebra I, students added, subtracted and multiplied polynomials. In Algebra II, students divide polynomials with remainders, leading to the factor and remainder theorems. This is the underpinning for much of advanced algebra, including the algebra of rational expressions. Themes from middle school algebra continue and 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 and 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) and 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, and 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 and 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 understanding of distance in coordinate systems and draw on their growing command of algebra to connect equations and 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 and apply trigonometry to model periodic phenomena. As students acquire mathematical tools from their study of algebra and 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 and the model function on the same coordinate axes. Fresno Unified Scope & Sequence Final Draft 2013-2014
Algebra II
Algebra II
**Algebra II extends functions from linear, exponential, quadratic, absolute value, and piecewise to include polynomial, simple rational, logarithmic, trigonometric
(sinusoid only), root/radical and step functions. Building functions limited to linear, exponential, e.g. 2 or 3 polynomial, simple rational and sinusoidal. Onevariable equations include linear, quadratic, exponential (irrational solutions included), and polynomial, rational and logarithmic. Systems limited to linear and
quadratic. Red standards and + standards are specific to honors sections subject to the limitations described here.
Scope and Sequence Year at a Glance SEMESTER 1 Functions & Equations Intro Unit 1 week Unit 1 Exponential Functions 5 weeks Unit 2 Trigonometric Functions 3 weeks Unit 3
Functions 3 weeks Extend the properties of exponents to rational exponents N‐RN 1,2 Write expressions in equivalent forms to solve problems (exponential and logarithmic) A‐SSE 3c, 4 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 Extend the properties of exponents to rational exponents N‐RN 1,2 Modeling and Projects 3 weeks Interpret the structure of expressions A‐SSE 1, 1a, 1b Solve equations and inequalities graphically A‐REI 11 Interpret functions in context F‐IF 4, 5, 6 Fresno Unified Scope & Sequence Final Draft 2013-2014
Functions and Equations Unit 4
Polynomial and Rational Expressions 4weeks Translate between the geometric description and equation for a conic (parabola and circle only) G‐GPE 3.1(CA) Use properties of rational and irrational numbers N‐RN 3 Perform arithmetic operations on complex numbers N‐CN 1, 2 WEMESTER 2 Statistics and Probability Unit 5
Probability 4 weeks Understand independence and conditional probability and 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 Random Processes 5 week Understand and evaluate random processes underlying statistical experiments S‐IC 1, 2 Make inferences and justify conclusions from sample surveys, experiments and observational studies S‐IC 3, 4, 5, 6 Summarize, represent and interpret data (normal distribution) S‐ID 4 Algebra II
Modeling and Projects 2 weeks Algebra II
Intro Unit 1 week Modeling and Projects Functions and Equations (continued) Unit 1 Exponential Functions 5 weeks Represent and solve equations and inequalities graphically (exponential and logarithmic) A‐REI 11 Interpret functions that arise in context (linear and exponential) F‐IF 4, 5 Understand the concept of a function and 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 Unit 2 Trigonometric Functions 3 weeks Unit 3
Functions 3 Weeks Functions and Equations Unit 4
Polynomial and Rational Expressions 4 weeks 3 weeks Fresno Unified Scope & Sequence Final Draft 2013-2014
(continued) continued)
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 and compare linear, quadratic, exponential models (include more general polynomials) F‐LE 3 Statistics and Probability Use complex numbers in polynomial (polynomial and rational) N‐CN 7, 8+, 9+ Interpret the structure of expressions (polynomial and rational) A‐SSE 1, 2 Perform arithmetic operations on polynomials (beyond quadratic) A‐APR 1 Understand the relationship between zeros and factors of polynomials A‐APR 2, 3 Unit 5
Probability 4 weeks Unit 6
Statistics Random Processes 5 weeks Use probability to evaluate outcomes of decisions S‐MD 6+, 7+ Use polynomial identities to solve problems A‐APR 4, 5+ Rewrite rational expressions A‐APR 6, 7+ Algebra II
Modeling and Projects 2 weeks Algebra II
Intro Unit 1 week Modeling and Projects Functions and Equations (continued) Unit 1 Exponential Functions 5 weeks Build new functions from existing functions F‐BF 4a Construct and compare linear, quadratic and exponential models (focus on exponential and logarithmic) F‐LE 1, 2, 3, 4, 5 Unit 2 Trigonometric Functions 3 weeks Unit 3
Functions 3 Weeks Create equations that describe numbers or relationships A‐CED 1, 2, 3 Functions and Equations Unit 4
Polynomial and Rational Expressions 4 weeks 3 weeks Rearrange formulas to highlight quantities of interest A‐CED 4 (continued) continued)
Statistics and Probability Understand solving equations as a process of reasoning and explain the reasoning (simple radical and rational) A‐REI 2 Unit 5
Probability 4 weeks Unit 6
Statistics Random Processes 5 weeks Represent and solve equations and inequalities graphically (combined types) A‐REI 11 Fresno Unified Scope & Sequence Final Draft 2013-2014
Modeling and Projects 2 weeks Algebra II
Algebra II
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
Modeling 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 and project based units built into the schedule. Modeling 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 and 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 and often serves to motivate students to become independent learners. Figure 1: The modeling cycle. Students examine a problem and 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, and report out. Fresno Unified Scope & Sequence Final Draft 2013-2014
Algebra II
Algebra II
EMPHASES BY CLUSTER (PARCC/SBAC Major Clusters (70%)
Supporting Clusters (20%)
Areas of intensive focus, where students need fluent understanding and application of the core Rethinking and linking; areas where some material is being covered, but in a way that applies core understandings Seeing the Structure in Expressioins o
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Interpret the structure of expressions. Write expressions in equivalent forms to solve problems. Arithmetic with Polynomials and Rational Exprressions o
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Rewrite rational expressions. Use polynomial identities to solve problems. Arithmetic with Polynomials and Rational Exprressions o Perform arthimetic operations on polynomials. o Understand the relationship between zeros and factors of polynomials. Creating Equations o Create equations that descrbe numbers or relationships. Reasoning with Eations and Inequalities o Understand solving equations as a process of reasoning and explain the reasoning. o Solve equations and inequalities in one variable. o Solve systems of equations. ALGEBRA Fresno Unified Scope & Sequence Final Draft 2013-2014
Algebra II
Algebra II
Major Clusters (70%)
Supporting Clusters (20%)
Areas of intensive focus, where students need fluent understanding and application of the core Rethinking and linking; areas where some material is being covered, but in a way that applies core understandings Interpreting Functions o
Interpret functions that arise in applications in terms of the context. Interpreting Functions o
Linear, Quadratics and Exponential Models NUMBER & QUANTITY STATISTICS & PROBABILTY FUNCTIONS Building Functions o
Analyze functions using different representaions. o
Construct and compare linear, quadratic, and exponential models and solve problems. Buils a function that models a relationship between two quantities. Building Functions . o Build new functions from exisiting functions. Making Inferences and Justifying Conclusions Making Inferences and Justifying Conclusions Trigonometric Functions o
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Make inferences and justify conclusions from sample surveys, experiments, and observational studies. o
Extend the domain of trogonometric functions using the unit circle.. Model periodic phenomena with trigonometric fuctions. Prove and apply trigonometric identies. Understand and evaluate random processes underlying statistical experiemnts. Interpreting Categorical and Quantatative Data o
Summarize, represent, and 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. Fresno Unified Scope & Sequence Final Draft 2013-2014
Algebra II