Mathematics Algebra I – Year at a Glance (SAMPLE) Math Algebra I Year Long Scope and Sequence Page 1 Mathematics Algebra I – Year at a Glance (SAMPLE) Univariate Statistics Unit 12 Bivariate Statistics: Linear, Quadratic, and Exponential Unit 11 Piecewise Functions Unit 10 Radical Functions Unit 9 Quadratic Functions Unit 8 Solving Quadratic Equations Unit 7 Arithmetic with Polynomials Unit 6 Linear and Exponential Functions Unit 5 Evaluating Functions Unit 4 Systems of Linear Equations and Inequalities Unit 3 Linear Equations and Inequalities Unit 2 Solving SingleVariable Equations and Inequalities Unit 1 12 days 10 days 15 days 15 days 10 days 15 days 20 days 20 days 14 days 15 days 12 days 12 days A-CED.A.1 A-REI.D.10 A-CED.A.3 F-IF.A.1 A-SSE.B.3c A-SSE.A.1 A-SSE.A.2 F-IF.B.4 F-IF.B.4 A-REI.11 S-ID.C.7 N-Q.A.3 A-CED.A.4 A-REI.D.12 A-REI.C.5 F-IF.A.2 A-CED.A.1 A-APR.A.1 A-REI.A.1 F-IF.B.5 F-IF.B.5 F-IF.B.4 S-ID.C.8 S-ID.A.1 A-REI.B.3 N-Q.A.1 A-REI.C.6 F-IF.A.3 A-CED.A.2 A-REI.B.4 F-IF.B.6 F-IF.B.6 F-IF.C.7b S-ID.C.9 S-ID.A.2 N-Q.A.1 F-IF.C.7a A-REI.D.12 F-IF.B.5 F-LE.A.1 A-SSE.B.3a A-APR.B.3 F-IF.C.7b F-BF.A.1a S-ID.A.3 F-BF.B.3 F-LE.A.2 A-SSE.B.3b F-IF.C.7a F-IF.C.9 N-Q.A.1 N-RN.B.3 Major Clusters A-SSE Seeing Structure in Expressions (1, 2) A-APR Arithmetic with Polynomials and Rational Expressions (1) A-CED Creating Equations (1, 2, 3, 4) A-REI Reasoning with Equations and Inequalities (1, 3, 4, 5, 6, 10, 11, 12) F-IF Interpreting Functions (1, 2, 3, 4, 5, 6) S-ID Interpreting Categorical and Quantitative Data (7, 8, 9) F-LE.A.3 F-IF.C.8a N-Q.A.2 F-LE.B.5 F-IF.C.9 S-ID.B.5 F-BF.B.3 S-ID.B.6 Supporting Clusters N-Q Quantities (1, 2, 3) A-SSE Seeing Structure in Expressions (3) A-APR Arithmetic with Polynomials and Rational Expressions (3) F-IF Interpreting Functions (7, 8, 9) F-BF Building Functions (1) F-LE Linear, Quadratic, and Exponential Models (1, 2, 3, 5) S-ID Interpreting Categorical & Quantitative Data (5, 6) Additional Clusters N-RN The Real Number System (3) F-BF Building Functions (3) S-ID Interpreting Categorical & Quantitative Data (1, 2, 3) Other MP – Standards for Mathematical Practice Page 2 Mathematics Algebra I – Year in Detail (SAMPLE) th Summary of Year for Algebra I Algebra I will begin where 8 grade ended: linear equations in one and two variables and linear functions. In Algebra I students will increase their procedural skill and fluency in solving linear equations and inequalities in one variable. Additionally, students will deepen their understanding of linear equations and inequalities in two variables. Also, the course will emphasize modeling with linear equations and inequalities, culminating with solving systems of both linear equations and inequalities. From there, the course shifts to developing a deeper understanding of functions. Students will focus on linear and exponential functions by exploring situations that could be modeled by either a linear function or an exponential function. Then students will move to studying quadratic equations and functions, including identifying key elements of graphs, transformations with functions, and identifying domain and range. Students will apply these newly developed understandings and skills with functions to simple radical functions and then to piecewise-defined functions. The course concludes with a study of bivariate and univariate statistics to develop necessary understandings and skills the students will need for their study of statistics in Algebra II. Standards Clarification for Algebra I • • Some standards may be revisited several times during the course; others may be only partially addressed in different units, depending on the focus of the unit. Comments are included throughout the document to clarify and provide additional background for each unit. Some standards are addressed in both Algebra I and Algebra II. The relevance to Algebra I is discussed in the Assessment Limits and Clarifications column. Mathematical Practices Recommendations for Algebra I Two overarching practices relevant to Algebra I are: • Make sense of problems and persevere in solving them (MP.1). • Model with mathematics (MP.4). Specific modeling standards appear throughout the high school standards indicated by a star (★) symbol. Indeed, other mathematical practices in Algebra I might be seen as contributing specific elements of these two. The intent of the following set is not to decompose the above mathematical practices into component parts but rather to show how the mathematical practices work together. • Reason abstractly and quantitatively (MP.2). This practice standard refers to one of the hallmarks of algebraic reasoning, the process of decontextualization and contextualization. Much of elementary algebra involves creating abstract algebraic models of problems (A-CED, F-BF) and then transforming the models via algebraic calculations (A-SSE, A-APR, F-IF) to reveal properties of the problems. • Use appropriate tools strategically (MP.5). Spreadsheets, a function modeling language, graphing tools, and many other technologies can be used strategically to gain understanding of the ideas expressed by individual content standards and to model with mathematics. • Attend to precision (MP.6). In algebra, the habit of using precise language is not only a mechanism for effective communication but also a tool for understanding and solving problems. Describing an idea precisely (A-CED, A-REI) helps students understand the idea in new ways. • Look for and make use of structure (MP.7). For example, writing a practice many teachers refer to as “chunking,’’ highlights the structural similarity between this expression and z2 + 5z + 6, leading to a factorization of the original: ( (7x) + 3) ( (7x) + 2) (A-SSE, A-APR). • Look for and express regularity in repeated reasoning (MP.8). Creating equations or functions to model situations is harder for many students than working with the resulting expressions. An effective way to help students develop the skill of describing general relationships is to work through several specific examples and then express what they are doing with algebraic symbolism (A-CED). For example, when comparing two different text messaging plans, many students who can compute the cost for a given number of minutes have a hard time writing general formulas that express the cost of each plan for any number of minutes. Constructing these formulas can be facilitated by methodically calculating the cost for several different input values and then expressing the steps in the calculation, first in words and then in algebraic symbols. Once such expressions are obtained, students can find the break-even point for the two plans, graph the total cost against the number of messages sent, and make a complete analysis of the two plans. Page 3 Mathematics Algebra I – Year in Detail (SAMPLE) Fluency Recommendations for Algebra I A/G: Algebra I students become fluent in solving characteristic problems involving the analytic geometry of lines, such as writing down the equation of a line given a point and a slope. Such fluency can support them in solving less routine mathematical problems involving linearity, as well as in modeling linear phenomena (including modeling using systems of linear inequalities in two variables). A-APR.1: Fluency in adding, subtracting, and multiplying polynomials supports students throughout their work in algebra, as well as in their symbolic work with functions. Manipulation can be more mindful when it is fluent. A-SSE.1b: Fluency in transforming expressions and chunking (parts of an expression as a single object) is essential in factoring, completing the square, and other algebraic calculations. Page 4 Mathematics Algebra I – Year in Detail (SAMPLE) Possible time frame: 12 days Unit 1: Solving Single-Variable Equations and Inequalities In this unit students will extend the work of the last three years and use linear equations and inequalities in one variable to model and solve real-world and mathematical problems. The ultimate goal of this unit is for students to be able to rearrange a formula to highlight a quantity of interest and understand this process as using the same reasoning as in solving equations. This will allow for students to use units as a way to understand problems and to guide their solution to multi-step problems by choosing and interpreting units consistently in formulas. Students will continue to use their conceptual understanding as well as their procedural skill and fluency in solving equations throughout the course to solve systems of linear equations and quadratic equations. Assessment Limits and Clarifications Major Cluster Standards Create equations that describe numbers or relationships HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. ★ HSA-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. ★ Modeling standard HAS-CED-A.1 i) Assessment tasks are limited to linear, quadratic, or exponential equations with integer exponents. ★ Solve equations and inequalities in one variable HSA-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Assessment Limits and Clarifications Supporting Cluster Standards Reason quantitatively and use units to solve problems. HSN-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. ★ Modeling standard HSN-Q.A.1 The skills associated with this standard can and should be used in nearly every unit of this course. This standard will culminate in Unit 11 when students study bivariate statistics. ★ HSN-Q.A.1 Does not include choosing and interpreting the scale and the origin in graphs and data displays. Additional Cluster Standards Use properties of rational and irrational numbers. HSN–RN.B.3 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 nonzero rational number and an irrational number is irrational. Assessment Limits and Clarifications N-RN.B.3 This standard can be taught through working with formulas involving rational and irrational numbers. Page 5 Mathematics Algebra I – Year in Detail (SAMPLE) Possible time frame: 10 days Unit 2: Linear Equations and Inequalities th th In 8 grade students developed a deep conceptual understanding of the equation for a line and slope as a rate of change. Additionally students will leave the 8 grade able to fluently graph linear equations in the form y=mx+b. In this unit students will extend their work in graphing linear equations to graphing linear inequalities. Students will also understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. This understanding can then be extended to the graph of a linear inequality in two variables. Assessment Limits and Clarifications Major Cluster Standards Represent and solve equations and inequalities graphically HSA-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). HSA-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. HSA-REI.D.12 Limited to a single linear inequality. Students will graph systems of linear inequalities in Unit 3. Assessment Limits and Clarifications Supporting Cluster Standards Reason quantitatively and use units to solve problems. HSN-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. ★ Analyze functions using different representations HSF-IF.C.7 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. Modeling standard ★ HSN-Q.A.1 The focus in this unit is choosing and interpreting the scale and the origin in graphs. HSF-IF.C.7a Limited to linear equations. Students will graph piece-wise defined functions in Unit 10. Page 6 Mathematics Algebra I – Year in Detail (SAMPLE) Possible time frame: 15 days Unit 3: Systems of Linear Equations and Inequalities th In 8 grade students were introduced to systems of linear equations and developed a conceptual understanding of the solution to a system of linear equations as the point of intersection of their graphs which is the point that satisfies both equations simultaneously. Students also solved systems of linear equations algebraically as well as estimated the solution to a system by graphing. This unit will extend the students work with systems of linear equations to include a focus on modeling with systems of linear equations and systems of linear inequalities. In this unit students will increase their procedural skill and fluency in solving systems of linear equations exactly and approximately (e.g., with graphs) and develop the ability to select a solution method based on the specific system they are trying to solve. Assessment Limits and Clarifications Major Cluster Standards Create equations that describe numbers or relationships HSA-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. ★ Solve systems of equations HSA-REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. HSA-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Represent and solve equations and inequalities graphically HSA-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Modeling standard ★ Students will use their procedural skill and fluency in graphing a linear inequality to graph a system of linear inequalities. (A-REI.D.12 from Unit 2) HSA-REI.C.6 This unit should include a heavy emphasis on using systems of equations (and inequalities) to model and solve real-world problems. i) Assessment tasks have a realworld context. ii) Assessment tasks have hallmarks of modeling as a mathematical practice (less defined tasks, more of the modeling cycle, etc.). Page 7 Mathematics Algebra I – Year in Detail (SAMPLE) Possible time frame: 15 days Unit 4: Evaluating Functions In this unit students extend their work with functions in grade 8 to develop a more sophisticated understanding of a function including function notation. After function notation is established, students will evaluate functions. As students evaluate functions using their graphs they will begin developing an understanding of function transformations which will be discussed more thoroughly in Unit 9. Also in this unit students will graph functions by creating a table of values. Function types are not limited in this unit but should focus on linear, exponential, quadratic, and radical functions. Through graphing their own functions, students will deepen their understanding of function transformations. Graphing will also provide students the opportunity to apply their conceptual understanding of domain and range as it relates to the function rule and its corresponding graph. Major Cluster Standards Understand the concept of a function and use function notation HSF-IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). HSF-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. HSF-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. Interpret functions that arise in applications in terms of the context HSF-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★ Additional Cluster Standards Build new functions from existing functions HSF-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Assessment Limits and Clarifications Modeling standard ★ HSF-IF.B.5 As students explore specific function types in Units 8, 9, and 10, they will continue to relate the domain of a function to its graph and, in context, to the quantitative relationship it describes. Assessment Limits and Clarifications HSF-BF.B.3 As students explore quadratic functions in Unit 8, they will formalize their understanding of function transformations and then use this understanding in Units 9 and 10. HSF-BF.B.3 i) Identifying the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k are limited to linear and quadratic functions. ii) Experimenting and illustrating an explanation of the effects on the graph using technology is limited to the functions in HSF-IF.B.4, HSF-IF.B.6, and HSFIF.C.9. iii) Assessment tasks do not involve recognizing even and odd functions. Page 8 Mathematics Algebra I – Year in Detail (SAMPLE) Unit 5: Linear and Exponential Functions Possible time frame: 10 days After a broad study of functions in Unit 4 that focused on the concept of a function and the relationship between inputs and outputs, students will take a closer look at linear and exponential functions. Students will be very familiar with relationships which grow linearly, but this will be their first exposure to relationships which grow exponentially. Students will recognize real-world situations can be modeled with linear and with exponential functions. Additionally, students will write linear and exponential functions, including arithmetic and geometric sequences, given a graph, a table, a description of a relationship, or two input-output pairs. Students will use the functions they create to solve problems and will interpret the parameters of both linear and exponential functions in terms of a context. Major Cluster Standards Write expressions in equivalent forms to solve problems HSA-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★ c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Create equations that describe numbers or relationships HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. ★ HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. ★ Supporting Cluster Standards Construct and compare linear, quadratic, and exponential models and solve problems HSF-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. ★ a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. HSF-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). ★ HSF-LE.A.3 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 HSF-LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context. ★ Assessment Limits and Clarifications Modeling standard HSA-SSE.B.3c Students will only evaluate and use exponential functions with domains in the integers. i) Assessments have a real-world context. There is interplay between the mathematical structure of the expression and the structure of the situation such that choosing and producing an equivalent form of the expression reveals something about the situation. ii) Tasks are limited to exponential expressions with integer exponents. HAS-CED.A.1 i) Assessments limited to linear, quadratic, or exponential equations with integer exponents. HSA-CED.A.2 Although students will be presented with graphs of exponential functions, students will not be expected to graph exponential functions. ★ Assessment Limits and Clarifications Modeling standard HSF-LE.A.2 Students will fit exponential and linear functions to data sets represented by scatter plots in Unit 11. i) Assessment tasks are limited to constructing linear and exponential functions in simple context (not multi-step). HSF-LE.B.5 i) Tasks have a real-world context. ii) Exponential functions are limited to those with domains in the integers. ★ Page 9 Mathematics Algebra I – Year in Detail (SAMPLE) Possible time frame: 15 days Unit 6: Arithmetic with Polynomials Units 6 and 7 will focus on developing the algebraic skills and fluencies that are essential to the work with quadratic functions beginning in Unit 8. This unit will focus on performing arithmetic operations with polynomials. Students will apply their understanding and skill in generating equivalent expressions involving combining like terms, the distributive property, and the properties of integer exponents to begin multiplying polynomials. Through their work with polynomial expressions with and without context, students will interpret parts of expressions including terms, factors, and coefficients; moreover, students will interpret complicated expressions in context by viewing one or more of their parts as a single entity. Assessment Limits and Clarifications Major Cluster Standards Interpret the structure of expressions HSA-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.★ a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. Perform arithmetic operations on polynomials HSA-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Modeling standard ★ HSA-APR.A.1 Multiplication of polynomials should extend beyond the product of two binomials. Page 10 Mathematics Algebra I – Year in Detail (SAMPLE) Unit 7: Solving Quadratic Equations Possible time frame: 20 days After learning in Unit 6 how to multiply polynomials, students will learn in this unit how to undo the process of multiplying polynomials to factor polynomials. Students th began factoring work in the 6 grade where they applied the distributive property to simplify an expression involving a variable as well as to factor out a greatest common factor of an expression involving a variable. This unit will build on this work by increasing the complexity of the expressions and the complexity of applying the distributive property. This unit will begin by teaching students the necessary conceptual understanding and procedural skill and fluency in factoring quadratics to enable them to finish the unit by solving quadratic equations. Students will also solve quadratic equations that cannot be solved by factoring requiring the introduction of completing the square and then further develop into the quadratic formula. Once these alternate methods have been established, students will reengage in solving quadratic equations and extend their work to include quadratics with irrational solutions. By the end of this unit, students will be able to select the most appropriate solution method based on the initial form of the equation. Major Cluster Standards Assessment Limits and Clarifications Interpret the structure of expressions 4 4 2 2 2 2 HSA-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x – y as (x ) – (y ) , 2 2 2 2 thus recognizing it as a difference of squares that can be factored as (x – y )(x + y ). Students will solve single variable equations to find the solutions for a quadratic equation. (A-REI.B.3 from Unit 1) Understand solving equations as a process of reasoning and explain the reasoning HSA-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Solve equations and inequalities in one variable HSA-REI.B.4 Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the 2 form (x – p) = q that has the same solutions. Derive the quadratic formula from this form. 2 b. Solve quadratic equations by inspection (e.g., for x = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Supporting Cluster Standards Write expressions in equivalent forms to solve problems HSA-SSE.B.3 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. HAS-SSE.A.2 i) Assessment tasks are limited to numerical expressions and polynomial expressions in one variable. ii) Examples: 2 2 Recognize 53 - 47 as a difference of squares and see an opportunity to rewrite it in the easier-to-evaluate form 2 (53-47)(53+47). See an opportunity to rewrite a + 9a + 14 as (a+7)(a+2). HAS-REI.A.1 i) Assessment tasks are limited to quadratic equations. HSA-REI.B.4 Students will use the skills associated with this standard in Unit 8 to graph quadratic functions. Assessment Limits and Clarifications HSA-SSE.B.3 Quadratic functions are not introduced until Unit 8; therefore the solutions for a quadratic function should not be referred to zeros until then. Page 11 Mathematics Algebra I – Year in Detail (SAMPLE) Possible time frame: 20 days Unit 8: Quadratic Functions 2 Students will begin this unit revisiting the function f(x)=x that they worked with in Unit 4 and graph the function by using by creating a table of values. From there students will 2 reengage with function transformations to graph quadratic functions and to generate a general form for a quadratic function (f(x)=a(x-h) +k). After developing their conceptual understanding of quadratic functions and procedural skill and fluency in graphing them, students will begin working with quadratic functions in different forms that will lead them to using other methods for graphing. Quadratic functions that model real-world relationships between quantities will be presented to students in this unit, and students will compare quadratic functions presented in different forms As students analyze and graph quadratic functions, the topic of domain should be revisited in this new context. Students will also identify key features of quadratic graphs and will interpret the meaning of these features in the context of the real-world relationship the graph models. Major Cluster Standards Assessment Limits and Clarifications Interpret functions that arise in applications in terms of the context HSF-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ HSF-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★ HSF-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.★ Modeling standard HSF-IF.B.4 i) Assessment tasks have a real-world context. ii) Assessment tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewisedefined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. ★ HSF-IF.B.6 i) Assessment tasks have a real-world context. ii) Assessment tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewisedefined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. Supporting Cluster Standards Assessment Limits and Clarifications Understand the relationship between zeros and factors of polynomials HSA-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Analyze functions using different representations HSF-IF.C.7 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. HSF-IF.C.8 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. HSF-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. Modeling standard HSA-APR.B.3 This standard should be addressed early in this unit when students start connecting solutions for the equation to zeros on the graph. Although cubic functions should be addressed as part of this standard, the main focus should still be quadratic functions. i) Assessment tasks are limited to quadratic and cubic polynomials in which linear and quadratic factors are available. For example, 2 find the zeros of (x - 2)(x - 9). HSF-IF.C.7a Limited to only quadratic functions. HSF-IF.C.9 i) Assessment tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. ★ Page 12 Mathematics Algebra I – Year in Detail (SAMPLE) Additional Cluster Standards Assessment Limits and Clarifications Build new functions from existing functions HSF-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. HSF-BF.B.3 i) Identifying the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x+k) for specific values of k (both positive and negative) is limited to linear and quadratic functions. ii) Experimenting with cases and illustrating an explanation of the effects on the graph using technology is limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. iii) Tasks do not involve recognizing even and odd functions. Page 13 Mathematics Algebra I – Year in Detail (SAMPLE) Possible time frame: 14 days Unit 9: Radical Functions Students will be coming in to this unit with a good understanding of how to use function transformations, how to evaluate functions, and how to graph linear and quadratic functions. Much of this conceptual understanding and procedural skill and fluency will be the foundation for this unit on radical functions. In Unit 4 students had limited exposure to radical functions as they explored the concepts of functions. In this unit students will be responsible for graphing square root and cube root functions. Students will graph the parent functions using a table of values and will again experiment with transformations to show that the familiar function transformation rules extend to radical functions. Radical functions that model real-world relationships between quantities will be presented. Students will compare radical functions presented in different forms. The topic of domain should be revisited in this new context. Students will also identify key features of radical graphs and will interpret the meaning of these features in the context of the real-world relationship the graph models. Major Cluster Standards Interpret functions that arise in applications in terms of the context HSF-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ HSF-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. ★ HSF-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.★ Assessment Limits and Clarifications Modeling standard HSF-IF.B.4 i) Assessment tasks have a real-world context. ii) Assessment tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewisedefined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. HSF-IF.B.6 i) Assessment tasks have a real-world context. ii) Assessment tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewisedefined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. ★ Supporting Cluster Standards Analyze functions using different representations HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. HSF-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. Assessment Limits and Clarifications Modeling standard HSF-IF.C.7b Limited to square root and cube root functions. Students do not solve square root or cube root equations in this course. HSF-IF.C.9 i) Assessment tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. ★ Page 14 Mathematics Algebra I – Year in Detail (SAMPLE) Possible time frame: 15 days Unit 10: Piecewise Functions This unit will continue to build on students’ conceptual understanding and procedural skill and fluency in working with functions. In this unit students will be responsible for graphing various piecewise-defined functions with an initial focus on linear functions leading to the concepts of step functions and absolute value functions. To begin this unit, students will explore real-world situations that require multiple function rules to model a given situation. Students will begin by graphing piecewise-linear functions and will extend this to include all previously graphed function types. Absolute value and step functions are among the various piecewise-defined functions students will graph in this unit. Students will again use their conceptual understanding of function transformations to graph various absolute value functions and step functions. Students will also identify key features of a graph and interpret the meaning of these features in the context of the real-world relationship the graph models. Major Cluster Standards Represent and solve equations and inequalities graphically HSA-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★ Interpret functions that arise in applications in terms of the context HSF-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ Assessment Limits and Clarifications Modeling standard ★ HAS-REI.D.11 i) Tasks that assess conceptual understanding of the indicated concept may involve any of the function types mentioned in the standard except exponential and logarithmic functions. ii) Finding the solutions approximately is limited to cases where f(x) and g(x) are polynomial functions. HSF-IF.B.4 i) Assessment tasks have a real-world context. ii) Assessment tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewisedefined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. Supporting Cluster Standards Analyze functions using different representations HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Assessment Limits and Clarifications Modeling standard ★ HSF-IF.C.7b Limited to piecewise-defined functions including, but not limited to, step functions and absolute value functions. The piece-wise functions that students will graph in this unit should include linear, quadratic, and radical functions as pieces. (F-IF.C.7 from Units 2, 8, and 9) Page 15 Mathematics Algebra I – Year in Detail (SAMPLE) Possible time frame: 15 days Unit 11: Bivariate Statistics: Linear, Quadratic, and Exponential This unit is designed to provide students the opportunity to apply their conceptual understanding of functions (emphasizing linear, quadratic, and exponential) in a statistics context. This unit will support and enhance the students’ conceptual understanding of functions and their ability to model real-world situations. The students will begin this unit by summarizing categorical data for two categories in two-way frequencies tables. From there students will represent data on two quantitative variables on a scatter plot and describe how the variables are related. They will extend their work to begin fitting a function to the data and using those functions to solve problems in the context of the data. Beyond informally assessing the fit of a function to a data set, students will compute (using technology) and interpret the correlation coefficient of a linear fit. Finally, students will establish and distinguish between correlation and causation in the context of the data. Major Cluster Standards Assessment Limits and Clarifications Interpret linear models HSS-ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. HSS-ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. HSS-ID.C.9 Distinguish between correlation and causation. Supporting Cluster Standards Build a function that models a relationship between two quantities HSF-BF.A.1 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. Reason quantitatively and use units to solve problems. HSN-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. ★ HSN-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. ★ Summarize, represent, and interpret data on two categorical and quantitative variables HSS-ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. HSS-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. Assessment Limits and Clarifications Modeling standard Students will write linear and exponential functions to model the relationship between two quantitative variables in a data set. (ACED.A.2 from Unit 5) HSF-BF.A.1 i) Assessment tasks have a real-world context. ii) Assessment tasks are limited to linear functions, quadratic functions, and exponential functions with domains in the integers. HSN-Q.A.1 This standard culminates in this unit by having students choose and interpret the scale and the origin in data displays. HSN-Q.A.2 This standard will be assessed in Algebra I by ensuring that some modeling tasks (involving Algebra I content or securely held content from grades 6-8) require the student to create a quantity of interest in the situation being described (i.e., a quantity of interest is not selected for the student by the task). For example, in a situation involving data, the student might autonomously decide that a measure of center is a key variable in a situation, and then choose to work with the mean. HSS-ID.B.6a i) Assessment tasks have a real-world context. ii) Exponential functions are limited to those with domains in the integers. ★ Page 16 Mathematics Algebra I – Year in Detail (SAMPLE) Possible time frame: 12 days Unit 12: Univariate Statistics In this unit students will use univariate statistics to compare and make inferences about two data sets. This unit will utilize statistical concepts and skills the students learned in th th the 6 and 7 grades and use them to develop more sophisticated statistical concepts and skills. Students will represent data sets graphically by plotting on the real number line including dot plots, histograms, and box plots and use their graphical displays to make observations about the shape and distribution of the data. Students will then use measures of center (median, mean) and spread (interquartile range, standard deviation) appropriate to the shape of the data distribution to compare two or more data sets. Additionally students will explore more closely the differences in shape, center, and spread accounting for possible effects of extreme data points (outliers) and interpret these differences within the context of the data sets. As students work with statistical measures in context they should choose levels of accuracy (and precision) appropriate to the limitations on measurement. Assessment Limits and Clarifications Supporting Cluster Standards Reason quantitatively and use units to solve problems. HSN-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. ★ Additional Cluster Standards Summarize, represent, and interpret data on a single count or measurement variable HSS-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots). HSS-ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. Modeling standard ★ Assessment Limits and Clarifications HSS-ID.A.2 Standard deviation is the only statistical measure the students have not been exposed to prior to this unit. HSS-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Page 17

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