Table of Contents 1 Table of Contents Weeks Unit CF Connecting Patterns and Functions 4-6 weeks CF.0 Culture Building - Launch section by building classroom culture. CF.1 Functions & Notation - Introduce function notation and review definitions of function through patterns. CF.2 Pattern Tasks- Find inputs and outputs of patterns and express in function notation. Write equations of linear functions with a discrete domain and range. Connect multiple representations of functions. CF.3 Expressing Quantities - Express quantities through converting units. CF.4 Domain & Range - Explicit understanding of domain and range. Unit LF Linear Functions 3-5 weeks LF.0 Linear Function Refresher - Launch unit by reviewing graphing and writing equations of lines. LF.1 Domain & Range - Domain and range of lines and line segments. LF.2 Apply Linear Functions - Solve problems of linear functions in a related context to write equations, express inputs and outputs in function notation, determine if functions are linear, find domain, range and inverse, and shift graphs of linear functions. Unit EX Exponential Functions and Equations 3 5 weeks EX.0 Introduce Exponential Functions - Launch unit by comparing linear and exponential growth. EX.1 Growth Rate Coefficients - Develop the concept of growth rate coefficients of exponential functions result in growth if they are greater than 1, and decay if they are between 0 and 1. EX.2 Exponential Growth - Exponential growth tasks EX.3 Exponential Decay - Exponential decay tasks EX.4 Synthesizing Exponential Functions - Synthesizing exponential functions by applying a linear or exponential growth or decay equation, and interpreting graphs with linear and exponential parts. Unit RE Regression Equations 3 5 weeks RE.0 Predictions - Launch unit by making predictions from scatterplots fitted by linear and exponential functions. RE.1 Draw Models - Make predictions from data modeled by linear and exponential functions (with a high correlation coefficient). RE.2 Model with Technology - Use technology to find linear and exponential equations that model data collected in experiments. Develop concept of 1 Adapted from Dana Center 5/29/15 Top of Document Page 1 of 28
influential points (outliers that change a linear regression equation and correlation coefficient). RE.3 Correlation - Interpret correlation coefficient and distinguish between correlation and causation. RE.4 The Best Equation Find residuals, construct residual plots and use them to determine the best regression equation. Unit SE Systems of Equations 3 5 weeks SE.0 Solve a Simple System - Launch the unit by having students determine the number of pennies and nickels in a box by reasoning. SE.1 Review of 8th Grade Systems Solve systems by graphing and substitution. SE.2 Linear Combinations Prove that you can use combinations of linear functions to solve a system of equations, and solve systems by using combinations. SE.3 Methods to Solve Systems Choose a method to solve a system: graphing, tables, substitution or combination methods SE.4 Systems Applications Write an solve systems of equations to solve problems. Unit IQ Inequalities 2 3 weeks IQ.0 Line Art Connect graphing line segments with restricted domains, and use technology to rewrite equations as inequalities. IQ.1 One Variable Inequalities Solve and graph one- variable inequalities. IQ.2 Compound Inequalities Solve and graph one- variable compound inequalities. IQ.3 Linear Inequalities Solve and graph two- variable inequalities. IQ.4 Systems of Inequalities Write, graph and find solutions to two- variable inequalities. Unit PA Parabolas & Applications 3 4 weeks PA.0 Rectangular Tile Patterns Launch the unit by representing rectangular tiles patterns with tables, equations and graphs. PA.1 Parabolas Solve problems using equations of parabolas superimposed on pictures. PA.2 Critical Points Using maximum, minimum, vertex, and intercepts of equations of parabolas written in standard form to solve problems. PA.3 Maximum Investigation Investigate equations parabolas that reveal the vertex. PA.4 Quadratic Regression Determining if data is best fit with linear, exponential or quadratic regression. Unit SQ Sequences and Functions 3 4 weeks SQ.1 Arithmetic Sequences Formalize explicit form of arithmetic sequences from the Connecting Patterns and Functions unit. Expand to recursive form and express both forms in sequence notation. SQ.2 Geo Sequences Write geometric sequences in explicit and recursive form and express both forms in sequence notation. 5/29/15 Top of Document Page 2 of 28
SQ.3 Compare Sequences - Determine if sequences are arithmetic or geometric. Beginning in the spring of 2015, Washington State high school students will take the 11th grade Smarter Balanced Comprehensive Assessments for English language arts and mathematics. The results of these assessments will be reported to the U.S. Department of Education for purposes of determining adequate yearly progress. A cut- score for each assessment indicating college and career readiness will be determined by the Smarter Balanced Assessment Consortium. The Washington State Board of Education may set an additional cut- score on the comprehensive assessments that could be used to satisfy the high school mathematics and English language arts assessment graduation requirements for the classes of 2015 and beyond. In addition to the comprehensive assessment for mathematics, students in the classes of 2015 through 2018 may take end- of- course exit exams (EOCs) when an appropriate academic course has been completed (Algebra, Geometry, Integrated I or Integrated II) to meet the high school mathematics assessment graduation requirement. Certificate of academic achievement options such as Collection of Evidence (COE), alternative assessment scores (SAT, ACT, IB, AP), or GPA comparison will also be available. The End- of- Course Exit Exam Specification documents (August 2014) contain information on the Common Core State Standards for Mathematics (CCSS- M) that are eligible to be assessed on the mathematics EOC Exams. A group of Washington State mathematics teachers was convened to determine these CCSS- M standards and the appropriate weighting of the content on the EOC exit exams. The mathematical content is listed by conceptual category, domain, cluster and standard as written in the CCSS- M for high school (http://www.k12.wa.us/corestandards/mathematics/pubdocs/ccssi_mathstandards.pdf ). Standards highlighted in gray are assessed and reported as course- specific content but are not used for purposes of graduation. Standards not highlighted are the standards common to Algebra 1/Integrated Mathematics 1 or Geometry/Integrated Mathematics 2 that are assessed for purposes of graduation. In addition, Smarter Balanced has developed four Mathematical Claims that state what students should know and be able to do in the domain of mathematics, and what the Smarter Balanced assessment system will provide data on. The content of the CCSS- M, as well as the eight Standards for Mathematical Practice within the CCSS- M that describe the mathematical habits mathematics educators should seek to develop in their students provide the basis of the claims that Smarter Balanced will use to report student progress. The EOC Exit Exams will report out on these same claims. The approximate emphasis on each of the claims for reporting purposes is indicated in the End- of- Course Exit Exam Specification documents. As with the Smarter Balanced comprehensive assessment, claims 2 and 4 will fall under one reporting strand. CCSS- M Standards for Mathematical Practice: 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. Smarter Balanced Mathematical Claims 5/29/15 Top of Document Page 3 of 28
Claim Explanation Practice Standard 1 Concepts & Procedures 5, 6, 7, 8 Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.* This claim addresses procedural skills and the conceptual understanding on which developing skills depend. It is important to assess student understanding of how concepts link together and why mathematical procedures work the way they do. This relates to the structural nature of mathematics. 2 Problem Solving Students can solve a range of complex well- posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies.* Assessments items and tasks focused on Claim 2 include problems in pure mathematics and problems set in context. Problems are presented as items and tasks that are well- posed (that is, problem formulation is not necessary) ad for which a solution path is n0t immediately obvious. These problems require students to construct their own solution pathway rather than follow a provided one. Such problems will therefore be unstructured, and students will need to select appropriate conceptual and physical tools to use. 3 Communicating Reasoning Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.* Claim 3 refers to a recurring theme in the CCSS- M content and practice standards the ability to construct and present a clear, logical, convincing argument. For older students, this may take the form of a rigorous, deductive proof based on clearly stated axioms. For younger students, this will involve more informal justifications. Assessment tasks that address this claim will typically present a claim and ask students to provide, for example, a justification or counterexample. 4 Modeling and Data Analysis Students can analyze complex, real- world scenarios and can construct and use mathematical models to interpret and solve problems.* Modeling links classroom mathematics and statistics to everyday life, work and decision- making. Students use modeling and data analysis to choose and use appropriate mathematics and statistics to analyze and understand situations, to make predictions, find solutions and improve decision making based on results from the model. CCS- M features modeling as both a mathematical practice at all grades a content focus in high school. 1, 5, 7, 8 3,6 2, 4, 5 * http://www.smarterbalanced.org/wordpress/wp- content/uploads/2012/09/smarter- Balanced- Mathematics- Claims/pdf 5/29/15 Top of Document Page 4 of 28
Major Content Connecting Patterns and Functions (CF) Target Practice 2 Reason abstractly and quantitatively. Practice 4 Model with mathematics. Practice 6 Attend to precision. Learning Target Learning Target Unit Emphasis Unit Emphasis A.CED I can create equations and inequalities that describe numbers or relationships.! Create a one variable equation.! Create equations from situation, tables, or graphs. Use equations to solve problems. A-CED.1! Create equations with 2 variables to represent relationships between two quantities with equations and tables.! Graph equations on coordinate axes with labels and scales.! Create equations from situation, tables, or graphs. Use equations to solve problems. A-CED.2 A-REIc I can represent and solve equations and inequalities graphically.! Understand that coordinate pairs that area solutions to a pattern function are all on the same line. A-REI.10 F-IFa I can understand the concept of a function and use function notation.! Identify x, the input, as a quantity of the domain.! Identify f(x) the output, as a quantity of the range.! Apply the definition of a function (a function assigns exactly one output for each input.) to determine if an equation, a table, or graph is a function.! Know the graph of the function is the graph of all (x, f(x)) coordinate points. F- IF.1! Use function notation to express an input with its connected output! Convert a table, graph, set of ordered pairs, or description into function notation by identifying the equation used to turn inputs into outputs and writing the equation.! Only evaluate functions with inputs in domain! Analyze the input and output values of a function based on a problem situation. F- IF.2 F-IFb I can interpret functions that arise in applications in terms of the context.! Given the graph or relationship of a pattern function, determine the practical domain of the function as it relates to the numerical relationship it describes. F- IF.5 Foundational Content Supporting Content F-IFc I can analyze functions using different representations.! Compare rate of change and intercepts of two pattern functions, each represented in a different way (equations, tables, graphs, and situations). F-IF.9 F-BF I can build a function.! Write a function that models a pattern. F-BF.1a N-Q I can reason quantitatively and use units to solve problems.! Interpret units in the context of the problem.! Choose and interpret both the scale and the origin in graphs.! Convert units to make sense of the problem. N-Q.1! Determine and interpret appropriate quantities when using modeling situations. N-Q.2 8.F.1.I understand that a function is a rule that assigns to each input exactly one output. 8.F.1 8.F.4 I can construct a linear function between two quantities. 8.F.4 7.EE I can use variables to represent quantities, and construct simple equations and inequalities to solve problems. 7.EE.4 5/29/15 Top of Document Page 5 of 28
Connecting Patterns and Functions Standards We start the year with a launch section to build classroom culture. The patterns in this unit are mostly arithmetic sequences (but they are not called that until the Sequences Unit) which is a subset of linear function with discrete domains and ranges. We build on the 8 th grade understanding of functions, and add function notation, domain and range. We expect students to find inputs, outputs and equations of patterns and express them in function notation. Then they connect multiple representations of functions. Throughout the unit, we expect students to attend to the precision of units. There is a section where students express quantities through converting units. MAJOR CONTENT 2 CREATING EQUATIONS (CED) " Create equations that describe numbers or relationships A- CED1 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. A- CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. REASONING WITH EQUATIONS AND INEQUALITIES (REI) " Represent and solve equations and inequalities graphically A- REI.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). INTERPRETING FUNCTIONS (IF) " Understand the concept of a function and use function notation. F- IF.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). # F- IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. " Interpret functions that arise in applications in terms of the context. 2 http://www.parcconline.org/parcc- model- content- frameworks 5/29/15 Top of Document Page 6 of 28
F- IF.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.# SUPPORTING CONTENT 3 QUANTITIES (QF) $ Reason quantitatively and use units to solve problems N- Q.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. N- Q.2 Define appropriate quantities for the purpose of descriptive modeling 4. INTERPRETING FUNCTIONS (IF) $ Analyze functions using different representations F- IF.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. BUILDING FUNCTIONS (BF) $ Build a function that models a relationship between two quantities F- BF.1 Write a function that describes a relationship between two quantities. 1.a Determine an explicit expression, a recursive process, or steps for calculation from a context. 3 http://www.parcconline.org/parcc- model- content- frameworks 4 In descriptive modeling, a model simply describes the phenomena or summarizes them in a compact form. Graphs of observations are a familiar descriptive model for example, graphs of global temperature and atmospheric CO2 over time. 5/29/15 Top of Document Page 7 of 28
Linear Functions (LF) Target Practice 1. Make sense of problems and persevere in solving them. Practice 3 Construct viable arguments and critique the reasoning of others. Practice 4. Model with mathematics. Practice 6 Attend to precision. Practice 7. Look for and make use of structure. Advancing content Learning Target Unit Emphasis A- REIb I can solve a system of equations. A- REI.6 Find where two functions have the same solution. Learning Target / Unit Emphasis Major Content A.CED I can create equations and inequalities that describe numbers or relationships.! Create equations with one variable to represent a quantity. A-CED1! Create equations with two variables to represent relationships between two quantities with equations and tables.! Graph equations on coordinate axes with labels and scales. A-CED.2! Rearrange an equation to find either the inputs outputs. A.CED.4 A-REIa I can solve equations and inequalities.! Show the steps used for solving an equation.! Use properties of numbers to justify each step of a solution! Properties: associative, commutative, distributive, identity, inverse and zero.! Verify a solution by substituting it into the original equation. A-REI.1 A-REIc I can represent and solve equations and inequalities graphically.! Understand that coordinate pairs that are solutions to a linear function are on the same line. A- REI.10 F-IFa I can understand the concept of a function and use function notation.! Identify x, the input, as a quantity of the domain.! Identify f(x) the output, as a quantity of the range.! Apply the definition of a function (a function assigns exactly one output for each input.) to determine if an equation, a table, or graph is a function.! Know the graph of the function is the graph of all (x, f(x)) coordinate points. F- IF.1! Use function notation to express an input with its connected output! Convert a table, graph, set of ordered pairs, or description into function notation by identifying the equation used to turn inputs into outputs and writing the equation.! Only evaluate functions with inputs in domain! Analyze the input and output values of a function based on a problem situation. F-IFb I can interpret functions that arise in applications in terms of the context. F- IF.2! For tables and graphs of functions, interpret rate of change and intercepts of linear functions! Sketch a graph of functions of a situation that shows rate of change and intercepts for linear functions. F-IF.4! Given the graph of a linear, determine the practical domain of the function as it relates to the numerical relationship it describes. F- IF.5! Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. F-IF.6 Supporting Content Additional F-IFc I can analyze functions using different representations.! Graph linear functions expressed symbolically and show intercepts and rate of change.! Graph by hand and using technology. F-IF.7! Compare rate of change and intercepts of two linear functions, each represented in a different way (equations, tables, graphs, and situations). F-IF.9 F-LE I can construct and compare linear, quadratic and exponential models and solve problems.! Show that quantities in a linear function change at a constant rate of change. F-LE.1! Write an equation of a linear function from a graph, table, inputoutput pairs, or a situation. F.LE.2! Interpret the rate of change and intercepts of a linear function in a situation. F-LE.5 F-BF I can build a function.! Write a function that models a linear relationship. F-BF.3! Identify the effect changing the output has on the graph of a function. F-BF.4 5/29/15 Top of Document Page 8 of 28
Foundational Understanding 8.EE Solve linear equations in one variable. 8.EE.7 Understand the connections between proportional relationships, lines, and linear equations. 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distancetime equation to determine which of two moving objects has greater speed. 8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.7- Solve linear equations in one variable. o 7.a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are o different numbers). 7.b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Define, evaluate, and compare functions. 8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 1 8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. 8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s 2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Use functions to model relationships between quantities. 8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 5/29/15 Top of Document Page 9 of 28
MAJOR CONTENT 5 CREATING EQUATIONS (CED) " Create equations that describe numbers or relationships A- CED1 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. A- CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A.CED.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. REASONING WITH EQUATIONS AND INEQUALITIES (REI) " Reasoning with Equations and Inequalities (A REI) A- REIa 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. " Represent and solve equations and inequalities graphically A- REI.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). INTERPRETING FUNCTIONS (IF) " Understand the concept of a function and use function notation. F- IF.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). # F- IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. " Interpret functions that arise in applications in terms of the context F- IF.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.# 5 http://www.parcconline.org/parcc- model- content- frameworks 5/29/15 Top of Document Page 10 of 28
F- IF.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.# F- IF.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.# [Linear functions only] BUILDING FUNCTIONS (BF) " Build a function that models a relationship between two quantities F- BF.1 Write a function that describes a relationship between two quantities. 1.a Determine an explicit expression, a recursive process,, or steps for calculation from a context. SUPPORTING CONTENT 6 INTERPRETING FUNCTIONS (IF) $ Analyze functions using different representations F- IF.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. [Linear functions only] F- IF.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. LINEAR, QUADRATIC AND EXPONENTIAL MODELS (LE) $ Construct and compare linear, quadratic and exponential models and solve problems F- LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. 1a. Prove that linear functions grow by equal differences over equal intervals; and that exponential functions grow by equal factors over equal intervals. 1b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 6 http://www.parcconline.org/parcc- model- content- frameworks 5/29/15 Top of Document Page 11 of 28
$ Interpret expressions for functions in terms of the situation they model F- LE.5 Interpret the parameters in a linear or exponential function in terms of a context. ADDITIONAL CONTENT 7 % Build new functions from existing functions (BF) F- BF.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. F- BF.4 Find inverse functions. 4a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. 7 http://www.parcconline.org/parcc- model- content- frameworks 5/29/15 Top of Document Page 12 of 28
Major Content Exponential Functions (EX) Targets Practice 1. Make sense of problems and persevere in solving them. Practice 2 Reason abstractly and quantitatively. Practice 4 Model with mathematics. Practice 5 use appropriate tools strategically. Practice 6 Attend to precision. Learning Target Unit Emphasis A-REIb I can solve a system of equations. A-REI.6 Find where two functions have the same solution. Learning Target Unit Emphasis A-SSE I can see structure in expressions.! Identify the initial value and growth rate of exponential expressions. A-SSE.1a! Write an equivalent form of an exponential function to reveal and explain specific information about its approximate rate of growth or decay. A-SSE.3c F-IFa I can understand the concept of a function and use function notation.! Identify x, the input, as a quantity of the domain.! Identify f(x) the output, as a quantity of the range.! Apply the definition of a function (a function assigns exactly one output for each input.) to determine if an equation, a table, or graph is a function.! Know the graph of the function is the graph of all (x, f(x)) coordinate points. F- IF.1! Use function notation to express an input with its connected output! Convert a table, graph, set of ordered pairs, or description into function notation by identifying the equation used to turn inputs into outputs and writing the equation.! Only evaluate functions with inputs in domain! Analyze the input and output values of a function based on a problem situation. F- IF.2 F-IFb I can interpret functions that arise in applications in terms of the context.! For tables and graphs of functions, interpret rate of change and intercepts of linear functions initial value, intercepts and growth rate of exponential functions.! Sketch a graph of functions of a situation that shows rate of change and intercepts for linear functions. initial value, intercepts and growth rate for exponential functions. F-IF.4! Given the graph of an exponential function, determine the practical domain of the function as it relates to the numerical relationship it describes. F-IF.5 Founational Understanding Supporting Content Learning Target Unit Emphasis F-IFc I can analyze functions using different representations.! Graph exponential functions expressed symbolically and show initial value, end behavior.! Graph by hand and using technology. F-IF.7e! Identify the initial value and growth rate of exponential expressions. F-IF.8b! Compare growth factor and initial value of two exponential functions, each represented in a different way (equations, tables, graphs, and situations). F-IF.9 F-LE I can construct and compare linear, quadratic and exponential models and solve problems.! Determine if a function is linear by finding the constant rate of change in at least 2 pairs of coordinate points using the slope formula f (b) f (a). b a! Find the ratio of two terms in a sequence using formula f (b) f (a)! Determine if a function is exponential by finding the constant ratio in at least 2 pairs of coordinate points using the slope formula f (b) f (a) b a! Given a contextual situation, describe whether the situation in question has o a linear pattern of change of repeated addition, or o an exponential pattern of change of repeated multiplication.! Students use technology such as graphing calculators or programs, spreadsheets, or computer algebra systems to model and compare linear and exponential functions. F-LE.1! Interpret the rate of change and intercepts of a linear function in a situation.! Interpret the growth rate and initial value of an exponential function in a situation. F-LE.5 F-BF I can build a function. Write a function that models a! linear relationship! exponential relationship F-BF.1ab. 5/29/15 Top of Document Page 13 of 28
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3 2 3 5 = 3 3 = 1/3 3 = 1/27. 8.EE.1 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. 8.EE.2 Describe qualitatively the functional relationship between two quantities by analyzing a graph Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 8.F.5 Write and evaluate numerical expressions involving whole-number exponents. 6.EE.1 Exponential Functions and Equations Standards MAJOR CONTENT 8 SEEING STRUCTURE IN EXPRESSIONS (SSE) " Interpret the structure of expressions A.SSE.1 Interpret expressions that represent a quantity in terms of its context. 1a Interpret parts of an expression, such as terms, factors, and coefficients. [Focus on Initial values and constant multipliers in exponential expressions.] REASONING WITH EQUATIONS AND INEQUALITIES (REI) Represent and solve equations and inequalities graphically. " A- REI.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.# INTERPRETING FUNCTIONS (IF) " Understand the concept of a function and use function notation. F- IF.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). # F- IF.2 Use function notation, evaluate function for inputs in their domains, and interpret statements that use function notation in terms of a context. " Interpret functions that arise in applications in terms of the context F- IF.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 8 http://www.parcconline.org/parcc- model- content- frameworks 5/29/15 Top of Document Page 14 of 28
maximums and minimums; symmetries (origin, x- axis, y- axis); end behavior; and periodicity. # F- IF.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. SUPPORTING CONTENT 9 SEEING STRUCTURE IN EXPRESSIONS (SSE) $ Write expressions in equivalent forms to solve problems A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. # 3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t 1.012 12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. INTERPRETING FUNCTIONS (IF) $ Analyze functions using different representations F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. # 7e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 8b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02) t, y = (0.97) t, y = (1.01) 12t, y = (1.2) t/10, and classify them as representing exponential growth or decay. F- IF.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. BUILDING FUNCTIONS (BF) $ Build a function that models a relationship between two quantities F- BF.1 Write a function that describes a relationship between two quantities. # 1.a Determine an explicit expression, a recursive process, or steps for calculation from a context. 9 http://www.parcconline.org/parcc- model- content- frameworks 5/29/15 Top of Document Page 15 of 28
1.b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate. LINEAR, QUADRATIC AND EXPONENTIAL MODELS (LE) $ Construct and compare linear, quadratic and exponential models and solve problems. F- LE.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. F.LE.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). F- LE.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. F- LE.5 Interpret the parameters in a linear or exponential function in terms of a context. 5/29/15 Top of Document Page 16 of 28
Regression Equations (RE) Targets Practice 1. Make sense of problems and persevere in solving them. Practice 3. Construct viable arguments and critique the reasoning of others. Practice 4. Model with mathematics. Practice 5. Use appropriate tools strategically. Practice 6. Attend to precision. Learning Target Sub- Targets Learning Target Sub- Targets Major Content S-IDc I can interpret linear models.! Use data to make predictions.! Write a sentence to explain the meaning of slope and y-intercept in terms of the units stated in the data and context of the situation.! Explain the meaning of the slope and y- intercept in context.! Explain the meaning of the growth rate and y-intercept in context. S-ID.7! Explain that the correlation coefficient, r, must be between -1 and 1 inclusive and explain what each of these values means.! Determine whether the correlation coefficient shows a positive or negative, and strong, moderate, weak, or no correlation.! Use the correlation coefficient to determine if a linear model may be a good fit for the data.! Understand and explain that a strong correlation does not mean causation. S-ID.8! Understand and explain that a strong correlation does not mean causation. S.ID.9 Foundational Content Supporting Content S-IDb I can summarize, represent and interpret data on two categorical and quantitative variables.! Construct a scatterplot on paper and with technology.! Describe the relationship between both variables. S.ID.6! Use technology to find a function (linear or exponential) that best models a set of data.! Use a function that model data to solve problems.! Determine which outliers distort a linear model. S.ID.6a! Describe the correlation of a scatterplot as strong, weak, no, positive or negative.! Use technology to create and analyze a residual plot to determine whether the function is an appropriate fit. S.ID.6b! Determine whether or not a linear function best fits the data.! Write the equation of the line of best fit with technology.! Estimate the equation of the line of best fit without technology. S.ID.6c N-Q I can reason quantitatively and use units to solve problems.! Choose and interpret both the scale and the origin in graphs and data displays.! Include units with answers. N-Q.1! Determine and interpret appropriate quantities when using descriptive modeling. N-Q.2! Determine the accuracy of values based on their limitations in the context of the situation. N-Q.3 Investigate patterns of association in bivariate data. & 8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. & 8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. & 8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. & 8.SP.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. 5/29/15 Top of Document Page 17 of 28
MAJOR CONTENT 10 Regression Equations Standards Interpreting categorical and quantitative data (ID) " Interpret linear models. S- ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. S- ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. S.ID.9 Distinguish between correlation and causation. " Make inferences and justify conclusions from sample surveys, experiments, and observational studies S- IC.6 Evaluate reports based on data.! Determine which outliers that distort a linear model. SUPPORTING CONTENT QUANTITIES (NQ) $ Reason quantitatively and use units to solve problems N- Q.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. N- Q.2 Define appropriate quantities for the purpose of descriptive modeling. N- Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. INTERPREITNG CATEGORICAL & QUANTITATIVE DATA (ID) $ Summarize, represent and interpret data on two categorical and quantitative variables. S.ID.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 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. 10 http://www.parcconline.org/parcc- model- content- frameworks 5/29/15 Top of Document Page 18 of 28
Systems of Equations (SE) Target Practice 1. Make sense of problems and persevere in solving them. Practice 4. Model with mathematics. Practice 5: Use appropriate tools strategically. Practice 6. Attend to precision. Major Content Learning Target Unit Emphasis Learning Target Unit Emphasis A.CED I can create equations and inequalities that describe numbers N-Q I can reason quantitatively or relationships. and use units to solve! Create equations with two variables to represent relationships problems. between two quantities with equations and tables.! Identify or choose the! Graph equations on coordinate axes with labels and scales. A-CED.2 appropriate unit of measure! Interpret solutions in the context of the situation and determine if they according to the context. are reasonable. A-CED3! Include units with answers.! Rearrange an equation to find either the inputs or outputs. A-CED.4 N-Q.1 A-REIa I can solve equations and inequalities.! Determine and interpret! Show the steps used for solving an equation. appropriate quantities when! Use properties of numbers to justify each step of a solution solving systems in context. Properties: associative, commutative, distributive, additive and N-Q.2 multiplicative identity, additive and multiplicative inverse and zero.! Determine the accuracy of Other justification: combine like terms A-REI.1 values based on their! Solve linear equations in one variable. A-REI.3 limitations in the context of A-REIb I can solve a system of equations. the situation. N-Q.3! Produce equivalent equations by multiplying or dividing each equation by the same constant.! Combining equivalent equations produces an equation that also passes through the solution of the original system. A-REI.5! Explain and identify why some linear systems have infinitely many solutions or no solution.! Solve a system of linear equations to find an exact solution.! Determine the approximate solution to a system by graphing both equations and estimating the point of intersection. A-REI.6 A-REIc I can represent and solve equations and inequalities graphically.! Understand that every point (x, y) on the line is a solution to the equation.! Verify that any point (x, y) on the line is a solution to the equation. A-REI.10! Set up a system of equations and use technology to find or approximate a solution to a one variable equation. A-REI.11 Foundational Understanding 8.EE.8- Analyze and solve pairs of simultaneous linear equations. & 8.a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. & 8.b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. & 8.c Solve real-world and mathematical problems leading to two linear equations in two variables. Supporting Content 5/29/15 Top of Document Page 19 of 28
MAJOR CONTENT 11 CREATING EQUATIONS (CED) Systems of Equations Standards Create equations that describe numbers or relationships " A- CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. " A- CED.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. # " A- CED.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. REASONING WITH EQUATIONS AND INEQUALITIES (REI) " Understand solving equations as a process of reasoning and explain the reasoning. A- REI.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 A- REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. " Solve systems of equations A- REI.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. A- REI.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 A- REI.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). A- REI.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.# 11 http://www.parcconline.org/parcc- model- content- frameworks 5/29/15 Top of Document Page 20 of 28