Domain: Create Equations* (*Modeling Domain) Cluster: Create equations that describe numbers or relationships Quarter 1: Identify the quantities in a mathematical problem or real-world situation that should be represented by distinct variables and describe what quantities the variables represent. one or more created equation on a coordinate axes with appropriate labels and scales. Create at two equations in two or more variables to represent relationships between quantities. Justify which quantities in a mathematical problem or real-world situation are dependent and independent of one another and which operations represent those relationships. Determine appropriate units for the labels and scale of a graph depicting the relationship between equations created in two or more variables. Notes from Appendix A: The targets are limited to linear and exponential equations, and, in the case of exponential equations, limited to situations requiring evaluation of exponential functions at integer inputs Quarter 2: Graph one or more created equation on a coordinate axes with appropriate labels and scales. Create at least two equations in two or more variables to represent relationships between quantities. Quarter 3: Quarter 4: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a contextual situation expressing a relationship Students know: Students understand/are able to: Level IV Students will: EEA-CED.2-4. Solve two-step inequalities with a variable. Ex. If I buy two movie tickets for $5 each and two drinks at $4 each, will $15 be enough money?
Domain: Creating Equations* Cluster: Create equations that describe numbers or relationships Quarter 1: Recognize when a modeling context involves constraints. Interpret solutions as viable or nonviable options in a modeling context. Represent constraints by equations , and by systems of equations From Appendix A: Limit targets to linear equations and inequalities. Quarter 2: Determine when a problem should be represented by equations, systems of equations and/ or inequalities. Quarter 3: Determine when a problem should be represented by equations, inequalities, systems of equations and/ or inequalities. Represent constraints by equations or inequalities.. Quarter 4: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities. Make of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a contextual situation involving constraints, Constraint Students know: When a particular system of two variable equations or inequalities accurately models Students understand/are able to: Level IV Students will: EEA-CED.2-4. Solve two-step inequalities with a variable. Ex. If I buy two movie tickets for $5 each and two drinks at $4 each, will $15 be enough money? Ex. I walked to the store to buy a book. I gave the cashier $10. She said, “You
Quarter 1: Recall properties of equality Quarter 2: Solve multi-step equations in one variable Solve equations with coefficients represented by letters. Quarter 3: multi-step inequalities in one variable Determine the effect that rational coefficients have on the inequality symbol and use this to find the solution set. Solve equations and inequalities with coefficients represented by letters. Quarter 4: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: linear equations and inequalities in one variable including being able to isolate Students know: Properties of equality and inequality (Tables 4 and 5). Students understand/are able to: Accurately use the properties of equality and inequality (Tables 4 EEA-REI.3-4. N/A (See EEA-ECED.1-2.)
Ax + By = C Dx + Ey = F Use the properties of equality (Table 4) (where A, B, C, D, E, and F are real numbers) to show that the same solution is produced by replacing one equation by the sum of that equation and a multiple of the other, For example, the system 2x + 3y = 18 4x + 2y = 20 is equivalent to 0x - 4y = -16 (multiply first equation by -2 add to the second equation). (Table 3) and equality (Table 4). Accurately perform the operations of multiplication and addition, and techniques for manipulating equations. When the properties of operations and equality are applied to systems of equations, the resulting equations have the same solution as the original. Grade Level/ Course: Algebra 1 Unit 2 Standard with code: A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Domain: Reasoning with Equations and Cluster: Solve system of equations
recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n - 1) for n ≥ 1. Domain: Interpreting Functions Cluster: Understand the concept of a function and use function notation. Quarter 1: 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. from Appendix A: Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of functions at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Draw examples from linear and exponential functions. In F.IF.3, draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences. Emphasize arithmetic and geometric sequences as examples of linear and exponential functions. Quarter 2: Quarter 3: Quarter 4: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors
Standard with code: 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.*(Modeling standard) Domain: Interpreting Functions Cluster: Interpret functions that arise in applications in terms of a context Quarter 1: Quarter 2: Recognize slope as an average rate of change. Interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Notes from Appendix A: Focus on linear functions and exponential functions whose domain is a subset of the integers. Quarter 3: Calculate 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 linear or exponential Interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Quarter 4: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given two functions represented differently (algebraically, graphically, numerically in tables, or by verbal descriptions), Use key features to compare the functions, Explain and justify the similarities and differences of the functions. Students know: Techniques to find key features of functions when presented in different ways, Techniques to convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions). Students understand/are able to: Accurately determine which key features are most appropriate for comparing functions, Manipulate functions algebraically to reveal key functions, Convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions) for the purpose of comparing it to another function. Functions can be in different but equivalent ways (algebraically, graphically, numerically in tables, or by verbal descriptions), Different representations of functions may aid in comparing key features of the functions. EEF-IF.9. N/A
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a function in algebraic form, Graph the function, f(x), conjecture how the graph of f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k(both positive and negative) will change from f(x), and test the conjectures, Describe how the graphs of the functions were affected (e.g., horizontal and vertical shifts, horizontal and vertical stretches, or reflections), Use technology to explain possible effects on the graph from adding or multiplying the input or output of a function by a constant value, Recognize if a function is even or odd. the graph of a function and the graph of a translation, stretch, or reflection of that function, Determine the value which was used to shift, stretch, or reflect the graph, Recognize if a function is even or odd. Even and odd functions Students know: Graphing techniques of functions, Methods of using technology to graph functions, Techniques to identify even and odd functions both algebraically and from a graph. Students understand/are able to: Accurately graph functions, Check conjectures about how a parameter change in a function changes the graph and critique the reasoning of others about such shifts, Identify shifts, stretches, or reflections between graphs, Determine when a function is even or odd. Graphs of functions may be shifted, stretched, or reflected by adding or multiplying the or output of a function by a constant value, Even and odd functions may be identified from a graph or algebraic form of a function. EEF-BF.3-4. N/A
Grade Level/ Course (HS): Algebra 1 Unit 2 Standard with code: F.LE.1a Distinguish between situations that can be modeled with linear functions and with exponential functions. a. that linear functions grow by equal differences over equal intervals; and that exponential functions grow by equal factors over equal intervals. Domain: Linear, Quadratic, and Exponential Models *(*Modeling Domain) Cluster: Construct and compare linear and exponential models and solve problems Quarter 1: Quarter 2: Recognize that linear functions grow by equal differences over equal intervals. Distinguish between situations that can be modeled with linear functions and with exponential functions to solve mathematical and real-world problems. Prove that linear functions grow by equal differences over equal intervals. Quarter 3: Quarter 4: Recognize that exponential functions grow by factors over equal intervals. Distinguish between situations that can be modeled with linear functions and with exponential functions to mathematical and real-world problems. that exponential functions grow by equal factors over equal intervals. Make of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors
Grade Level/ Course (HS): Algebra 1 Unit 2 Standard with code: F.LE.1b Distinguish between situations that can be modeled with linear functions and with exponential functions. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Domain: Linear, Quadratic, and Exponential Models *(*Modeling Domain) Cluster: Construct and compare linear, quadratic, and exponential models and solve problems Quarter 1: Quarter 2: Recognize situations in which one quantity changes at a constant rate per unit (equal differences) interval relative to another to solve mathematical and real-world problems. Quarter 3: Quarter 4: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Domain: Linear, Quadratic, and Exponential Models *(Modeling Domain) Cluster: Construct and compare linear, quadratic, and exponential models and solve problems Quarter 1: Informally define the concept of “end behavior”. Quarter 2: Compare tables and graphs of linear functions to observe that a quantity increasing exponentially exceeds all others to solve mathematical and real-world problems. Note from Appendix A: Limit to comparisons between linear and exponential models. Quarter 3: Quarter 4: Compare tables and graphs of linear and exponential functions to observe that a quantity increasing exponentially exceeds all others to mathematical and real-world problems. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a quantity increasing exponentially and a quantity increasing as a polynomial Increasing exponentially Students know: Students understand/are able to: Level IV Students will: EEF-LE.1-4. Plot points using pictures in first quadrant on a graph using whole numbers and explain how y increases/decreases as x changes. Ex. If you go to the store where every item is one dollar, students should state y
Quarter 1: Quarter 2: Recognize the parameters in a linear function including: vertical and horizontal shifts, vertical and horizontal dilations. Recognize of change and intercepts as “parameters” in linear functions. Interpret the parameters in a linear function in of a context. Quarter 3: Quarter 4: Recognize the parameters in a linear or exponential function including: vertical and horizontal shifts, vertical and horizontal dilations. Recognize rates of change and intercepts as “parameters” in linear or exponential functions. Interpret the parameters in a linear or exponential function in terms of a context. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: a contextual situation that may be modeled by a linear or exponential function, Parameters Students know: Key components of linear and exponential functions. Students understand/are able to: Communicate the meaning of defining values (parameters and variables) in functions used to model contextual EEF-LE.5. N/A
Quarter 1: Quarter 2: Quarter 3: Represent data with on the real number line using various display types by creating dot plots, histograms and box plots. Quarter 4: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given numerical data in any form (e.g., all real numbers), Dot Histograms Box Students know: Techniques for constructing dot plots, histograms, and box plots from a set of data. Students understand/are able to: Level IV Students will: EES-ID.1-2. Collect and organize data in simple graphs and use findings to draw conclusions from the data. Ex. Ask 10 people how many hours of TV they watch a day. Put the findings into a graph and tell which person watches the most and least TV.
Domain: Interpreting Categorical and Quantitative Data Cluster: Summarize, represent, and interpret data on a single count or measurement variable Quarter 1: Quarter 2: Quarter 3: Define “the context of data sets” as meaning the specific nature of the attributes under investigation. Interpret differences in shape, center and spread in the context of data sets. Describe the possible effects the presence of outliers in a set of data can have on shape, center, and spread in the context of the data sets. Quarter 4: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given multiple data sets, Outliers Center Students know: Students understand/are able to: Level IV Students will: EES-ID.3. Extend a graph or chart to make a prediction. Ex. If the weatherman says there is a 60% chance of rain, should you wear a rain coat?
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Descriptors Students: Given a data set of two quantitative variables, Create a scatter plot (with and without technology), Create a function best fits the data (linear, quadratic, and exponential models), Compare the graphs of the scatter plot and function to see the fit to the original data, Fit a linear function to the data if the scatter plot indicates a linear association. Quantitative variables Scatter plot Residuals Students know: Techniques for creating a scatter plot, Techniques for fitting various functions (linear, quadratic, exponential) to data, Methods for residuals to judge the closeness of the fit of the function to the original data. Students understand/are able to: Accurately create a scatter plot of data, Correctly choose a function to fit the scatter plot, Make reasonable assessments on the fit of the function to the data by examining residuals, Accurately fit a linear function to data when there is evidence of a linear association. Functions are used to create equations representative of ordered pairs of data, Residuals may be examined to analyze how well a function fits the data, When a linear association is suggested, a linear function can be fit to the scatter plot to aid in modeling the relationship. EES-ID.6. N/A
Standard with code: S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. (Statistics and Probability is a Modeling Conceptual Category.) Domain: Interpreting Categorical and Quantitative Data Cluster: Interpret linear models. Quarter 1: Quarter 2: Quarter 3: Compute (using technology) the correlation coefficient of a linear fit. Define the correlation coefficient. Interpret the correlation coefficient of a linear fit as a measure of how well the data fit the relationship. From Appendix A: on students’ work with linear relationships in eighth grade and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship. The important distinction between a statistical relationship and a cause-and-effect relationship arises in S.ID.9. Quarter 4: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors
Quarter 1: Quarter 2: Quarter 3: Quarter 4: Identify ways to rewrite expressions, such as difference of squares, factoring out a common monomial, regrouping, etc. Identify various structures of expressions (e.g. an exponential monomial multiplied by a scalar of the same base, difference of squares in other than just x) Use the structure of an expression to identify ways to rewrite it. Classify expressions by structure and develop strategies to assist in classification. from Appendix A: Focus on quadratics and exponential expressions Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Make sense of algebraic expressions by identifying structures within the expression Students know: Students understand/are able to: Use algebraic properties to produce equivalent forms of the same EEA-SSE.2. N/A
Level/ Course: Algebra 1 Unit 4 Standard with code: 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. Domain: Creating Equations* (*Modeling Domain) Cluster: Create equations that describe numbers and relationships Quarter 1: Quarter 2: Quarter 3 Define a “quantity of interest” to mean any numerical or algebraic quantity (e.g., in which 2 is the quantity of interest showing that d must be even; and showing that ) Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g. π * r2 can be as (π *r)*r which makes the form of this expression resemble b*h.) From Appendix A: Extend A.CED.4 to formulas involving squared variables. Quarter 4: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Students know: Students understand/are able to: Level IV Students will: EEA-CED.2-4. Solve two-step inequalities with a variable. Ex. If I buy two movie tickets for $5 each and two drinks at $4 each, will $15 written
evident square roots by inspection, Transform quadratic equations to a form where the square root of each side of the equation may be taken, including completing the square, Use the method of completing the square on the equation in standard form (ax2+bx+c=0) to derive the quadratic formula, Identify quadratic equations which may be solved efficiently by factoring, and then use factoring to solve the equation, Use the quadratic formula to solve quadratic equations, Explain when the roots are real or complex for a given quadratic equation, and when complex write them as a ± bi for real numbers a and b, Demonstrate that a proposed solution to a quadratic equation is truly a solution by making the original true. Quadratic equations Quadratic formula Complex solutions Inspection square root of a real number then so is -a, The method for completing the square, Notational methods for expressing complex numbers, A quadratic equation in standard form (ax2+bx+c=0) has real roots when b2-4ac is greater than or equal to zero and complex roots when is less than zero. manipulations including taking square roots of both sides of an equation, Accurately complete the square on a quadratic polynomial as a strategy for finding solutions to quadratic equations, Factor quadratic polynomials as a strategy for finding solutions to quadratic equations, Rewrite solutions to quadratic equations in useful forms including a ± bi and simplified radical expressions, Make strategic choices which procedures (inspection, completing the square, factoring, and quadratic formula) to use to reach a solution to a quadratic equation. Solutions to a quadratic equation must make the original equation true and this should be verified, When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically, Different procedures for solving quadratic equations are necessary under different conditions, If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then
Solve quadratic equations where both sides of the equation have evident square roots by inspection, Transform quadratic equations to a form the square root of each side of the equation may be taken, including completing the square, Use the method of completing the square on the equation in standard form (ax2+bx+c=0) to derive the quadratic formula, Identify quadratic equations which may be solved efficiently by factoring, and then use factoring to solve the equation, Use the quadratic formula to solve quadratic equations, Explain when the roots are real or complex for a given quadratic equation, and when complex write them as a ± bi for real numbers a and b, Demonstrate that a proposed solution to a quadratic equation is truly a solution by making the original true. Quadratic equations Quadratic formula Complex solutions Inspection Any real number has two square roots, that is, if a is the square root of a real number then so is -a, The method for completing the square, Notational methods for expressing complex numbers, A quadratic equation in standard form (ax2+bx+c=0) has real roots when b2-4ac is greater than or equal to zero and complex roots when is less than zero. Accurately use properties of equality (Table 4) and other algebraic manipulations including taking square roots of both sides of an equation, Accurately complete the square on a quadratic polynomial as a strategy for finding solutions to quadratic equations, Factor quadratic polynomials as a strategy for finding solutions to quadratic equations, Rewrite solutions to quadratic equations in useful forms including a ± bi and simplified radical expressions, Make strategic choices about which procedures (inspection, completing the square, factoring, and quadratic formula) to use to reach a solution to a quadratic equation. Solutions to a quadratic equation must make the original equation true and this should be verified, When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically,
nonzero rational number and an irrational number is irrational. Domain: The Real Number System Cluster: Use properties of rational and irrational numbers. Quarter 1: Quarter 2: Quarter 3: Quarter 4: Find the sums and products of rational and irrational numbers. Recognize that the sum of a rational number and an irrational number is irrational. Recognize that the product of a nonzero rational number and an irrational number is irrational. Explain why rational numbers are closed under addition or multiplication. Note from Appendix A: Connect N.RN.3 to physical situations, e.g., finding the perimeter of a square of area 2. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors
Students: the definition of rational numbers, Show that the sum or product of two rational numbers is rational and explain their reasoning, Use repeated reasoning from examples to demonstrate that the sum (or product) of a rational number and an irrational number appears to always result in an infinite non-repeating decimal, for example 2 + π = 2 + 3.14159... = 5.14159... Irrational number Students know: The distinction between rational and irrational numbers. Students understand/are able to: Convert between fraction and decimal form of rational numbers. Reasoning from repeated examples is not a proof but can lend credibility to a statement when there is reasoning to why the regularity continues. The decimal approximation of an irrational will never produce an exact or infinite repeating decimal, Making the same change to digit in an infinite non-repeating decimal will not make the decimal repeat, Reasoning from repeated examples is not a proof but can lend credibility to a statement when there is reasoning to why the regularity continues. EEN-RN.3. N/A Grade Level/ Course: Algebra 1 Unit 4 Standard with code: A.REI.7 a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of
Standard with code: 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.*(Modeling standard) Domain: Interpreting Functions Cluster: Interpret functions that arise in applications in terms of the context. Quarter 1: Quarter 2: Quarter 3: Recognize as an average rate of change. Calculate 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 linear, exponential, or quadratic graph. Interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Note from Appendix A: Focus on quadratic functions; compare with linear and exponential functions studied in Unit 2 of the Traditional Algebra 1 Pathway. Quarter 4: Recognize slope as an average rate of change. Calculate the average rate of change of a function (presented symbolically or as a over a specified interval. Estimate the rate of change from a linear, exponential, or quadratic graph. Interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Note from Appendix A: on quadratic functions; compare with linear and exponential functions studied in Unit 2 of the Traditional Algebra 1 Pathway. Make of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a symbolic representation of a function (including linear, quadratic, square root, cube root, piecewise-defined functions, polynomial, exponential, logarithmic, trigonometric, and (+) rational), Produce an accurate graph (by hand in simple cases and by technology in more complicated cases) and justify that the graph is an alternate representation of the symbolic function, Identify key features of the graph and connect these graphical features to the symbolic function, specifically for special functions: quadratic or linear (intercepts, maxima, and minima), square root, cube root, and piecewise-defined functions, including step functions and absolute value functions (descriptive features such as the values that are in the range of the function and those that are not), polynomial (zeros when suitable factorizations are available, end behavior), (+) rational (zeros and asymptotes when suitable factorizations are available, end behavior), exponential and logarithmic (intercepts and end behavior), trigonometric functions (period, midline, and amplitude). Students know: Techniques for graphing, Key features of graphs of functions. Students understand/are able to: Identify the type of function from the symbolic representation, Manipulate expressions to reveal important features for identification in the function, Accurately any relationship. Key features are different depending on the function, Identifying key features of functions aid in graphing and interpreting the function. EEF-IF.7. N/A (See EEF-IF.1-3)
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a contextual situation containing a function defined by an expression, Use algebraic properties to rewrite the expression in a form that key features of the function easier to find, Manipulate a quadratic function by factoring and completing the square to show zeros, extreme values, and symmetry of the graph, Explain and justify the meaning of zeros, extreme values, and symmetry of the graph in terms of the contextual situation, Apply exponential properties to expressions and explain and justify the meaning in a contextual situation. Extreme values Symmetry Exponential growth or Students know: Techniques to factor and complete the square, Properties of exponential expressions, Algebraic properties of equality (Table 4). Students understand/are able to: Accurately manipulate (e.g., factoring, completing the square) expressions using appropriate technique(s) to reveal key properties of a function. An expression may be in various equivalent forms, Some forms of the expression are more beneficial for revealing key properties of the function. EEF-IF.8. N/A
Standard with code: F.BF.1a Write a function that describes a relationship between two quantities.*(Modeling standard) a. Determine an explicit expression, a recursive process, or steps for calculation from a context. Domain: Building Functions Cluster: a function that models a relationship between two quantities Quarter 1: Quarter 2: Quarter 3 Quarter 4: Define “explicit function” and “recursive process”. Write a function that describes a relationship between two quantities by determining an explicit expression, a recursive process, or steps for calculation from a context. Note from Appendix A: Focus on situations that exhibit a quadratic relationship. This standard builds from Algebra 1 Unit 2. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors