help develop Topic #1. functions; connect this standard to F.LE.4. i.e. use inverses to show the connection between exponential and logarithmic functions. 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). (supporting) (cross-cutting) F.LE.4 For exponential models, express as a logarithm the solution to a bct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. (supporting) Note: Consider extending this unit to include the relationship between properties of logarithms and properties of exponents, such as the connection between the properties of exponents and the basic logarithm property that log xy = log x +log y. F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context. (additional) (cross-cutting) Note: Limit exponential functions to those of the form f(x) = bx + k. Algebra II Unit 3: Modeling with Functions Topic #2 Arithmetic and Geometric Sequences
Standard Essential Skills and Knowledge Clarification/Teacher Notes A.CED.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. (supporting) (cross-cutting) Ability to connect experience from Algebra I with creating linear, exponential and quadratic equations in one variable to include creating simple rational functions Ability to distinguish between linear, quadratic, exponential, root and simple rational relationships given the verbal, numeric and/or graphic representations Students need to be able to create and solve the type of exponential equations in one variable that require the use of logarithms in the solving process. Exponential equations can include rational or real exponents. Integer exponents were explored in Algebra 1. In this course simple rational functions refers to rational expressions whose numerators are of degree at most one and whose denominators are of degree at most 2. Students would need to be able to create and solve simple rational equations. Tasks should include real-world contexts. A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. ★ (major) Note: Consider extending this standard to infinite geometric series in curricular implementations of this course description. Knowledge of the difference between an infinite and a finite series Ability to apply the formula for the sum of a finite geometric series : Sum of Finite Geometric Progression The sum in geometric progression (also called geometric series) is given by Equation (1) Multiply both sides of Equation (1) by will have Equation (2) Subtract Equation (2) from Equation (1)
F.IF.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. (supporting/fluency) (cross-cutting) Note: 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. See the skills and knowledge that are stated in the Standard. In Algebra 1, students identified arithmetic sequences as linear functions and geometric sequences as exponential functions. Students need to be able to identify arithmetic and geometric sequences in both explicit and recursive forms. Students should be able to see the relationships between linear functions and arithmetic sequences. They should recognize that the domain of an arithmetic sequence is discrete, while the domain of a linear function is continuous. Students should be able to see the relationships between exponential functions and geometric sequences. They should recognize that the domain of a geometric sequence is discrete, while the domain of a exponential function is continuous.
F.IF.4 For a function that models a relationship between two quantities, interpret key features of the graph and the table in terms of the quantities, and sketch the graph 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. ★ (major) (cross-cutting) Ability to translate from algebraic representations to graphic or numeric representations and identify key features using the various representations Functions should include the following types: polynomial, exponential, logarithmic, and trigonometric. This standard connects well with F.IF.7 Tasks should include real-world contexts. 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. ★ (major) (cross-cutting) Knowledge that the rate of change of a function can be positive, negative or zero Ability to identify the rate of change from multiple representations Functions should include the following types: polynomial, exponential, logarithmic, and trigonometric. Tasks should include real-world contexts.
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. ★ e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. (supporting) Ability to connect experience with graphing linear and quadratic functions from Algebra I to graphing exponential and logarithmic functions Ability to produce a rough graph of the parent function for each type of function Knowledge of how parameters introduced into a function alter the shape of the graph of the parent function Connects well with standard F.IF.4
A description of prior learning that should support the learning of the concepts in this unit A description of how the concepts studied in this unit will support the learning of other mathematical concepts. Previous Math Courses Algebra II Unit 3 Future Mathematics Concepts developed in previous mathematics course/units which serve as a foundation for the development of the “Key Concept” Key Concept(s) Concepts that a student will study either later in Algebra II or in future mathematics courses for which this “Key Concept” will be a foundation. In Units 2 and 5 of Algebra I, students have started to develop an understanding of transformations of graphs of linear, quadratic, absolute value and exponential functions In Unit 1 of Geometry students experiment with transformations of geometric figures in the plane. Transformations of graphs The knowledge of the general principle of transformations will make work with functions in future courses more fluent. In Unit 1 of Algebra I, students have started to develop an understanding of patterns and sequences by studying linear functions Arithmetic Sequences Connect arithmetic sequences to linear functions. Write arithmetic sequences both recursively and with an explicit formula Develop models for more complex or sophisticated situations than in previous courses. In Calculus, students will develop the concept of area under the curve from the concept of sequences. This understanding is the basis of the integral. In Unit 5 of Algebra I, students have started to develop an understanding of patterns and sequences by studying exponential functions Geometric Sequences Connect geometric sequences to exponential functions. Write geometric sequences both recursively and with an explicit formula In Calculus, students will develop the concept of area under the curve from the concept of sequences. This understanding is the basis of the integral.
terms of context key features of the graph and the table in terms of the quantities, and sketch the graph showing key features given a verbal description of the relationship square root functions, cube root functions, piecewise-defined functions (including step functions absolute value functions) and exponential functions with domains in the integers. The functions listed here are the same as those listed in the Algebra I column for Standards F.IF.6 and F.IF.9. trigonometric functions. The functions listed here are the same as those listed in the Algebra II column for Standards F.IF.6 and F.IF.9. Interpret functions that arise in applications in terms of a context. 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. ★ Note: Focus on linear functions and exponential functions whose domain is a subset of the integers. Unit 5 in this course and the Algebra II course address other types Tasks have a real-world context. Tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions absolute value functions) and exponential functions with domains in the integers. The functions listed here are the same as those listed in the Algebra I column for Standards F.IF.4 and F.IF.9. Tasks have a real-world context. Tasks may involve polynomial, exponential, logarithmic, and trigonometric functions. The functions listed here are the same as those listed in the Algebra II column for Standards F.IF.4 and F.IF.9.
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. 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. Tasks do not involve recognizing even and odd functions The functions listed in note (i) are the same as those listed in the Algebra II column for F.IF.4, F.IF.6 and F.IF.9. Construct and compare linear, quadratic and exponential models and solve problems. 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). Tasks are limited to constructing linear and exponential functions in simple context (not multi-step). Tasks will include solving problems by constructing linear and exponential functions. Interpret expressions for functions in terms of the F.LE.5 F.LE.5 Interpret the parameters in a linear or exponential function in terms of a Tasks have a real-world context. Exponential functions are limited to those with domains in the Tasks have a real-world context. Tasks are limited to exponential functions with domains not in the