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In Algebra 1, students demonstrate fluency by creating, comparing, analyzing, solving, providing multiple representations, and identifying the structures of exponential & quadratic functions building on the understanding of linear functions from grade 8 to apply linear, quadratic and exponential models to data and unfamiliar problems.
The Story of Alg. 1 - Unit 4: Exponential Functions (Up to 27 days)
This unit begins with creating and using exponential functions in order to be sure students can use context as a scaffold to understand exponential functions. From there, students can dive more deeply into them by rearranging formulas, graphing them, and comparing two functions of the same type and two functions of different types, linear & exponential. Comparing two functions leads to graphing & solving a system in the context of f(x)=g(x) versus linear systems in the last unit. Throughout the unit explicit & recursive relationships should be identified and examined as well as operations on functions when possible. By the end of the unit, students should have a sound understanding of the difference between linear and exponential growth and be able to distinguish between them given a context. The ideas of systems as being a focus of this course comes up in this way yet again as students explore linear & exponential systems in order to fulfill F-IF.9 and learn F-LE.1 & F.LE.5. Finally, the discussion of exponential growth itself leads to building on the properties of exponents from eighth grade (where they had numerical bases), and fractional exponents are introduced as students explore rational & radical expressions. Time permitting, additional depth in this unit can be found by diving deeper explicitly into equivalent exponential expressions, rational versus irrational numbers and constructing linear and exponentials including sequences.
Unit 4 Notes
Sequences could act as bridge from linear to exponential functions. The idea of systems can carry through this unit as students compare linear & exponential growth (F-LE.1 & F-LE.5). This unit also includes work with rational exponents & generalizes exponent rules to variables along with rational & radical expressions while revisiting irrational numbers from 8th grade.
Vocabulary, Tools & Developmental Notes from SBAC
A-CED.A Create equations that describe numbers or relationships. (Target G)
A-CED.A Create equations that describe numbers or relationships. (Target G)
Tasks for this target will require students to create equations and inequalities in one variable to solve problems. Other tasks will require students to create and graph equations in two variables to represent relationships between quantities.
F-IF.C Analyze functions using different representations. (Target M)
F-IF.C Analyze functions using different representations. (Target M)
Tasks for this target will ask students to graph functions (linear... exponential... ) by hand or using technology and compare properties of two functions represented in different ways. Some tasks will focus on understanding equivalent forms that can be used to explain properties of functions and may be associated with AlgebraTarget H.
A-REI.D Represent and solve equations and inequalities graphically. (Target J)
A-REI.D Represent and solve equations and inequalities graphically. (Target J)
Tasks for this target will require students to interpret a line or curve as a solution set of an equation in two variables, including tasks that tap student understanding of points beyond the displayed portion of a graph as part of the solution set. Some of these tasks should explicitly focus on non-integer solutions (e.g., give three points on the graph of y = 7x + 2 that have x-values between 1 and 2).
Other tasks for this target will require students to approximate solutions to systems of equations represented graphically, including linear... exponential... (often paired with Target L - Interpret functions that arise in applications in terms of the context).
F.BF.A Build a function that describes a relationship between two quantities. (Target N)
F.BF.A Build a function that describes a relationship between two quantities. (Target N)
Tasks for this target will require students to write a function (recursive or explicit, as well as translate between the two forms) to describe a relationship between two quantities.
A-SSE.B Write expressions in equivalent forms to solve problems. (Target E)
A-SSE.B Write expressions in equivalent forms to solve problems. (Target E)
Tasks for this target will require students to choose or produce an equivalent form of an expression, including factoring a quadratic expression, completing the square, and using properties of exponents. Some of these tasks will connect the form of the expression to a property of the quantity represented by the expression.
N-RN-A Extend the properties of exponents to rational exponents. (Target A)
N-RN-A Extend the properties of exponents to rational exponents. (Target A)
Tasks for this target will require students to use the definition of radicals and rational exponents and identify equivalent numeric and algebraic expressions involving rational exponents and radicals. Tasks for this target will require students to find the exact value of numbers expressed in terms of rational exponents or estimate their magnitude, where appropriate. Claim 3 tasks will tap student understanding of the properties of exponents and their ability to identify flawed reasoning applied to this target.
F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. (Claim 4)
F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. (Claim 4)
Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems.
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