This unit bundles student expectations that address transformations, characteristics, and applications of exponential functions. Student expectations also address formulating, solving, and justifying solutions for exponential equations and equations involving rational exponents. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.
Prior to this unit, in Algebra I Unit 06, Unit 09 and Unit 11, students analyzed exponential functions and applied laws of exponents in mathematical and real-world problem situations. In Algebra II Unit 04, students applied laws of exponents. In Units 06 – 08, students solved equations involving rational exponents.
During this unit, students graph the exponential function, f(x) = b^{x}, where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, and asymptotic behavior. Students determine the effects on the attributes on the graph of f(x) = b^{x} when f(x) is replaced by af(x), f(x) + d, and f(x - c) for specific positive and negative values of a, c, and d, and investigate parameter changes and key attributes in terms of real-world problem situations. Students solve equations involving rational exponents, and conclude with an equation with a variable in the exponent that cannot be solved by the same methods. Students solve exponential equations of the form y = ab^{x} where a is a nonzero real number and b is greater than zero and not equal to one, where solutions can be determined without the use of logarithms. Students formulate exponential equations that model real-world situations, including exponential relationships written in recursive notation, solve the exponential equations without the use of logarithms, and justify the solution in terms of the problem situation.
After this unit, in Algebra II Unit 10, students will make connections between exponential and logarithmic functions. In Algebra II Unit 11, students will distinguish between linear, quadratic, and exponential functions to model problem situations. In Algebra II Unit 12, students will review concepts of exponential functions. In subsequent courses in mathematics, these concepts will continue to be applied to problem situations involving exponential functions and equations.
In Algebra II, graphing, transforming, and analyzing key attributes of exponential functions are identified in STAAR Readiness Standards 2A.2A and 2A.5A. Solving exponential equations and equations with rational exponents are identified in STAAR Readiness Standards 2A.5D and 2A.7H. These Readiness Standards are subsumed under STAAR Reporting Category 1: Number and Algebraic Methods, STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses, and STAAR Reporting Category 5: Exponential and Logarithmic Functions and Equations. Formulating exponential functions to model problem situations is identified in STAAR Supporting Standard 2A.5B, and is subsumed under STAAR Reporting Category 5: Exponential and Logarithmic Functions and Equations. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, C1, D1, D2; III. Geometric Reasoning B1, C1; VII. Functions A1, A2, B1, B2, C2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.
According to the National Council of Teachers of Mathematics (2000), Principles and Standards for School Mathematics, students should develop an understanding of the algebraic properties that govern manipulation of symbols in expressions, equations, and inequalities. According to Navigating through Algebra in Grades 9 – 12, “High school students continue to develop fluency with mathematical symbols and become proficient in operating on algebraic expressions in solving problems. Their facility with representation expands to include equations, inequalities, systems of equations, graphs, matrices, and functions, and they recognize and describe the advantages and disadvantages of various representations for a particular situation. Such facility with symbols and alternative representations enables them to analyze a mathematical situation, choose an appropriate model, select an appropriate solution method, and evaluate the plausibility of their solutions” (NCTM, 2002, p. 3). Research found in National Council of Teachers of Mathematics also states, “Using a variety of representations can help make functions more understandable to a wider range of students than can be accomplished by working with symbolic representations alone” (as cited by NCTM, 2009, p. 41). This unit places particular emphasis on multiple representations. State and national mathematics standards support such an approach. The Texas Essential Knowledge and Skills repeatedly require students to relate representations of functions, such as algebraic, tabular, graphical, and verbal descriptions. This skill is mirrored in the Principles and Standards for School Mathematics (NCTM, 1989). Specifically, this work calls for instructional programs that enable all students to understand relations and functions and select, convert flexibly among, and use various representations for them. More recently, the importance of multiple representations has been highlighted in Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics (NCTM, 2007). According to this resource, students should be able to translate among verbal, tabular, graphical, and algebraic representations of functions and describe how aspects of a function appear in different representations as early as Grade 8. Also, in research summaries such as Classroom Instruction That Works: Research-Based Strategies for Increasing Student Achievement such concept development is even cited among strategies that increase student achievement. Specifically, classroom use of multiple representations, referred to as nonlinguistic representations, and identifying similarities and differences has been statistically shown to improve student performance on standardized measures of progress (Marzano, Pickering & Pollock) (2001),.
Marzano, R. J., Pickering, D. J., & Pollock, J. E. (2001). Classroom instruction that works: Research-based strategies for increasing student achievement. Alexandria, VA: Association for Supervision and Curriculum Development. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc. National Council of Teachers of Mathematics. (2002). Navigating through algebra in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc. National Council of Teachers of Mathematics. (2007). Curriculum focal points for prekindergarten through grade 8 mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc. National Council of Teachers of Mathematics. (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: National Council of Teachers of Mathematics, Inc. Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/collegereadiness/crs.pdf |