Glastonbury Public Schools Mathematics Curriculum

Algebra II

The Big Idea of Algebra II: Features of Functions

How do function families behave and how can understanding the behaviors help us to make predictions and solve problems?

Mathematics is learned through questions that arise while solving well-constructed problems. Our students begin with problems, they use strategies to solve the problems, and they learn the necessary mathematics along the way. Many classroom investigations are designed so that students will collaboratively or individually discover the mathematical properties. The properties are then discussed in class, summarized, and become part of the students’ mathematical knowledge to be applied to future problems.

The discovery of the mathematics is an essential part of the development of each student’s confidence as a mathematician. Knowledge that is gained through inquiry is more likely to be remembered for the long term. Teachers and parents work together to promote this discovery of math through investigation, problem solving, and reasoning. Our goal is for students to understand that mathematics makes sense.

Building on the work with linear, quadratic and exponential functions in grade 8 and Algebra I, students extend their repertoire of functions to include polynomial, rational, and radical functions. Students work closely with the expressions that define the functions, and continue to expand and hone their abilities to model situations and to solve equations, including solving quadratic equations over the set of complex numbers and solving exponential equations using the properties of logarithms. The Mathematical Practice Standards apply throughout each course and ,together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes us of their ability to make sense of problem situations. The units of the Glastonbury Public School Algebra II curriculum are below. (Common Core Standards, Appendix A)

I. Linear Functions Extension Topics

Students will discover how to extend their knowledge of solving two variable systems of equations into three variable systems. Graphing systems of linear inequalities will be used as constraints in the process of linear programming allowing students to develop a process how a real life application with limited resources can be optimized. Students will also discover that even though functions are versatile, not all real life applications can be modeled by a single function in order to develop the need for piecewise functions. Students will identify the domain and range of piecewise functions as they write and graph these functions. Through these investigations the students will understand that linear programming uses linear modeling to find optimal outcomes given a set of constraints. Students will also understand when describing real world relationships, this often requires a representation of several equations in different sections of one graph.

Common Core State Standards: SSE.A.1.a, SSE.A.1.b, IF.C.7a, IF.C.7b, CED.A.3, CED.A.2, BF.A.1a, IF.B.5, REI.D.11

II. Transformations

Students will investigate the shapes of the basic parent functions of quadratics, square roots and absolute values. They will explore how adding and multiplying parameters to these parent functions will induce transformations of the graph as a translations, stretches, compressions and reflections. The concept of domain and range of functions will be consistently defined. Students will also extend their knowledge of piecewise functions to writing and graphing nonlinear piecewise functions.

Common Core State Standards: BF.B.3, REI.D.11, SSE.A.1.a, SSE.A.1.b, IF.C.7ab, IF.B.4, IF.B.5, IF.B.6

III. Polynomial Functions

Building on the prior knowledge of quadratics and real solutions, we will explore imaginary solutions and complex numbers in all polynomials. Students will discover how complex solutions can be found utilizing completing the square and the quadratic formula. End behavior, maximum and minimum points, and zeros will be discovered graphically and used to identify polynomials of varying degrees. For a given polynomial function, students will be able to identify end behavior and the maximum number of turning points. Students will learn different methods for dividing polynomials. Students will extend their knowledge of factoring by factoring special case polynomials. Students will use the rational zero theorem and the fundamental theorem of algebra to find all zeros of a function. Modeling and real world applications will be used throughout the unit.

Common Core State Standards: CN.A.1-2, CN.A.3(+), CN.C.7, IF.B.5, SSE.A.1.a, SSE.A.1.b, IF.B.6,IF.C.7a, IF.C.9, CED.A, IF.B.4, BF.A.1a, SSE.A.2, CN.C.8-9 (+), IF.B.5, APR.A.1, REI.D.11, APR.B.2-3, IF.B.6, IF.C.7c, IF.C.8a, APR.D.6, BF.A.1a, APR.D.7(+), APR.C.5(+), SSE.A.2

IV. Powers, Roots, and Radicals

Students will extend their knowledge of exponents as they develop the understanding of rational exponents and how they can be used to solve equations including real life applications. Graphing of power functions with rational exponents will be explored by using transformations of the parent functions. Students will discover the domain and range of these functions while performing various operations, noting the difference between even and odd radicals. Students will also develop the idea of an inverse of a function and use this to find the equation and graph of the inverse of power functions with rational exponents. Through these investigations, the students will understand the relationship between a function and its inverse, using this relationship to determine the domain and range of functions.

Common Core State Standards: SSE.A.1.a, SSE.A.1.b, IF.B.4, IF.B.5, IF.B.6, IF.C.7b, REI.A.2, CED.A.1, BF.A.1b, BF.A.1c(+) , BF.B.4a, BF.B.4b(+ ), BF BF.B.4c(+) .B.4d(+)

V. Exponential and Logarithmic Functions

Students will learn about how a geometric sequence ties to an exponential function and how those functions and patterns differ from other families of functions. Students will explore exponential growth and decay graphs and applications while also exploring the concept of the number e. Students will graph exponential functions. Real-life applications involving compound interest, population growth and decay, and intensity will be explored throughout the unit. Logarithms will be examined through the lens of inverses and the definition of a logarithm will be used to start to build an understanding of logarithmic equations.

Common Core State Standards: IF.B.5,SSE.A.1a,SSE.A.1b, REI.D.11, IF.B.6, IF.C.7e, IF.C.8b, IF.C.9,REI.A.2, CED.A, BF.A.1a, BF.A.1b, BF.B.4a, BF.B.4b(+), BF.B.4c(+), IF.B.4, LE.A.4, BF.B.5(+)

VI. Modeling with Functions

Students will be given sets of data and using their knowledge of the families of functions, students will use technology to find the function that best models the data. Students will then use this model to make predictions.

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