Unit 1: Exploring Patterns in Linear and Quadratic Relationships

Unit Context: Exploring Patterns in Linear and Quadratic Relationships contains three topics: Extending Linear Relationships, Exploring and Analyzing Patterns and Applications of Quadratics. The module extends what students know about linear functions by introducing new strategies for solving systems of equations, graphing and solving inequalities, and absolute value functions. The topics in this module extend students’ familiarity with linear relationships to derive absolute value functions and to solve problems involving linear systems with more than two equations and two variables.

Extending Linear Relationships, students review what they have learned about solving a system comprising two equations and two variables by solving graphically and algebraically. Systems are extended to include three equations and three variables, and students learn to use Gaussian elimination to solve. They review systems of linear inequalities and use linear programming to determine optimal solutions to a real-world problem. Finally, students explore matrices. Then, students are reminded of what they know about the absolute value of a number. A number and its opposite are reflections across x = 0 and therefore, have the same absolute value. Students build from this to understand that taking the absolute value of a function reflects the negative y-values across the x-axis, or the line y = 0. The visual representation of an absolute value function allows students to see that there are two x-values associated with each y-value. From this intuitive understanding, students combine what they know about solving equations and inequalities to solve absolute value equations and inequalities. Students then explore how the A- and D-values affect the graph of an absolute value function in the same way they affect the graph of a linear function. However, now they add a constandy to the argument and explore how the C-value transforms the graph.

Exploring and Analyzing Patterns focuses on different representations of functions. Students begin the topic exploring three different patterns that are modeled by functions from three different function families. After describing and extending the patterns, students represent them numerically, graphically and algebraically. They determine whether different expressions representing each function are equivalent both graphically and algebraically. Students then dive deep into the structure of degree-2 polynomials. They write quadratic equations given two or three points, and solve quadratic equations using Properties of Equality, factoring, completing the square, and the Quadratic Formula. Students explore the complex number system, operate with complex numbers, and learn to solve quadratic equations with complex solutions.

Applications of Quadratics, students apply what they know about inequalities, systems, regressions, and inverses to equations in the quadratic function family. Throughout the topic, students solve problems in context that require a combination of these skills and make sense of their solutions in terms of the problem situation.

Standards: *2A.2A(absolute value only), *2A.2C, 2A.3A, 2A.3B, 2A.3C, 2A.3D, 2A.3E, *2A.3F, 2A.3G, *2A.4A, 2A.4B, 2A.4D, 2A.4E, 2A.4H, *2A.4F (NOT SQUARE ROOT), 2A.5B(exponential only), 2A.6C, 2A.6D, 2A.6E, 2A.6F, 2A.7A, 2A.7B, 2A.7I, 2A.8A, 2A.8B (ONLY LINEAR & QUAD), *2A.8C (ONLY LINEAR & QUAD)

Unit 2: Analyzing Structure

Unit Context: Analyzing Structure contains two topics: Composing and Decomposing Functions and Characteristics of Polynomial Functions. In this module, students perform arithmetic with complex numbers and use complex numbers in polynomial equations. The scenarios in the module provide modeling opportunities for students to use area volume formulas in an algebraic setting.

Composing and Decomposing Functions, students expand their knowledge of degree-1 and degree-2 polynomials to build degree-3 polynomials. They begin by using their intuition about geometric measurement as well as their knowledge of linear factors to build a degree-2 polynomial graphically. Then, using what they know about function transformation, students transform linear and quadratic functions by non-constant factors to create higher-degree functions. Returning to geometric measurement and concrete representations, students build degree-3 polynomials from linear and quadratic factors to solve volume problems. They discover the two shapes of cubic functions, which they learn to sketch when given specific zeros and multiplicity.

Characteristics of Polynomial Functions takes students from their concrete understanding of polynomials to the formalization of key characteristics of cubics and quartics. The topic begins with an exploration of the power functions. Students are reminded of transformation function form, and they use this notation to transform functions. Given a graph, students identify the A-, B-, C-, and D-values of a transformed polynomial. They characterize functions in terms of real and imaginary zeros, extrema, intervals of increase or decrease, and intercepts. They use the structure of polynomials to make sense of a real-world context modeled by a higher-order polynomial, and they compare polynomials represented in different forms. Students complete this module understanding how to compose and decompose polynomial functions.

Standards:  *2A.2A (only cubic),  2A.3E,  2A.4D, 2A.5A, 2A.6A (ONLY CUBIC), 2A.7B, 2A.7C, 2A.7I

Unit 3: Developing  Structural Similarities

Unit Context: Developing Structural Similarities contains two topics: Relating Factors and Zeros, and Polynomial Models.

Relating Factors and Zeros, students build upon what they know about the key characteristics of polynomial functions. They have composed functions from a product of factors and can identify the number and type of zeros. This topic focuses on identifying zeros from algebraic representations. Students begin with an expansion of their previous work on factoring quadratics to now include polynomials. They use the factors to determine zeros and sketch a graph of the polynomial. They then consider polynomials that cannot be solved using known factoring strategies and require division. Using long division and synthetic division, students combine strategies to write polynomials in factored form over the complex numbers.      

Standards: 2A.7C, 2A.7D, 2A.7E

Unit 4: Extending Beyond Polynomials

Module Context: Extending Beyond Polynomials contains two topics: Rational Functions and Radical Functions. In the same way that integers are not closed under division and lead to rational numbers, polynomials are not closed under division and lead to rational functions.

Rational Functions requires students to synthesize their understanding of rational numbers and polynomial functions to investigate rational functions. They explore the key characteristics of the functions and discover why vertical and horizontal asymptotes exist in rational functions. Students learn to identify any asymptotes from algebraic representations. They then operate with rational expressions, using similar strategies to the ones they use when operating with rational numbers. Students then write and solve rational equations and list restrictions, considering efficient ways to operate with rational expressions and to solve rational equations based on the structure of the original equation. The topic closes with problems related to work, mixture, cost, and distance.

Radical Functions, students build from what they established about power functions in the previous module, Developing Structural Similarities. They begin by reflecting power functions across the line y = x and identifying key characteristics. After establishing the graphical inverses of the power functions, students learn to invert functions algebraically, and they define the square root and cube root functions. Students transform radical functions, again recognizing that transformations behave in the same way with all function types. They then consider how to rewrite radical expressions; this includes extracting roots and operating with addition, subtraction, multiplication, and division. When solving radical equations, students discover that extraneous solutions may arise and analyze the circumstances under which this occurs.They conclude Radical Functions by solving real-world problems modeled by radical equations.

Standards: *2A.2A(rational function, square and cubic root functions), 2A.2B, *2A.2C, 2A.2D, 2A.4C, *2A.4F, 2A.4G, 2A.6B, 2A.6G, 2A.6H, *2A.6I, 2A.6J, 2A.6K, 2A.6L, 2A.7C, *2A.7F, 2A.7G, *2A.7H, 2A.7I

Unit 5: Inverting Functions

Module Context: Inverting Functions contains three topics: Exponential and Logarithmic Functions, exponential and Logarithmic Equations, and Applications of Exponential Functions. Students use structure to rewrite expressions and write equations based on the constraints of a problem situation. Students deepen their understanding through building functions from existing functions through inverses. They graph these functions and identify their key characteristics. Students determine regressions for data sets that can be modeled by exponential and logarithmic functions.

Exponential and Logarithmic Functions begins by reviewing the graphic and algebraic characteristics of exponential functions. Students invert exponential functions and define logarithm, natural logarithm, and logarithmic function. They connect the key characteristics of logarithmic functions to their inverses, exponential functions. Students investigate continuously compounding interest and derive the irrational number e. They use Euler’s number to solve compound interest and population growth problems. Students again expand on their understanding of transformations to now include exponential and logarithmic functions and make generalizations about the effect of transformations on an inverse function.

Exponential and Logarithmic Equations, students build on their knowledge to solve exponential and logarithmic equations. To start, they develop fluency converting between exponential and logarithmic equations. They learn strategies to solve these equations, including the Change of Base Formula. They apply these processes to solve real-world problems that are modeled by exponential and logarithmic equations.

Applications of Exponential Functions, provides students with opportunities to use their creativity to solve problems. Students start by exploring geometric series, connecting to their understanding of geometric sequences and exponential functions. They derive two formulas to calculate the sum of a geometric series and use them to solve real-world and mathematical problems. They then integrate their knowledge of all of the function types they have considered thus far to analyze and create images on the coordinate plane. The topic concludes with an exploration of fractals. Students investigate characteristics of self-similar objects and solve related area and volume problems using exponential models.

Standards:*2A.2A (ONLY EXP & LOG), 2A.2B, *2A.2C (ONLY EXP & LOG), 2A.5A, 2A.5B, 2A.5C, *2A.5D, 2A.5E, 2A.7I, 2A.8A (NOT QUADRATIC)), 2A.8B (NOT QUADRATIC), *2A.8C (NOT QUADRATIC)