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GEORGIA OAS

COURSE OVERVIEW

SEMESTER 1

Unit 1:  Extending the Number System
Students extend the laws of exponents to rational exponents and explore distinctions between rational and irrational numbers by considering their decimal representations. In Unit 5, students learn that when quadratic equations do not have real solutions the number system must be extended so that solutions exist, analogous to the way in which extending the whole numbers to the negative numbers allows x + 1 = 0 to have a solution.  Students explore relationships between number systems: whole numbers, integers, rational numbers, real numbers, and complex numbers.  The guiding principle is that equations with no solutions in one number system may have solutions in a larger number system.  Students also focus on the structure of expressions, rewriting expressions to clarify and reveal aspects of the relationship they represent in working with polynomial arithmetic.


Unit 2:  Quadratic Functions
Students consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions. They select from among these functions to model phenomena. Students learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions. In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function. When quadratic equations do not have real solutions, students learn that that the graph of the related quadratic function does not cross the horizontal axis.

  Unit 3:  Modeling Geometry
In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center, and the equation of a parabola with vertical axis when given an equation of its directrix and the coordinates of its focus. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations to determine intersections between lines and circles or a parabola and between two circles.

Unit 4:  Applications of Probability
Building on probability concepts that began in the middle grades, students use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible.


Unit 5:  Inferences and Conclusions from Data
In this unit, students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions.  They identify different ways of collecting data - including sample surveys, experiments, and simulations - and the role that randomness and careful design play in the conclusions that can be drawn.  

SEMESTER 2

Unit 6:  Polynomial Functions
This unit develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base‐ten computation,focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi‐digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials and make connections between zeros of polynomials and solutions of polynomial equations. The unit culminates with the fundamental theorem of algebra.

Unit 7: Rational and Radical Relationships
Rational numbers extend the arithmetic of integers by allowing division by all numbers except 0. Similarly, rational expressions extend the arithmetic of polynomials by allowing division by all polynomials except the zero polynomial. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Similarly,radical expressions follow the rules governed by irrational numbers.  

Unit 8: Exponential and Logarithmic Functions
Students extend their work with exponential functions to include solving exponential equations with logarithms. They analyze the relationship between these two functions.

Unit 9: Radians, Unit Circle and Graphing Basic Trigonometric Functions
Building on their previous work with functions, and on their work with trigonometric ratios and circles in
Analytic Geometry,students now use the coordinate plane to extend trigonometry to model periodic phenomena.

Unit 10:  Mathematical Modeling
In this unit students synthesize and generalize what they have learned about a variety of function families. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying functions. They identify appropriate types of functions to model a situation,they adjust parameters to improve the model, and they compare models by analyzing appropriateness of
fit and making judgments about the domain over which a model is a good fit. They determine whether it is best to model with multiple functions creating a piece wise function. The description of modeling as “the process of choosing and using mathematics and statistics to analyze empirical situations,to understand them better, and to make decisions” is at the heart of this unit. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions,statistics, and geometry is applied in a modeling context.