GPS+Algebra+Standards

//**GPS Algebra**//
//**This is the first course in a sequence of courses designed to provide students with a rigorous** **program of study in mathematics. It includes radical, polynomial and rational expressions, basic** **functions and their graphs, simple equations, complex numbers; quadratic and piecewise** **functions, sample statistics, and curve fitting.** **(Prerequisite: Successful completion of 8th Grade Mathematics.)**// //**Instruction and assessment should include the appropriate use of manipulatives and technology.** **Topics should be represented in multiple ways, such as concrete/pictorial, verbal/written,** **numeric/data-based, graphical, and symbolic. Concepts should be introduced and used, where** **appropriate, in the context of realistic phenomena.**//

NUMBER AND OPERATIONS
**Students will use the complex number system.** 

MM2N1. Students will represent and operate with complex numbers.
a. Write square roots of negative numbers in imaginary form. b. Write complex numbers in the form //a + bi.// c. Add, subtract, multiply, and divide complex numbers. d. Simplify expressions involving complex numbers. 

ALGEBRA
**Students will explore functions, solve equations and operate with radical, polynomial and** **rational expressions. Students will investigate piecewise and quadratic functions, using** **numerical, analytical, and graphical approaches, focusing on the use of these functions in** **problem-solving situations.**  ===MM1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques.=== a. Represent functions using function notation. b. Graph the basic functions //f//(//x//) = //x^n//, where //n// = 1 to 3, //f//(//x//) = //x//, //f//(//x//) = |//x//|, and //f//(//x//) =1/x <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the //x//- and //y//-axes. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

<span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">e. Relate to a given context the characteristics of a function, and use graphs and tables to investigate its behavior. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">f. Recognize sequences as functions with domains that are whole numbers. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">g. Explore rates of change, comparing constant rates of change (i.e., slope) versus variable rates of change. Compare rates of change of linear, quadratic, square root, and other function families. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">h. Determine graphically and algebraically whether a function has symmetry and whether it is even, odd, or neither. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">i. Understand that any equation in //x// can be interpreted as the equation f(//x//) = //g//(//x//), and interpret the solutions of the equation as the //x//-value(s) of the intersection point(s) of the graphs of //y// = //f//(//x//) and //y// = //g//(//x//). <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;"> ===MM1A2. Students will simplify and operate with radical expressions, polynomials, and rational expressions.=== <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Simplify algebraic and numeric expressions involving square root. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Perform operations with square roots. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Add, subtract, multiply, and divide polynomials. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">d. Expand binomials using the Binomial Theorem. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">e. Add, subtract, multiply, and divide rational expressions. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">f. Factor expressions by greatest common factor, grouping, trial and error, and special products. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">g. Use area and volume models for polynomial arithmetic. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

MM1A3. Students will solve simple equations.
<span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Solve quadratic equations in the form //ax^//2+ //bx// + //c// = 0, where //a// = 1, by using factorization and finding square roots where applicable. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Solve equations involving radicals such as //x// + //b// = //c//, using algebraic techniques. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Use a variety of techniques, including technology, tables, and graphs to solve equations resulting from the investigation of //x^//2+ //bx// + //c// = 0//.// <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">d. Solve simple rational equations that result in linear equations or quadratic equations with leading coefficient of 1. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;"> ===MM2A1. Students will investigate step and piecewise functions, including greatestinteger and absolute value functions.=== <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Write absolute value functions as piecewise functions. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Investigate and explain characteristics of a variety of piecewise functions including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, points of discontinuity, intervals over which the function is constant, intervals of increase and decrease, and rates of change. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Solve absolute value equations and inequalities analytically, graphically, and by using appropriate technology. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

===MM2A3. Students will analyze quadratic functions in the forms //f//(//x//) = //ax^//2+ //bx// + //c// and //f//(//x//) = //a//(//x// – //h//)^2+ //k//.=== <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Convert between standard and vertex form. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

<span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Graph quadratic functions as transformations of the function f(//x//) = //x^//2. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase and decrease, and rates of change. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">d. Explore arithmetic series and various ways of computing their sums. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">e. Explore sequences of partial sums of arithmetic series as examples of quadratic functions. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

MM2A4. Students will solve quadratic equations and inequalities in one variable.
<span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Solve equations graphically using appropriate technology. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Analyze the nature of roots using technology and using the discriminant. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">d. Solve quadratic inequalities both graphically and algebraically, and describe the solutions using linear inequalities. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

DATA ANALYSIS AND PROBABILITY
<span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">**Students will use counting techniques and determine probability. Students will demonstrate** **understanding of data analysis by posing questions to be answered by collecting data. Students** **will organize, represent, investigate, interpret, and make inferences from data. They will use** **regression to analyze data and to make inferences.** <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

MM1D1. Students will determine the number of outcomes related to a given event.
<span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Apply the addition and multiplication principles of counting. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Calculate and use simple permutations and combinations. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

MM1D2. Students will use the basic laws of probability.
<span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Find the probabilities of mutually exclusive events. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Find the probabilities of dependent events. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Calculate conditional probabilities. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">d. Use expected value to predict outcomes. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

MM1D3. Students will relate samples to a population.
<span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Compare summary statistics (mean, median, quartiles, and interquartile range) from one sample data distribution to another sample data distribution in describing center and variability of the data distributions. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Compare the averages of the summary statistics from a large number of samples to the corresponding population parameters. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Understand that a random sample is used to improve the chance of selecting a representative sample. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;"> ===MM1D4. Students will explore variability of data by determining the mean absolute deviation (the average of the absolute values of the deviations).=== ===MM2D2. Students will determine an algebraic model to quantify the association between two quantitative variables.=== <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Gather and plot data that can be modeled with linear and quadratic functions. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Examine the issues of curve fitting by finding good linear fits to data using simple methods such as the median-median line and “eyeballing.” <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Understand and apply the processes of linear and quadratic regression for curve fitting using appropriate technology. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">d. Investigate issues that arise when using data to explore the relationship between two variables, including confusion between correlation and causation. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

Process Standards
<span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">**The following process standards are essential to mastering each of the mathematics content** **standards. They emphasize critical dimensions of the mathematical proficiency that all students** **need.** <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

MM1P1. Students will solve problems (using appropriate technology).
<span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Build new mathematical knowledge through problem solving. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Solve problems that arise in mathematics and in other contexts. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Apply and adapt a variety of appropriate strategies to solve problems. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">d. Monitor and reflect on the process of mathematical problem solving. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

MM1P2. Students will reason and evaluate mathematical arguments.
<span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Recognize reasoning and proof as fundamental aspects of mathematics. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Make and investigate mathematical conjectures. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Develop and evaluate mathematical arguments and proofs. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">d. Select and use various types of reasoning and methods of proof. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

MM1P3. Students will communicate mathematically.
<span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Organize and consolidate their mathematical thinking through communication. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Analyze and evaluate the mathematical thinking and strategies of others. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">d. Use the language of mathematics to express mathematical ideas precisely. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;"> ===MM1P4. Students will make connections among mathematical ideas and to other disciplines.=== <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Recognize and use connections among mathematical ideas. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Recognize and apply mathematics in contexts outside of mathematics. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

MM1P5. Students will represent mathematics in multiple ways.
<span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">a. Create and use representations to organize, record, and communicate mathematical ideas. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">b. Select, apply, and translate among mathematical representations to solve problems. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">c. Use representations to model and interpret physical, social, and mathematical phenomena. <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif;">

//<span style="font-family: Tahoma,Geneva,sans-serif;">All standards above were copied from the georgiastandards.org website. //