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Math Course: PRE - ALGEBRA Grade: 7 MATH 100 Pre-Algebra No graduation credit 5 days per week; 1 year Taught in English th This is a required class for all 7 grade students in the Mexican and/or U.S. Diploma program. Course content includes positive and negative numbers, rational numbers, geometric figures, ratios/proportions/percents, areas and volumes. Students will learn to evaluate variable expressions, do operations with positive and negative numbers, solve equations, identify fundamental concepts of geometry, find areas and volumes, and to interpret data from graphs. Textbook: Randall, Charles I., et.al. Pre-Algebra. Pearson/Prentice Hall. Upper Saddle River, NJ (2004 Edition) Prerequisite: NONE Benchmark Code Subject (M, S, SS, LA).Subject #.Strand#.Standard#. Benchmark# Example: PA.3.1.4.3 –Pre-Algebra, Strand 1, Standard 4, Benchmark 3 Strand 1: Strand 2: Strand 3: Strand 4: Strand 5: Strand 6: Strand 7: Strand 8: Strand 9: Real Numbers, Exponents, Roots and the Pythagorean Theorem. Variables and Expressions. Variables and Expressions. Functions Equations and Identities Circles Ratios, Rates, Scaling, and Similarity Number Representation and Computation Probability Number Systems Strand 1: Real Numbers, Exponents, Roots and the Pythagorean Theorem. Standard 1: Categorize real numbers as either rational or irrational and know that, by definition, these are the only two possibilities; extend the properties of computation with rational numbers to real number computation. Benchmark Code Benchmark PA.1.1.1 Approximately locate any real number on the number line. Apply the definition of irrational number to identify examples and recognize approximations. PA.1.1.2 Recognize and use 22/7 and 3.14 as approximations for the irrational number represented by pi. PA.1.1.3 Approximately locate any real number on the number line. Standard 2: Apply the Pythagorean theorem to solve problems. Benchmark Code Benchmark Determine distances between points in the Cartesian coordinate plane and relate the Pythagorean Theorem to this process. Determine the lengths of sides of right angle triangles. Strand 2: Variables and Expressions. Standard 1: Interpret and compare the different uses of variables and describe patterns, properties of numbers, formulas, and equations using variables. Benchmark Code Benchmark PA.2.1.1 Compare the different uses of variables. PA.2.1.2 Express patterns, properties, formulas and equation using and defining variables appropriately for each case. Standard 2: Analyze and identify characteristics of algebraic expressions; evaluate, interpret, and construct simple algebraic expressions; identify and transform expressions into equivalent expressions; determine whether two algebraic expressions are equivalent. Benchmark Code Benchmark PA.2.2.1 Use commutative, associative, and distributive properties of number operations to transform simple expressions into equivalent forms in order to collect like terms or to reveal or emphasize a particular characteristic. PA.2.2.2 Rewrite linear expressions n the form ax + b for constants a and b. PA.1.2.1 Strand 3: Functions. Standard 1: Determine whether a relationship is or is not a function; represent and interpret functions using graphs, tables, words, and symbols. Benchmark Code Benchmark PA.3.1.1 PA.2.1.2 Identify the independent (input) and dependent (output) quantities/variables of a function. Make tables of inputs x and outputs f(x) for a variety of rules that take numbers as inputs and produce numbers outputs. Define functions algebraically, e.g. g(x) = 3 + 2(x – x2). Create the graph of a function f by plotting and connecting a sufficient number of ordered pairs (x, f(x)) in the coordinate plane. Construct and interpret functions that describe simple problem situations using expressions, graphs, tables, and verbal descriptions and move flexibly among these multiple representations. Solve one-step inequality, two-step inequality., Standard 2: Analyze and identify linear functions of one variable; know the definitions of x and y- intercepts and slope, know how to find them and use them to solve problems. Benchmark Code Benchmark PA.3.2.1 Explain why any function defined by a linear algebraic expression has a constant rate of change. Know that a line with slope equal to zero is horizontal and represents a function while the slope of a vertical line is undefined and cannot represent a function. Standard 3: Express a linear function in several different forms for different purposes. Benchmark Code Benchmark PA.3.3.1 Recognize that in the form f(x) = mx + b, m is the slope, or constant rate of change of the graph of f, that b is the y-intercept and that in many applications of linear functions, b defines the initial state of a situation; express a function in this form when this information is given or needed. Recognize that in the form f(x) = m(x – xo) + yo, the graph of f(x) passes through the point (xo, yo); express a function in this form when this information is given or needed. Standard 4: Express a linear situation in terms of a linear function f(x) = mx + b and interpret the slope (m) and the y-intercept (b) in terms of the original linear context. Recognize, graph, and use direct proportional relationships. PA.3.4.1 Common examples of linear phenomena include distance traveled over time for objects traveling at constant speed; shipping costs under constant incremental cost per pound; conversion of measurement units (e.g., pounds to kilograms or degrees Celsius to degrees Fahrenheit); cost of gas in relations to gallons used; the height and weight of a stack of identical chairs. PA.3.4.2 Show that the graph of a direct proportional relationship is a line that passes through the origin (0, 0) whose slopes is the constant of proportionality. PA.3.4.3 Compare and contrast the graphs of x= k, y = k, and y = kx, where k is a constant. Strand 4: Equations and Identities. Standard 1: Solve linear equations and solve and graph the solution of linear inequalities in one variable. Benchmark Code Benchmark Solve equations using the fact that equals added to equals are equal and that equals multiplied by equals are equal. More formally, if A PA.4.1.1 = B and C = D, then A + C = B + D and AC = BD. Strand 5: Circles Standard 1: Identify and explain the relationships among the radius, diameter, circumference, and area of a circle; know and apply formulas for the circumference and area of a circle, semicircle, and quarter-circle. Benchmark Code Benchmark PA.5.1.1 Identify the relationship between the circumference of a circle and its radius or diameter as a direct proportion and between the area of a circle and the square of its radius or the square of its diameter as a direct proportion. Identify and describe methods for approximating π. Show that for any circle, the ratio of the circumference to the diameter is the same as the ratio of the area to the square of the radius and that these ratios are the same for different circles; identify the constant ratio A/r2= 2Cr/r2=C/2r=C/d as the number π and know that although the rational number 3.14, or 22/7= 31/7 are often used to approximate , they are not the actual values of the irrational number pi. Strand 6: Ratios, Rates, Scaling, and Similarity. Standard 1: Use ratios, rates, and derived quantities to solve problems. PA.6.1.1 Interpret and apply measures of change such as percent change and rates of growth. PA.6.1.2 Calculate with quantities that are derived as ratios and products. Create and interpret scale drawings as a tool for solving problems. Strand 7: Number Representation and Computation Standard 1: Extend and apply understanding about rational numbers; translate among different representations of rational numbers. Benchmark Code Benchmark PA.7.1.1 Use inequalities to compare rational numbers and locate them on the number line; apply basic rules of inequalities to transform numeric expressions involving rational numbers. Standard 2: Apply the properties of computation (e.g., commutative property, associative property, distributive property) to positive and negative rational number computation; know and apply effective methods of calculation with rational numbers. Benchmark Code Benchmark PA.7.2.1 Demonstrate understanding of the algorithms for additions, subtraction, multiplication, and division (non-zero divisor) of numbers expressed as fractions, terminating decimals, or repeating decimals by applying the algorithms and explaining why they work. Add, subtract, multiply, and divide (non-zero divisor) rational numbers and explain why these operations always produce another rational number. Interpret parentheses and employ conventional order of operations in a numerical expression, recognizing that conventions are universally agreed upon rules for operating on expressions. Solve practical problems involving rational numbers. Strand 8: Probability Standard 1: Represent probabilities using ratios and percents; use sample spaces to determine the (theoretical) probabilities of events; compare probabilities of two or more events and recognize when certain events are equally likely. Benchmark Code PA.8.1.1 Benchmark Calculate theoretical probabilities in simple models (e.g., number cubes, coins, spinners). Know and use the relationship between probability and odds. Analyze and interpret actual data to estimate probabilities and predict outcomes. Example: In a sample of 100 randomly selected students, 37 of them could identify the difference in two brands of soft drink. Based on these data, what is the best estimate of how many of the 2,352 students in the school could distinguish between the brands of soft drink? Standard 2: Describe the relationship between probability and relative frequency; use a probability distribution to assess the likelihood of the occurrence of an event. Benchmark Code Benchmark PA.8.2.1 Recognize and use relative frequency as an estimate for probability. PA.8.2.2 PA.8.2.3 Recognize and use relative frequency as an estimate for probability. Analyze and interpret actual data to estimate probabilities and predict outcomes. Example: In a sample of 100 randomly selected students, 37 of them could identify the difference in two brands of soft drink. Based on these data, what is the best estimate of how many of the 2,352 students in the school could distinguish between the brands of soft drink? Strand 9: Number Systems Standard 1: Identify the key features of the Ancient numbers systems. SEP Benchmark Code Benchmark PA.9.1.1 Be able to identify the symbols of the five ancient cultures, Mayan, Egyptian, Aztec, Babylonian and Roman. Be able to complete basic arithmetic calculations using 1 and 2 digits. Math Course: ALGEBRA 1 Grade: 8 MATH 200 Algebra I 1 credit 5 days per week; 1 year Taught in English th This is a required class for all 8 grade students in the Mexican and/or U.S. diploma program. Course content includes real numbers, polynomials, fractions, linear equations and systems, rational and irrational numbers, and quadratic functions. Students will learn to simplify and evaluate numerical and algebraic expressions, to do operations with positive and negative numbers, to solve operations with and factor polynomials, to solve and graph quadratic equations and equations with two variables. Students will also learn to use negative exponents and to simplify radicals. Textbook: Bellman, Allan E., et. Al. Algebra 1. Prentice/Hall. Upper Saddle River, NJ (2004 Edition) Prerequisite: MATH 100 Strand 1 = Application of Numbers. Strand 2 = Linear equations and inequalities and systems of linear equations and Inequalities. Strand 3 = Linear, Proportional And Piecewise-Linear Functions. Strand 4 = Exponents And Simple Exponential Functions. Strand 5 = Patterns Of Growth Through Iteration. Strand 6 = Quadratic functions and equations over the real numbers. Strand 7 = Data Analysis. Strand 8 = Probability. Strand 9 = Classification and operations with polynomials. Benchmark code: Example: A1.1.2.3 Subject: Number of Number of Number of Algebra 1 : Strand : Standard : Benchmark : A1 1 2 3 Strand 1 : APPLICATION OF NUMBERS Standard 1: The Student will use variables to transform Mathematical sentences into an algebraic expression. Benchmark Code Benchmark A1.1.1.1 The Student will define the concept of variable. The Student will transform mathematical sentences into algebraic expressions using Mathematical basic symbols as addition, subtraction, division, multiplication, exponents and inequalities’ symbols. Standard 2 : The Student will apply the properties of computation (commutative, associative, distributive properties) to positive and negative rational number computation; Know and apply effective methods of calculation with rational numbers). Benchmark Code Benchmark The Student will interpret parentheses and employ conventional order in a numerical expression, recognizing that conventions are universally agreed A1.1.2.1 upon rules for evaluating expressions at specified values of their variables. Standard 3: The Student will classify Real Numbers and represent them in different forms; Compare their magnitude on the number line. Benchmark Code Benchmark A1.1.3.1 The Student will identify finite (terminating) decimals, and repeating decimals as rational numbers. The Student will express a percent having a finite number of digits as a rational number by expressing it as a ratio whose numerator is an integer and whose denominator is 100 (or, more generally, whose denominator is a power of 10) The Student will transform rational numbers from one form (fractions, decimals, percents and mixed numbers) to another. Standard 4: Locate rational numbers on the number line and explain the significance of these locations. The Student will show that a number and its opposite are mirror images A1.1.4.1 with respect to the point 0. The Student will identify one or more rational numbers that lie between two given rational numbers and explain how this can be done no matter how close together the given numbers are. Standard 5: The Student will graph and analyze data on the Coordinate plane. Benchmark Code Benchmark A1.1.5.1 The Student will identify the independent and dependent variable. The Student will identify the quadrants of the Coordinate plane. The Student will graph data and will appreciate the trend of the data regarding the graphed values. (“Scatter plot”) A1.1.5.2 The Student will relate non-linear graphs to real world-events through a description of the graphs by sections according to its inflexion points and emphasizing how both variables change regarding each other. Standard 6: The Student will extend and apply understanding about rates, ratios, and unit rates, percents and percents of change. Benchmark Code Benchmark The Student will identify applications that can be expressed using rates, ratios and unit rates. The Student will convert rates to different units. A1.1.6.1 The Student will solve and write a percent equation for a given real world-situation. The Student will calculate the percent of change for the increment or decrement of a quantity. Standard 7: The Student will interpret; compare numbers involving significant figures, orders of magnitude and scientific notation. Benchmark Code Benchmark The Student will use scientific notation with positive and negative powers of 10, with appropriate treatment of significant digits, to solve A1.2.7.1 real-world and math problems. Strand 2: LINEAR EQUATIONS AND INEQUALITIES AND SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES Standard 1: The Student will solve linear equations involving several variables for one variable in terms of others. Benchmark Code Benchmark A1.2.1.1 The Student will solve one-step equations, two-step equations, and multi-step equations. The Student will solve equations with variables on both sides. The Student will write the corresponding equation solve a given real world-situation. A1.2.1.2 The Student will solve one-step inequality, two-step inequality, and multi-step inequality. The Student will solve inequalities with variables on both sides. Standard 2: The Student will solve linear equations’ systems involving two variables using algebraic procedures. Benchmark Code Benchmark A1.2.2.1 The Student will solve systems of equations by graphing. A1.2.2.2 The Student will solve systems of equations by graph substitution. A1.2.2.3 The Student will solve systems of equations by elimination. Standard 3: The Student will graph the solution of linear inequalities and systems of linear inequalities in two variables. Benchmark Code Benchmark A1.2.3.1 The Student will Know what it means to be a solution of a linear inequality in two variables, represent solutions algebraically and graphically and provide examples of ordered pairs that lie in the solution set. A1.2.3.2 The Student will graph a linear inequality in two variables and explain why the graph is always a half-plane (open or closed). Standard 4 The Student will create, interpret and apply Mathematical models to solve problems arising from contextual situations that involve linear relationships. Benchmark Code Benchmark A1.2.4.1 The Student will distinguish relevant from irrelevant information, identify missing information and find what is needed or make appropriate estimates. A1.2.4.2 The Student will recognize and solve problems that can be modeled using linear inequalities in two variables or a system of linear equations in two variables; interpret the solution(s) in terms of the context of the problem. A1.2.4.3 The Student will represent linear relationships using tables, graphs, verbal statements; translate among these forms to extract information about the relationship. Strand 3 : LINEAR, PROPORTIONAL AND PIECEWISE-LINEAR FUNCTIONS. Standard 1: The Student will recognize, graph and use direct proportional relationships. Benchmark Code Benchmark A1.3.1.1 The Student will analyze the graph of direct proportional relationships f(x) = kx and identify its key characteristics. The Student will compare and contrast the graphs x=k, y=k and y=kx where k is a constant. The Student will recognize quantities that are directly proportional and express their relationship symbolically. Standard 2: The Student will recognize, graph and use inversely proportional relationships. Benchmark Code A1.3.2.1 Benchmark The Student will recognize quantities that are inversely proportional and express their relationship symbolically. The Student will analyze the graph of inversely proportional relationships f(x) = kx and identify its key characteristics. Standard 3: The Student will distinguish between direct proportional and inversely proportional relationships. Benchmark Code Benchmark A1.3.3.1 The Student will identify whether a table, graph, formula or context suggests a direct or inversely proportional relationship. The Student will create graphs of direct proportional and inversely functions. The Student will distinguish practical situations that can be represented by direct proportional and inversely proportional relationships. The Student will analyze and use the characteristics of these relationships to answer questions about a given situation. Standard 4: The Student will explain and illustrate the effect of varying the parameters m and b in the function f(x) = mx + b Benchmark Code Benchmark A1.3.4.1 The Student will define the concept of slope (m) and connect it to the concept of constant of variation (k). The Student will define the concept of y-intercept (b) The Student will graph the function f(x) = mx + b using a table of values with a positive slope value. The Student will graph the function f(x) = mx + b using a table of values with a negative slope value. The Student will conduct the deduction of relating the slope’s sign and the position of the graph. The Student will write, evaluate and graph the function rule f(x) = mx + b from a real world-situation. Strand 4: EXPONENTS AND SIMPLE EXPONENTIAL FUNCTIONS. Standard 1: The Student will build knowledge on the use of the properties of exponents. Benchmark Code Benchmark A1.4.1.1 The Student will simplify expressions with zero and negative exponents. A1.4.1.2 The Student will simplify algebraic expressions using the multiplication’s properties of exponents. The Student will simplify algebraic expressions using the division’s properties of exponents. Standard 2: The Student will apply the properties of exponents to transform variable expressions involving integral and rational exponents. Benchmark Code Benchmark A1.4.2.1 The Student will translate between rational exponents and notation involving integral powers. A1.4.2.2 The Student will factor out common factors with exponents. Standard 3: The Student will graph and analyze exponential functions and identify their characteristics. Benchmark Code Benchmark A1.4.3.1 The Student will identify functions having the general form f(x) = abx + c A1.4.3.2 The Student will recognize and represent the graphs of exponential functions. Standard 4: The Student will recognize problems that can be modeled using exponential functions; interpret the solution(s) in terms of the context of the problem. Benchmark Code Benchmark A1.4.4.1 The Student will use exponential functions to represent growth and decay functions such as f(x) = abx and f(x) = ab-x A1.4.1.3 A1.4.4.1 The Student will use the laws of exponents to determine exact solutions for problems involving exponential functions where possible; Otherwise approximate the solutions graphically or numerically. Strand 5: PATTERNS OF GROWTH THROUGH ITERATION. Standard 1: The Student will generate and describe sequences having specific characteristics. Benchmark Code Benchmark A1.5.1.1 The Student will generate and describe arithmetic sequences expressed recursively. A1.5.1.2 The Student will generate and describe Geometric sequences expressed recursively. Standard 2: The Student will demonstrate the effect of compound interest, exponential decay or Exponential growth using iteration. Benchmark Code Benchmark A1.5.2.1 The Student will identify the diminishing effect of increasing the number of times per year that interest is compounded and relate this to the notion of instantaneous compounding. Strand 6 : QUADRATIC FUNCTIONS AND EQUATIONS OVER THE REAL NUMBERS. Standard 1: The Student will identify quadratic functions expressed in multiple forms; identify the specific information each form clarifies. Benchmark Code Benchmark A1.6.1.1 The Student will express a quadratic function as a polynomial f(x) = ax2 +bx+c. .A1.6.1.2 The Student will express a quadratic function having integral roots in factored form f(x) = (x – r) (x – s) = 0. Standard 2: The Student will graph quadratic functions and use the graph to help locate zeros. Benchmark Code Benchmark A1.6.2.1 The Student will sketch graphs of quadratic functions using a table of values. A1.6.2.2 The Student will estimate the real zeros of a quadratic function from its graph and identify quadratic functions that do not have real zeros by the behavior of their graphs. Standard 3: The Student will solve quadratic equations with integral solutions; The Student will use quadratic equations to represent and solve problems involving quadratic behavior. Benchmark Code Benchmark A1.6.3.1 The Student will solve quadratic equations that can be easily transformed into the form (x –a) (x – b) = 0 or (x + a)2 = b. A1.6.3.2 The Student will estimate the roots of a quadratic equation from the graph of the corresponding function. Standard 5: The Student will rewrite quadratic functions and interpret their graphical forms. Benchmark Code Benchmark A1.6.5.1 The Student will write a quadratic function in polynomial or standard form f(x) = ax2 +bx +c to identify the y-intercept of the function’s parabolic graph or the x-coordinate of its vertex x = - b 2a A1.6.5.2 The Student will write a quadratic function into factored form f(x) = a(x – r) (x – s) to identify its roots. A1.6.5.3 The Student will write a quadratic function in vertex form f(x) = a(x – h)2 + k (x – s) to identify the vertex and axis of symmetry of the function’s parabolic graph. A1.6.5.4 The Student will determine the axis of symmetry, maximum and minimum on a parabola. Standard 6: The Student will graph quadratic equations and solve those with real solutions using a variety of methods. Benchmark Code Benchmark A1.6.6.1 The Student will solve quadratic equations having real solutions by factoring, by completing the square and by using the quadratic formula. A1.6.6.2 The Student will recognize and solve practical problems that can be expressed using quadratic equations having real solutions; Interpret their solutions in terms of the context of the situation. Standard 7: The Student will make regular fluent use of basic algebraic identities such as (a + b)2 = a2 + 2ab + b2; (a - b)2 = a2 - 2ab + b2. Benchmark Code Benchmark A1.6.7.1 The Student will use geometric constructions to illustrate these formulas. Strand 7: DATA ANALYSIS Standard 1: The Student will collect and record; Display data using tables, charts or graphs. Benchmark Code Benchmark A1.7.1.1 The Student will understand that data are numbers in context and identify appropriate units. A1.7.1.2 The Student will organize written data records using a chart. A1.7.1.3 The Student will define measurements that are relevant to the questions posed. Standard 2: The Student will analyze and interpret categorical and quantitative data. Benchmark Code Benchmark .A1.7.2.1 The Student will make use of frequency and relative frequency tables and bar graphs. Strand 8: Standard 1 : PROBABILITY. The Student will represent probabilities using ratios and percents. The Student will use spaces to determine the (theoretical) probabilities of Events. The Student will compare probabilities of two or more events and recognize when certain events are equally likely. Benchmark Code Benchmark A1.8.1.1 The Student will calculate theoretical probabilities in simple models (number cubes, coins, spinners). A1.8.1.2 The Student will know and use the relationship between probability and odds. Standard 2 : The Student will use a probability distribution to assess the likelihood of the occurrence of an event. Benchmark Code Benchmark A1.8.2.1 The Student will analyze and interpret actual data to estimate probabilities and predict outcomes. Strand 9 : CLASSIFICATION AND OPERATIONS WITH POLYNOMIALS. Standard 1: The Student will identify the key characteristics of monomials, binomials and polynomials. Benchmark Code Benchmark A1.9.1.1 The Student will classify algebraic expressions according to the number of terms. The Student will name algebraic expressions based on their degree. Standard 2: The Student will solve basic operations with polynomials. Benchmark Code Benchmark A1.9.2.1 The Student will addition and subtraction of polynomials. .A1.9.2.2 The Student will multiplication and division of polynomials. Math Course: Mathematics 300 Plane Geometry and Introduction to Algebra Grade 9 MATH 300 Geometry 1 credit 5 days per week; 1 year Taught in English th This is a required class for all 9 grade students in the Mexican and/or U.S. diploma program. Course content includes the study of: points lines and planes, parallel lines and planes, congruent triangles, similar polygons, right triangles, circles, and coordinate geometry. Proofs will be introduced but not emphasized. Textbook: Bass, Laurie E., et. al. Geometry. Prentice/Hall. Upper Saddle River, NJ (2004 Edition) Prerequisite: MATH 200 Strand 1: Geometric Representations Strand 2: Reasoning in Geometric Situations Strand 3: Using Perpendicular and Perpendicular Lines Strand 4: Similarity, Congruence and Right Triangles Trigonometry Strand 5: Quadrilaterals Strand 6: Area Strand 7: Three-Dimensional Geometry Strand 8: Circles Strand 9: Tools of Algebra Strand 10: Functions, Equations and Graphs Strand 11: Linear Systems Benchmark Code Example: GEO.9.1.4.3 – Geometry, Grade 9th, Strand 1, Standard 4, Benchmark 3 Strand 1: Geometric Representations Standard 1: Students will review the coordinate plane to help them make logical transition from algebra to geometry and to identify, describe, and compare points, lines and planes to be used to define other geometric terms such as angles, segments and rays. Benchmark Code Benchmark GEO.9.1.1.1 GEO.9.1.1.2 GEO.9.1.1.3 GEO.9.1.1.4 GEO.9.1.1.5 GEO.9.1.1.6 GEO.9.1.1.7 Graph ordered pairs on a coordinate plane. Identify collinear points. Identify and model points, lines and planes. Identify coplanar points and intersecting lines and planes. Solve problems by listing the possibilities and by using formulas. Calculate maximum area of a rectangle for a given perimeter. Calculate distance between two points on a number line and between two points in a coordinate plane. GEO.9.1.1.8 GEO.9.1.1.9 GEO.9.1.1.10 GEO.9.1.1.11 GEO.9.1.1.12 Use the Pythagorean theorem to find the length of the hypotenuse. Identify and classify angles. Use the angle addition postulate to find the measures of angles. Identify and use congruent angles and the bisector of an angle. Identify and use adjacent, vertical, complementary, supplementary and linear pairs of angles and perpendicular lines. GEO.9.1.1.13 Make inferences to determine what information can and cannot be assumed from a diagram. GEO.9.1.1.14 GEO.9.1.1.15 Learn basic constructions. Find and calculate Perimeter, Circumference and Area of polygons and circles. Strand 2: Reasoning in Geometric Situations Standard 1: Explore inductive and deductive reasoning strategies leading to a study of if-then statements and their logic and conditional statements, their inverses, their converses, and contrapositives. Benchmark Code Benchmark GEO.9.2.1.1 GEO.9.2.1.2 Make conjectures based on inductive reasoning. Write a statement in "if-then" form with the converse, inverse, and contrapositive of the same if-then statement. GEO.9.2.1.3 Identify and use basic postulates about points, lines, and planes. GEO.9.2.1.4 Prove that angles are congruent. Strand 3: Using Perpendicular and Parallel Lines Standard 1: State the relationship between points, lines, angles, planes, spheres, parallel lines, and slopes as well as the angle measures formed when lines intersect. Benchmark Code Benchmark GEO.9.3.1.1 Solve problems by drawing a diagram and identifying the relationships between two lines or two planes and naming angles formed by a pair of lines and a transversal. GEO.9.3.1.2 Find the measures of the angles formed by two parallel lines and a transversal. GEO.9.3.1.3 GEO.9.3.1.4 GEO.9.3.1.5 Use the properties of parallel lines to determine angle measures. Find the slopes of lines and use it to identify parallel and perpendicular lines. Recognize angle conditions that produce parallel lines and prove two lines are parallel based on given angle relationships. GEO.9.3.1.6 Recognize and use distance relationships among points, lines, and planes. GEO.9.3.1.7 Find a distance between a point and a line. GEO.9.3.1.8 Use the polygon angle-sum theorem to find internal and external angles. GEO.9.3.1.9 Construct parallel and perpendicular lines. Strand 4: Similarity, Congruence and Right Triangle Trigonometry Standard 1: Students will use Algebra skills to solve proportions and identify similar figures and solve problems using proportions. Benchmark Code Benchmark GEO.9.4.1.1 Recognize and use ratios and proportions and apply their properties. GEO.9.4.1.2 GEO.9.4.1.3 GEO.9.4.1.4 Identify similar figures and use them to solve problems. Identify and use similar triangles to solve problems. Use the properties of similarity in right triangles to find missing sides or angles. GEO.9.4.1.5 Use proportional parts of triangles to solve problems and divide a segment into congruent parts. GEO.9.4.1.6 Recognize and use the proportional relationships of corresponding perimeters, altitudes, angle bisectors, and medians of similar triangles. Standard 2: Students will identify, classify and apply theorems relating to triangles. Also apply congruence to different types and parts of triangles while identifying and using medians, altitudes, angle bisectors, perpendicular bisectors. Indirect reasoning and indirect proofs will be used to reach conclusions and solve problems. Benchmark Code Benchmark GEO.9.4.2.1 Identify congruent figures. GEO.9.4.2.2 Prove triangle congruence by Side-Side-Side , Side-Angle-Side, Angle-SideAngle and Angle-Angle-Side theorems. GEO.9.4.2.3 Use congruent triangles to prove the CPCTC (corresponding parts of congruent triangles are congruent). GEO.9.4.2.4 Identify isosceles and equilateral triangles. GEO.9.4.2.5 Prove congruence in right triangles. GEO.9.4.2.6 Use corresponding parts of congruent triangles. GEO.9.4.2.7 Identify and use medians, altitudes, angle bisectors, and perpendicular bisectors in a triangle. GEO.9.4.2.8 Use indirect reasoning and indirect proof to draw conclusions. GEO.9.4.2.9 Recognize and apply properties of inequalities to the measures of segments and angles. GEO.9.4.2.10 Apply the Triangle Inequality Theorem to verify possible sides of triangles. Standard 3: Investigate and use the Pythagorean theorem and find geometric mean which will be used to solve triangles using the altitude to the hypotenuse. Benchmark Code Benchmark GEO.9.4.3.1 Use paper folding to develop the Pythagorean Theorem. GEO.9.4.3.2 Use tangent, sine and cosine ratios to find missing sides or angles. GEO.9.4.3.3 Use trigonometry to solve problems involving angles of elevation or depression. GEO.9.4.3.4 Apply properties of a right triangle to solve magnitude and direction of vectors. GEO.9.4.3.5 Use trigonometric functions to calculate area in figures. Strand 5: Quadrilaterals Standard 1: Students will be introduce to various quadrilaterals and use technology to investigate some while employing the properties of parallelograms, rhombi, rectangles, squares and trapezoids to solve problems. Benchmark Code Benchmark GEO.9.5.1.1 Classify quadrilaterals. GEO.9.5.1.2 Identify properties of parallelograms. GEO.9.5.1.3 Prove that a quadrilateral is a parallelogram by analyzing their unique properties. GEO.9.5.1.4 Recognize and apply the properties of rectangles and squares. GEO.9.5.1.5 Recognize and apply the properties of rhombi and trapezoids. GEO.9.5.1.6 Place figures on a coordinate plane. GEO.9.5.1.7 Write proofs using coordinate geometry. Strand 6: Area Standard 1: Identify and name polygons and investigate interior and exterior angle measures of convex and regular polygons. Also identify types of tessellations. Benchmark Code Benchmark GEO.9.6.1.1 Find area of parallelograms and triangles. GEO.9.6.1.2 Apply Pythagorean theorem and its converse to find area. GEO.9.6.1.3 Apply the properties of special right triangles to find area and missing sides. GEO.9.6.1.4 Calculate area of trapezoids, rhombi and kites. GEO.9.6.1.5 Calculate area of regular polygons. GEO.9.6.1.6 Measure circles and arcs. GEO.9.6.1.7 Find areas of circles and sectors. GEO.9.6.1.8 Apply geometric probability to situational problems . Strand 7: Three-Dimensional Geometry Standard 1: Recognize and draw many three-dimensional figures and to flatten these figure to look at nets and surface area; to determine surface area and volume of prisms, cylinders, pyramids, cones and spheres. Benchmark Code Benchmark GEO.9.7.1.1 Create cross sections and other slices of solids. GEO.9.7.1.2 Use top, front, side, and corner views of three-dimensional solids to make models and describe and draw cross sections and other slices of threedimensional figures. GEO.9.7.1.3 Construct a tetrahedron kite. GEO.9.7.1.4 Draw three-dimensional figures on isometric dot paper and make twodimensional nets. GEO.9.7.1.5 GEO.9.7.1.6 GEO.9.7.1.7 GEO.9.7.1.8 GEO.9.7.1.9 GEO.9.7.1.10 GEO.9.7.1.11 GEO.9.7.1.12 GEO.9.7.1.13 Find surface area of three-dimensional solids . Find the lateral area and surface area of a right prism and a right cylinder. Find the lateral area and surface area of a regular pyramid and a right circular cone. Find the volume of a right prism and a right cylinder. Compare the volumes of prisms and pyramids and the volumes of cylinders and cones. Find the volume of a pyramid and circular cone. Recognize and define basic properties of spheres. Find the surface area and the volume of a sphere. Identify congruent or similar solids and state the properties of congruent solids. Strand 8: Circles Standard 1: Define radius, diameter and circumference; use major and minor arcs and their relationships to central angles; and develop theorems involving inscribed angles and intercepted arcs using chords, secants, tangents and their properties. Benchmark Code Benchmark GEO.9.8.1.1 Identify and use parts of circles. GEO.9.8.1.2 Solve problems involving the circumference of a circle. GEO.9.8.1.3 Recognize major arcs, minor arc, semicircles, and central angles. GEO.9.8.1.4 Find measures of arcs and central angles and solve problems by making circle graphs. GEO.9.8.1.5 Recognize and use relationships among arcs, chords, and diameters. GEO.9.8.1.6 Recognize and find measures of inscribed angles and apply properties of inscribed figures. Recognize tangents and use their properties. Find the measures of angles formed by intersecting secants and tangents in relation to intercepted arcs GEO.9.8.1.7 GEO.9.8.1.8 GEO.9.8.1.9 Use the properties of chords, secants and tangents to solve segment measure problems. GEO.9.8.1.10 Write and use the equation of a circle in the coordinate plane. Strand 9: Tools of Algebra Standard 1: Students will evaluate and simplify expressions, solve equations, solve and graph inequalities Benchmark Code Benchmark ALG.9.9.1.1 Use the order of operations to evaluate expressions and use formulas. ALG.9.9.1.2 Determine the sets of numbers to which a number belongs. ALG.9.9.1.3 Translate verbal expressions and sentences into algebraic expressions and equations. ALG.9.9.1.4 Solve equations by using the properties of equalities. ALG.9.9.1.5 Solve equations for a specific variable ALG.9.9.1.6 Solve equations containing absolute value and problems by using a table. ALG.9.9.1.7 Solve inequalities and graph the solution sets. ALG.9.9.1.8 Solve compound inequalities using the properties of “and/or”. ALG.9.9.1.9 Solve inequalities involving absolute value and graph the solution. Strand 10: Functions, Equations and Graphs Standard 1: Students will identify different types of relations and functions, graph relations and functions on the coordinate plane, look for patterns to solve problems, model real-world data using scatter plots, and graph inequalities on the coordinate plane. Benchmark Code Benchmark ALG.9.10.1.1 Graph relations, state its domain and range, and determine if it’s a function. ALG.9.10.1.2 Identify and graph linear equations. ALG.9.10.1.3 Write linear equations in standard form. ALG.9.10.1.4 Determine the intercepts of a line and use them to graph an equation. ALG.9.10.1.5 Determine the slope of a line and use slope and a point to graph an equation. ALG.9.10.1.6 Identify and use patterns points or linear equations to make conclusions. ALG.9.10.1.7 Write an equation of a line in slope-intercept form given the slope and/or two points. ALG.9.10.1.8 Write an equation of a line that is parallel or perpendicular to the graph of a given equation. ALG.9.10.1.9 Draw scatter plots to find and use prediction equations. ALG.9.10.1.10 Draw graphs of inequalities with two variables. Strand 11: Linear Systems Standard 1: Solve systems of equations in two or three variables, solve system of inequalities, use linear programming to find maximum and minimum values of functions, and solve problems by solving simpler problems. Benchmark Code Benchmark ALG.9.11.1.1 Solve systems of equations by graphing. ALG.9.11.1.2 Use the substitution and elimination methods to solve system of equations. ALG.9.11.1.3 Find the values of second-order determinants. ALG.9.11.1.4 Solve system of equations by using Cramer’s rule. ALG.9.11.1.5 Solve system of inequalities by graphing. ALG.9.11.1.6 Find the maximum and minimum values of a function over a region using linear programming techniques. ALG.9.11.1.7 Apply algebraic rules to solve word problem. ALG.9.11.1.8 Solve problems involving maximum and minimum values by using linear programming techniques. ALG.9.11.1.9 Solve a system of three equations with three variables. ALG.9.11.1.10 ALG.9.11.1.11 Determine the octant in which a point in space is located. Graph linear equations in space and determine the intercepts. Math Course: Algebra II Grade 10 MATH 401 Algebra II 1/2 credit 5 times per week (1st Semester) Taught in English th This is a required class for all 10 grade students in the Mexican and/or U.S. Diploma program. Major content areas in the class include properties of real numbers, solving equations in one variable, solving inequalities and absolute value problems, graphs of linear equations, slope, solving systems of equations and linear inequalities, properties of polynomials and operations involving polynomials, and factoring polynomials, word and application problems are emphasized nd throughout the course. In the 2 Semester the course will cover laws of exponents, operations using rational expressions, solving fractional equations, properties of radicals, simplifying radicals, real and complex numbers, solving quadratic equations, graphing quadratic equations, direct and inverse variation, operations involving polynomial equations, and solving polynomial equations. Textbook: Bellman, Allan E., Algebra 2. Prentice/Hall: Upper Saddle River, NJ (2004 Edition) Prerequisite: MATH 300 Benchmark Code – Subject: Algebra II = ALG2 Strand 1: Matrices Strand 2: Quadratic Equations and Functions Strand 3: Polynomials and Polynomial Functions Strand 4: Radical Functions and Rational Exponents Strand 5: Exponential and Logarithmic Functions Strand 6: Rational Functions Strand 7: Sequences and Series Subject.Grade.Strand#.Standard#. Benchmark# Example: ALG2.10.1.1.3 – Algebra 2, Grade 10, Strand 1, Standard 1, Benchmark 3 Strand 1: Matrices Standard 1: Students will create matrices to represent data, solve problems by using matrix logic, perform operations with matrices, use matrices to achieve transformations of geometric figures, and use matrices to solve systems of equations. Benchmark Code Benchmark Use a graphing calculator to perform operations with matrices and find ALG2.10.1.1.1 variables. Perform scalar multiplication on a matrix and solve matrices for ALG2.10.1.1.2 determinants and inverses. Solve problems using matrix logic. ALG2.10.1.1.3 Add, subtract, and multiply matrices. ALG2.10.1.1.4 Evaluate the determinant of a 3 x 3 matrix. ALG2.10.1.1.5 Find the inverse of a 2 x 2 matrix. ALG2.10.1.1.6 Solve systems of linear equations by using inverse matrices. ALG2.10.1.1.7 Solve systems of linear equations by using augmented matrices. ALG2.10.1.1.8 ALG2.10.1.1.9 Use a graphing calculator to solve systems of linear equations. ALG2.10.1.1.10 Preparation for PSAT and SAT. Strand 2: Quadratic Equations and Functions Standard 1: Students will graph quadratic functions, solve quadratic equations, solve problems using the guess-and-check strategy, and analyze graphs of quadratic functions and inequalities. Benchmark Code Benchmark ALG2.10.2.1.1 Use a graphing calculator to graph and solve quadratic equations. ALG2.10.2.1.2 Write functions in quadratic form, graph, and solve by graphing. ALG2.10.2.1.3 Solve problems by using the guess-and-check strategy. ALG2.10.2.1.4 Solve quadratic equations by factoring, completing the square, and by using the quadratic formula. ALG2.10.2.1.5 Use discriminants to determine the nature of the roots of quadratic equations. ALG2.10.2.1.6 Find the sum and product of the roots of quadratic equations. ALG2.10.2.1.7 Find a quadratic equation to fit a given condition. ALG2.10.2.1.8 Use a graphing calculator to graph and explore similarities between parabolas. ALG2.10.2.1.9 Determine the equation of a parabola by using points on its graph. ALG2.10.2.1.10 Use a graphing calculator to graph and solve quadratic inequalities. ALG2.10.2.1.11 Graph quadratic inequalities and solve inequalities in one variable. ALG2.10.2.1.12 Solve non-quadratic equations by using quadratic techniques. ALG2.10.2.1.13 Preparation for PSAT and SAT. Strand 3: Polynomials and Polynomial Functions Standard 1: Students will simplify expressions containing polynomials, factor polynomials; and solve problems by identifying and achieving sub-goals, evaluate polynomial functions, and identify general shapes of the graphs of polynomial functions. Benchmark Code Benchmark ALG2.10.3.1.1 Multiply and divide monomials. ALG2.10.3.1.2 Represent numbers in scientific notation and multiply and divide expressions written in scientific notation. ALG2.10.3.1.3 Add, subtract, and multiply polynomials; divide using long division. ALG2.10.3.1.4 Divide polynomials by binomials using synthetic division. ALG2.10.3.1.5 Factor polynomials and simplify polynomial quotients by factoring. ALG2.10.3.1.6 Solve problems by identifying and achieving sub-goals. ALG2.10.3.1.7 Evaluate polynomial functions and identify general shapes of the graphs of polynomial functions. ALG2.10.3.1.8 Find factors of polynomials by using the factor theorem and synthetic division. ALG2.10.3.1.9 Use a graphing calculator to graph polynomial functions and approximate the real zeros of the functions. ALG2.10.3.1.10 Find the number and type of zeros of a polynomial function. ALG2.10.3.1.11 Identify all possible rational zeros of a polynomial function by using the rational zero theorem. ALG2.10.3.1.12 Find zeros of polynomial functions. ALG2.10.3.1.13 Solve non-quadratic equations. ALG2.10.3.1.14 Find the composition of functions. ALG2.10.3.1.15 Graph the iterations of a function. ALG2.10.3.1.16 Preparation for PSAT and SAT. Strand 4: Radical Functions and Rational Exponents Standard 1: Students will simplify expressions containing radicals, complex numbers, or rational exponents; and they solve equations containing radicals. Benchmark Code Benchmark ALG2.10.4.1.1 Simplify radicals having various indices. ALG2.10.4.1.2 Use a calculator to estimate roots of numbers. ALG2.10.4.1.3 Simplify radical expressions and add, subtract, multiply, and divide radical expressions ALG2.10.4.1.4 Rationalize the denominator of a fraction containing a radical expression. ALG2.10.4.1.5 Write expressions with radical exponents in simplest radical form and vice versa. ALG2.10.4.1.6 Evaluate expressions in either exponential or radical form. ALG2.10.4.1.7 Solve equations containing radicals. ALG2.10.4.1.8 Simplify square roots containing negative radicands. ALG2.10.4.1.9 Solve quadratic equations that have pure imaginary solutions ALG2.10.4.1.10 Add, subtract, and multiply complex numbers. ALG2.10.4.1.11 Simplify rational expressions containing complex numbers in the denominator. ALG2.10.4.1.12 Determine the inverse of a function or relation. ALG2.10.4.1.13 Graph functions and their inverses. ALG2.10.4.1.14 Work backward to solve problems. ALG2.10.4.1.15 Graph and analyze square root functions and graph square root inequalities. ALG2.10.4.1.16 Preparation for PSAT and SAT. Strand 5: Exponential and Logarithmic Functions Standard 1: Students will simplify expressions and solve equations involving real exponents, write exponential equations in logarithmic form and vice versa, evaluate expressions and solve equations involving logarithms, find common and natural logarithms and antilogarithms, and solve equations with variable exponents by using logarithms. Benchmark Code Benchmark ALG2.10.5.1.1 Use a graphing calculator to draw graphs of exponential functions. ALG2.10.5.1.2 Simplify expressions and solve equations and inequalities involving real exponents. ALG2.10.5.1.3 Use a graphing calculator to fit a curve to a scatter plot of real-world data. ALG2.10.5.1.4 Write exponential equations in logarithmic form and vice versa. ALG2.10.5.1.5 Evaluate logarithmic expressions. ALG2.10.5.1.6 Solve equations and inequalities involving logarithmic functions. ALG2.10.5.1.7 Simplify and evaluate expressions using properties of logarithms. ALG2.10.5.1.8 Solve equations involving logarithms. ALG2.10.5.1.9 Identify the characteristic and the mantissa of a logarithm. ALG2.10.5.1.10 Find common logarithms and antilogarithms. ALG2.10.5.1.11 Find natural logarithms of numbers. ALG2.10.5.1.12 Solve equations with variable exponents by using logarithms. ALG2.10.5.1.13 Use logarithms to solve problems involving growth and decay. ALG2.10.5.1.14 Preparation for PSAT and SAT. Strand 6: Rational Functions Standard 1: Students will graph rational functions, solve problems involving direct, inverse, and joint variation, simplify rational expressions, solve rational equations and solve problems by organizing data. Benchmark Code Benchmark ALG2.10.6.1.1 Use a graphing calculator to explore graphs of rational functions. ALG2.10.6.1.2 Graph rational functions. ALG2.10.6.1.3 Solve problems involving direct, inverse, and joint variation. ALG2.10.6.1.4 Simplify rational expressions. Simplify complex fractions. ALG2.10.6.1.5 Find the least common denominator of two or more algebraic expressions. ALG2.10.6.1.6 Add and subtract rational expressions. ALG2.10.6.1.7 Solve rational equations inequalities. ALG2.10.6.1.8 Preparation for PSAT and SAT. Strand 7: Sequences and Series Standard 1: Students will be introduced to arithmetic and geometric sequences, and arithmetic and geometric series. Benchmark Code Benchmark ALG2.10.7.1.1 Identify and generate arithmetic sequences and geometric sequences. ALG2.10.7.1.2 Evaluate arithmetic series and geometric series. ALG2.10.7.1.3 Use rectangles to approximate the area under a curve. ALG2.10.7.1.4 Preparation for PSAT and SAT. Math Course: Pre-Calculus (Trigonometry) Grade 11 MATH 501 Trigonometry 1/2 credit 5 times per week (1st Semester) Taught in English th This is a required class for all 11 grade students in the Mexican and/or U.S. Diploma program. Major content areas in the class include angles, arcs, and sectors of a circle, trigonometric functions, trigonometric equations and their applications, and triangle trigonometry [including the study of the Pythagorean theorem and the laws of sine and cosine]. Students will develop an understanding of the different units of angles. They will also understand the trigonometric functions and be able to apply them to solve problems. In addition, students will develop the ability to manipulate trigonometric expressions and equations. Textbook: Prerequisite: Larson, Ron and Hostetler, Robert P. Pre-Calculus. Houghton/Mifflin Company. Boston, MA (2004 Edition) MATH 402 Benchmark Code – Subject: Pre-Calculus 1 = PC1 Strand 1: Trigonometry Strand 2: Analytic Trigonometry Strand 3: Additional Topics in Trigonometry Subject.Grade.Strand#.Standard#. Benchmark# Example: PC1.11.1.1.3 – Pre-Calculus 1, Grade 11, Strand 1, Standard 1, Benchmark 3 Strand 1: Trigonometry Standard 1: Students will measure angles in radians and degrees, evaluate the trigonometric functions, graph the trigonometric functions and the inverse trigonometric functions and solve right triangles. Benchmark Code Benchmark PC1.11.1.1.1 Describe an angle and convert between radian and degree measure. PC1.11.1.1.2 Find angles that are coterminal with a given angle. PC1.11.1.1.3 Find the reference angle for a given angle. PC1.11.1.1.4 Find the length of an arc given the measure of the central angle. PC1.11.1.1.5 Find the linear and angular velocities. PC1.11.1.1.6 Find the area of a sector. PC1.11.1.1.7 Identify a unit circle and its relationship to real numbers. PC1.11.1.1.8 Evaluate trigonometric functions of any angle. PC1.11.1.1.9 Use fundamental trigonometric identities. PC1.11.1.1.10 Sketch the graphs of trigonometric functions and translations of graphs of sine and cosine functions. PC1.11.1.1.11 Find the amplitude, period, and phase shift for a trigonometric function. PC1.11.1.1.12 Evaluate the inverse trigonometric functions. PC1.11.1.1.13 Evaluate the compositions of trigonometric functions and inverse trigonometric functions. PC1.11.1.1.14 Solve right triangles. PC1.11.1.1.15 Solve problems involving simple harmonic motion. PC1.11.1.1.16 Preparation for SAT. Strand 2: Analytic Trigonometry Standard 1: Students will use the fundamental trigonometric identities and will verify them. Benchmark Code Benchmark PC1.11.2.1.1 Use the fundamental trigonometric identities to evaluate trigonometric functions and simplify trigonometric expressions. PC1.11.2.1.2 Identify and use reciprocal identities, quotient identities, Pythagorean identities, and symmetry identities. PC1.11.2.1.3 Verify trigonometric identities. PC1.11.2.1.4 Preparation for SAT. Standard 2: Students will be able to solve trigonometric equations and to recognize and use the sum and difference, multiple-angle and product-to-sum formulas. Benchmark Code Benchmark PC1.11.2.2.1 Use standard algebraic techniques and inverse trigonometric functions to solve trigonometric equations PC1.11.2.2.2 Use the sum and difference formulas, multiple angle formulas, power reducing formulas, half–angle formulas, and product-tosum formulas to rewrite and evaluate trigonometric functions. PC1.11.2.2.3 Preparation for SAT. Strand 3: Additional Topics in Trigonometry Standard 1: Students will be able to solve oblique triangles using the Law of Sines and Law of Cosines. Benchmark Code Benchmark PC1.11.3.1.1 Use the Law of Sines and the Law of Cosines to solve oblique triangles. PC1.11.3.1.2 Determine whether a triangle has zero, one, or two solutions. PC1.11.3.1.3 Find the areas of oblique triangles. PC1.11.3.1.4 Preparation for SAT. Standard 2: Students will be able to draw vectors in a plane, find their components and perform vector operations. Benchmark Code Benchmark PC1.11.3.2.1 Represent a vector in its polar form, as a sum of unit vectors and as an ordered pair or an ordered triple. PC1.11.3.2.2 Write the component forms of vectors and perform basic vector operations. PC1.11.3.2.3 PC1.11.3.2.4 Find the direction angles of vectors and the angle between two vectors. Preparation for SAT. Math Course: Pre-Calculus 2 (Analytic Geometry) Grade 11 MATH 502 Analytical Geometry 1/2 credit 5 days per week (2nd Semester) Taught in English th This is a required class for all 11 grade students in the Mexican and/or U.S. diploma program. Major content areas in the course include Cartesian coordinates, points and lines, conic sections, and the translation of axes of a graph. Students will be expected to work extensively with graphs; in addition, students will be able to identify equations of conic sections. Textbook: Prerequisite: Larson, Ron and Hostetler, Robert P. Pre-Calculus. Houghton/Mifflin Company. Boston, MA (2004 Edition) MATH 501 Benchmark Code – Subject: Pre-Calculus 2 = PC2 Strand 1: Functions and Their Graphs Strand 2: Polynomial and Rational Functions Strand 3: Exponential and Logarithmic Functions Strand 4: Topics in Analytic Geometry Subject.Grade.Strand#.Standard#. Benchmark# Example: PC2.11.1.1.3 – Pre-Calculus 2, Grade 11, Strand 1, Standard 1, Benchmark 3 Strand 1: Functions and Their Graphs Standard 1: Students will graph functions, solve equations, solve problems using functions, and analyze graphs of functions. Benchmark Code Benchmark Solve linear and quadratic equations using a variety of methods. PC2.11.1.1.1 Solve equations involving radicals and absolute values. PC2.11.1.1.2 Plot points, relations and functions in Cartesian planes. PC2.11.1.1.3 Find x and y-intercepts of a function. PC2.11.1.1.4 Sketch graphs of equations; use symmetry to sketch graphs. PC2.11.1.1.5 Graph linear equations using different methods. PC2.11.1.1.6 Find the slope of a line. Use the slope to determine parallel and PC2.11.1.1.7 perpendicular lines. Determine whether relations are functions or not. Domain and range of PC2.11.1.1.8 functions. Vertical line test. PC2.11.1.1.9 Zeros of functions. Increasing or decreasing intervals. Even and odd functions. PC2.11.1.1.10 Identify and graph cubic, square root and reciprocal functions. PC2.11.1.1.11 Combinations of functions. Add, subtract, multiply and divide functions. Composition of functions. PC2.11.1.1.12 Inverse functions. Find them and graph them. PC2.11.1.1.13 Preparation for SAT. Strand 2: Polynomial and Rational Functions Standard 1: Students will simplify expressions containing polynomial and rational functions, factor polynomials, evaluate polynomial functions, and graph polynomial and rational functions. Benchmark Code Benchmark PC2.11.2.1.1 Sketch graphs of quadratic functions. PC2.11.2.1.2 Add, subtract, and multiply polynomials; divide using long division. PC2.11.2.1.3 Divide polynomials by binomials using synthetic division. PC2.11.2.1.4 Factor polynomials and simplify polynomial quotients by factoring. PC2.11.2.1.5 Evaluate polynomial functions and identify general shapes of the graphs of polynomial functions. PC2.11.2.1.6 Find factors of polynomials by using the factor theorem and synthetic division. PC2.11.2.1.7 Sketch graphs of polynomial functions of higher degree. PC2.11.2.1.8 Use a graphing calculator to graph polynomial functions and approximate the real zeros of the functions. PC2.11.2.1.9 Find the number and type of zeros of a polynomial function. PC2.11.2.1.10 Identify all possible rational zeros of a polynomial function by using the rational zero theorem. PC2.11.2.1.11 Find zeros of polynomial functions. PC2.11.2.1.12 Operations with complex numbers. PC2.11.2.1.13 Sketch graphs of rational functions. PC2.11.2.1.14 Partial fraction decomposition. PC2.11.2.1.15 Preparation for SAT. Strand 3: Exponential and Logarithmic Functions Standard 1: Students will recognize, graph, evaluate and solve logarithmic functions. Benchmark Code Benchmark PC2.11.3.1.1 Recognize and evaluate exponential and logarithmic functions. PC2.11.3.1.2 Graph exponential and logarithmic functions. PC2.11.3.1.3 Use the change-of-base formula to rewrite and evaluate logarithmic expressions PC2.11.3.1.4 Solve exponential and logarithmic equations PC2.11.3.1.5 Use exponential growth models, exponential decay models, Gaussian models, logistic growth models, and logarithmic models to solve real-life problems. PC2.11.3.1.6 Preparation for SAT. Strand 4: Topics in Analytic Geometry Standard 1: Students will learn about Conics and conic sections (parabolas, ellipses, hyperbolas), rotation of conics, parametric equations, polar coordinates. Benchmark Code Benchmark PC2.11.4.1.1 Graph parabolas, ellipses, and hyperbolas. PC2.11.4.1.2 Identify the equation of a specific conic sections. PC2.11.4.1.3 Use conic sections to model real-world problems. PC2.11.4.1.4 Graph polar equations. PC2.11.4.1.5 Preparation for SAT. Mathematics Course: Probability and Statistics Grade: 12 MATH 601 Probability and Statistics 1/2 credit 5 days per week (1st Semester) Taught in English th This is a required class for all 12 grade students in the Mexican and/or U.S. diploma program. In this course students will study sequences and series, permutations and combinations, discrete mathematics, and data analysis. Students will develop the knowledge and skills necessary to be able to organize, analyze and graph data, as well as make decisions based on the data. Students will also be able to model problems and solve them. Matrices are introduced. Textbook: Brase, Charles Henry and Corrinne Pellillo Brase. Understandable Statistics, Concepts & Methods, Houghton/Mifflin Company. Boston, MA (2003 Edition). Prerequisite: MATH 502 Subject : Probability and Statistics = PS Strand 1: Introduction to Statistics. Strand 2: Organizing Data. Strand 3 : Averages and Variation Strand 4: Elementary Probability Theory. Strand 5: The Binomial Probability Distribution and Related Topics. Strand 6: Normal Distributions. Strand 7: Introduction to Sample Distributions. Benchmark Code Subject (M, S, SS, LA).Grade#.Strand#.Standard#. Benchmark# Example: PS.1.4.3 – Probability and Statistics, Strand 1, Standard 4, Benchmark 3 Strand: 1 INTRODUCTION TO STATISTICS. Standard 1: Introduction to statistical data. Students will be able to state the importance of the study of Statistics, the nature of the statistical data, what a sample is, what the sampling methods are and how to design ways to collect data. Benchmark Code Benchmark PS.1.1.1 Identify variables in a statistical study. PS. 1.1.2 Distinguish between quantitative and qualitative variables. PS. 1.1.3 Identify populations and samples. PS.1.1.4 Determine the levels of measurement. PS. 1.1.5 Compare descriptive and inferential statistics. Standard 2: Random samples. Students will be able to explain why random sampling is important to the study of Statistics, they will know how to use a their calculators o a random number table to make a simulation, and describe different sampling strategies and how to use them. Benchmark Code PS.1.2.1 PS.1.2.2 PS.1.2.3 Benchmark Explain the importance of random samples. Construct a simple random sample using random numbers. Simulate a random process. PS.1.2.4 Describe stratified sampling, cluster sampling, systematic sampling and convenience sampling. Standard 3: Introduction to Experimental Design. Students will learn the basics for planning a statistical study understanding the differences between observations and experiments. Benchmark Code Benchmark PS. 1.3.1 Be able to explain the term census. PS.1.3.2 PS.1.3.3 Describe simulations, observational studies and experiments. Identify control groups, placebo effects and randomized twotreatment design. PS.1.3.4 Discuss potential pitfalls that might make the data unreliable. Strand: 2 ORGANIZING DATA Standard 1: Graphing statistical data. Students should be able to display effectively information using a variety graphs. Benchmark Code Benchmark PS.2.1.1 Determine types of graphs appropriate for specific data. PS. 2.1.2 Construct bar graphs, Pareto charts, circle graphs, and time plots. PS.2.1.3 Interpret information displayed in graphs. Standard 2: Frequency distributions and histograms. Students will be able to organize data in a frequency table and construct a histogram or a frequency polygon. Benchmark Code Benchmark PS.2.2.1 Organize raw data using a frequency table. PS.2.2.2 Construct histograms, relative-frequency histograms, frequency polygons and orgies. PS.2.2.3 Recognize basic distribution shapes: uniform, symmetric, bimodal, and skewed. PS.2.2.4 Interpret graphs in the context of the data setting. Standard 3: Stem-and-leaf displays. Students will be able to construct a stem-and-leaf display. Benchmark Code Benchmark PS.2.3.1 Construct a stem-and –leaf display from raw dataPS.2.3.2 Use a stem-and-leaf display to visualize data distribution. PS.2.3.3 Compare a stem-and leaf display to a histogram. Strand: 3 AVERAGES AND VARIATION Standard 1: Measures of central tendency. Mean, Median, Mode. Students will be able to understand, state and/or compute the measures of central tendency. Benchmark Code Benchmark PS.3.1.1 Compute mean, median and mode of raw data. PS.3.1.2 PS.3.1.3 Interpret what mean, mode and median tell us. Explain how mean, median and mode can be affected by extreme data values. PS.3.1.4 Compute a trimmed means and explains why is used. Standard 2: Measures of variation. Students will be able to understand, state and/or compute the measures of variation. Benchmark Code Benchmark PS.3.2.1 Find the range, variance, and standard deviation. PS.3.2.2 Compute the coefficient of variation from raw data and understand its importance. PS.3.2.3 Apply Chebyshev’s theorem to raw data. PS.3.2.4 Understand the information given by the Chebyshev’s theorem. Standard 3: Mean and standard deviation of grouped data. Students will be able to understand, state and/or compute the mean, variance and standard deviation of grouped data. Benchmark Code Benchmark PS.3.3.1 Estimate the mean, variance and standard deviation from grouped data. PS.3.3.2 Compute a weighted average. PS.3.3.3 Understand the applications of weighted averages. Standard 4: Percentiles and box-and whisker plots. Students will be able to understand, state and/or compute percentiles, the five number summary and construct a box-and-whiskers plot. Benchmark Code Benchmark PS.3.4.1 Interpret the meaning of percentile scores. PS.3.4.2 Compute the median, quartiles, and five-number number from raw data. PS.3.4.3 Make a box-and-whisker plot and interpret its results. PS.3.4.4 Describe how a box-and-whisker plot indicates spread of data around the median. Strand: 4 ELEMENTARY PROBABILITY THEORY Standard 1: Introduction to Probability. Basic concepts. Students will be able to understand the methods to assign probabilities and apply them to basic problems. Benchmark Code Benchmark PS.4.1.1 Assign probabilities to events. Relative frequency. Law of large numbers. Equally likely outcomes. PS.4.1.2 PS.4.1.3 Explain how the law of large numbers relates to relative frequencies. Apply basic rules of probability in everyday life. Sample space. Complement of an event. PS.4.1.4 Explain the relationship between statistics and probability. Standard 2: Some probability rules. Compound events. Students will be able to understand and apply basic probability rules. Benchmark Code Benchmark PS.4.2.1 Compute probabilities of general compound events. What is and independent event. What is a compound event? PS.4.2.2 Compute probabilities involving independent events or mutually exclusive events. Multiplication rule. Addition rule. PS.4.2.3 Use results to compute conditional probabilities. Standard 3: Trees and counting techniques. Students will be able to construct tree diagrams, organize the outcomes of a series of event and assign probabilities to these outcomes. Benchmark Code Benchmark PS.4.3.1 Organize outcomes in a sample space using tree diagrams. PS.4.3.2 Compute number of ordered arrangements of outcomes using permutations. PS.4.3.4 Compute number of (non ordered) groupings of outcomes using combinations. PS.4.3.4 Explain how counting techniques relate to probability in everyday life. Strand: 5 THE BINOMIAL PROBABILITY DISTRIBUTION AND RELATED TOPICS Standard 1: Introduction to random variables and probability distributions. Students will learn the difference between continuous and discrete random variables. They will graph discrete distributions and compute their parameters. Benchmark Code Benchmark PS.5.1.1 Distinguish between continuous and random variables. PS.5.1.2 Graph discrete probability distributions. PS.5.1.3 Compute the mean μ and standard deviation σ for a discrete probability distribution. PS.5.1.4 Compute the mean μ and standard deviation σ for a linear function of a random variable x. PS.5.1.5 Compute the mean μ and standard deviation σ for a linear combination of two independent random variables. Standard 2: Binomial probabilities. Students will learn the characteristics of a Binomial probability distribution and they will learn to compute Binomial probabilities. Benchmark Code Benchmark PS.5.2.1 List the defining features of a binomial experiment. PS.5.2.2 Compute binomial probabilities using the formula for this probability distribution. PS.5.2.3 Use a binomial table to find the probability of an event P(r). PS.5.2.4 Use the binomial probability distribution to solve real-world situations. Standard 3 : Additional properties of the binomial distribution. Students will graph Binomial distributions and compute their parameters. Benchmark Code Benchmark PS.5.3.1 Make histograms for binomial distributions. PS.5.3.2 Compute the mean μ and standard deviation σ for a binomial distribution. PS.5.3.3 Compute minimal number of trials n to achieve a given probability of success P(r) Standard 4 : The Geometric and Poisson Probability distributions. Students will learn the characteristics of Geometric and Poisson probability distributions and they will learn to compute Geometric and Poisson probabilities. Benchmark Code Benchmark PS.5.4.1 Use the Geometric probability distribution to compute the probability that the nth trial is the first success. PS.5.4.2 Use the Poisson distribution to compute the probability of the occurrence of events spread out over time or space. PS.5.4.3. Use the Poisson distribution to approximate the binomial distribution when the number of trials is large and the probability of success is small. Strand: 6 NORMAL DISTRIBUTION Standard 1 : Graphs of Normal Probability Distributions. Students will graph normal distributions and understand its properties. Benchmark Code Benchmark PS.6.1.1 Graph a normal curve and summarize its important properties. PS.6.1.2 Apply the empirical rule to solve real-life problems. PS.6.1.3 Use control limits to construct control charts. Examine the chart for out of control Standard 2: Standard units and areas under the standard normal distribution. Students will learn to compute z scores and find the area under the standard normal curve. Benchmark Code Benchmark PS.6.2.1 Given μ and σ, convert raw data to z scores. PS.6.2.2 Given μ and σ, convert z scores to raw data. PS.6.2.3 Graph the standard normal distribution, and find areas under the standard normal curve. Standard 3: Areas under any normal curve. Students will learn to compute probabilities of “standardized events” and they will solve guarantee problems. Benchmark Code Benchmark PS.6.3.1 Compute the probability of “standardized events” PS.6.3.2 Find a z score from a given normal probability (inverse normal). PS.6.3.3 Use the inverse normal to solve guarantee problems. Standard 4: Normal approximation to the Binomial distribution. Students will learn to use the normal approximation of the Binomial distribution. Benchmark Code Benchmark PS.6.4.1 State the assumptions needed for the normal approximation to the binomial. PS.6.4.2 Compute μ and σ for the normal approximation. PS.6.4.3 Use the continuity correction to convert a range of r values to a corresponding range of normal x values. PS.6.4.4. Convert the x values to a range of standardized z scores and find desired probabilities. Strand: 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS Standard 1: Sampling distributions. Students will review statistical terms and will construct frequency distributions. Benchmark Code Benchmark PS:7.1.1 Review commonly used terms as random sample, relative frequency, parameter, statistic, and sampling distribution. PS.7.1.2 From raw data, construct a relative frequency distribution for mean values or expected values and compare the result to a theoretical sampling distribution. Standard 2: The Central Limit Theorem. Students will learn to use the mean and the standard deviation to construct sampling distributions. Benchmark Code Benchmark PS.7.2.1 For a normal distribution, use μ and σ to construct the theoretical sampling distribution for the statistic x-bar PS.7.2.2 For large samples, use sample estimates to construct a good approximate sampling distribution for the statistic x-bar. PS.7.2.3 Learn the statement and underlying meaning of the central limit theorem well enough to explain it. Standard 3: Sampling Distributions for Proportions. Students will learn to compute the mean and standard deviation for a given proportion and they will be able to compute probabilities for proportions. Benchmark Code Benchmark PS.7.3.1 Compute the mean and standard deviation for the proportion p=r/n PS.7.3.2 Use the normal approximation to compute the probabilities for proportions p= r/n. PS.7.3.3 Construct P-Charts and interpret what they tell us.