Courses of Study : Mathematics

Students:

• Explain how the meaning of rational exponents follows from extending the properties of integer exponents.
• Use rational exponent notation to represent radicals.

Students know:

• Techniques for applying the properties of exponents.

Students are able to:

• Correctly perform the manipulations of rational exponents by analyzing and applying the properties of integer exponents.
• Use mathematical reasoning and prior knowledge of integer exponents rules to develop rational exponent notation for radicals.

Students understand that:

• The properties of exponents apply to rational exponents as well as integer exponents.

Learning Objectives:
ALGI. 1.1: Define exponent, integer, rational number, and radicals.
ALGI. 1.2: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values.
ALGI. 1.3: Use notation for radicals in terms of rational exponents.
ALGI. 1.4: Apply the properties of integer exponents to generate equivalent numerical expressions.
ALGI. 1.5: Know the properties of integer exponents.
ALGI. 1.6: Write numerical expressions involving whole-number exponents.
ALGI. 1.7: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.

Students:

• Given expressions involving radicals and rational exponents, use reasoning and properties of exponents to demonstrate that various forms of radicals and roots actually represent the same quantity,

Students know:

• Properties of exponents.
• The meaning of algebraic symbols such as radicals and rational exponents.

Students are able to:

• Use mathematical reasoning to justify the equality of various forms of radical expressions.
• Correctly perform the manipulations of exponents which apply properties of exponents.

Students understand that:

• The properties of exponents are true regardless of the type of numbers being used.

Learning Objectives:
ALGI. 2.1: Rewrite expressions involving radicals using the properties of exponents.
ALGI. 2.2: Rewrite expressions involving rational exponents using the properties of exponents.
ALGI. 2.3: Recognize the properties of exponents.

Students:
Given an equation where x² is less than zero,

• Explain by repeated reasoning from square roots in the positive numbers what conditions a solution must satisfy, how defining a number i by the equation i² = -1 would satisfy those conditions, and extend the real numbers to a set called the complex numbers.
• Explain how adding and/or multiplying i by any real number results in a complex number and is real when the multiplier is zero.

Students know:

• Which manipulations of radicals produce equivalent forms.
• The extension of the real numbers which allows equations such as x² = -1 to have solutions is known as the complex numbers and the defining feature of the complex numbers is a number i, such that i² = -1.

Students are able to:

• Perform manipulations of radicals, including those involving square roots of negative numbers, to produce a variety of forms, for example, √(-8) = i√(8) = 2ⁱ√(2).

Students understand that:

• When quadratic equations do not have real solutions, the number system must be extended so that solutions exist. and the extension must maintain properties of arithmetic in the real numbers.

Learning Objectives:
ALGI. 3.1: Define rational and irrational numbers.
ALGI. 3.2: Identify the product of a nonzero rational number and an irrational number as irrational.
ALGI. 3.3: Identify the sum of a rational number and an irrational number is irrational.
ALGI. 3.4: Discuss why the product of two rational numbers is rational.
ALGI. 3.5: Discuss why the sum of two rational numbers is rational.
ALGI. 3.6: Describe the properties of addition and multiplication.
ALGI. 3.7: Apply properties of fractions to add, subtract, multiply, and divide rational numbers.
ALGI. 3.8: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

Students:

• Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful ways to assist in the solution of given problems.
• Interpret the meaning of the parts of an expression. For example, see that 3 + (x-2)² is a sum of a constant and a square, that the square contains the expression x-2, and that the value of the expression is always greater than 3.
• Justify their selection of a form for an expression by explaining which features of the expression are revealed by the particular form and how these features aid in resolving a problem situation.

Students know:

• How to recognize the parts of linear, quadratic and exponential expressions and what each part represents.
• When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem.
• That one or more parts of an expression can be viewed as a single entity.

Students are able to:

• Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
• Interpret expressions in terms of a context.
• View one or more parts of an expression as a single entity and determine the impact parts of the expression have in terms of the context.

Students understand that:

• Making connections among the parts of an expression reveals the roles of important mathematical features of a problem.

Learning Objectives:
ALGI.4.1: Define linear, quadratic and exponential functions.
ALGI.4.2: Classify an expression as linear, quadratic or exponential from a table.
ALGI.4.3: Classify an expression as linear, quadratic or exponential from an equation.
ALGI.4.4: Classify an expression as linear, quadratic or exponential from a graph.
ALGI.4.5: Define terms, factors, and coefficients.
ALGI.4.6: Identify factors in linear, exponential and quadratic expressions.
ALGI.4.7: Identify coefficients in linear, exponential and quadratic expressions.
ALGI.4.8: Identify terms in linear, exponential and quadratic expressions.
ALGI.4.9: Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient).
ALGI.4.10: Recognize one or more parts of an exponential expression as a single entity.
ALGI.4.11: Recognize one or more parts of a quadratic expression as a single entity.
ALGI.4.12: Recognize one or more parts of a linear expression as a single entity.

Students:

• Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful and more efficient ways.

Students know:

• Algebraic properties.
• When one form of an algebraic expression is more useful than an equivalent form of that same expression.

Students are able to:

• Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures.

Students understand that:

• Generating equivalent algebraic expressions facilitates the investigation of more complex algebraic expressions.

Learning Objectives:
ALGI.5.1: Define equivalent expressions.
ALGI.5.2: Rewrite an exponential expression in an alternative way.
ALGI.5.3: Rewrite a quadratic expression in an alternative way.
ALGI.5.4: Rewrite a linear expression in an alternative form.
ALGI.5.5: Understand that rewriting an expression in different forms in a problem context can shed light on the problem.
ALGI.5.6: Recall properties of exponents.

Students:

• Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful ways to assist in the solution of given problems.
• Produce the useful equivalent forms of expressions,
• Factor a quadratic expression with leading coefficient of one to reveal the zeros of the function it defines and complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
• Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines.
• Justify their selection of a form for an expression by explaining which features of the expression are revealed by the particular form and how these features aid in resolving a problem situation.

Students know:

• Techniques for generating equivalent forms of an algebraic expression, including factoring and completing the square for quadratic expressions and using properties of exponents.
• When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem.

Students are able to:

• Use algebraic properties including properties of exponents to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
• Factor quadratic expressions.
• Complete the square in quadratic expressions.
• Use the vertex form of a quadratic expression to identify the maximum or minimum and the axis of symmetry.

Students understand that:

• Making connections among equivalent expressions reveals the roles of important mathematical features of a problem.

Learning Objectives:
ALGI.6.1: Convert an expression to an alternative format.
ALGI.6.2: Recognize the best format for a specific application.
ALGI.6.3: Match equivalent expressions written in different forms.

a.
ALGI.6.4: Define factor, quadratic expression and zero product property.
ALGI.6.5: Factor a quadratic expression.
ALGI.6.6: Use the zero product property to reveal the zeros in the function.
ALGI.6.7: Solve a one-step equation.
ALGI.6.8: Solve a two-step equation.
ALGI.6.9: Determine the Greatest Common Factor (GCF).

b.
ALGI.6.10: Define maximum and minimum value.
ALGI.6.11: Explain the steps for completing the square.
ALGI.6.12: Given a quadratic expression in which the square has already been completed, determine the maximum or minimum values.

c.
ALGI.6.13: Define roots.
ALGI.6.14: Find the equation using the distributive property.
ALGI.6.15: Locate and identify the roots on a graph using the x-intercepts.
ALGI.6.16: Take given roots and convert into a one-step equation set equal to zero.

Students:

• Use the repeated reasoning from prior knowledge of properties of arithmetic on integers to progress consistently to rules for arithmetic on polynomials.
• Accurately perform combinations of operations on various polynomials.

Students know:

• Corresponding rules of arithmetic of integers, specifically what it means for the integers to be closed under addition, subtraction, and multiplication, and not under division.
• Procedures for performing addition, subtraction, and multiplication on polynomials.

Students are able to:

• Communicate the connection between the rules for arithmetic on integers and the corresponding rules for arithmetic on polynomials.
• Accurately perform combinations of operations on various polynomials.

Students understand that:

• There is an operational connection between the arithmetic on integers and the arithmetic on polynomials.

Learning Objectives:
ALGI.7.1: Combine like terms of a given expression
ALGI.7.2: Define monomial, term, binomial, trinomial and polynomial.
ALGI.7.3: Multiply polynomial expressions (quadratic).
ALGI.7.4: Multiply polynomial expressions (linear).
ALGI.7.5: Subtract polynomial expressions.
ALGI.7.6: Add polynomial expressions.
ALGI.7.7: Use order of operations to evaluate and simplify algebraic and numerical expressions.
ALGI.7.8: Identify the terms in a polynomial expressions.
ALGI.7.9: Explain the distributive property.

Students:

• Solve problems involving absolute value equations.
• Identify extraneous solutions to these equations if any.
• Produce examples of equations that would or would not have extraneous solutions and communicate the conditions that lead to the extraneous solutions.

Students know:

• Algebraic rules for manipulating absolute value equations.
• Conditions under which a solution is considered extraneous.

Students are able to:

• Accurately rearrange absolute value equations to produce a set of values to test against the conditions of the original situation and equation, and determine whether or not the value is a solution.
• Explain with mathematical reasoning from the context (when appropriate) why a particular solution is or is not extraneous.

Students understand that:

• Values which arise from solving absolute value equations may not satisfy the original equation.
• Values which arise from solving the equations may not exist due to considerations in the context.

Learning Objectives:
ALGI. 8.1: Define integers.
ALGI. 8.2: Demonstrate the location of positive and negative numbers on a vertical and horizontal number line.
ALGI. 8.3: Give examples of positive and negative numbers to represent quantities having opposite directions in real-world contexts.
ALGI. 8.4: Discuss the measure of centering of 0 in relationship to positive and negative numbers.
ALGI. 8.5: Substitute for the variable to find the value of a given expression.
ALGI. 8.6: Explain the meaning of absolute value and determine the absolute value of rational numbers in real-world contexts.
ALGI. 8.7: Compare and order rational numbers and absolute value of rational numbers with and without a number line in order to solve real-world and mathematical problems.

Students:

• Solve quadratic equations where both sides of the equation have evident square roots by inspection.
• Transform quadratic equations to a form where the square root of each side of the equation may be taken, including completing the square.
• Use the method of completing the square on the equation in standard form ax²+bx+c=0 to derive the quadratic formula.
• Identify quadratic equations which may be solved efficiently by factoring, and then use factoring to solve the equation.
• Use the quadratic formula to solve quadratic equations.
• Explain when the roots are real or complex for a given quadratic equation, and when complex write them as a ± bi.
• Demonstrate that a proposed solution to a quadratic equation is truly a solution by making the original true.

Students know:

• Any real number has two square roots, that is, if a is the square root of a real number then so is -a.
• The method for completing the square.
• Notational methods for expressing complex numbers.
• A quadratic equation in standard form (ax²+bx+c=0) has real roots when b²-4ac is greater than or equal to zero and complex roots when b²-4ac is less than zero.

Students are able to:

• Accurately use properties of equality and other algebraic manipulations including taking square roots of both sides of an equation.
• Accurately complete the square on a quadratic polynomial as a strategy for finding solutions to quadratic equations.
• Factor quadratic polynomials as a strategy for finding solutions to quadratic equations.
• Rewrite solutions to quadratic equations in useful forms including a ± bi and simplified radical expressions.
• Make strategic choices about which procedures (inspection, completing the square, factoring, and quadratic formula) to use to reach a solution to a quadratic equation.

Students understand that:

• Solutions to a quadratic equation must make the original equation true and this should be verified.
• When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically.
• Different procedures for solving quadratic equations are necessary under different conditions.
• If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation.
• Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax²+bx+c=0).

Learning Objectives:
ALGI.9.1: Define quadratic equation and zero product property.
ALGI.9.2: Solve one-step equations using addition and subtraction that are set equal to zero.
ALGI.9.3: Solve two-step equations using addition and subtraction that are set equal to zero.

a.
ALGI.9.4: Define completing the square.
ALGI.9.5: Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)²= q that has the same solutions.
ALGI.9.6: Derive the quadratic formula from the form (x - p)= q.

b.
ALGI.9.7: Define quadratic formula, factoring, square root, complex number, and real number.
ALGI.9.8: Solve quadratic equations by completing the square.
ALGI.9.9: Solve quadratic equations by the quadratic formula.
ALGI.9.10: Solve quadratic equations by factoring.
ALGI.9.11: Solve quadratic equations by taking square roots.
ALGI.9.12: Recognize when the quadratic formula gives complex solutions.
ALGI.9.13: Write complex solutions as a ±bi for real numbers a and b.

Students:

• Choose an appropriate method for solving a system of two linear equations (e.g., substitution, addition, tables, graphing).
• Solve and justify solutions.
• Contrast solutions to a system of two linear equations to determine which method is more efficient.
• Understand that tables and graphs of systems of equations my produce estimates rather than exact solutions.
• Provide reasonable approximations when appropriate in a graph or table.

Students know:

• Appropriate use of properties of addition, multiplication and equality.
• Techniques for producing and interpreting graphs of linear equations.
• Techniques for producing and interpreting tables of linear equations.
• The conditions under which a system of linear equations has 0, 1, or infinitely many solutions.

Students are able to:

• Accurately perform the operations of multiplication and addition, and techniques for manipulating equations.
• Graph linear equations precisely.
• Create tables and locate solutions from the tables for systems of linear equations.
• Use estimation to find approximate solutions on a graph.
• Contrast solution methods and determine efficiency of a method for a given problem situation.

Students understand that:

• The solution of a linear system is the set of all ordered pairs that satisfy both equations.
• Solving a system by graphing or with tables can sometimes lead to approximate solutions.
• A system of linear equations will have 0, 1, or infinitely many solutions.

Learning Objectives:
ALGI.10.1: Solve a system of equations using three methods (Substitution, Elimination, and Graphing.
ALGI.10.2: Distinguish the similarities and differences between the three methods of solving systems of equations.

Students:

• Given a contextual situation that may include linear, quadratic, exponential, or rational functional relationships in one variable.
• Model the relationship with equations or inequalities and solve the problem presented in the contextual situation for the given variable.

Students know:

• When the situation presented in a contextual problem is most accurately modeled by a linear, quadratic, exponential, or rational functional relationship.

Students are able to:

• Write equations in one variable that accurately model contextual situations.

Students understand that:

• Features of a contextual problem can be used to create a mathematical model for that problem.

Learning Objectives:
ALGI.11.1: Solve the equation represented by the real-world situation.
ALGI.11.2: Set up an equation to represent the given situation, using correct mathematical operations and variables.
ALGI.11.3: Given a contextual situation, interpret and defend the solution in the context of the original problem.
ALGI.11.4: Define equation, expression, variable, equality and inequality.
ALGI.11.5: Create inequalities with one variable (Exponential, Quadratic, Linear).
ALGI.11.6: Create equalities with one variable (Exponential, Quadratic, Linear).
ALGI.11.7: Solve two-step equations and inequalities.
ALGI.11.8: Solve one-step equations and inequalities using the four basic operations.
ALGI.11.9: Compare and contrast equations and inequalities.
ALGI.11.10: Recognize inequality symbols including greater than, less than, greater than equal to and less than equal to.

Students:
Given a contextual situation expressing a relationship between quantities with two or more variables,

• Model the relationship with equations and graph the relationship on coordinate axes with labels and scales.
• Make predictions about the contextual situation using the graphs of the equations.

Students know:

• When a particular two variable equation accurately models the situation presented in a contextual problem.

Students are able to:

• Write equations in two variables that accurately model contextual situations.
• Graph equations involving two variables on coordinate axes with appropriate scales and labels.
• Make predictions about the context using the graph.

Students understand that:

• There are relationships among features of a contextual problem, a created mathematical model for that problem, and a graph of that relationship.

Learning Objectives:
ALGI.12.1: Solve the equations represented by real-world situations.
ALGI.12.2: Set up an equation to represent the given situation, using correct mathematical operations and variables.
ALGI.12.3: Given a contextual situation, interpret and defend the solution in the context of the original problem.
ALGI.12.4: Explain how to draw informal inferences from data distributions.
ALGI.12.5: Define ordered pair and coordinate plane.
ALGI.12.6: Create equations with two variables (exponential, quadratic and linear).
ALGI.12.7: Graph equations on coordinate axes with labels and scales (exponential, quadratic, and linear).
ALGI.12.8: Identify an ordered pair and plot it on the coordinate plane.

Students:
Given a contextual situation involving constraints,

• Write equations or inequalities or a system of equations or inequalities that model the situation and justify each part of the model in terms of the context.
• Solve the equation, inequalities or systems and interpret the solution in the original context including discarding solutions to the mathematical model that cannot fit the real-world situation (e.g., distance cannot be negative).
• Solve a system by graphing the system on the same coordinate grid and determine the point(s) or region that satisfies all members of the system.
• Determine the point(s) of the region satisfying all members of the system that maximizes or minimizes the variable of interest in the case of a system of inequalities.

Students know:

• When a particular system of two variable equations or inequalities accurately models the situation presented in a contextual problem.
• Which points in the solution of a system of linear inequalities need to be tested to maximize or minimize the variable of interest.

Students are able to:

• Graph equations and inequalities involving two variables on coordinate axes.
• Identify the region that satisfies both inequalities in a system.
• Identify the point(s) that maximizes or minimizes the variable of interest in a system of inequalities.
• Test a mathematical model using equations, inequalities, or a system against the constraints in the context and interpret the solution in this context.

Students understand that:

• A symbolic representation of relevant features of a real-world problem can provide for resolution of the problem and interpretation of the situation and solution.
• Representing a physical situation with a mathematical model requires consideration of the accuracy and limitations of the model.

Learning Objectives:
ALGI.13.1: Define systems of equations, constraints, viable solution, and nonviable solution.
ALGI.13.2: Create a system of equations or inequalities to represent the given constraints (linear).
ALGI.13.3: Create an equation or inequality to represent the given constraints (linear).
ALGI.13.4: Determine if a solution to a system of equations or inequalities is viable or nonviable.
ALGI.13.5: Determine if there is one solution, infinite solutions, or no solutions to a system of equations or inequalities.

Students:
Given a relation defined by an equation in two variables,

• Verify that any ordered pair in the relation that makes the equation true is a point on the graph.
• Show that there are an infinite number of ordered pairs that satisfy the equation.

Students know:

• Appropriate methods to find ordered pairs that satisfy an equation.
• Techniques to graph the collection of ordered pairs to form a curve.

Students are able to:

• Accurately find ordered pairs that satisfy an equation.
• Accurately graph the ordered pairs and form a curve.

Students understand that:

• An equation in two variables has an infinite number of solutions (ordered pairs that make the equation true), and those solutions can be represented by a curve in the coordinate plane.

Learning Objectives:
ALGI.14.1: Understand that the graph of an equation is the solution of an equation.
ALGI.14.2: Graph a linear equation and use the graph to determine the solution set.
ALGI.14.3: Use a given graph to determine the solution set.
ALGI.14.4: Plot given points from a table.

Students:

• Given input/output relations between two variables in graphical form, verbal description, set of ordered pairs, or algebraic model, distinguish between those that are functions and non-functions.

Using functional notation,

• Evaluate functions for inputs.
• Interpret statements in terms of context.

Given a contextual relationship that may be represented as a function,

• Determine that exactly one element of the range (output) is assigned to each element of the domain (input) by the function.
• Relate the domain to its graph and to the quantitative relationship it describes.

Students know:

• Distinguishing characteristics of functions.
• Conventions of function notation.
• In graphing functions the ordered pairs are (x,f(x)) and the graph is y = f(x).

Students are able to:

• Evaluate functions for inputs in their domains.
• Interpret statements that use function notation in terms of context.
• Accurately graph functions when given function notation.
• Accurately determine domain and range values from function notation.

Students understand that:

• A function is a mapping of the domain to the rangeFunction notation is useful in contextual situations to see the relationship between two variables when the unique output for each input relation is satisfied.

Learning Objectives:
ALGI.15.1: Define domain, range, relation, function, table of values, input, and output.
ALGI.15.2: Understand the graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
ALGI.15.3: Understand that a function is a rule that assigns to each input exactly one output.
ALGI.15.4: Identify the equation of a function, given its graph.
ALGI.15.5: Find the range of a function given its domain.
ALGI.15.6: Recognize that f(x) and y are the same.
ALGI.15.7: Recall how to complete input/output tables.
ALGI.15.8: Recall how to read/interpret information from a table.
ALGI.15.9: Define function notation.
ALGI.15.10: Translate a simple word problem into function notation.
ALGI.15.11: Evaluate function when given x-value.

Students:

• Can identify similarities and differences between relations and functions represented by equations, graphs or tables.
• Determine if a relation is a function and explain that a function is a special kind of relation.

Students know:

• How to represent relations and functions by equations, graphs or tables and can compare and contrast the different representations.
• A function is a special kind of relation.

Students are able to:

• Compare and contrast relations and functions given different representations.
• Identify which relations are functions and which are not.

Students understand that:

• All functions are relations, but that some relations are not functions.
• Equations, graphs, and tables are useful representations for comparing and contrasting relations and functions.

Learning Objectives:
ALGI.16.1: Define functions, relations (ordered pairs), input, output.
ALGI.16.2: Recall how to complete input/output tables.
ALGI.16.3: Recall how to read/interpret information from a table.
ALGI.16.4: Identify algebraic expressions.
ALGI.16.5 Recall how to name points from a graph (ordered pairs).
ALGI.16.6: Recall how to name points on a Cartesian plane using ordered pairs.

Students:
Given different types of standard functions

• Use arithmetic operations to combine functions in context.
• Use function composition to combine functions in context.
• Write, evaluate, and interpret combined functions in context.

Students know:

• Techniques to combine functions using arithmetic operations.
• Techniques for combining functions using function composition.

Students are able to:

• Accurately develop a model that shows the functional relationship between two quantities.
• Accurately create a new function through arithmetic operations of other functions.
• Present an argument to show how the function models the relationship between the quantities.

Students understand that:

• Arithmetic combinations of functions may be used to improve the fit of a model.

Learning Objectives:
ALGI.17.1: Define functions, relations (ordered pairs), input, output.
ALGI.17.2: Recall how to complete input/output tables.
ALGI.17.3: Recall how to read/interpret information from a table.
ALGI.17.4: Identify algebraic expressions.
ALGI.17.5: Recall how to name points from a graph (ordered pairs).
ALGI.17.6: Recall how to name points on a Cartesian plane using ordered pairs.

a.
ALGI.17.7: Identify, represent, and analyze two quantities that change in relationship to one another in real-world or mathematical situations.
ALGI.17.8: Set up an equation to represent the given situation, using correct mathematical operations and variables.

b.
ALGI.17.9: Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtration, and multiplication.

Students:
Given a system of a linear equation and a quadratic equation,

• Solve the system algebraically by substitution.
• Graph the linear equation and the quadratic equation on the same Cartesian plane, and identify the intersection point(s).
• Make sense of the existence of 0, 1, or 2 solutions to the system by explaining the relationship of the solutions to the graph.
• Verify that the proposed solutions satisfy both equations.

Students know:

• Appropriate use of properties of equality.
• Techniques to solve quadratic equations.
• The conditions under which a linear equation and a quadratic equation have 0, 1, or 2 solutions.
• Techniques for producing and interpreting graphs of linear and quadratic equations.

Students are able to:

• Accurately use properties of equality to solve a system of a linear and a quadratic equation.
• Graph linear and quadratic equations precisely and interpret the results.

Students understand that:

• Solutions of a system of equations is the set of all ordered pairs that make both equations true simultaneously.
• A system consisting of a linear equation and a quadratic equation will have 0,1, or 2 solutions.

Learning Objectives:
ALGI.18.1: Use the substitution method to replace a variable in the quadratic equation.
ALGI.18.2: Solve for the variables in a system of equations. (Algebraically).
ALGI.18.3: Graph a quadratic equation.
ALGI.18.4: Graph a linear equation.
ALGI.18.5: Identify the point(s) of intersection when given graphs.
ALGI.18.6: Use digital tools to defend solutions to authentic problems.
ALGI.18.7: Use digital tools to formulate solutions to authentic problems (Ex: electronic graphing tools, probes, spreadsheets).

Students:
Given two functions (where f(x) is linear, quadratic ,absolute value, or exponential and g(x) is constant or linear) that intersect,

• Graph each function and identify the intersection point(s).
• Explain solutions for f(x) = g(x) as the x-coordinate of the points of intersection of the graphs, and explain solution paths.
• Use technology, tables, and successive approximations to produce the graphs, as well as to determine the approximation of solutions.

Students know:

• Defining characteristics of linear, polynomial, absolute value, and exponential graphs.
• Methods to use technology and tables to produce graphs and tables for two functions.

Students are able to:

• Determine a solution or solutions of a system of two functions.
• Accurately use technology to produce graphs and tables for linear, quadratic, absolute value, and exponential functions.
• Accurately use technology to approximate solutions on graphs.

Students understand that:

• By graphing y=f(x) and y=g(x) on the same coordinate plane, the x-coordinate of the intersections of the two equations is the solution to the equation f(x) = g(x)

Learning Objectives:
ALGI.19.1: Define function, function notation, linear, polynomial, rational, absolute value, exponential, and logarithmic functions, and transitive property.
ALGI.19.2: Explain, using the transitive property, why the x-coordinates of the points of the graphs are solutions to the equations.
ALGI.19.3: Find solutions to the equations y = f(x) and y = g(x) using the graphing calculator.
ALGI.19.4: Solve equations for y.
ALGI.19.5: Demonstrate use of a graphing calculator, including using a table, making a graph, and finding successive approximations.

Students:

• Given a linear inequality in two variables or a system of linear inequalities, graph solutions and solution sets using the appropriate notation (dotted or solid line).

Students know:

• When to include and exclude the boundary of linear inequalities.
• Techniques to graph the boundaries of linear inequalities.
• Methods to find solution regions of a linear inequality and systems of linear inequalities.

Students are able to:

• Accurately graph a linear inequality and identify values that make the inequality true (solutions).
• Find the intersection of multiple linear inequalities to solve a system.

Students understand that:

• Solutions to a linear inequality result in the graph of a half-plane.
• Solutions to a system of linear inequalities are the intersection of the solutions of each inequality in the system.

Learning Objectives:
ALGI.20.1: Define the half-plane as the shaded region.
ALGI.20.2: Determine the intersecting shaded region is the solution to the system.
ALGI.20.3: Graph the lines of the systems and shade the appropriate region.
ALGI.20.4: Determine the shaded region is the solution to the inequality.
ALGI.20.5: Graph an inequality and shade the appropriate region.
ALGI.20.6: Determine whether a line should be solid or dotted, depending on the inequality symbol.
ALGI.20.7: Recognize inequality symbols >, < .

Students:
Given relations or functions represented in various ways(algebraically, graphically, numerically in tables, or by verbal descriptions),

• Use key features to compare the functions.
• Explain and justify the similarities and differences of the functions.

Students know:

• Techniques to find key features of functions when presented in different ways.
• Techniques to convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions).

Students are able to:

• Accurately determine which key features are most appropriate for comparing functions.
• Manipulate functions algebraically to reveal key functions.
• Convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions) for the purpose of comparing it to another function.

Students understand that:

• Functions can be written in different but equivalent ways (algebraically, graphically, numerically in tables, or by verbal descriptions).
• Different representations of functions may aid in comparing key features of the functions.

Learning Objectives:
ALGI.21.1: Define function, function notation, (linear, polynomial, rational, absolute value, exponential, piecewise, and logarithmic) functions, and transitive property.
ALGI.21.2: Explain, using the transitive property, why the x-coordinates of the points of the graphs are solutions to the equations.
ALGI.21.3: Find solutions to the equations y = f(x) and y = g(x) using the graphing calculator.
ALGI.21.4: Solve equations for y.
ALGI.21.5: Demonstrate use of a graphing calculator, including using a table, making a graph, and finding successive approximations.

Students:

• Given a sequence, generate and justify a function that relates the number of the term to the value of the term in the sequence.

Students know:

• Distinguishing characteristics of a function.
• Distinguishing characteristics of function notation.
• Distinguishing characteristics of generating sequences.

Students are able to:

• Relate the number of the term to the value of the term in a sequence and express the relation in functional notation.

Students understand that:

• Each term in the domain of a sequence defined as a function is unique and consecutive.

Learning Objectives:
ALGI.22.1: Define proportions and proportional relationships.
ALGI.22.2: Write equations to represent a proportional relationship.
ALGI.22.3: Discuss the use of variables in proportional relationships.
ALGI.22.4: Define sequences and recursively-defined sequences.
ALGI.22.5: Recognize that sequences are functions whose domain is the set of all positive integers and zero.

Students:
Given a function in algebraic form,

• Graph the function, f(x), conjecture how the graph of f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k(both positive and negative) will change from f(x), and test the conjectures.
• Describe how the graphs of the functions were affected (e.g., horizontal and vertical shifts, horizontal and vertical stretches, or reflections).
• Use technology to explain possible effects on the graph from adding or multiplying the input or output of a function by a constant value.

• Given the graph of a function and the graph of a translation, stretch, or reflection of that function, determine the value which was used to shift, stretch, or reflect the graph.

Students know:

• Graphing techniques of functions.
• Methods of using technology to graph functions

Students are able to:

• Accurately graph functions.
• Check conjectures about how a parameter change in a function changes the graph and critique the reasoning of others about such shifts.
• Identify shifts, stretches, or reflections between graphs.

Students understand that:

• Graphs of functions may be shifted, stretched, or reflected by adding or multiplying the input or output of a function by a constant value.

Learning Objectives:
ALGI.23.1: Define dilation, rotation, reflection, translation, congruent and sequence.
ALGI.23.2: Identify rotations.
ALGI.23.3: Identify reflections.
ALGI.23.4: Identify translations.
ALGI.23.5: Use digital tools to formulate solutions to authentic problems (Ex: electronic graphing tools, probes, spreadsheets).

Students:
Given a linear or exponential function,

• Create a sequence from the functions and examine the results to demonstrate that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.
• Use slope-intercept form of a linear function and the general definition of exponential functions to justify through algebraic rearrangements that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.

• Given a contextual situation modeled by functions, determine if the change in the output per unit interval is a constant being added or multiplied to a previous output, and appropriately label the function as linear, exponential, or neither.

Students know:

• Key components of linear and exponential functions.
• Properties of operations and equality

Students are able to:

• Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity changes at a constant rate per unit interval relative to another (linear).
• Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity grows or decays by a constant percent rate per unit interval relative to another (exponential).

Students understand that:

• Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.
• Distinguishing key features of and categorizing functions facilitates mathematical modeling and aids in problem resolution.

Learning Objectives:
ALGI.24.1: Define linear function and exponential function.
ALGI.24.2: Distinguish between graphs of a line and an exponential function.
ALGI.24.3: Identify the graph of an exponential function.
ALGI.24.4: Identify the graph of a line.
ALGI.24.5: Plot points on a coordinate plane from a given table of values. a.
ALGI.24.6: Divide each y-value in a table of values by its successive y-value to determine if the quotients are the same, to prove an exponential function.
ALGI.24.7: Subtract each y-value in a table of values by its successive y-value to determine if the differences are the same, to prove a linear function.
ALGI.24.8: Apply rules for adding, subtracting, multiplying, and dividing integers. b.
ALGI.24.9: Define constant rate of change as slope.
ALGI.24.10: Subtract each y-value in a table of values by its successive y-value to determine if the differences are the same, to prove a linear function.
ALGI.24.11: Recognize the calculated difference is the constant rate of change.
ALGI.24.12: Apply rules for adding, subtracting, multiplying, and dividing integers. c.
ALGI.24.13: Define exponential growth and decay.
ALGI.24.14: Divide each y-value in a table of values by its successive y-value to determine if the quotients are the same, to prove an exponential function.
ALGI.24.15: Apply the rules of multiplication and division of integers.

Students:
Given a contextual situation shown by a graph, a description of a relationship, or two input-output pairs,

• Create a linear or exponential function that models the situation.
• Create arithmetic and geometric sequences from the given situation.
• Justify the equality of the sequences and the functions mathematically and in terms of the original sequence.

Students know:

• That linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
• Properties of arithmetic and geometric sequences.

Students are able to:

• Accurately recognize relationships within data and use that relationship to create a linear or exponential function to model the data of a contextual situation.

Students understand that:

• Linear and exponential functions may be used to model data that is presented as a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
• Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.

Learning Objectives:
ALGI.25.1: Define linear function and exponential function.
ALGI.25.2: Define arithmetic sequence, geometric sequence, and input-output pairs.
ALGI.25.3: Define sequences and recursively-defined sequences.
ALGI.25.4: Recognize that sequences are functions whose domain is the set of all positive integers and zero.
ALGI.25.5: Given a chart, write an equation of a line.
ALGI.25.6: Given a graph, write an equation of a line.
ALGI.25.7: Given two ordered pairs, write an equation of a line.

Students:
Given a quantity increasing exponentially and a quantity increasing as a polynomial function (e.g., linearly, quadratically),

• Construct graphs and tables that demonstrate the exponential function will exceed the polynomial function at some point.
• Present a convincing argument that this must be true for all polynomial functions.

Students know:

• Techniques to graph and create tables for exponential and polynomial functions.

Students are able to:

• Accurately create graphs and tables for exponential and polynomial functions.
• Use the graphs and tables to present a convincing argument that the exponential function eventually exceeds the polynomial function.

Students understand that:

• Exponential functions grow at a faster rate than polynomial functions after some point in their domain.

Learning Objectives:
ALGI.26.1: Define a polynomial function and parabola.
ALGI.26.4: Compare the y-values by looking at the same x-value in a variety of tables or graphs.
ALGI.26.3: Identify the graph of an exponential function.
ALGI.26.4: Identify the graph of a line.
ALGI.26.5: Plot points on a coordinate plane from a given table of values.
ALGI.26.6: Identify the graph of a quadratic function.

Students:
Given a contextual situation that may be modeled by a linear or exponential function,

• Create a function that models the situation.
• Define and justify the parameters (all constants used to define the function) in terms of the original context.

Students know:

• Key components of linear and exponential functions.

Students are able to:

• Communicate the meaning of defining values (parameters and variables) in functions used to model contextual situations in terms of the original context.

Students understand that:

• Sense making in mathematics requires that meaning is attached to every value in a mathematical expression.

Learning Objectives:
ALGI.27.1: Recall the formula of an exponential function.
ALGI.27.2: Recall the slope-intercept form of a linear function.
ALGI.27.3: Define b as growth or decay factor in the context of an exponential problem.
ALGI.27.4: Define k as the initial amount in the context of an exponential problem.
ALGI.27.5: Define m as the rate of change in the context of a linear problem.
ALGI.27.6: Define b as the initial amount in the context of a linear problem.

Students:

• Given a function that models a relationship between two quantities, produce the graph and table of the function and show the key features (intercepts. intervals where the function is increasing, decreasing, positive, or negative. relative maximums and minimums. symmetries. end behavior. and periodicity) that are appropriate for the function.

Given key features from verbal description of a relationship,

• Sketch a graph with the given key features.
• Know periodicity.

Students know:

• Key features of function graphs (i.e., intercepts. intervals where the function is increasing, decreasing, positive, or negative. relative maximums and minimums. symmetries. end behavior. and periodicity).
• Methods of modeling relationships with a graph or table.

Students are able to:

• Accurately graph any relationship.
• Interpret key features of a graph.

Students understand that:

• The relationship between two variables determines the key features that need to be used when interpreting and producing the graph.

Learning Objectives:
ALGI.28.1: Define intercepts, intervals, relative maxima, relative minima, symmetry, end behavior, and periodicity.
ALGI.28.2: For a function that models a relationship between two quantities, find the periodicity.
ALGI.28.3: For a function that models a relationship between two quantities, find the end behavior.
ALGI.28.4: For a function that models a relationship between two quantities, find the symmetry.
ALGI.28.5: For a function that models a relationship between two quantities, find the intervals where the function is increasing, decreasing, positive, or negative.
ALGI.28.6: For a function that models a relationship between two quantities, find the relative maxima and minima.
ALGI.28.7: For a function that models a relationship between two quantities, find the x and y intercepts.

Students:

• Given an interval on a graph or table, calculate the average rate of change within the interval.
• Given a graph of contextual situation, estimate the rate of change between intervals that are appropriate for the summary of the context.

Students know:

• Techniques for graphing.
• Techniques for finding a rate of change over an interval on a table or graph.
• Techniques for estimating a rate of change over an interval on a graph.

Students are able to:

• Calculate rate of change over an interval in a table or graph.
• Estimate a rate of change over an interval on a graph.

Students understand that:

• The average provides information on the overall changes within an interval, not the details within the interval (an average of the endpoints of an interval does not tell you the significant features within the interval).

Learning Objectives:
ALGI.29.1: Identify equivalent ratios.
ALGI.29.2: Define average rate of change as slope.
ALGI.29.3: Estimate the rate of change from a graph (rise/run).
ALGI.29.4: Interpret the average rate of change.
ALGI.29.5: Calculate the average rate of change.
ALGI.29.6: Compute the slope of a line given two ordered pairs.
ALGI.29.7: Identify the slope, given slope-intercept form.

Students:
Given a symbolic representation of a function (including linear, quadratic, absolute value, piecewise-defined functions, and exponential,

• Produce an accurate graph (by hand in simple cases and by technology in more complicated cases) and justify that the graph is an alternate representation of the symbolic function.
• Identify key features of the graph and connect these graphical features to the symbolic function, specifically for special functions:
  quadratic or linear (intercepts, maxima, and minima) and piecewise-defined functions, including step functions and absolute value functions (descriptive features such as the values that are in the range of the function and those that are not).
  1. Exponential (intercepts and end behavior).

Students know:

• Techniques for graphing.
• Key features of graphs of functions.

Students are able to:

• Identify the type of function from the symbolic representation.
• Manipulate expressions to reveal important features for identification in the function.
• Accurately graph any relationship.

Students understand that:

• Key features are different depending on the function.
• Identifying key features of functions aid in graphing and interpreting the function.

Learning Objectives:
ALGI.30.1: Define piecewise-defined functions and step functions.
ALGI.30.2: Graph functions expressed symbolically by hand in simple cases.
ALGI.30.3: Graph functions expressed symbolically using technology for a more complicated case.

a.
ALGI.30.4: Graph quadratic functions showing maxima and minima.
ALGI.30.5: Graph quadratic functions showing intercepts.
ALGI.30.6: Graph linear functions showing intercepts.

b.
ALGI.30.7: Define square root, cube root, and absolute value function.
ALGI.30.8: Graph piecewise-defined functions.
ALGI.30.9: Graph step functions.
ALGI.30.10: Graph cube root functions.
ALGI.30.11: Graph square root functions.
ALGI.30.12: Graph absolute value functions.

c.
ALGI.30.13 Identify exponential numbers as repeated multiplication.
ALGI.30.14 Rewrite exponential numbers as repeated multiplication.

Students:

• Engage in the Mathematical Modeling Cycle (Appendix E) to solve contextual problems involving linear, quadratic, exponential, absolute value and linear piecewise functions.

Students know:

• The Mathematical Modeling Cycle.
• When to use the Mathematical Modeling Cycle to solve problems.

Students are able to:

• Make decisions about problems, evaluate their decisions, and revisit and revise their work.
• Determine solutions to problems that go beyond procedures or prescribed steps.
• Make meaning of problems and their solutions.

Students understand that:

• Mathematical modeling uses mathematics to answer real-world, complex problems.

Note: One does not need to move through the modeling cycle in the same order, aspects of the cycle may be repeated.
The Mathematical Modeling Cycle:

• Define the problem.
• Make assumptions/Define variables.
• Do the math/Get solutions.
• Assess the model and solutions.
• Iterate to refine and extend model.
• Implement and report results.

Students:
By using mathematical and statistical reasoning,

• Draw conclusions about bivariate categorical data.
• Assess risk related to bivariate categorical data

Students know:

• Key features of bivariate categorical data.
• Strategies for drawing conclusions.
• Strategies for assessing risk.

Students are able to:

• Analyze bivariate categorical data,
• Draw conclusions from real-life bivariate categorical data,
• Assess risk given real-life bivariate categorical data.

Students understand that:

• Real-life situations often require drawing conclusions and assessing risk.
• Quantitative literacy is important for making informed decisions.

Learning Objectives:
ALGI.32.1: Define proportional relationships, unit rate, and slope.
ALGI.32.2: Define probability of chance, outcomes and events.
ALGI.32.3: Define bivariate scatter plot, outlier, cluster, linear, nonlinear, and positive and negative association.
ALGI.32.4: Define relative frequency, bivariate, and frequency.
ALGl.32.5: Calculate frequency as it pertains to the data for a two-way table.

Students:

• Design and implement an investigation using the Statistical Problem-Solving Cycle (Appendix E).
• Use elements of the design to determine if there appears to be an association between two categorical variables.
• Write a persuasive argument based on the results of the investigation.

Students know:

• Techniques for designing and conducting an investigation between categorical variables.
• Strategies for determining associations between categorical variables.
• Effective elements of a persuasive argument.

Students are able to:

• Design an investigation related to two categorical variables.
• Carry out their investigation.
• Determine if an association exists between two categorical variables.
• Write an argument persuading readers based on the results of the investigation.

Students understand that:

• Knowledge of the statistical investigation process (Appendix E) gives them the tools to make informed decisions.

Learning Objectives:
ALGI.33.1: Write arguments to support claims with clear reasons and relevant evidence.
ALGI.33.2: Write a persuasive argument to justify the solution.
ALGI.33.3: Introduce claim(s) and organize the reasons and evidence clearly.
AGLI.33.4: Support claim(s) with clear reasons and relevant evidence, using credible sources and demonstrating an understanding of the topic or text.
ALGI.33.5: Summarize numerical data sets in relation to their context.
ALGI.33.6: Infer information from data distributions.

Students:

• Compare quantitative and categorical data.
• Compare the techniques that may be used to analyze quantitative and categorical data.

Students know:

• Characteristics of quantitative data.
• Characteristics of categorical data.
• Techniques for analyzing categorical data.
• Techniques for analyzing quantitative data.

Students are able to:

• Analyze quantitative and categorical data.
• Appropriately summarize categorical data.
• Appropriately summarize categorical data.

Students understand that:

• Methods for summarizing categorical and quantitative data.

Learning Objectives:
ALGl.34.1: Define categorical and quantitative data.
ALGI.34.2: Calculate frequency as it pertains to the data for a two-way table, graph or given context within the problem.
ALGI.34.3: Investigate to determine whether there is an association between two categorical variables.

Students:
Given categorical data for two categories,

• Create two-way frequency tables.
• Create segmented bar graphs.
• Find and interpret relative frequencies using ratios.
• Recognize and justify possible relationships and patterns in the data by examining the joint, marginal, and conditional relative frequencies.

Students know:

• Characteristics of a two-way frequency table.
• Methods for converting frequency tables to relative frequency tables.
• That the sum of the frequencies in a row or a column gives the marginal frequency.
• Techniques for finding conditional relative frequency.
• Techniques for finding the joint frequency in table.
• How to identify possible associations and trends in categorical data.

Students are able to:

• Accurately construct frequency tables and segmented bar graphs.
• Accurately construct relative frequency tables.
• Accurately find the joint, marginal, and conditional relative frequencies.
• Recognize and explain possible associations and trends in the data.

Students understand that:

• Two-way frequency tables may be used to represent categorical data.
• Relative frequency tables show the ratios of the categorical data in terms of joint, marginal, and conditional relative frequencies.
• Two-way frequency or relative frequency tables may be used to aid in recognizing associations and trends in the data.

Learning Objectives:
ALGI.35.1: Define categorical data, two-way frequency table, relative frequency, joint frequency, marginal frequency, and conditional relative frequency.
ALGI.35.2: Recognize possible associations and trends in the data.
ALGI.35.3: Interpret conditional relative frequencies in the context of the data.
ALGI.35.4: Interpret marginal frequencies in the context of the data.
ALGI.35.5: Analyze data from tables.

Students:
Given a situation in which it is meaningful to collect categorical data for two categories,

• Collect data and create two-way categorical tables.
• Analyze two-way categorical tables based on aggregate data from several groups .
• Use a two-way table to find an overall association between two categorical data.
• Examine situations where the association between two variables is reversed when a third variable is introduced.

Students know:

• Techniques for constructing and analyzing two-way frequency tables.
• The impact of considering a third variable on the association of two existing variables.

Students are able to:

• Accurately construct a two-way frequency table.
• Aggregate data from several groups to find an overall association.
• Use Simpson's Paradox.

Students understand that:

• Real-world categorical data can be represented using a two-way table.
• The association between two categorical may be reversed when a third variable is considered.

Learning Objectives:
ALGl.36.1: Define categorical and quantitative data.
ALGI.36.2: Calculate frequency as it pertains to the data for a two-way table, graph or given context within the problem.
ALGI.36.3: Put the data into a two-way categorical table and analyze the data for relationships.
ALGI.36.4: Investigate to determine whether there is an association between two categorical variables.
ALGI.36.5: Recognize possible associations and trends in the data.
ALGI.36.6: Summarize categorical data for two categories in two-way frequency tables.
ALGI.36.7: Analyze data from tables.

Students:
Given scenarios involving chance,

• Determine the sample space and a variety of simple and compound events that may be defined from the sample space.
• Use the language of union, intersection, and complement appropriately to define events.

Students know:

• Methods for describing events from a sample space using set language (subset, union, intersection, complement).

Students are able to:

• Interpret the given information in the problem.
• Accurately determine the probability of the scenario.

Students understand that:

• Set language can be useful to define events in a probability situation and to symbolize relationships of events.

Learning Objectives:
ALGI.37.1: Define sample, validity, population, inference, random sampling, statistic, and generalization.
ALGI.37.2: Identify the nature of the attribute, how it was measured, and its unit of measure.
ALGI.37.3: Discuss real-world examples of valid sampling and generalizations.
ALGI.37.4: Compare sample size with population to check for validity.
ALGI.37.5: Analyze attributes of sample size.
ALGI.37.6: Differentiate between appropriate sampling methods.
ALGI.37.7: Explain the validity of random sampling.
ALGI.37.8: Given a contextual situation, interpret and defend the solution in the context of the original problem.

Students:
Given two-way tables or tree diagrams,

• Explain the meaning of independence from a formula perspective P(A and B) = P(A) x P(B) and from the intuitive notion that P(A) occurring has no impact on whether P(B) occurs or not.
• Compare these two interpretations within the context of the scenario.

Students know:

• Methods to find probability of simple and compound events.

Students are able to:

• Interpret the given information in the problem.
• Accurately determine the probability of simple and compound events.
• Accurately calculate the product of the probabilities of two events.

Students understand that:

• Events are independent if one occurring does not affect the probability of the other occurring, and that this may be demonstrated mathematically by showing the truth of P(A and B) = P(A) x P(B).

Learning Objectives:
ALGI.38.1: Define probability, ratio, simple event, compound event, and independent event.
ALGI.38.2: Determine the probability of a compound event.
ALGI.38.3: Determine the probability of an independent event.
ALGI.38.4: Determine the probability of a simple event by expressing the probability as a ratio, percent, or decimal.
ALGI.38.5: Identify the probability of an event that is certain as 1 or impossible as 0.
ALGI.38.6: Solve word problems involving probability.
ALGI.38.7: Use proportional relationships to solve multi-step ratio and percent problems.
ALGI.38.8: Recognize and represent proportional relationships as ratios between two quantities.

Students:

• Using two-way tables or tree diagrams, compute the probability of event A given event B.

Students know:

• Methods to find probability using two-way tables or tree diagrams.
• Techniques to find conditional probability.

Students are able to:

• Accurately determine the probability of events from a two-way table or tree diagram.

Students understand that:

• The independence of two events is determined by the effect that one event has on the outcome of another event.
• The occurrence of one event may or may not influence the likelihood that another event occurs.

Learning Objectives:
ALGI.39.1: Define likelihood, probability and event.
ALGI.39.2: Construct a graphic representation of all outcomes (probability model-tree diagram, organized list, table, etc.).
ALGI.39.3: Compare and contrast probability of chance and probability of observed frequency.
ALGI.39.4: Write the probability as a fraction, with likely outcomes as the numerator and possible outcomes as the denominator.

Students:
Given a contextual situation and scenarios involving two events,

• Explain the meaning of independence from a formula perspective P(A∩B) = P(A) x P(B) and from the intuitive notion that A occurring has no impact on whether B occurs or not.
• Compare these two interpretations within the context of the scenario.

Students know:

• Possible relationships and differences between the simple probability of an event and the probability of an event under a condition.

Students are able to:

• Accurately determine the probability of simple and compound events.
• Accurately determine the conditional probability P(A given B) from a sample space or from the knowledge of P(A and B) and the P(B).

Students understand that:

• Conditional probability is the probability of an event occurring given that another event has occurred.

Learning Objectives:
ALGI.40.1: Define probability using everyday language.
ALGI.40.2: Compare and contrast probability of chance and probability of observed frequency.
ALGI.40.3: Explain the difference between possible outcomes and likely outcomes.
ALGI.40.4: Cite evidence to support analysis of what the data says explicitly as well as inferences drawn from the data.

Students:
Given a contextual situation consisting of two events,

• Determine the probability of each individual event, then limit the sample space to those outcomes where B has occurred and calculate the probability of A compare the P(A)and the P(A given B), and explain the equality or difference in the original context of the problem.
• Determine the probability of each individual event, then limit the sample space to those outcomes where B has occurred and calculate the P(A and B), compare the ratio of P(A and B) and P(B) to P(A given B), and explain the equality or difference in the original context of the problem.

Students know:

• Possible relationships and differences between the simple probability of an event and the probability of an event under a condition.

Students are able to:

• Accurately determine the probability of simple and compound events.
• Accurately determine the conditional probability P(A given B) from a sample space or from the knowledge of P(A and B) and the P(B).

Students understand that:

• Conditional probability is the probability of an event occurring given that another event has occurred.

Learning Objectives:
ALGI.41.1: Define likelihood, probability and event.
ALGI.41.2: Construct a graphic representation of all outcomes (probability model-tree diagram, organized list, table, etc.).
ALGI.41.3: Compare and contrast probability of chance and probability of observed frequency.
ALGI.41.4: Write the probability as a fraction, with likely outcomes as the numerator and possible outcomes as the denominator.
ALGI.41.5: Explain the difference between possible outcomes and likely outcomes.
ALGI.41.6: Cite evidence to support analysis of what the data says explicitly as well as inferences drawn from the data.