Courses of Study : Mathematics

Students:

• Produce equivalent expressions in the form a + bi, where a and b are real for combinations of complex numbers by using addition, subtraction, and multiplication and justify that these expressions are equivalent through the use of properties of operations and equality

Students know:

• Combinations of operations on complex number that produce equivalent expressions.
• Properties of operations and equality that verify this equivalence.

Students are able to:

• Perform arithmetic manipulations on complex numbers to produce equivalent expressions.

Students understand that:

• Complex number calculations follow the same rules of arithmetic as combining real numbers and algebraic expressions.

Learning Objectives:
ALGII.1.1: Define real and complex numbers, commutative, associative, and distributive properties.
ALGII.1.2: Apply commutative, associative, and distributive properties to using multiplication with complex numbers.
ALGII.1.3: Apply commutative, associative, and distributive properties to using addition and subtraction with complex numbers.
ALGII.1.4: Use commutative, associative, and distributive properties.
ALGII.1.5: Identify imaginary numbers.

Students:

• Given a contextual situation, represent the data in a matrix and interpret the value of each entry.

Students know:

• The aspects of a matrix with regard to entries, rows, columns, dimensions, elements, and subscript notations.

Students are able to:

• Translate data into a matrix.

Students understand that:

• A matrix is a tool that can help to organize, manipulate, and interpret data.

Learning Objectives:
ALGII.2.1: Define matrix.
ALGII.2.2: Organize data into a matrix using rows and columns.

Students:

• Given a matrix, use scalar multiplication to produce a new matrix and interpret the value of the new entries.

Students know:

• The aspects of a matrix with regard to entries, rows, columns, dimensions, elements, and subscript notations.

Students are able to:

• Write percents as decimals.
• Increase or decrease an amount by multiplying by a percent (i.e., increase of 10% would multiply by 1.1).

Students understand:

• Multiplying a matrix by a scalar affects every element in the matrix equally.
• Scalar multiplication is a tool that allows all elements of a matrix to be changed in a simple manner.

Learning Objectives:
ALGII.3.1: Define scalar.
ALGII.3.2: Multiply a matrix by a scalar.

Students:
Given two matrices,

• Determine whether arithmetic operations (add, subtract, multiply) are defined.
• Perform arithmetic (add, subtract, multiply) operations to form new matrices.

Students know:

• The aspects of a matrix with regard to entries, rows, columns, dimensions, elements, and subscript notations.

Students are able to:

• Strategically choose and apply appropriate representations of matrices on which arithmetic operations can be performed.

Students understand that:

• Matrix addition and subtraction may be performed only if the matrices have the same dimensions.
• Matrix multiplication can be performed only when the number of columns in the first matrix equal the number of rows in the second matrix.
• There are many contextual situations where arithmetic operations on matrices allow us to solve problems.

Learning Objectives:
ALGII.4.1: Multiply matrices of appropriate dimensions.
ALGII.4.2: Subtract matrices of appropriate dimensions.
ALGII.4.3: Add matrices of appropriate dimensions.

Students:
Given a matrix,

• Add the zero matrix to show that the matrix does not change.
• Multiply by the identity matrix to show that the matrix does not change.
  
  Given a square matrix,
  • Find the determinant.
  • If the determinant is non-zero, find the multiplicative inverse.
  • Find the multiplicative inverse, if it is defined, and show that the determinant is not zero.
  • If the multiplicative inverse is not defined, show that the determinant is equal to zero

Students know:

• The additive and multiplicative identity properties for real numbers.
• The aspects of the zero and identity matrices.
• A matrix multiplied by its multiplicative inverse equals the identity matrix.

Students are able to:

• Find the determinant of a square matrix.
• Find the multiplicative inverse of a square matrix.
• Add and multiply matrices.

Students understand that:

• Identity properties that apply to other number systems apply to matrices.
• The multiplicative inverse property that applies to other number systems applies to matrices.
• A matrix with a determinant equal to zero does not have a multiplicative inverse analogous to zero in the real number system not having a multiplicative inverse.
• Division by zero in the real number system is undefined.

Learning Objectives:
ALGII.5.1: Define zero matrix and identity matrix.
ALGII.5.2: Multiply an identity matrix by any other matrix will result in the other matrix.
ALGII.5.3: Know that identity matrices have a diagonal of 1's, starting at the top left-hand corner and going down. All other entries are zeroes.
ALGII.5.4: Recognize that when the zero matrix is added to any other matrix, the sum is the other matrix.

Students:
Given any polynomial,

• Analyze and determine if suitable factorizations exist,
• Use the roots determined by these factorizations to determine the zeros of the function.

Students know:

• Common factoring techniques.
• When a factorization of a polynomial reveals a root of that polynomial.
• When a rearrangement of the terms of a polynomial expression can reveal a recognizable factorable form of the polynomial.
• Relationships of roots to points on the graph of the polynomial.

Students are able to:

• Use techniques for factoring polynomials.
• Use factors of polynomials to find zeros.

Students understand that:

• Important features of the graph of a polynomial can be revealed by its zeros and by inputting values between the identified roots of the given polynomial.

Learning Objectives:
ALGII.6.1: Define factor, monomial, binominal, trinomial, polynomial, quadratic expression and zero product property.
ALGII.6.2: Factor a quadratic expression (Greatest Common Factor, completing the square, difference of two squares, perfect square trinomials, and quadratic formula).
ALGII.6.3: Use the zero-product property to reveal the zeros in the function.
ALGII.6.4: Solve a two-step equation.
ALGII.6.5: Solve a one-step equation. ALGII 6.6: Determine the Greatest Common Factor (GCF).

Students:

• Use properties of operations on polynomials to justify identities, such as
  1. (a+b)²=a²+2ab+b²
  2. (a+b)(c+d)=ac+ad+bc+bd
  3. A²-b²=(a+b)(a-b)
  4. x²+(a+b)x+ab=(x+a)(x+b)
  5. (x²+y² )²=(x²-y² )²+(2xy)²
• Use these identities to describe numerical relationships (e.g., identity 3 can be used to mentally compute 79 x 81, or identity 5 can be used as a generator for Pythagorean triples).

Students know:

• Distributive Property of multiplication over addition.

Students are able to:

• Accurately perform algebraic manipulations on polynomial expressions.

Students understand that:

• Reasoning with abstract polynomial expressions reveals the underlying structure of the Real Number System.
• Justification of generalizations is necessary before using these generalizations in applied settings.

Learning Objectives:
ALGII.7.1: Define polynomial identities.
ALGII.7.2: Identify the polynomial identities used to manipulate numerical relationships.

Students:

• Solve problems involving radical equations in one variable.
• 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 radical equations.
• Conditions under which a solution is considered extraneous.

Students are able to:

• Explain with mathematical and reasoning from the context (when appropriate) why a particular solution is or is not extraneous.

Students understand that:

• Values which arise from solving 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:
ALGII.8.1: Define rational, irrational, and radical expressions and extraneous solutions.
ALGII.8.2: Simplify rational and radical equations.
ALGII.8.3: Apply properties of exponents.
ALGII.8.4: Evaluate solutions by substituting into the original equation.

Students:
Given a contextual situation involving exponential growth or decay,

• Develop an exponential function which models the situation.
• Rewrite the exponential function as an equivalent logarithmic function.
• Use logarithmic properties to rearrange the logarithmic function, to isolate the variable, and use technology to find an approximation of the solution.

Students know:

• Methods for using exponential and logarithmic properties to solve equations.
• Techniques for rewriting algebraic expressions using properties of equality.

Students are able to:

• Accurately use logarithmic properties to rewrite and solve an exponential equation.
• Use technology to approximate a logarithm.

Students understand that:

• Logarithmic and exponential functions are inverses of each other, and may be used interchangeably to aid in the solution of problems.

Learning Objectives:
ALGII.9.1: Define logarithmic and exponential function.
ALGII.9.2: Recognize the inverse relationship of logarithmic function and exponential functions.
ALGII.9.3: Calculate the change of base formula for logarithms.
ALGII.9.4: Apply the properties of logarithms.
ALGII.9.5: Discuss the appropriateness of the solution.
ALGII.9.6: Recall laws of exponents.

Students:

• Given a contextual situation that may include polynomial, exponential, logarithmic, trigonometric (sine and cosine), radical, and piecewise 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 polynomial, exponential, logarithmic, trigonometric, radical, or piecewise functional relationship.

Students are able to:

• Write equations or inequalities 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:
ALGII.10.1: Define equation, expression, variable, equality, and inequality.
ALGII.10.2: Create inequalities with one variable.
ALGII.10.3: Create equations with one variable.
ALGII.10.4: Solve two-step equations and inequalities.
ALGII.10.5: Solve one-step equations and inequalities.
ALGII.10.6: Compare and contrast equations and inequalities.
ALGII.10.7: Recognize inequality symbols including, £, and ³.

Students:

• Given a contextual situation in which a quadratic solution is necessary find all solutions real or complex.

Students know:

• strategies for solving quadratic equations

Students are able to:

• apply the quadratic equation.
• provide solutions in complex form.

Students understand that:

• all quadratic equations have two solutions: real or imaginary.
• Some contextual situations are better suited to quadratic solutions.

Learning Objectives:
ALGII.11.1: Solve quadratic equations with real coefficients that have simple solutions.
ALGII.11.2: Review quadratic formula, completing the square, and factoring.
ALGII.11.3: Review the zero-product property.

Students:
Given a contextual situation,

• Develop an appropriate model for the situation (exponential, radical, logarithmic, or trigonometric).
• Rewrite the function as an equivalent inverse function.
• Use algebraic properties to rearrange the function, to isolate the variable, and to find a solution (using technology as needed).

Students know:

• Techniques for rewriting algebraic expressions using properties of equality.
• Methods for solving exponential, logarithmic, radical, and trigonometric equations.

Students are able to:

• Accurately use properties of inverse to rewrite and solve an exponential, logarithmic, radical, or trigonometric equation.
• Use technology to approximate solutions to equation, if necessary.

Students understand that:

• The inverse of exponential, logarithmic, radical, and trigonometric functions may be used to aid in the solution of problems.

Learning Objectives:
ALGII.12.1: Define function, function notation, linear, polynomial, rational, radical, absolute value, exponential, and logarithmic functions, and transitive property.
ALGII.12.2: Solve an equation of the form f(x) = c for a simple linear function f that has an inverse.
ALGII.12.3: Write an expression for the inverse of a simple linear function f of the form f(x) = c.
ALGII.12.4: Apply the properties of multiplicative inverses.
ALGII.12.5: Apply the properties of exponentials.
ALGII.12.6: Apply the substitution principle.
ALGII.12.7: Solve a multi-step equation.

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 and use them to make predictions.

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, using it to make predictions.

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 which is useful in making predictions.

Learning Objectives:
ALGII.13.1: Define ordered pair, coordinate plane, polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
ALGII.13.2: Create equations with two variables (polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions).
ALGII.13.3: Graph equations on coordinate axes with labels and scales (polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.).
ALGII.13.4: Identify an ordered pair and plot it on the coordinate plane.

Students:
Given two functions (linear, polynomial, rational, absolute value, exponential, logarithmic, radical, trigonometric and piecewise) 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, rational, absolute value, exponential, logarithmic graphs, radical, trigonometric (sine and cosine), and general piecewise functions.
• Methods to use technology, tables, and successive approximations to produce graphs and tables for linear, polynomial, rational, absolute value, exponential, logarithmic, radical, trigonometric (sine and cosine), and general piecewise 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, polynomial, rational, absolute value, exponential, logarithmic, radical, trigonometric (sine and cosine) and general piecewise functions.
• Accurately use technology to approximate solutions on graphs.

Students understand that:

• When two functions are equal, the x coordinate(s) of the intersection of those functions is the value that produces the same output (y-value) for both functions.
• Technology is useful to quickly and accurately determine solutions and produce graphs of functions.

Learning Objectives:
ALGII.14.1: Define function, function notation, linear, polynomial, trigonometric (sine and cosine), rational, radical, absolute value, exponential, and logarithmic functions, general piecewise functions, and transitive property.
ALGII.14.2: Explain, using the transitive property, why the x-coordinates of the points of the graphs are solutions to the equations.
ALGII.14.3: Find solutions to the equations y = f(x) and y = g(x) using graphing technology.
ALGII.14.4: Solve equations for y.
ALGII.14.5: Apply the properties of multiplicative inverses.
ALGII.14.6: Apply the properties of exponents.
ALGII.14.7: Demonstrate use of a graphing technology, including using a table, making a graph, and finding successive approximations.

Students:
Given two functions represented differently (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:
ALGII.15.1: Compare properties of two functions each represented in a different way.
ALGII.15.2: Identify properties of functions defined algebraically.
ALGII.15.3: Identify properties of functions defined by verbal description.
ALGII.15.4: Identify properties of functions defined graphically.
ALGII.15.5: Identify properties of functions defined numerically in tables.

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.
  • Recognize if a function is even or odd.

Students know:

• Graphing techniques of functions.
• Methods of using technology to graph functions.
• Techniques to identify even and odd functions both algebraically and from a graph.

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:
ALGII.16.1: Recognize even and odd functions from algebraic expressions for them.
ALGII.16.2: Recognize even and odd functions from their graphs.
ALGII.16.3: Experiment with various cases of functions and illustrate an explanation of the effects on the graph using technology.
ALGII.16.4: Find the value of k given the graphs of f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k.
ALGII.16.5: Identify the effect on the graph of replacing f(x) by k f(x) and f(kx) for specific values of k.
ALGII.16.6: Identify the effect on the graph of replacing f(x) by f(x) + k and f(x +k) for specific values of k.

Students:
Given a symbolic representation of a function (including polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise),

• 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.
• Recognize if a function is even or odd.

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.
• Determine when a function is even or odd.

Students understand that:

• Key features are different depending on the function.
• Identifying key features of functions aid in graphing and interpreting the function.
• Even and odd functions may be identified from a graph or algebraic form of a function.

Learning Objectives:
ALGII.17.1: Define intercepts, intervals, relative maxima, relative minima, symmetry, end behavior, and periodicity.
ALGII.17.2: For a function that models a relationship between two quantities, find the periodicity.
ALGII.17.3: For a function that models a relationship between two quantities, find the end behavior.
ALGII.17.4: For a function that models a relationship between two quantities, find the symmetry.
ALGII.17.5: For a function that models a relationship between two quantities, find the intervals where the function is increasing, decreasing, positive, or negative.
ALGII.17.6: For a function that models a relationship between two quantities, find the relative maxima and minima.
ALGII.17.7: For a function that models a relationship between two quantities, find the x- and y-intercepts.

Students:

• Given a contextual situation that is functional, model the situation with a graph and construct the graph based on the parameters given in the domain of the context.

Students know:

• Techniques for graphing functions.
• Techniques for determining the domain of a function from its context.

Students are able to:

• Interpret the domain from the context.
• Produce a graph of a function based on the context given.

Students understand that:

• Different contexts produce different domains and graphs.
• Function notation in itself may produce graph points which should not be in the graph as the domain is limited by the context.

Learning Objectives:
ALGII.18.1: Define domain, range, relation, function, table of values, and mappings.
ALGII.18.2: Determine the appropriate domain for a given function.
ALGII.18.3: Identify functions from information in tables, sets of ordered pairs, and mappings.
ALGII.18.4: Translate verbal phrases into a function.
ALGII.18.5: Arrange data given as ordered pairs into a table and a table of values into ordered pairs.
ALGII.18.6: Identify the x- and y-values in an ordered pair.

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 on 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:
ALGII.19.1: Define average rate of change as slope.
ALGII.19.2: Estimate the rate of change from a graph (rise/run).
ALGII.19.3: Interpret the average rate of change.
ALGII.19.4: Calculate the average rate of change.
ALGII.19.5: Compute the slope of a line given two ordered pairs.
ALGII.19.6: Identify the slope, given slope-intercept form.

Students:

• Given a symbolic representation of a function (including polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, and radical), 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.

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.
• Find the inverse of a function algebraically and/or graphically.

Students understand that:

• Key features are different depending on the function.
• Identifying key features of functions aid in graphing and interpreting the function.
• A function and its inverse are reflections over the line y = x.

Learning Objectives:
ALGII.20.1: Graph functions expressed symbolically by hand in simple cases.
ALGII.20.2: Graph functions expressed symbolically using technology for more complicated cases.
ALGII.20.3: Solve polynomial function for their zeros.
ALGII.20.4: Plot the zeros on a coordinate plane.
ALGII.20.5: Recognize end behavior on a graph.
ALGII.20.6: Review multiplicity of zeros.
ALGII.20.7: Graph trigonometric functions showing period, midline, and amplitude.
ALGII.20.8: Graph logarithmic functions showing intercepts and end behavior.
ALGII.20.9: Graph reciprocal functions, identifying horizontal and vertical asymptotes.
ALGII.20.10: Define square root and cube root.
ALGII.20.11: Graph cube root functions.
ALGII.20.12: Graph square root functions.
ALGII.20.13: Define exponential function, logarithmic function, trigonometric function, intercepts, end behavior, period, midline, and amplitude.
ALGII.20.14: Graph exponential functions showing intercepts and end behavior.

Students:

• Given a unit circle and an angle in radians in the first quadrant of the coordinate plane, use right triangles and the definitions of the trigonometric functions to find the trigonometric ratios for the angle in radian measure when the angle is defined on the unit circle.
• Given a unit circle and an angle measured in radians traversed counterclockwise around the circle in any quadrant, determine a reference angle (smallest angle from the x-axis to the terminal ray of the original angle), and use congruent triangles and trigonometric ratios of the reference angle to define trigonometric ratios for angles of any real number size with an adjustment in sign.

Students know:

• Trigonometric ratios for right triangles.
• The appropriate sign for coordinate values in each quadrant of a coordinate graph.

Students are able to:

• Accurately find relationships of trigonometric functions for an acute angle of a right triangle to measures within the unit circle.
• Justify triangle similarity.
• Find the reference angle for any angle found by a revolution on a ray in the coordinate plane.
• Relate the trigonometric ratios for the reference angle to those of the original angle.
• Determine the appropriate sign for trigonometric functions of angles of any given size.

Students understand that:

• Trigonometric functions may be extended to all real numbers from being defined only for acute angles in right triangles by using the unit circle, reflections, and logical reasoning.

Learning Objectives:
ALGII.21.1: Define unit circle, trigonometric functions, periodic functions and radians.
ALGII.21.2: Apply special right triangles to trigonometric ratios.
ALGII.21.3: Demonstrate periodicity of trigonometric functions.
ALGII.21.4: Recall Law of Sines and Law of Cosines.
ALGII.21.4: Recall Pythagorean Theorem.

Students:
Given a contextual situation that may include polynomial, exponential, logarithmic, trigonometric (sine and cosine), radical, and piecewise functional relationships in one variable,

• Model the relationship with equations and solve the problem presented in the contextual situation for the given variable.
• Interpret solutions in the context of the problem.

Students know:

• When the situation presented in a contextual problem is most accurately modeled by a polynomial, exponential, logarithmic, trigonometric (sine and cosine), radical, or general piecewise functional relationship.

Students are able to:

• Accurately model contextual situations.

Students understand that:

• There are relationships among features of a contextual problem and a created mathematical model for that problem.
• Different contexts produce different domains and feasible solutions.

Learning Objectives:
ALGII.22.1: Define the real-world problem. (i.e., what is the problem asking).
ALGII.22.2: Make assumptions and define the variables (independent, dependent).
ALGII.22.3: Assess the model and identify which function will be used (i.e.; polynomial, trigonometric (sine and cosine), logarithmic, radical and general piecewise functions).
ALGII.22.4: Find the solution.
ALGII.22.5: Interpret the results.

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:

• Use mathematical reasoning about normal distributions.
• Draw conclusions and assess risk based on this reasoning.

Students know:

• Properties of a normal distribution.
• Empirical Rule

Students are able to:

• Draw accurate conclusions and assess risk using their knowledge of the normal distribution.

Students understand that:

• For a normal distribution, nearly all of the data will fall within three standard deviations of the mean.
• The empirical rule can be broken down into three parts: 68% of data falls within the first standard deviation from the mean, 95% fall within two standard deviations. and 99.7% fall within three standard deviations.

Learning Objectives:
ALGII.23.1: Define margin of error and confidence interval.
ALGII.23.2: Justify the mathematical and statistical reasoning.

Students:

• Design experiments or surveys.
• Write persuasive arguments.

Students know:

• Techniques to design an experiment or survey

Students are able to:

• Develop a statistical question.
• Design and carry out an experiment or survey.
• Accurately interpret the results of an experiment or survey.

Students understand that:

• A statistical question is one that can be answered by collecting data and where there will be variability in that data.
• An experiment is a controlled study in which the researcher attempts to understand cause-and-effect relationships. Based on the analysis, the researcher draws a conclusion about whether the treatment ( independent variable ) had a causal effect on the dependent variable.
• Statistical surveys are collections of information about items in a population and may be grouped into numerical and categorical types.

Learning Objectives:
ALGII.24.1: Determine your question of interest.
ALGII.24.2: Design your study (experiment, survey, etc.).
ALGII.24.3: Collect data.
ALGII.24.4: Analyze results.
ALGII.24.5: Interpret results.
ALGII.24.6: Develop an informal persuasive argument.

OBJECTIVES FOLLOW THE STEPS OF THE STATISTICAL PROBLEM-SOLVING CYCLE.

Students:
Given a set of data,

• Find the mean and standard deviation of the set and use them to fit data to a normal distribution (when appropriate) to estimate population percentages,
• Estimate areas under the normal curve using calculators, spreadsheets, and standard normal distribution tables.

Students know:
From a normal distribution,

• Techniques to find the mean and standard deviation of data sets using technology.
• Techniques to use calculators, spreadsheets, and standard normal distribution tables to estimate areas under the normal curve.

Students are able to:

• From a normal distribution, accurately find the mean and standard deviation of data sets using technology.
• Make reasonable estimates of population percentages from a normal distribution.
• Read and use normal distribution tables and use calculators and spreadsheets to accurately estimate the areas under a normal curve.

Students understand that:
Under appropriate conditions,

• The mean and standard deviation of a data set can be used to fit the data set to a normal distribution.
• Population percentages can be estimated by areas under the normal curve using calculators, spreadsheets, and standard normal distribution tables.

Learning Objectives:
ALGII.25.1: Define normal distribution, mean, standard deviation, and empirical rule.
ALGII.25.2: Use technology to calculate mean and standard deviation.
ALGII.25.3: Use technology (ex. calculator, Microsoft Excel, etc.) to estimate areas under the normal curve.
ALGII 25.4: Analyze data sets to determine if appropriate.

Students:
Given a scenario in which a statistical question needs to be investigated,

• Select an appropriate method of data collection (sample surveys, experiments, and observational studies) and justify the selection.
• Show randomization leads to more accurate inferences to the population.

Students know:

• Key components of sample surveys, experiments, and observational studies.
• Procedures for selecting random samples.

Students are able to:

• Use key characteristics of sample surveys, experiments, and observational studies to select the appropriate technique for a particular statistical investigation.

Students understand that:

• Sample surveys, experiments, and observational studies may be used to make inferences made about the population.
• Randomization is used to reduce bias in statistical procedures.

Learning Objectives:
ALGII.26.1: Define sample surveys, experiments, randomization and observational studies.

Students:
Given a statistical question about a population,

• Describe and justify a data collection process that would result in a random sample from which inferences about the population can be drawn.
• Explain and justify their reasoning concerning data collection processes that do not allow generalizations (e.g., non-random samples) from the sample to the population.

Students know:

• Techniques for selecting random samples from a population.

Students are able to:

• Accurately compute the statistics needed.
• Recognize if a sample is random.
• Reach accurate conclusions regarding the population from the sample.

Students understand that:

• Statistics generated from an appropriate sample are used to make inferences about the population.

Learning Objectives:
ALGII.27.1: Define statistic, parameter, statistical process, and random sample.

Students:

• Describe differences in randomly selecting samples and randomly assigning subjects to experimental treatment.
• Explain how these differences effect inferences drawn regarding the population in regards to cause and effect.

Students know:

• Techniques for selecting random samples from a population.
• Techniques for randomly assigning subjects to experimental treatment groups.

Students are able to:

• Recognize if a sample is random.
• Reach accurate conclusions regarding the population from the sample.
• Reach accurate conclusions regarding the cause and effect of an experimental treatment.

Students understand that:

• Random selection is essential to external validity, or the extent to which the researcher can generalize the results of the study to the larger population.
• Random assignment is central to internal validity, which allows the researcher to make causal claims about the effect of the treatment.

Learning Objectives:
ALGII.28.1: Define random selecting, random assigning, experimental treatment group, and control group.
ALGII.28.2: Use data from a random sample to make an inference about a population.
ALGII.28.3: Distinguish between random selecting and random assigning and between control group and experimental treatment group.

Students:

• Explain consequences that arise from non-randomized assignment to experimental groups.

Students know:

• Differences between random and non-random assignment.
• The definition of independent and dependent variables.

Students are able to:

• Conclude whether a causal relationship exists in an experiment based on the type of assignment of subjects in the experiment.
• Identify uncontrolled variable that may be responsible for observed difference when non-randomized assignment is used in an experiment.

Students understand that:

• Uncontrolled variables are characteristic factors that are not regulated or measured by the investigator during an experiment or study, so that they are not the same for all participants in the research.
• Randomized selection of subjects to groups in experiments is the only type of study able to establish causation.

Learning Objectives:
ALGII.29.1: Define non-randomized assignment and categorical outcomes.
ALGII.29.2: Analyze the data and explain the outcome.

Students:
Given the description of a survey,

• Evaluate bias.
• Determine if results are representative of a population.

Students know:

• Techniques for conducting surveys.
• Techniques to identify bias

Students are able to:

• Given the description of a survey,
  • Evaluate bias that may occur in the survey.
  • Determine whether a bias precludes results of the survey from being generalized to the population.

Students understand that:

• Bias is the intentional or unintentional favoring of one group or outcome over other potential groups or outcomes in the population.
• A common cause of sampling bias lies in the design of the study or in the data collection procedure, both of which may favor or disfavor collecting data from certain classes or individuals or in certain conditions.
• Response bias (also called survey bias) is the tendency of a person to answer questions on a survey untruthfully or misleadingly.
• Nonresponse bias is the bias that results when respondents differ in meaningful ways from nonrespondents.

Learning Objectives:
ALGII.30.1: Define bias (sampling, response, or nonresponse bias).
ALGII.30.2: Interpret survey results.
ALGII.30.3: Determine where bias may occur.

Students:

• Simulate sampling distribution.
• Evaluate effects of sample size on variability in sampling distribution (mean and standard deviation).

Students know:

• Techniques to find the mean and standard deviation.

Students are able to:

• Accurately compute the statistics needed.
• Reach accurate conclusions regarding the population from the the sampling distribution of a sample statistic.

Students understand that:

• The center is not affected by sample size. The mean of the sample means is always approximately the same as the population mean.
• As the sample size increases, the standard deviation of the means decreases. and as the sample size decreases, the standard deviation of the sample means increases.

Learning Objectives:
ALGII.31.1: Define data, random sampling, population, variation, prediction, estimation, standard deviation and inference.
ALGII.31.2: Calculate standard deviation of the samples.
ALGII.31.3: Compare and contrast the random sampling data to the population.
ALGII.31.4: Predict an outcome of the entire population based on random samplings.
ALGII.31.5: Collect data from population randomly, choosing same size samples.

Students:

• Produce sampling distribution, using technology to generate a large number of samples.
• Verify, that when sample size is large, sampling distribution is centered at population mean and approximately normal.
• Create a 95% confidence interval.
• Interpret a confidence interval.

Students know:

• Techniques for producing a sampling distribution.
• Properties of a normal distribution.

Students are able to:

• Produce a sampling distribution.
• Reach accurate conclusions regarding the population from the sampling distribution.
• Accurately create and interpret a confidence interval based on observations from the sampling distribution.

Students understand that:

• The central limit theorem in statistics states that, given a sufficiently large sample size, the sampling distribution of the mean for a variable will approximate a normal distribution regardless of that variable's distribution in the population.
• A 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population. With large samples, you know that mean with much more precision than you do with a small sample, so the confidence interval is quite narrow when computed from a large sample.

Learning Objectives:
ALGII.32.1: Define data, random sampling, population, variation, prediction, estimation, normal, empirical rule, standard deviation and inference.
ALGII.32.2: Calculate standard deviation of the samples by hand and using technology to justify the empirical rule.
ALGII.32.3: Predict an outcome of the entire population based on random samplings.
ALGII.32.4: Collect data from population randomly, choosing same size samples.

Students:
Given a scenario in which a statistical question may be investigated by a randomized experiment,

• Design and conduct a randomized experiment to evaluate differences in two treatments based on data from randomized experiments.
• Interpret and explain the results in the context of the original scenario.
• Design, conduct, and use simulations to generate data simulating application of the two treatments.
• Use results of the simulation to evaluate significance of differences in the parameters of interest.

Students know:

• Techniques for conducting randomized experiments.
• Techniques for conducting simulations of randomized experiment situations.

Students are able to:

• Design and conduct randomized experiments with two treatments.
• Draw conclusions from comparisons of the data of the randomized experiment.
• Design, conduct, and use the results from simulations of a randomized experiment situation to evaluate the significance of the identified differences.

Students understand that:

• Differences of two treatments can be justified by a significant difference of parameters from a randomized experiment.
• Statistical analysis and data displays often reveal patterns in data or populations, enabling predictions.

Learning Objectives:
ALGII.33.1: Define randomized experiment, simulation, and parameter.
ALGII.33.2: Determine if differences in two parameters are significant.

Students:
Given a unit circle and an angle that is defined in terms of a fractional part of a revolution,

• Use the definition of one radian as the measure of the central angle of a unit circle which subtends (cuts off) an arc of length one to determine measures of other central angles as a fraction of a complete revolution (2π for the unit circle).
• Create a circle in the coordinate plane other than a unit circle, and show that an arc equal in length to the radius defines a triangle inside the circle, similar to one in the unit circle for an arc of length one, so the angle must have the same measure.

Students know:

• The circumference of any circle is 2πr and therefore, the circumference of a unit circle is 2π.

Students are able to:

• Translate between arc length and central angle measures in circles.

Students understand that:

• Radians measure angles as a ratio of the arc length to the radius.
• The unit circle has a circumference of 2π which aids in sense making for angle measure as revolutions (one whole revolution measures 2π radians) regardless of radius.
• Use of the unit circle gives a one-to-one ratio between arc length and the measure of the central angle, putting the angle in direct proportion to the arc length, and that the circle can then be divided up to find the radian measure of other angles.

Learning Objectives:
ALGII.34.1: Define arc length, radian measure, and sector.
ALGII.34.2: Prove the length of the arc intercepted by an angle is proportional to the radius by similarity.
ALGII.34.3: Discuss the relationship between arc length and angles.
ALGII.34.4: Apply the arc length formula.

Students:
Given a contextual situation of a periodic phenomenon that may be modeled by a trigonometric function,

• Create a trigonometric function to model the phenomena.
• Use features such as the specified amplitude, frequency, and midline of the function to justify the model.

Students know:

• Key features of trigonometric functions (e.g., amplitude, frequency, and midline).
• Techniques for selecting functions to model periodic phenomena.

Students are able to:

• Determine the amplitude, frequency, and midline of a trigonometric function.
• Develop a trigonometric function to model periodic phenomena.

Students understand that:

• Trigonometric functions are periodic and may be used to model certain periodic contextual phenomena.
• Amplitude, frequency, and midline are useful in determining the fit of the function used to model the phenomena.

Learning Objectives:
ALGII.35.1: Define amplitude, frequency, period, vertical and horizontal translation, and midline.
ALGII.35.2: Calculate amplitude, frequency, period, vertical and horizontal translations, and midline from given data.
ALGII.35.3: Graph the trigonometric function (sine and cosine) that model periodic phenomena.
ALGII.35.4: Graph the sine and cosine parent functions.

Students:

• Given a right triangle, use the Pythagorean Theorem and the extensions of the trigonometric ratios using the unit circle to prove the Pythagorean identity sin² (θ) + cos² (θ) = 1.
• Given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle, apply the Pythagorean identity sin²(θ) + cos²(θ) = 1 to find the other two ratios (sin(θ), cos(θ), or tan(θ)).

Students know:

• Methods for finding the sine, cosine, and tangent ratios of a right triangle.
• The Pythagorean Theorem.
• Properties of equality.
• The signs of the sine, cosine, and tangent ratios in each quadrant.

Students are able to:

• Use the unit circle, definitions of trigonometric functions, and the Pythagorean Theorem to prove the Pythagorean Identity sin² (θ) + cos²(θ) = 1.
• Accurately use the Pythagorean identity sin² (θ) + cos²(θ) = 1 to find the sin(θ), cos(θ), or tan(θ) when given the quadrant and one of the values.

Students understand that:

• The sine and cosine ratios and Pythagorean Theorem may be used to prove that sin² (θ) + cos² (θ) = 1.
• The sine, cosine, or tangent value of an angle and a quadrant location provide sufficient information to find the other trigonometric ratios.

Learning Objectives:
ALGII.36.1: Define Pythagorean identity sin2 (θ) + cos2 (θ) = 1.
ALGII.36.2: Identify the sine and cosine of special angles.
ALGII.36.3: Identify trigonometric ratios (sine, cosine and tangent).
ALGII.36.4: Square fractions.

Students:

• Given any triangle, derive the given formula for the area using the lengths of two sides of the triangle and the included angle.

Students know:

• The auxiliary line drawn from the vertex perpendicular to the opposite side forms an altitude of the triangle.
• The formula for the area of a triangle (A = 1/2 bh).
• Properties of the sine ratio.

Students are able to:

• Properly label a triangle according to convention.
• Perform algebraic manipulations.

Students understand that:

• Given the lengths of the sides and included angle of any triangle the area can be determined.
• There is more than one formula to find the area of a triangle.

Learning Objectives:
ALGII.37.1: Define the formula A = ½·ab·sin(C) for the area of a triangle.
ALGII.37.2: Derive the formula A = ½·ab·sin(C) for the area of a triangle when given base and height.
ALGII.37.3: Apply the formula A = ½·ab·sin(C) for the area of a triangle.

Students:
Given any triangle,

• Derive the Law of Sines and the Law of Cosines.
• Use the Law of Sines or the Law of Cosines to find unknown lengths of sides and measures of angles.

• Given a contextual situation, choose the appropriate law and apply it to determine the measures of unknown quantities.

Students know:

• The auxiliary line drawn from the vertex perpendicular to the opposite side forms an altitude of the triangle.
• Properties of the Sine and Cosine ratios.
• Pythagorean Theorem.
• Pythagorean Identity.
• The Laws of Sines and Cosines can apply to any triangle, right or non-right.
• Laws of Sines and Cosines.
• Vector quantities can represent lengths of sides and angles in a triangle.
• Values of the sin(90 degrees) and cos(90 degrees).

Students are able to:

• Label triangles in context and by convention.
• Perform algebraic manipulations.
• Find inverse sine and cosine values.

Students understand that:

• The given information will determine whether it is appropriate to use the Law of Sines or the Law of Cosines.
• Proof is necessary to establish that a conjecture about a relationship in mathematics is always true and may provide insight into the mathematics being addressed.
• Proven laws allow us to solve problems in contextual situations.

Learning Objectives:
ALGII.38.1: Define the Law of Sines and the Law of Cosines.
ALGII.38.2: Solve real-world problems using the Law of Sines and the Law of Cosines.
ALGII.38.3: Apply the Law of Sines and the Law of Cosines
ALGII.38.4: Create an equation using the given information.