Alg2 Standards

Human Code,Full Statement,Supp. Info MA19.A2,Algebra II with Statistics, MA19.A2.NQ,Number and Quantity, MA19.A2.NQ.A,"Together, irrational numbers and rational numbers complete the real number system, representing all points on the number line, while there exist numbers beyond the real numbers called complex numbers.",

MA19.A2.1,"Identify numbers written in the form $a + bi$, where $a$ and $b$ are real numbers and $i^2 = -1$, as complex numbers.","

Teacher Vocabulary

Associative property
Commutative property
Complex number
Distributive property

Knowledge

Students know:

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

Skills

Students are able to:

Perform arithmetic manipulations on complex numbers to produce equivalent expressions.

Understanding

Students understand that:

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

"

MA19.A2.1a,"Add, subtract, and multiply complex numbers using the commutative, associative, and distributive properties.", MA19.A2.NQ.B,Matrices are a useful way to represent information.,

MA19.A2.2,Use matrices to represent and manipulate data.,"

Teacher Vocabulary

Columns
Data
Dimensions
Elements
Matrix/matrices
Rows
Subscript notation

Knowledge

Students know:

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

Skills

Students are able to:

Translate data into a matrix.

Understanding

Students understand that:

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

" MA19.A2.3,Multiply matrices by scalars to produce new matrices.,"

Teacher Vocabulary

Scalars

Knowledge

Students know:

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

Skills

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).

Understanding

Students understand that:

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.

" MA19.A2.4,"Add, subtract, and multiply matrices of appropriate dimensions.","

Teacher Vocabulary

Appropriate dimensions

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

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.

" MA19.A2.5,"Describe the roles that zero and identity matrices play in matrix addition and multiplication, recognizing that they are similar to the roles of 0 and 1 in the real numbers.","

Teacher Vocabulary

Determinant
Identity matrix
Multiplicative inverse
Zero matrix

Knowledge

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.

Skills

Students are able to:

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

Understanding

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.

"

MA19.A2.5a,"Find the additive and multiplicative inverses of square matrices, using technology as appropriate.", MA19.A2.5b,Explain the role of the determinant in determining if a square matrix has a multiplicative inverse., MA19.A2.AF,Algebra and Functions, MA19.A2.AF.1,Focus 1: Algebra, MA19.A2.AF.1.A,"Expressions can be rewritten in equivalent forms by using algebraic properties, including properties of addition, multiplication, and exponentiation, to make different characteristics or features visible.",

MA19.A2.6,"Factor polynomials using common factoring techniques, and use the factored form of a polynomial to reveal the zeros of the function it defines.","

Teacher Vocabulary

Factorization
Polynomial
Zeros

Knowledge

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.

Skills

Students are able to:

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

Understanding

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.

" MA19.A2.7,Prove polynomial identities and use them to describe numerical relationships.,"

Teacher Vocabulary

Polynomial identity

Knowledge

Students know:

Distributive property of multiplication over addition.

Skills

Students are able to:

Accurately perform algebraic manipulations on polynomial expressions.

Understanding

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.

"

MA19.A2.AF.1.B,"Finding solutions to an equation, inequality, or system of equations or inequalities requires the checking of candidate solutions, whether generated analytically or graphically, to ensure that solutions are found and that those found are not extraneous.",

MA19.A2.8,Explain why extraneous solutions to an equation may arise and how to check to be sure that a candidate solution satisfies an equation. **Extend to radical equations.**,"

Teacher Vocabulary

Extraneous solutions
Radical
Radical equations

Knowledge

Students know:

Algebraic rules for manipulating radical equations.
Conditions under which a solution is considered extraneous.

Skills

Students are able to:

Accurately rearrange radical 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.

Understanding

Students understand that:

Values that arise from solving equations may not satisfy the original equation.
Values that arise from solving the equations may not exist due to considerations in the context.

"

MA19.A2.AF.1.C,"The structure of an equation or inequality (including, but not limited to, one-variable linear and quadratic equations, inequalities, and systems of linear equations in two variables) can be purposefully analyzed (with and without technology) to determine an efficient strategy to find a solution, if one exists, and then to justify the solution.",

MA19.A2.9,"For exponential models, express as a logarithm the solution to $ab^{ct} = d$, where $a$, $c$, and $d$ are real numbers and the base $b$ is 2 or 10; evaluate the logarithm using technology to solve an exponential equation.","

Teacher Vocabulary

Exponential equation
Exponential model
Logarithm
Logarithmic base

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A2.AF.1.D,"Expressions, equations, and inequalities can be used to analyze and make predictions, both within mathematics and as mathematics is applied in different contexts--in particular, contexts that arise in relation to linear, quadratic, and exponential situations.",

MA19.A2.10,"Create equations and inequalities in one variable and use them to solve problems. **Extend to equations arising from polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.**","

Teacher Vocabulary

Exponential functions
Logarithmic functions
Piecewise functions
Polynomial functions
Radical functions
Trigonometric functions

Knowledge

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.

Skills

Students are able to:

Write equations or inequalities in one variable that accurately model contextual situations.

Understanding

Students understand that:

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

" MA19.A2.11,Solve quadratic equations with real coefficients that have complex solutions.,"

Teacher Vocabulary

Complex solution
Quadratic equation
Real coefficients

Knowledge

Students know:

Strategies for solving quadratic equations.

Skills

Students are able to:

Apply the quadratic equation.
Provide solutions in complex form.

Understanding

Students understand that:

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

" MA19.A2.12,"Solve simple equations involving exponential, radical, logarithmic, and trigonometric functions using inverse functions.","

Teacher Vocabulary

Exponential equations
Inverse functions
Logarithmic equations
Radical equations
Trigonometric equations

Knowledge

Students know:

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

Skills

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 equations, if necessary.

Understanding

Students understand that:

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

" MA19.A2.13,"Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales and use them to make predictions. **Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.**","

Teacher Vocabulary

Exponential functions
Logarithmic functions
Piecewise functions
Polynomial functions
Radical functions
Reciprocal functions
Trigonometric functions

Knowledge

Students know:

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

Skills

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.

Understanding

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.

"

MA19.A2.AF.2,Focus 2: Connecting Algebra to Functions, MA19.A2.AF.2.A,"Graphs can be used to obtain exact or approximate solutions of equations, inequalities, and systems of equations and inequalities--including systems of linear equations in two variables and systems of linear and quadratic equations (given or obtained by using technology).",

MA19.A2.14,Explain why the _x_-coordinates of the points where the graphs of the equations $y = f(x)$ and $y = g(x)$ intersect are the solutions of the equation $f(x) = g(x)$.,"

Teacher Vocabulary

Absolute value functions
Exponential functions
Functions
General piecewise functions
Linear functions
Logarithmic functions
Polynomial functions
Radical functions
Rational functions
Successive approximations
Trigonometric (sine and cosine) functions

Knowledge

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.

Skills

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.

Understanding

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.

"

MA19.A2.14a,"Find the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate. **Extend to cases where $f(x)$ and/or $g(x)$ are polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.**", MA19.A2.AF.3,Focus 3: Functions, MA19.A2.AF.3.A,"Functions can be described by using a variety of representations: mapping diagrams, function notation (e.g., $f(x) = x^2$), recursive definitions, tables, and graphs.",

MA19.A2.15,"Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). **Extend to polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.**","

Teacher Vocabulary

Algebraic expressions
Exponential functions
General piecewise functions
Logarithmic functions
Polynomial functions
Radical functions
Trigonometric functions (sine and cosine)

Knowledge

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).

Skills

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.

Understanding

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.

"

MA19.A2.AF.3.B,Functions that are members of the same family have distinguishing attributes (structure) common to all functions within that family.,

MA19.A2.16,"Identify the effect on the graph of replacing $f(x)$ by $f(x) + k$, $k \cdot f(x)$, $f(k \cdot x)$, and $f(x + k)$ for specific values of $k$ (both positive and negative); find the value of $k$ given the graphs. **Experiment with cases and illustrate an explanation of the effects on the graph using technology. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.**","

Teacher Vocabulary

General piecewise functions
Logarithmic functions
Polynomial functions
Radical functions
Reciprocal functions
Trigonometric (sine and cosine) functions

Knowledge

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.

Skills

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.

Understanding

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.

"

MA19.A2.AF.3.C,"Functions can be represented graphically, and key features of the graphs, including zeros, intercepts, and, when relevant, rate of change and maximum/minimum values, can be associated with and interpreted in terms of the equivalent symbolic representation.",

MA19.A2.17,"For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. _Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; and periodicity._ **Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.**","

Teacher Vocabulary

Amplitude
End behavior
Even and odd
Intercepts
Intervals
Intervals
Logarithmic function
Maximum
Midline
Minimum
Period
Piecewise function
Polynomial function
Radical function
Reciprocal function
Symmetry
Trigonometric (sine and cosine) function

Knowledge

Students know:

Techniques for graphing.
Key features of graphs of functions.

Skills

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.

Understanding

Students understand that:

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

" MA19.A2.18,"Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. **Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.**","

Teacher Vocabulary

Domain
Function
Quantitative

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A2.19,"Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. **Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.**","

Teacher Vocabulary

Average rate of change
Specified interval

Knowledge

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.

Skills

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.

Understanding

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).

" MA19.A2.20,"Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. **Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.**","

Teacher Vocabulary

Amplitude
End behavior
Horizontal asymptote
Intervals
Inverse functions
Logarithmic function Trigonometric (sine and cosine) function
Maximum
Midline
Minimum
Period
Polynomial function
Radical function
Reciprocal function
Vertical asymptote

Knowledge

Students know:

Techniques for graphing.
Key features of graphs of functions.

Skills

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.

Understanding

Students understand that:

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

"

MA19.A2.20a,"Graph polynomial functions expressed symbolically, identifying zeros when suitable factorizations are available, and showing end behavior.", MA19.A2.20b,"Graph sine and cosine functions expressed symbolically, showing period, midline, and amplitude.", MA19.A2.20c,"Graph logarithmic functions expressed symbolically, showing intercepts and end behavior.", MA19.A2.20d,"Graph reciprocal functions expressed symbolically, identifying horizontal and vertical asymptotes.", MA19.A2.20e,Graph square root and cube root functions expressed symbolically., MA19.A2.20f,"Compare the graphs of inverse functions and the relationships between their key features, including but not limited to quadratic, square root, exponential, and logarithmic functions.",

MA19.A2.21,"Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle, building on work with non-right triangle trigonometry.","

Teacher Vocabulary

Quadrantal
Radian measure
Traversed
Unit circle

Knowledge

Students know:

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

Skills

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.

Understanding

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.

"

MA19.A2.AF.3.D,"Functions model a wide variety of real situations and can help students understand the processes of making and changing assumptions, assigning variables, and finding solutions to contextual problems.",

MA19.A2.22,"Use the mathematical modeling cycle to solve real-world problems involving polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions, from the simplification of the problem through the solving of the simplified problem, the interpretation of its solution, and the checking of the solution's feasibility.","

Teacher Vocabulary

Feasibility
Mathematical modeling cycle

Knowledge

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.

Skills

Students are able to:

Accurately model contextual situations.

Understanding

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.

"

MA19.A2.DA,"Data Analysis, Statistics, and Probability", MA19.A2.DA.1,Focus 1: Quantitative Literacy, MA19.A2.DA.1.A,Mathematical and statistical reasoning about data can be used to evaluate conclusions and assess risks.,

MA19.A2.23,Use mathematical and statistical reasoning about normal distributions to draw conclusions and assess risk; limit to informal arguments.,"

Teacher Vocabulary

Margin of error
Normal distribution

Knowledge

Students know:

Properties of a normal distribution.
Empirical rule.

Skills

Students are able to:

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

Understanding

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% falls within two standard deviations, and 99.7% falls within three standard deviations.

"

MA19.A2.DA.1.B,Making and defending informed data-based decisions is a characteristic of a quantitatively literate person.,

MA19.A2.24,"Design and carry out an experiment or survey to answer a question of interest, and write an informal persuasive argument based on the results.","

Teacher Vocabulary

Experiment
Survey

Knowledge

Students know:

Techniques to design an experiment or survey.

Skills

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.

Understanding

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.

"

MA19.A2.DA.2,Focus 2: Visualizing and Summarizing Data, MA19.A2.DA.2.A,"Distributions of quantitative data (continuous or discrete) in one variable should be described in the context of the data with respect to what is typical (the shape, with appropriate measures of center and variability, including standard deviation) and what is not (outliers), and these characteristics can be used to compare two or more subgroups with respect to a variable.",

MA19.A2.25,"From a normal distribution, use technology to find the mean and standard deviation and estimate population percentages by applying the empirical rule.","

Teacher Vocabulary

Empirical rule
Mean
Normal curve
Normal distribution
Population percentages
Standard deviation

Knowledge

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.

Skills

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, calculators, and spreadsheets to accurately estimate the areas under a normal curve.

Understanding

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.

"

MA19.A2.25a,Use technology to determine if a given set of data is normal by applying the empirical rule., MA19.A2.25b,"Estimate areas under a normal curve to solve problems in context, using calculators, spreadsheets, and tables as appropriate.", MA19.A2.DA.3,Focus 3: Statistical Inference, MA19.A2.DA.3.A,"Study designs are of three main types: sample survey, experiment, and observational study.",

MA19.A2.26,"Describe the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.","

Teacher Vocabulary

Experiments
Observational studies
Randomization
Sample surveys

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A2.DA.3.B,The role of randomization is different in randomly selecting samples and in randomly assigning subjects to experimental treatment groups.,

MA19.A2.27,Distinguish between a statistic and a parameter and use statistical processes to make inferences about population parameters based on statistics from random samples from that population.,"

Teacher Vocabulary

Inferences
Population parameters
Random samples

Knowledge

Students know:

Techniques for selecting random samples from a population.

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A2.28,Describe differences between randomly selecting samples and randomly assigning subjects to experimental treatment groups in terms of inferences drawn regarding a population versus regarding cause and effect.,"

Teacher Vocabulary

Cause and effect
Inference
Non-randomized
Randomly
Treatments

Knowledge

Students know:

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

Skills

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.

Understanding

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.

"

MA19.A2.DA.3.C,The scope and validity of statistical inferences are dependent on the role of randomization in the study design.,

MA19.A2.29,"Explain the consequences, due to uncontrolled variables, of non-randomized assignment of subjects to groups in experiments.","

Teacher Vocabulary

Non-randomized
Uncontrolled variables

Knowledge

Students know:

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

Skills

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 variables that may be responsible for the observed difference when non-randomized assignment is used in an experiment.

Understanding

Students understand that:

Uncontrolled variables are characteristic factors that are not regulated or measured by the investigator during an experiment or study, so 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.

"

MA19.A2.DA.3.D,"Bias, such as sampling, response, or nonresponse bias, may occur in surveys, yielding results that are not representative of the population of interest.",

MA19.A2.30,"Evaluate where bias, including sampling, response, or nonresponse bias, may occur in surveys, and whether results are representative of the population of interest.","

Teacher Vocabulary

Bias
Nonresponse bias
Response bias
Sampling

Knowledge

Students know:

Techniques for conducting surveys.
Techniques to identify bias

Skills

Students are able to:

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

Understanding

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 non-respondents.

"

MA19.A2.DA.3.E,"The larger the sample size, the less the expected variability in the sampling distribution of a sample statistic.",

MA19.A2.31,Evaluate the effect of sample size on the expected variability in the sampling distribution of a sample statistic.,"

Teacher Vocabulary

Sample size
Sampling distribution
Standard deviation
Variability

Knowledge

Students know:

Techniques to find the mean and standard deviation.

Skills

Students are able to:

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

Understanding

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.

"

MA19.A2.31a,"Simulate a sampling distribution of sample means from a population with a known distribution, observing the effect of the sample size on the variability.", MA19.A2.31b,Demonstrate that the standard deviation of each simulated sampling distribution is the known standard deviation of the population divided by the square root of the sample size., MA19.A2.DA.3.F,"The sampling distribution of a sample statistic formed from repeated samples for a given sample size drawn from a population can be used to identify typical behavior for that statistic. Examining several such sampling distributions leads to estimating a set of plausible values for the population parameter, using the margin of error as a measure that describes the sampling variability.",

MA19.A2.32,"Produce a sampling distribution by repeatedly selecting samples of the same size from a given population or from a population simulated by bootstrapping (resampling with replacement from an observed sample). Do initial examples by hand, then use technology to generate a large number of samples.","

Teacher Vocabulary

Approximately normal
Bootstrapping
Confidence interval
Population mean
Standard deviation

Knowledge

Students know:

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

Skills

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.

Understanding

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 within which you can be 95% confident that the true population mean lies. With larger samples, this interval is more precise, meaning it tends to be narrower compared to an interval calculated from a smaller sample.

"

MA19.A2.32a,Verify that a sampling distribution is centered at the population mean and approximately normal if the sample size is large enough., MA19.A2.32b,Verify that 95% of sample means are within two standard deviations of the sampling distribution from the population mean., MA19.A2.32c,Create and interpret a 95% confidence interval based on an observed mean from a sampling distribution.,

MA19.A2.33,"Use data from a randomized experiment to compare two treatments; limit to informal use of simulations to decide if an observed difference in the responses of the two treatment groups is unlikely to have occurred due to randomization alone, thus implying that the difference between the treatment groups is meaningful.","

Teacher Vocabulary

Parameters
Randomized experiment
Significant

Knowledge

Students know:

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

Skills

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.

Understanding

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.

"

MA19.A2.GM,Geometry and Measurement, MA19.A2.GM.1,Focus 1: Measurement, MA19.A2.GM.1.A,"When an object is the image of a known object under a similarity transformation, a length, area, or volume on the image can be computed by using proportional relationships.",

MA19.A2.34,"Define the radian measure of an angle as the constant of proportionality of the length of an arc it intercepts to the radius of the circle; in particular, it is the length of the arc intercepted on the unit circle.","

Teacher Vocabulary

Constant of proportionality
Intercepted arc
Radian measure
Unit circle

Knowledge

Students know:

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

Skills

Students are able to:

Translate between arc length and central angle measures in circles.

Understanding

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 the circle can then be divided up to find the radian measure of other angles.

"

MA19.A2.GM.2,Focus 2: Transformations (Note: There are no _Algebra II with Statistics_ standards in Focus 2), MA19.A2.GM.3,"Focus 3: Geometric Argument, Reasoning, and Proof (Note: There are no _Algebra II with Statistics_ standards in Focus 3)", MA19.A2.GM.4,Focus 4: Solving Applied Problems and Modeling in Geometry, MA19.A2.GM.4.A,"Recognizing congruence, similarity, symmetry, measurement opportunities, and other geometric ideas, including right triangle trigonometry in real-world contexts, provides a means of building understanding of these concepts and is a powerful tool for solving problems related to the physical world in which we live.",

MA19.A2.35,"Choose trigonometric functions (sine and cosine) to model periodic phenomena with specified amplitude, frequency, and midline.","

Teacher Vocabulary

Amplitude
Frequency
Midline
Periodic phenomena
Trigonometric functions

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

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.

" MA19.A2.36,Prove the Pythagorean identity $sin^2 (\theta) + cos^2 (\theta) = 1$ and use it to calculate trigonometric ratios.,"

Teacher Vocabulary

Pythagorean Identity

Knowledge

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.

Skills

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

Understanding

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.

" MA19.A2.37,"Derive and apply the formula $A = \frac{1}{2} \cdot ab \cdot sin(C)$ for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side, extending the domain of sine to include right and obtuse angles.","

Teacher Vocabulary

Auxiliary line
Perpendicular
Vertex

Knowledge

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.

Skills

Students are able to:

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

Understanding

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.

" MA19.A2.38,Derive and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. **Extend the domain of sine and cosine to include right and obtuse angles.**,"

Teacher Vocabulary

Law of Cosines
Law of Sines
Resultant force

Knowledge

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).

Skills

Students are able to:

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

Understanding

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.

"