Alg1 Standards: Difference between revisions

Latest revision as of 21:58, 5 August 2025

Human Code,Full Statement,Supp. Info MA19.A1,Algebra I with Probability, MA19.A1.NQ,Number and Quantity, MA19.A1.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.A1.1,"Explain how the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for an additional notation for radicals using rational exponents.","

Teacher Vocabulary

Exponent
Radical -nth root
Rational exponent
Rational Exponent
Root

Knowledge

Students know:

Techniques for applying the properties of exponents.

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.2,Rewrite expressions involving radicals and rational exponents using the properties of exponents.,"

Teacher Vocabulary

Negative exponent
Power of a power
Power of a product
Product of a power
Quotient of a power
Rational exponent
Zero exponent

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.3,Define the imaginary number i such that $i^2 = -1$.,"

Teacher Vocabulary

Complex number

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.AF,Algebra and Functions, MA19.A1.AF.1,Focus 1: Algebra, MA19.A1.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.A1.4,"Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.","

Teacher Vocabulary

Equivalent expressions
Exponential expression
Linear expression
Quadratic expression

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.5,Use the structure of an expression to identify ways to rewrite it.,"

Teacher Vocabulary

Difference of squares
Difference of two squares
Equivalent expressions
Factor
Linear expressions
Terms

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.6,Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.,"

Teacher Vocabulary

Complete the square
Equivalent form
Exponents
Factor
Maximum value
Minimum value
Quadratic expression
Roots
Roots
Solutions
Vertex form of a quadratic expression
X-intercepts
Zeros

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.6a,"Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.", MA19.A1.6b,Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one., MA19.A1.6c,Use the properties of exponents to transform expressions for exponential functions.,

MA19.A1.7,"Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.","

Teacher Vocabulary

Analogous system
Closure
Polynomials

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.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.A1.8,Explain why extraneous solutions to an equation involving absolute values may arise and how to check to be sure that a candidate solution satisfies an equation.,"

Teacher Vocabulary

Absolute value
Equations
Extraneous solution

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.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.A1.9,Select an appropriate method to solve a quadratic equation in one variable.,"

Teacher Vocabulary

Binomials
Completing the square
Imaginary numbers
Inspection
Quadratic equations
Quadratic formula
Trinomials

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.9a,Use the method of completing the square to transform any quadratic equation in $x$ into an equation of the form $(x - p)^2 = q$ that has the same solutions. Explain how the quadratic formula is derived from this form., MA19.A1.9b,"Solve quadratic equations by inspection (such as $x^2 = 49$), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.",

MA19.A1.10,Select an appropriate method to solve a system of two linear equations in two variables.,"

Teacher Vocabulary

Elimination method
Graphically solve
Solution of a system of linear equations
Solving systems by addition
Substitution method
System of linear equations
Tabular methods

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.10a,Solve a system of two equations in two variables by using linear combinations; contrast situations in which use of linear combinations is more efficient with those in which substitution is more efficient., MA19.A1.10b,Contrast solutions to a system of two linear equations in two variables produced by algebraic methods with graphical and tabular methods., MA19.A1.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.A1.11,"Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. **Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions.**","

Teacher Vocabulary

Approximate solutions
Equation
Identity
Inequality
No solution for a given domain
Solution set Variable

Knowledge

Students know:

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

Skills

Students are able to:

Write equations in one variable that accurately models contextual situations.

Understanding

Students understand that:

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

" MA19.A1.12,"Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. **Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.**","

Teacher Vocabulary

Piecewise 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.
Make predictions about the context using the graph.

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.

" MA19.A1.13,"Represent constraints by equations and/or inequalities, and solve systems of equations and/or inequalities, interpreting solutions as viable or nonviable options in a modeling context. **Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.**","

Teacher Vocabulary

Boundary
Closed half-plane
Consistent
Constraint
Dependent
Half plane
Inconsistent
Independent
Open half-plane
Profit
Region
System of equations
System of inequalities

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.AF.2,Focus 2: Connecting Algebra to Functions, MA19.A1.AF.2.A,Functions shift the emphasis from a point- by-point relationship between two variables (input/output) to considering an entire set of ordered pairs (where each first element is paired with exactly one second element) as an entity with its own features and characteristics.,

MA19.A1.14,"Given a relation defined by an equation in two variables, identify the graph of the relation as the set of all its solutions plotted in the coordinate plane. _Note: The graph of a relation often forms a curve (which could be a line)._","

Teacher Vocabulary

Curve (which could be a line)
Graphically finite solutions
Infinite solutions
Relation

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.15,Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range.,"

Teacher Vocabulary

Domain
Function
Function notation
Range
Relation
Set notation

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.15a,"Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. _Note: If f is a function and x is an element of its domain, then_ $f(x)$ _denotes the output of_ $f$ _corresponding to the input_ $x$.", MA19.A1.15b,"Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. **Limit to linear, quadratic, exponential, and absolute value functions.**",

MA19.A1.16,"Compare and contrast relations and functions represented by equations, graphs, or tables that show related values; determine whether a relation is a function. Explain that a function f is a special kind of relation defined by the equation $y = f(x)$.","

Teacher Vocabulary

Function
Relation

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.17,"Combine different types of standard functions to write, evaluate, and interpret functions in context. **Limit to linear, quadratic, exponential, and absolute value functions.**","

Teacher Vocabulary

Function composition

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.17a,Use arithmetic operations to combine different types of standard functions to write and evaluate functions.,

MA19.A1.17b,Use function composition to combine different types of standard functions to write and evaluate functions.,"

"

MA19.A1.AF.2.B,"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.A1.18,"Solve systems consisting of linear and/or quadratic equations in two variables graphically, using technology where appropriate.","

Teacher Vocabulary

Cartesian plane
Elimination method
Solving systems of equations
Substitution method
System of equations

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.19,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
Intersection
Linear functions

Knowledge

Students know:

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

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, quadratic, absolute value, and exponential functions.
Accurately use technology to approximate solutions on graphs.

Understanding

Students understand that:

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

"

MA19.A1.19a,"Find the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate. _Note: Include cases where_ $f(x)$ _is a linear, quadratic, exponential, or absolute value function and_ $g(x)$ _is constant or linear._",

MA19.A1.20,"Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes, using technology where appropriate.","

Teacher Vocabulary

Boundaries
Closed half-plane
Half-planes
Open half-plane
System of linear inequalities

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.AF.3,Focus 3: Functions, MA19.A1.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.A1.21,"Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). **Extend from linear to quadratic, exponential, absolute value, and general piecewise.**","

Teacher Vocabulary

Absolute value function
Exponential function
Linear function
Linear Piecewise
Quadratic function

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.A1.22,"Define sequences as functions, including recursive definitions, whose domain is a subset of the integers.","

Teacher Vocabulary

Arithmetic sequence
Domain
Geometric sequence
Recursively
Sequence

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.22a,Write explicit and recursive formulas for arithmetic and geometric sequences and connect them to linear and exponential functions., MA19.A1.AF.3.B,Functions that are members of the same family have distinguishing attributes (structure) common to all functions within that family.,

MA19.A1.23,"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 explain the effects on the graph, using technology as appropriate. **Limit to linear, quadratic, exponential, absolute value, and linear piecewise functions.**","

Teacher Vocabulary

Composite functions
Horizontal and vertical shifts
Horizontal and vertical stretch
Reflections
Translations

Knowledge

Students know:

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

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.A1.24,Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.,"

Teacher Vocabulary

Constant percent rate of change
Constant rate of change
Exponential functions
Intervals
Linear functions
Percentage of decay
Percentage of growth

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.24a,"Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.", MA19.A1.24b,Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another., MA19.A1.24c,Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.,

MA19.A1.25,"Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).","

Teacher Vocabulary

Arithmetic and geometric sequences
Arithmetic sequence
Exponential function
Geometric sequence

Knowledge

Students know:

Linear functions grow by equal differences over equal intervals, and exponential functions grow by equal factors over equal intervals.
Properties of arithmetic and geometric sequences.

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.26,Use graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.,"

Teacher Vocabulary

Increasing exponentially
Increasing linearly
Polynomial functions

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.27,"Interpret the parameters of functions in terms of a context. **Extend from linear functions, written in the form $mx + b$, to exponential functions, written in the form $ab^x$.**","

Teacher Vocabulary

Parameters

Knowledge

Students know:

Key components of linear and exponential functions.

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.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.A1.28,"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; and end behavior._ **Extend from relationships that can be represented by linear functions to quadratic, exponential, absolute value, and linear piecewise functions.**","

Teacher Vocabulary

End behavior
Function
Function is negative
Function is positive
Intervals of decreasing
Intervals of Increasing
Origin symmetry
Periodicity
Relative maximum
Relative minimum
X-intercepts
Y-axis symmetry
Y-intercepts

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.29,"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. **Limit to linear, quadratic, exponential, and absolute value functions.**","

Teacher Vocabulary

Average rate of change
Intervals

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 the rate of change over an interval in 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.A1.30,"Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.","

Teacher Vocabulary

Absolute value function
Amplitude
Domain
End behavior
Exponential function
Factorization
Intercepts
Linear function
Maximum
Midline
Minimum
Period
Piecewise function
Quadratic function
Range
Step function
X-intercept
Y-intercept
Zeros

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.

Understanding

Students understand that:

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

"

MA19.A1.30a,"Graph linear and quadratic functions and show intercepts, maxima, and minima.", MA19.A1.30b,"Graph piecewise-defined functions, including step functions and absolute value functions.", MA19.A1.30c,"Graph exponential functions, showing intercepts and end behavior.", MA19.A1.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.A1.31,"Use the mathematical modeling cycle to solve real-world problems involving linear, quadratic, exponential, absolute value, and linear piecewise functions.","

Teacher Vocabulary

Iterate to refine and extend a model
Assess a model and solutions
Define a problem
Define variables
Do the math and get solutions
Implement and report results
Mathematical modeling cycle

Knowledge

Students know:

The mathematical modeling cycle.
When to use the mathematical modeling cycle to solve problems.

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.DA,"Data Analysis, Statistics, and Probability", MA19.A1.DA.1,Focus 1: Quantitative Literacy, MA19.A1.DA.1.A,Mathematical and statistical reasoning about data can be used to evaluate conclusions and assess risks.,

MA19.A1.32,Use mathematical and statistical reasoning with bivariate categorical data in order to draw conclusions and assess risk.,"

Teacher Vocabulary

Bivariate data
Categorical data
Quantitative literacy
Risk

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.33,"Design and carry out an investigation to determine whether there appears to be an association between two categorical variables, and write a persuasive argument based on the results of the investigation.","

Teacher Vocabulary

Association
Categorical variables
Persuasive argument

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.DA.1.B,"Making and defending informed, databased decisions is a characteristic of a quantitatively literate person.", MA19.A1.DA.2,Focus 2: Visualizing and Summarizing Data, MA19.A1.DA.2.A,"Data arise from a context and come in two types: quantitative (continuous or discrete) and categorical. Technology can be used to ""clean"" and organize data, including very large data sets, into a useful and manageable structure--a first step in any analysis of data.",

MA19.A1.34,Distinguish between quantitative and categorical data and between the techniques that may be used for analyzing data of these two types.,"

Teacher Vocabulary

Categorical data
Frequency
Mean
Median
Mode
Quantitative data

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

Methods for summarizing categorical and quantitative data.

"

MA19.A1.DA.2.B,The association between two categorical variables is typically represented by using two-way tables and segmented bar graphs.,

MA19.A1.35,Analyze the possible association between two categorical variables.,"

Teacher Vocabulary

Categorical data
Conditional relative frequency
Joint frequency
Marginal frequency
Relative frequency
Segmented bar graphs
Two-way frequency tables

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.35a,Summarize categorical data for two categories in two-way frequency tables and represent using segmented bar graphs., MA19.A1.35b,"Interpret relative frequencies in the context of categorical data (including joint, marginal, and conditional relative frequencies).", MA19.A1.35c,Identify possible associations and trends in categorical data., MA19.A1.DA.2.C,Data analysis techniques can be used to develop models of contextual situations and to generate and evaluate possible solutions to real problems involving those contexts.,

MA19.A1.36,Generate a two-way categorical table in order to find and evaluate solutions to real-world problems.,"

Teacher Vocabulary

Aggregate data
Association between two variables
Categorical data
Simpson's Paradox
Two-way categorical table

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.36a,Aggregate data from several groups to find an overall association between two categorical variables., MA19.A1.36b,Recognize and explore situations where the association between two categorical variables is reversed when a third variable is considered (Simpson's Paradox)., MA19.A1.DA.3,Focus 3: Statistical Inference (Note: There are no _Algebra I with Probability_ standards in Focus 3), MA19.A1.DA.4,Focus 4: Probability, MA19.A1.DA.4.A,Two events are independent if the occurrence of one event does not affect the probability of the other event. Determining whether two events are independent can be used for finding and understanding probabilities.,

MA19.A1.37,"Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (""or,"" ""and,"" ""not"").","

Teacher Vocabulary

Complements
Intersections
Sample space
Subsets
Unions

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.38,"Explain whether two events, A and B, are independent, using two-way tables or tree diagrams.","

Teacher Vocabulary

Independent
Probability
Tree diagram

Knowledge

Students know:

Methods to find the probability of simple and compound events.

Skills

Students are able to:

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

Understanding

Students understand that:

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

"

MA19.A1.DA.4.B,"Conditional probabilities -- that is, those probabilities that are ""conditioned"" by some known information -- can be computed from data organized in contingency tables. Conditions or assumptions may affect the computation of a probability.",

MA19.A1.39,"Compute the conditional probability of event A given event B, using two-way tables or tree diagrams.","

Teacher Vocabulary

Conditional probability
Independence
Probability

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.40,Recognize and describe the concepts of conditional probability and independence in everyday situations and explain them using everyday language.,"

Teacher Vocabulary

Conditional probability
Probability

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

" MA19.A1.41,"Explain why the conditional probability of A given B is the fraction of B's outcomes that also belong to A, and interpret the answer in context.","

Teacher Vocabulary

Compound events
Conditional probability
Probability
Sample space
Simple events

Knowledge

Students know:

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

Skills

Students are able to:

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

Understanding

Students understand that:

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

"