FR: Functional Relationships

FR.1.BTAII: Interpret the structure of expressions, write expressions in equivalent forms to solve problems, perform arithmetic operations on functions, and understand the relationship between zeros and factors of polynomials.

FR.1.BTAII.1: Interpret expressions that represent a quantity in terms of its context:

FR.1.BTAII.1.1: interpret parts of an expression, such as terms, factors, and coefficients

 Compound Interest
 Operations with Radical Expressions
 Simplifying Algebraic Expressions I
 Simplifying Algebraic Expressions II
 Solving Equations on the Number Line

FR.1.BTAII.1.2: interpret complicated expressions by viewing one or more of their parts as a single entity [e.g., interpret P (1 + r)ⁿ as the product of P and a factor not depending on P]

 Compound Interest
 Simplifying Algebraic Expressions I
 Simplifying Algebraic Expressions II

FR.1.BTAII.2: Use the structure of an expression to identify ways to rewrite it [e.g., see x to the 4th power - y to the 4th power as (x²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² = y²)]

 Dividing Exponential Expressions
 Equivalent Algebraic Expressions I
 Equivalent Algebraic Expressions II
 Exponents and Power Rules
 Modeling the Factorization of ax2+bx+c
 Modeling the Factorization of x2+bx+c
 Multiplying Exponential Expressions
 Simplifying Algebraic Expressions I
 Simplifying Algebraic Expressions II
 Simplifying Trigonometric Expressions

FR.1.BTAII.3: Understand that polynomials form a system analogous to the integers and exhibit closure under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials

 Addition and Subtraction of Functions
 Addition of Polynomials

FR.1.BTAII.4: Use various methods to factor quadratic polynomials; understand the relationship between the factored form of a quadratic polynomial and the zeros of a function

 Modeling the Factorization of ax2+bx+c
 Modeling the Factorization of x2+bx+c
 Quadratics in Factored Form

FR.1.BTAII.5: Identify zeros of linear and quadratic polynomials when suitable factorizations are available; use the zeros to construct a rough graph of the function defined by the polynomial

 Modeling the Factorization of x2+bx+c
 Quadratics in Factored Form

FR.1.BTAII.6: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters

 Absolute Value Equations and Inequalities
 Area of Triangles
 Compound Inequalities
 Exploring Linear Inequalities in One Variable
 Modeling One-Step Equations
 Modeling and Solving Two-Step Equations
 Solving Algebraic Equations II
 Solving Equations on the Number Line
 Solving Formulas for any Variable
 Solving Linear Inequalities in One Variable
 Solving Two-Step Equations

FR.1.BTAII.7: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions

 Solving Equations by Graphing Each Side
 Solving Linear Systems (Standard Form)

FR.1.BTAII.8: Interpret the parameters in a linear or exponential function in terms of a context

 Arithmetic Sequences
 Compound Interest
 Introduction to Exponential Functions

RF: Representing Functions

RF.2.BTAII: Represent and solve equations and inequalities graphically and analyze functions using different representations.

RF.2.BTAII.1: 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); find the solutions approximately [e.g., using technology to graph the functions, make tables of values or find successive approximations; include cases where f(x) and/or g(x) are linear, quadratic, absolute value, and exponential functions]

 Cat and Mouse (Modeling with Linear Systems)
 Point-Slope Form of a Line
 Solving Equations by Graphing Each Side
 Solving Linear Systems (Matrices and Special Solutions)
 Solving Linear Systems (Slope-Intercept Form)
 Standard Form of a Line

RF.2.BTAII.2: Graph polynomial functions identifying real zeros from the factored form; show end behavior by hand in simple cases and by technology in more complicated cases

 Graphs of Polynomial Functions
 Polynomials and Linear Factors
 Quadratics in Factored Form

RF.2.BTAII.5: Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally, as a polynomial function

 Exponential Functions

RF.2.BTAII.7: Solve quadratic equations in one variable:

RF.2.BTAII.7.1: 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; derive the quadratic formula from this form

 Circles
 Roots of a Quadratic

RF.2.BTAII.7.2: solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation; recognize when the quadratic formula gives complex solutions

 Modeling the Factorization of x2+bx+c
 Points in the Complex Plane
 Quadratics in Factored Form
 Roots of a Quadratic

FM: Function Modeling

FM.3.BTAII: Create equations that describe numbers or relationships, interpret functions that arise in applications in terms of a context, analyze functions using different representations, build a function that models a relationship between two quantities, and build new functions from existing functions.

FM.3.BTAII.1: Create equations and inequalities in one variable and use them to solve problems, including equations arising from linear, quadratic, and exponential functions

 Absolute Value Equations and Inequalities
 Arithmetic Sequences
 Exploring Linear Inequalities in One Variable
 Geometric Sequences
 Modeling One-Step Equations
 Modeling and Solving Two-Step Equations
 Solving Equations on the Number Line
 Solving Linear Inequalities in One Variable
 Solving Two-Step Equations
 Using Algebraic Equations

FM.3.BTAII.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales

 Absolute Value Equations and Inequalities
 Circles
 Linear Functions
 Point-Slope Form of a Line
 Points, Lines, and Equations
 Solving Equations on the Number Line
 Standard Form of a Line
 Using Algebraic Equations

FM.3.BTAII.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context (e.g., represent inequalities describing nutritional and cost constraints on combinations of different foods)

 Linear Programming
 Systems of Linear Inequalities (Slope-intercept form)

FM.3.BTAII.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations (e.g., rearrange Ohm’s law V = IR to highlight resistance R)

 Area of Triangles
 Solving Formulas for any Variable

FM.3.BTAII.5: 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: key features include intercepts; intervals where the function is increasing, decreasing, positive or negative; maximums and minimums; symmetries; and end behavior

 Absolute Value with Linear Functions
 Cat and Mouse (Modeling with Linear Systems)
 Exponential Functions
 Function Machines 3 (Functions and Problem Solving)
 General Form of a Rational Function
 Graphs of Polynomial Functions
 Introduction to Exponential Functions
 Logarithmic Functions
 Point-Slope Form of a Line
 Points, Lines, and Equations
 Quadratics in Factored Form
 Quadratics in Polynomial Form
 Radical Functions
 Roots of a Quadratic
 Slope-Intercept Form of a Line
 Standard Form of a Line

FM.3.BTAII.6: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes [e.g., if the function ℎ(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function]

 Exponential Functions
 Introduction to Functions
 Logarithmic Functions
 Radical Functions

FM.3.BTAII.7: 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

 Cat and Mouse (Modeling with Linear Systems)
 Point-Slope Form of a Line
 Slope

FM.3.BTAII.8: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and by technology in more complicated cases:

3.1.8.1: graph exponential functions, showing intercepts and end behavior

 Exponential Functions
 Introduction to Exponential Functions
 Logarithmic Functions

3.1.8.2: graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions

 Absolute Value with Linear Functions
 Radical Functions
 Translating and Scaling Functions

FM.3.BTAII.9: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function:

FM.3.BTAII.9.2: use the properties of exponents to interpret expressions for exponential functions

 Compound Interest
 Exponential Functions

FM.3.BTAII.11: Write a function that describes a relationship between two quantities:

FM.3.BTAII.11.1: combine standard function types using arithmetic operations (e.g., build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential and relate these functions to the model)

 Addition and Subtraction of Functions

FM.3.BTAII.11.2: determine an explicit expression, a recursive process, or steps for calculation from a context

 Arithmetic Sequences
 Arithmetic and Geometric Sequences
 Estimating Population Size
 Geometric Sequences

FM.3.BTAII.12: Identify the effect on the graph of replacing f(x) by f(x) + k, k f (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; include recognizing even and odd functions from their graphs and algebraic expressions for them

 Absolute Value with Linear Functions
 Addition and Subtraction of Functions
 Exponential Functions
 Function Machines 3 (Functions and Problem Solving)
 Introduction to Exponential Functions
 Rational Functions
 Translating and Scaling Functions
 Translating and Scaling Sine and Cosine Functions
 Translations
 Zap It! Game

FM.3.BTAII.14: Define appropriate quantities for the purpose of descriptive modeling

 Estimating Population Size

FM.3.BTAII.15: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities

 Polling: Neighborhood

FM.3.BTAII.16: Graph the solutions to a linear inequality in two variables as a half-plane, excluding the boundary in the case of a strict inequality; graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes

 Linear Inequalities in Two Variables
 Linear Programming
 Systems of Linear Inequalities (Slope-intercept form)

FM.3.BTAII.19: Construct linear and exponential functions, including arithmetic sequences and geometric sequences, given a graph, a description of a relationship, or two input-output pairs; read linear and exponential functions from a table

 Absolute Value with Linear Functions
 Arithmetic Sequences
 Arithmetic and Geometric Sequences
 Compound Interest
 Exponential Functions
 Function Machines 1 (Functions and Tables)
 Function Machines 2 (Functions, Tables, and Graphs)
 Function Machines 3 (Functions and Problem Solving)
 Geometric Sequences
 Introduction to Exponential Functions
 Linear Functions
 Logarithmic Functions
 Points, Lines, and Equations
 Slope-Intercept Form of a Line

FM.3.BTAII.20: Use the properties of exponents to transform expressions for exponential functions

 Exponential Functions

SP: Statistics and Probability

SP.4.BTAII: Summarize, represent, and interpret data on a single count or a measurement variable and use probability to evaluate outcomes of decisions.

SP.4.BTAII.1: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets

 Box-and-Whisker Plots
 Describing Data Using Statistics
 Mean, Median, and Mode
 Polling: City
 Populations and Samples
 Reaction Time 1 (Graphs and Statistics)
 Real-Time Histogram

SP.4.BTAII.2: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator)

 Probability Simulations
 Theoretical and Experimental Probability

SP.4.BTAII.3: Represent data on two quantitative variables on a scatter plot and describe how the variables are related:

SP.4.BTAII.3.1: fit a function to the data; use functions fitted to data to solve problems in the context of the data; use given functions or choose a function suggested by the context; emphasize linear and exponential models

 Correlation
 Least-Squares Best Fit Lines
 Solving Using Trend Lines
 Trends in Scatter Plots
 Zap It! Game

SP.4.BTAII.3.2: informally assess the fit of a function by plotting and analyzing residuals

 Correlation
 Least-Squares Best Fit Lines
 Solving Using Trend Lines

SP.4.BTAII.4: Compute and interpret the correlation coefficient of a linear fit using technology

 Correlation

Correlation last revised: 1/19/2017

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