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
Using Algebraic 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
Quadratics in Vertex 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.6: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines

Quadratics in Vertex Form

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
Quadratics in Polynomial Form
Quadratics in Vertex Form
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
Quadratics in Vertex 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