# California Common Core (CACC) - Integrated Pathway Mathematics 2 Skills Practice

Skills available for California Common Core (CACC) - Integrated Pathway Mathematics 2 Skills Practice
Number and Quantity
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The Real Number System
Extend the properties of exponents to rational exponents.
N-RN1 - Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^1/3 to be the cube root of 5 because we want (5^1/3)^3= 5^(1/3)^3 to hold, so (5^1/3)^3 must equal 5.
• Converting Between Radical Form and Exponent Form
N-RN2 - Rewrite expressions involving radicals and rational exponents using the properties of exponents.
• Simplifying a Radical Expression with an Even Exponent
• Finding the Square Root of a Perfect Square Monomial
• Evaluating Rational Exponents with Unit Fraction Exponents & Whole Number Bases
• Evaluating Rational Exponents with Non-unit Fraction Exponent with a Whole Number Base
• How to Evaluate a Non-unit Fraction Rational Exponent with a Whole Number Base
• How to Evaluate a Non-unit Fraction Rational Exponent with a Fractional Base
• How to Evaluate a Negative Rational Exponent with a Whole Number Base
• How to Evaluate a Negative Rational Exponent with a Fractional Base
• Using the Product Rule For Rational Exponents
• Understanding the Quotient Rule For Rational Exponents
• Using the Power of a Power Rule for Rational Exponents
• Using the Power of a Power Rule with Negative Exponents and a Whole Number Base
• How To Use the Power of a Power Rule with Negative Exponents
• Understanding the Power of a Power Rule
• How to Simplify a Square Root
• How to Simplify the Square Root of a Whole Number Less Than 100
• Simplifying the Square Root of a Whole Number Greater than 100
• Simplifying a Radical Expression with an Odd Exponent
• Simplifying a Higher Root of a Whole Number
• Addition and Subtraction of Square Roots
• Calculating the Addition and Subtraction of Square Roots
• How to Solve Square Root Multiplication
• How to Solve Basic Square Root Multiplication
• How to Solve Advanced Square Root Multiplication
• How to Simplify a Product Involving Square Roots Using the Distributive Property
• Simplifying a Basic Product Involving Square Roots Using the Distributive Property
• Simplifying an Advanced Product Involving Square Roots Using the Distributive Property
• How to Simplify a Quotient of Square Roots
• How to Simplify a Quotient Involving a Sum or Difference with a Square Root
• Rationalizing the Denominator of a Quotient Involving Square Roots
Use properties of rational and irrational numbers.
N-RN3 - Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
• Classifying Sums and Products As Rational Or Irrational
The Complex Number System
Perform arithmetic operations with complex numbers.
N-CN1 - Know there is a complex number i such that i^2 = -1, and every complex number has the form a + bi with a and b real.
• Using I to Rewrite Square Roots of Negative Numbers
N-CN2 - Use the relation i^2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
• How to Simplify a Product Involving Square Roots of Negative Numbers
• How to Simplify a Quotient Involving Square Roots of Negative Numbers
• How to Multiply Complex Numbers
• How to Find a Complex Conjugate
• How to Divide Complex Numbers
• How to Simplify a Power of I
Use complex numbers in polynomial identities and equations.
N-CN7 - Solve quadratic equations with real coefficients that have complex solutions.
• How to Solve a Quadratic Equation with Complex Roots
• Solving an Equation Written in Factored Form
• How to Solve a Basic Quadratic Equation Using the Square Root Property
N-CN8 - Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x - 2i).
• How to Multiply Expressions Involving Complex Conjugates
N-CN9 - Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
• How to Apply the Fundamental Theorem of Algebra
Algebra
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Seeing Structure in Expressions
Interpret the structure of expressions.
A-SSE1 - Interpret expressions that represent a quantity in terms of its context.
A-SSE1a - Interpret parts of an expression, such as terms, factors, and coefficients.
• How to Rewrite a Quadratic Function to Find Its Vertex and Sketch Its Graph
• How to Identify the Degree & Leading Coefficient of a Univariate Polynomial
• Identifying Parts of an Expression Using Mathematical Terms
• Identifying Terms in an Algebraic Expression
A-SSE1b - Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)^n as the product of P and a factor not depending on P.
• Finding the Initial Amount & Rate of Change with an Exponential Function
• How to Rewrite a Quadratic Function to Find Its Vertex and Sketch Its Graph
• How to Rewrite a Quadratic Function to Find Its Vertex and Sketch Its Graph
• How to Find the Initial Value Given an Exponential Function
A-SSE2 - Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)2 - (y^2)2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2).
• Factoring Out a Monomial From a Polynomial: Univariate
• Factoring Out a Monomial from a Multivariate Polynomial
• Factoring Out a Binomial from a Polynomial using Greatest Common Factor
• Factoring a Univariate Polynomial by Grouping with Positive Terms
• Factoring a Univariate Polynomial by Grouping with Some Negative Terms
• Factoring Out a Constant Before Factoring a Quadratic
• Factoring a Quadratic with Leading Coefficient Greater Than 1 Using Trial and Error
• Factoring a Quadratic with Leading Coefficient Greater Than 1 Using the AC Method
• Factoring a Perfect Square Trinomial with a Leading Coefficient of 1
• Factoring a Perfect Square Trinomial with a Leading Coefficient Greater Than 1
• Factoring a Difference of Squares in One Variable with a Leading Coefficient of 1
• Factoring a Difference of Squares in One Variable with a Leading Coefficient Not Equal To 1
• Factoring a Polynomial Involving a GCF & a Difference of Squares with Univariate Terms
• How to Find the Roots of a Product of Polynomials
Write expressions in equivalent forms to solve problems.
A-SSE3 - Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A-SSE3a - Factor a quadratic expression to reveal the zeros of the function it defines.
• Finding the Roots of a Quadratic Equation of the Form ax^2 + ax = 1
• Finding the Roots of a Quadratic Equation with a Leading Coefficient of 1
• Finding the Roots of a Quadratic Equation with a Leading Coefficient Greater Than 1
• Finding the Vertex, Intercepts, & Axis of Symmetry From the Graph of a Parabola
• How to Find the Zeros of a Quadratic Function Given Its Equation
• Finding the X-intercept(s) and Vertex of a Parabola
A-SSE3b - Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
• How to Rewrite a Quadratic Function to Find Its Vertex and Sketch Its Graph
• How to Rewrite a Quadratic Function to Find Its Vertex and Sketch Its Graph
A-SSE3c - Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15t can be rewritten as (1.15^1/12)^12t ? 1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
• Understanding the Product Rule of Exponents
• Using the Power of a Power Rule with Negative Exponents and a Whole Number Base
• Using the Power of a Power Rule with Negative Exponents and a Whole Number Base
• Using the Product Rule with Positive Exponents & a Whole Number Base
• How To Use the Power of a Power Rule with Negative Exponents
• How To Use the Power of a Power Rule with Negative Exponents
• Using the Product Rule with Negative Exponents and a Whole Number Base
• How to Use the Product Rule with Positive Exponents & Univariate Terms
• How to Use the Product Rule with Positive Exponents & Multivariate Terms
• Understanding the Power Rules of Exponents
• How to Use the Power Rule with Positive Exponents & a Whole Number Base
• Using Power Rules with Positive Exponents Resulting in Multivariate Products
• Using Power Rules with Positive Exponents Resulting in Multivariate Quotients
• Simplifying a Ratio of Multivariate Monomials with Exponents Equal To 1
• Understanding the Quotient Rule of Exponents
• Using the Quotient Rule with Positive Exponents and a Whole Number Base
• Simplifying a Ratio of Univariate Monomials
• Calculating the Quotient of Expressions Involving Exponents
• Simplifying a Ratio of Multivariate Monomials with Exponents Not Equal To 1
• How to Use Power and Quotient Rules with Positive Exponents
• How to Evaluate Expressions with Exponents of Zero
• Determining Positive Powers of 10
• How to Solve the Power of 10 with a Negative Exponent
• How to Evaluate an Expression with a Negative Exponent & a Whole Number Base
• How to Evaluate an Expression with a Negative Exponent & a Positive Fraction Base
• How to Evaluate an Expression with a Negative Exponent & a Negative Integer Base
• How To Rewrite an Algebraic Expression Without a Negative Exponent
• How To Use the Product Rule with Negative Exponents
• Using the Quotient Rule with Negative Exponents & a Whole Number Base
• Using Power Rules with Negative Exponents
• Applying Power and Quotient Rules with Negative Exponents
Arithmetic with Polynomials and Rational Expressions
Perform arithmetic operations on polynomials.
A-APR1 - Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
• Simplifying a Sum or Difference of Two Univariate Polynomials
• Multiplying a Univariate Polynomial by a Monomial with a Positive Coefficient
• Multiplying a Univariate Polynomial by a Monomial with a Negative Coefficient
• Multiplying Binomials with Leading Coefficients of 1
• Multiplying Binomials with Leading Coefficients Greater Than 1
• Multiplying Binomials in Two Variables
• Multiplying Conjugate Univariate Binomials
• Multiplying Multivariate Conjugate Binomials
• Squaring a Multivariate Binomial
• Squaring a Univariate Binomial
• Multiplying Binomials with Negative Coefficients
• Multiplication Involving Binomials and Trinomials in One Variable
• Using Closure Properties of Integers & Polynomials
• Dividing Polynomials Using Long Division with No Remainder
• Dividing Polynomials Using Long Division with a Remainder
• Dividing a Polynomial by a Univariate Monomial
• How to do Polynomial Long Division
Creating Equations
Create equations that describe numbers or relationships.
A-CED1 - Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CA
• How to Translate a Sentence into a 1-Step Equation
• Solving a Decimal Word Problem Using a Linear Equation with the Variable on Both Sides
• How to Translate a Sentence into a Multi-Step Equation
• How to Solve a Fraction Word Problem Using a Linear Equation of the Form AX = B
• How to Write an Equation of the Form A(X + B) = C to Solve a Word Problem
• Solving a Decimal Word Problem Using a Linear Equation with the Variable on Both Sides
• Solving a Word Problem with 2 Unknowns Using a Linear Equation
• How to Write an Equation with the Variable on Both Sides to Represent a Real-World Problem
• Solving a Decimal Word Problem Using a 2-Step Linear Inequality
• How to Solve a Word Problem with 3 Unknowns Using a Linear Equation
• Solving a Value Mixture Problem Using a System of Linear Equations
• Solving a Word Problem Involving the Average Rate of Change
• Solving a 1-Step Word Problem Using the Formula D = Rt
• How to Solve a Distance, Rate, Time Problem Using a Linear Equation
• Finding Side Lengths of Rectangles Given One Dimension & an Area
• How to Find the Side Lengths of Rectangles Given One Dimension & an Area or a Perimeter
• How to Find the Perimeter or Area of a Rectangle Given 1 of These Values
• Finding a Side Length Given the Perimeter & Side Lengths with Variables
• Solving a Word Problem on Proportions
• Solving Word Problems on Proportions Using a Unit Rate
• Translating a Sentence by Using an Inequality Symbol
• Translating a Sentence into a 1-Step Inequality
• Writing an Inequality for a Real-World Situation
• Solving a Word Problem Using a 1-Step Linear Inequality
• Translating a Sentence into a Multi-Step Inequality
• How to Solve a Word Problem Using a 2-Step Linear Inequality & Describing the Solution
• Solving a Word Problem Using a 2-Step Linear Inequality
• Solving a Word Problem Using a Quadratic Equation with Rational Roots
• How to Solve Word Problems with Rates for a Variable in Terms of Other Variables in a Rational Equation
• How to Solve a Word Problem Using a Rational Equation
• How to Solve a Distance, Rate & Time Problem Using a Rational Equation
• How to Solve a Word Problem Using a Quadratic Equation with Irrational Roots
• How to Solve Absolute Value Equations
• How to Solve Absolute Value Inequalities
• Understanding When an Absolute Value Equation Has No Solution
• Describing Absolute Value as Magnitude for a Positive or Negative Quantity in Real World Situations
• Finding the Final Value in a Word Problem on Exponential Growth or Decay
A-CED2 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
• Writing & Evaluating a Function with 1 Variable for a Real-world Situation
• Writing & Evaluating a Function with 2 Variables for a Real-world Situation
• Writing a Basic Equation & Drawing a Graph for a Real-World Situation
• Writing an Advanced Equation & Drawing its Graph for a Real-World Situation
• Writing a Quadratic Function Given Its Zeros
• How to Write a Quadratic Equation Given the Roots & the Leading Coefficient
• Writing an Equation that Models Exponential Growth Or Decay
• Writing an Exponential Function Rule Given a Table of Ordered Pairs
A-CED4 - Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
• Solving for a Variable in Terms of Other Variables Using Addition or Subtraction
• Solving for a Variable in Terms of Other Variables Using Multiplication or Division
• How to Solve Word Problems with Rates for a Variable in Terms of Other Variables in a Rational Equation
• Solving for a Variable in Terms of Other Variables Using Addition or Subtraction with Division
• Solving for a Variable Inside Parentheses in Terms of Other Variables
• How to Solve for a Variable in Terms of Other Variables Using Addition
• How to Solve for a Variable in Terms of Other Variables Using Multiplication
• Solving for a Variable in Terms of Other Variables in a Linear Equation with Fractions
• Solving for a Variable in Terms of Other Variables in a Rational Equation
Reasoning with Equations and Inequalities
Solve equations and inequalities in one variable.
A-REI4 - Solve quadratic equations in one variable. (Quadratics with real coefficients)
A-REI4a - 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.
• Solving a Quadratic Equation by Completing the Square with Decimal Answers
• How to Solve a Quadratic Equation by Completing the Square
• How to Apply the Quadratic Formula
A-REI4b - Solve quadratic equations by inspection (e.g., for x^2 = 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 and write them as a +/- bi for real numbers a and b.
• How to Solve an Advanced Quadratic Equation Using the Square Root Property
• Finding the Roots of a Quadratic Equation of the Form ax^2 + ax = 1
• Solving a Quadratic Equation by Completing the Square with Decimal Answers
• How to Find the Zeros of a Quadratic Function Given Its Equation
• How to Solve a Quadratic Equation with Complex Roots
• Finding the Roots of a Quadratic Equation of the Form ax^2 + ax = 1
• Solving an Equation in x^2 = a Using the Square Root Property
• Finding the Roots of a Quadratic Equation with a Leading Coefficient of 1
• Finding the Roots of a Quadratic Equation with a Leading Coefficient of 1
• Solving a Quadratic Equation Needing Simplification
• Finding the Roots of a Quadratic Equation with a Leading Coefficient Greater Than 1
• How to Solve a Quadratic Equation by Completing the Square
• Finding the Roots of a Quadratic Equation with a Leading Coefficient Greater Than 1
• Understanding the Discriminant of a Quadratic Equation
• How to Apply the Quadratic Formula
• Solving a Quadratic Equation by Completing the Square with Decimal Answers
• How to Solve a Quadratic Equation by Completing the Square
• How to Apply the Quadratic Formula
• How to Find the Zeros of a Quadratic Function Given Its Equation
• How to Solve a Quadratic Equation with Complex Roots
Solve systems of equations.
A-REI7 - Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x^2 + y^2 = 3.
• Graphically Solving a System of Linear and Quadratic Equations
• Solving a System of Linear and Quadratic Equations
Functions
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Interpreting Functions
Interpret functions that arise in applications in terms of the context.
F-IF4 - 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; relative maximums and minimums; symmetries; end behavior; and periodicity.
• Finding Intercepts of a Nonlinear Function Given its Graph
• How to Solve a Word Problem Using a Quadratic Equation with Irrational Roots
• Finding the Vertex, Intercepts, & Axis of Symmetry From the Graph of a Parabola
• Finding the Vertex, Intercepts, & Axis of Symmetry From the Graph of a Parabola
• Finding Local Maxima of a Function Given the Graph
• Finding the X-intercept(s) and Vertex of a Parabola
• Finding the X-intercept(s) and Vertex of a Parabola
• Finding Local Minima of a Function Given the Graph
• How to Find Where a Function is Increasing, Decreasing, or Constant Given the Graph
• Choosing a Graph to Fit a Narrative with Linear Graphs of Consistent Slope
• Choosing a Graph to Fit a Narrative with Graphs Including Linear Sections with Different Slopes
• Choosing a Graph to Fit a Narrative with Graphs Including Linear and Non-linear Sections of Differing Slopes
• How to Find the Vertex from the Graph of a Parabola
• How to Find the Maximum Or Minimum of a Quadratic Function
• How to Determine the End Behavior of the Graph of a Polynomial Function
• How to Determine End Behavior & Intercepts to Graph a Polynomial Function
• How to Determine the End Behavior of a Rational Function
• Finding Axis of Symmetry from the General Form of an Equation
• Determine Equivalent Equations Given Symmetry & Periodicity of Trigonometric functions.
• Determining if Graphs Have Symmetry with Respect to the X-axis, Y-axis, or Origin
F-IF5 - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
• How to Find the Domain and Range From the Graph of a Parabola
• How to Find the Domain & Range from a Linear Graph in a Real World Problem
• Finding Domain & Range from the Graph of a Continuous Function
• How to Get the Domain and Range from the Graph of a Piecewise Function
• Finding Domain and Range From the Graph of an Exponential Function
• How to Graph an Exponential Function & Finding its Domain & Range
F-IF6 - 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.
• How to Find the Average Rate of Change of a Function Given Its Equation
• How to Find the Average Rate of Change of a Function Given Its Graph
Analyze functions using different representations.
F-IF7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF7a - Graph linear and quadratic functions and show intercepts, maxima, and minima.
• How to Graph a Line Given its Equation in Slope-Intercept Form with an Integer Slope
• Finding the Vertex, Intercepts, & Axis of Symmetry From the Graph of a Parabola
• Finding the Vertex, Intercepts, & Axis of Symmetry From the Graph of a Parabola
• Finding the X-intercept(s) and Vertex of a Parabola
• How to Graph a Line Given its Equation in Slope-Intercept Form with a Fractional Slope
• Finding the X-intercept(s) and Vertex of a Parabola
• How to Graph a Line Given its Equation in Standard Form
• How to Graph a Vertical or Horizontal Line
• How to Find X- & Y-intercepts Given the Graph of a Line on a Grid
• How to Find X- and Y-intercepts of a Line Given the Equation
• How to Graph a Line by First Finding its X- and Y-intercepts
• Graphing a Line Given its Slope and Y-intercept
• Graphing a Line Through a Given Point with a Given Slope
• How to Graph a Line Given its X- and Y-Intercepts
• How to Graph a Parabola of the Form Y = Ax2
• How to Graph a Parabola of the Form y = ax^2 + c
• How to Graph a Function of the Form F(x) = Ax2
• How to Graph a Parabola of the Form Y = (x-h)2 + K
• Translating the Graph of a Parabola with 1 Translation
• How to Graph a Parabola of the Form f(x) = ax^2 + bx + c with Integer Coefficients
• How to Find the X-intercept(s) & Vertex of a Quadratic Function on a Graphing Calculator
• Solving a Quadratic Equation by Graphing
F-IF7b - Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
• How to Graph Absolute Value Equation of the Form Y = A|x|
• How to Graph an Absolute Value Equation of the Form Y = A|x+b|in the Plane
• How to Graph an Absolute Value Equation of the Form Y = A|x-b|+c in the Plane
• How to Graph a Piecewise-defined Function: F(x) = a for Each Defined Region of X
• How to Graph a Piecewise-defined Function: F(x) = Ax + B for Each Defined Region of X
• How to Graph a Basic Square Root Function
• How to Graph an Advanced Square Root Function
• Graphing a Cube Root Function
F-IF8 - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F-IF8a - Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
• Rewriting an Exponential Expression as A x B^t
• How to Find the Maximum Or Minimum of a Quadratic Function
• Finding the X-intercept(s) and Vertex of a Parabola
• How to Find the Maximum Or Minimum of a Quadratic Function
• How to Find the Zeros of a Quadratic Function Given Its Equation
• Finding the X-intercept(s) and Vertex of a Parabola
• Finding Equivalent Forms of Exponential Expressions
• How to Rewrite a Quadratic Function to Find Its Vertex and Sketch Its Graph
• How to Rewrite a Quadratic Function to Find Its Vertex and Sketch Its Graph
• Finding the Time to Reach a Limit in a Word Problem on Exponential Growth
• How to Find the Zeros of a Quadratic Function Given Its Equation
• Finding the Time to Reach a Limit in a Word Problem on Exponential Decay
• Finding the Time to Reach a Limit Given an Exponential Function with Base e for a Real-world Situation
F-IF8b - Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^12t, and y = (1.2)^t/10, and classify them as representing exponential growth or decay.
• Finding the Initial Amount & Rate of Change with an Exponential Function
• Finding the Initial Amount & Rate of Change with an Exponential Function
F-IF9 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
• Comparing Properties of Linear Functions in Different Forms
• Comparing Properties of Quadratic Functions Given in Equation & Table Forms
• Comparing Properties of Quadratic Functions Given in Equation & Graph Forms
• Comparing Properties of Quadratic Functions Given in Graph & Table Forms
• Comparing Properties of 2 Functions that are Represented Algebraically & Graphically
• Comparing Properties of Functions Represented Algebraically & Numerically in Tables
• Comparing Properties of 2 Functions Each Represented Algebraically or by Verbal Descriptions
Building Functions
Build a function that models a relationship between two quantities.
F-BF1 - Write a function that describes a relationship between two quantities.
F-BF1a - Determine an explicit expression, a recursive process, or steps for calculation from a context.
• How to Write the Equation of a Quadratic Function Given Its Graph
• How to Write a Quadratic Equation Given the Roots & the Leading Coefficient
• How to Write a Quadratic Equation Given the Roots & the Leading Coefficient
• Identifying an Arithmetic Sequence and Writing an Explicit Rule
• Writing a Quadratic Function Given Its Zeros
• Writing a Quadratic Function Given Its Zeros
• Identifying an Geometric Sequence and Writing an Explicit Rule
• Writing an Equation that Models Exponential Growth Or Decay
• Writing an Equation that Models Exponential Growth Or Decay
• Writing Recursive Rules For Arithmetic Sequences
• Writing an Exponential Function Rule Given a Table of Ordered Pairs
• Writing an Exponential Function Rule Given a Table of Ordered Pairs
• Writing Recursive Rules For Geometric Sequences
F-BF1b - Combine standard function types using arithmetic operations.
• Combining Functions to Write a New Function that Models a Real-World Situation
• How to Find the Sum, Difference, and Product of Two Functions
• Finding the Quotient of Two Functions
Build new functions from existing functions
F-BF3 - Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), 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.
• How to Compare Linear Functions to the Parent Function Y=X
• Translating the Graph of a Parabola with 1 Translation
• Translating the Graph of a Parabola with 1 Translation
• Translating the Graph of a Parabola with 2 Translations
• Understanding How the Leading Coefficient Affects the Shape of a Parabola
• How to Graph Quadratic Functions Y=Ax^2 & Y=(Bx)^2 by Transforming the Parent Graph Y=X^2
• Translating the Graph of an Absolute Value Function with 1 Translation
• Translating the Graph of an Absolute Value Function with 2 Translations
• Understanding How the Leading Coefficient Affects the Graph of an Absolute Value Function
• Writing an Equation for a Function After a Vertical Translation
• How to Write an Equation for a Quadratic Function After a Vertical and Horizontal Translation
• How to Reflect the Graph of a Function Vertically
• How to Reflect the Graph of a Function Horizontally
• How to Change the Scale of the Graph of a Function Vertically
• How to Change the Scale of the Graph of a Function Horizontally
• How to Write an Equation for a Cubic Function After a Vertical and a Horizontal Translation
• How to Tell if a Function is Even or Odd
• How to Determine if a Non-Polynomial Function is Even or Odd
F-BF4 - Find inverse functions.
• How to Find the Inverse of a Quadratic Function & Square Root Function
• How to Find the Inverse of a Cubic Function & a Cube Root Function
• How to Find the Inverse of a Rational Function
• Determining Whether 2 Functions Are Inverses
• Finding the Inverse Function of a Linear Relationship
• Evaluating & Interpreting The Inverse Function of a Linear Relationship
F-BF4a - Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2x^3.
Construct and compare linear, quadratic, and exponential models and solve problems.
F-LE3 - Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly,
• Comparing Linear, Polynomial and Exponential Functions
Interpret expressions for functions in terms of the situation they model.
F-LE6 - Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity. CA
• Solving a Word Problem Using a Quadratic Equation with Rational Roots
• How to Solve a Word Problem Using a Quadratic Equation with Irrational Roots
Trigonometric Functions
Prove and apply trigonometric identities.
F-TF8 - Prove the Pythagorean identity sin^2(x) + cos^2(x) = 1 and use it to find sin(x), cos(x), or tan(x) given sin(x), cos(x), or tan(x) and the quadrant of the angle.
• How to Find the Values of Trigonometric Functions Given Information About an Angle
• How to Find the Values of Trigonometric Functions & their Reciprocals
• How to Find the Values of Trigonometric Functions in any Quadrant
• Using the Pythagorean Identity to Solve For Angles
• How to Find the Values of Trigonometric Functions and Determine the Quadrant
Geometry
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Congruence
Prove geometric theorems.
G-CO9 - Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
• How to Justify a Statement in a Geometric Proof
• How to Write Proofs Involving Segment Congruence
• Identifying Facts About the Angles Created When Parallel Lines are Cut by a Transversal
• How to Solve Proofs Involving Parallel Lines
• Completing Proofs Involving Points on the Perpendicular Bisector of a Line Segment
• How to Find the Interior Angles of a Triangle
G-CO10 - Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
• How to Find the Exterior Angles of a Triangle
• Completing Proofs Involving Congruent Triangles Using SSS
• Completing Proofs Involving Congruent Triangles Using SAS
• How to Prove Triangles are Congruent Using SSS
• How to Prove Triangles are Congruent Using SAS
• Completing Proofs Involving Congruent Triangles Using ASA or AAS
• How to Prove Triangles are Congruent Using ASA or AAS
• Completing Proofs Involving Congruent Triangles & Segment or Angle Bisectors
• Completing Proofs Involving Congruent Triangles that Overlap
• Completing Proofs Involving Congruent Triangles with Parallel or Perpendicular Segments
• How to Prove Triangles are Congruent Using the HL Property
• Completing Proofs Involving Congruent Triangles and CPCTC
• How to do Proofs of Theorems Involving Isosceles Triangles
• Proving the Triangle Midsegment Theorem in the Coordinate Plane
G-CO11 - Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
• Completing Proofs of Theorems Involving Sides of a Parallelogram
• Completing Proofs of Theorems Involving Angles of a Parallelogram
Similarity, Right Triangles, and Trigonometry
Understand similarity in terms of similarity transformations.
G-SRT1 - Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
• How to Dilate a Line Segment & Give the Coordinates of its Endpoints
G-SRT1a - A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
• How to Dilate a Figure
• Determining the Effect of Dilation on Side Length
G-SRT1b - The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
• Identifying Similar or Congruent Shapes on a Grid
G-SRT2 - Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
• How to Find the Measure of an Angle of a Triangle Given 2 Angles of a Similar Triangle
• Examining Triangle Similarity in Terms of Similarity Transformations
• Examining Triangle Similarity in Terms of Similarity Transformations
• Examining Triangle Similarity in Terms of Similarity Transformations
• Determining if Figures are Congruent & Related by a Transformation
• Determining if Figures are Congruent & Related by a Sequence of Transformations
G-SRT3 - Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.
• Finding Missing Angles in Similar Triangles
Prove theorems involving similarity.
G-SRT4 - Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
• Completing Proofs Involving the Triangle Proportionality Theorem
• Proving the Pythagorean Theorem Using Similar Triangles
• How to Use an Informal Proof of the Pythagorean Theorem
G-SRT5 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
• How to Identify & Name Congruent Parts of Congruent Triangles
• Completing Proofs Involving Congruent Triangles Using SSS
• How to Find the Measure of an Angle of a Triangle Given 2 Angles of a Similar Triangle
• Identifying and Naming Congruent Triangles
• How to Find the Measure of an Angle of a Triangle Given 2 Angles of a Similar Triangle
• Completing Proofs Involving Congruent Triangles Using SAS
• Completing Proofs Involving Congruent Triangles Using SSS
• Finding Angle Measures & Side Ratios to Determine if Two Triangles are Similar
• How to Prove Triangles are Congruent Using SSS
• Completing Proofs Involving Congruent Triangles Using SAS
• How to Identify Similar Polygons
• How to Prove Triangles are Congruent Using SAS
• How to Prove Triangles are Congruent Using SSS
• Solving Similar Triangles Given 2 Similar Triangles, Sides, and Angles
• Completing Proofs Involving Congruent Triangles Using ASA or AAS
• How to Prove Triangles are Congruent Using SAS
• Solving Similar Polygons Given Two Similar Figures and Sides and Angles
• How to Prove Triangles are Congruent Using ASA or AAS
• Completing Proofs Involving Congruent Triangles Using ASA or AAS
• How to Identify Similar Right Triangles
• How to Prove Triangles are Congruent Using ASA or AAS
• Identifying Similar Right Triangles that Overlap
• How to Solve the Geometric Mean with Right Triangles
• Using Trigonometric Ratios with Similar Right Triangles
• How to Use Indirect Measurements
• Solving Proportional Parts in Triangles and Parallel Lines
• Identifying and Naming Similar Triangles
• How to Find Trigonometric Ratios in Similar Right Triangles
Define trigonometric ratios and solve problems involving right triangles.
G-SRT6 - Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
• Using Trigonometric Ratios with Similar Right Triangles
• Using Trigonometric Ratios with Similar Right Triangles
• How to Find Trigonometric Ratios in Similar Right Triangles
• How to Find Trigonometric Ratios in Similar Right Triangles
G-SRT7 - Explain and use the relationship between the sine and cosine of complementary angles.
• Using the Relationship Between the Sines & Cosines of Complementary Angles
• Using Cofunction Identities
G-SRT8 - Use trigonometric ratios and Pythagorean Theorem to solve right triangles in applied problems.
• How to Use the Pythagorean Theorem
• How to Use the Converse of the Pythagorean Theorem
• How to Solve a Word Problem Involving the Pythagorean Theorem
• Using the Pythagorean Theorem Repeatedly
• How to Identify Side Lengths that Give Right Triangles
• How to Find the Area of a Right Triangle Using the Pythagorean Theorem
• How to Use the Pythagorean Theorem to Find a Trigonometric Ratio
• How to Find Trigonometric Ratios Given a Right Triangle
• Using a Trigonometric Ratio to Find a Side Length in a Right Triangle
• Using the Pythagorean Theorem to Find Distance on a Grid
• How to Find a Length in a Word Problem with 1 Right Triangle Using Trigonometry
• How to Find a Length in a Word Problem with 2 Right Triangles Using Trigonometry
• How to Find Angles of Elevation in a Word Problem Using Trigonometry
• How to Find Angles of Depression in a Word Problem Using Trigonometry
• How to Solve a Right Triangle
• How to Solve a Right Triangle in a Word Problem
• Using Trigonometry to Find the Area of a Right Triangle
G-SRT8.1 - Derive and use the trigonometric ratios for special right triangles (30 degrees, 60 degrees, 90 degrees and 45 degrees, 45 degrees, 90 degrees). CA
• How to Solve Special Right Triangles
• How to Find the Area of a Regular Polygon Using Special Right Triangles
Circles
Understand and apply theorems about circles.
G-C1 - Prove that all circles are similar.
G-C2 - Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
• Identifying Chords, Secants, and Tangents of a Circle
• How to Find the Tangent of a Circle
• Naming & Finding Measures of Central Angles of a Circle
• Naming & Finding Measures of Inscribed Angles of a Circle
• Naming & Finding Measures of Arcs of a Circle
• Applying Properties of Radii, Diameters & Chords
• How to Find the Central Angles and Inscribed Angles of a Circle
• How to Find Central Angles & Angles Involving Chords & Tangents of a Circle
• Finding Inscribed Angles in Relation to a Diameter or to a Polygon Inscribed in a Circle
• Using the Inscribed Angle Theorem with Chords & Tangents of a Circle
• Measuring Angles of Intersecting Secants & Tangents
• Finding the Lengths of Two Chords Intersecting in the Interior of a Circle
• Finding Lengths of Two Secants Intersecting in the Exterior of a Circle
• Finding Lengths of a Secant & a Tangent Intersecting in the Exterior of a Circle
G-C3 - Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
• Establishing Facts About the Angles of a Quadrilateral Inscribed in a Circle
• How to Inscribe a Circle in a Triangle
• How to Construct a Circle Circumscribed About a Triangle
• How to Solve Inscribed Quadrilaterals
G-C4 - Construct a tangent line from a point outside a given circle to the circle.
• How to Construct a Tangent of a Circle
Find arc lengths and areas of sectors of circles.
G-C5 - Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Convert between degrees and radians. CA
• How to Find Arc Length and Area of a Sector of a Circle
• How to Find Arc Length
• How to Find Arc Length in Radians
• How to Find the Area of a Sector of a Circle with Exact Answers in Terms of pi
• Converting Ratios of Arc Lengths to Radii & Describing the Result
• How to Convert Degrees to Radians
• How to Convert Radians to Degrees
Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic section.
G-GPE1 - Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
• Identifying the Center & Radius to Graph a Circle Given its Equation in Standard Form
• Identifying the Center and Radius to Graph a Circle Given its Equation in General Form
• How to Derive the Equation of a Circle Using the Pythagorean Theorem
G-GPE2 - Derive the equation of a parabola given a focus and directrix.
• How to Derive the Equation of a Parabola Given its Focus & Directrix
• How to Write the Equation of a Parabola in Vertex Form Given a Vertex & a Point on the Graph
Use coordinates to prove simple geometric theorems algebraically.
G-GPE4 - Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, sqrt (3)) lies on the circle centered at the origin and containing the point (0, 2). [Include simple circle theorems.]
• How to Identify Right Triangles from Coordinates
• Proving the Triangle Midsegment Theorem in the Coordinate Plane
• Proving the Triangle Midsegment Theorem in the Coordinate Plane
• How to Identify Scalene, Isosceles & Equilateral Triangles from Coordinates
• Proving that a Quadrilateral with Given Vertices is a Parallelogram
• Classifying Parallelograms in the Coordinate Plane
• How to Write the Equation of a Circle & Identify Points That Lie on the Circle
G-GPE6 - Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
• Finding a Point that Partitions a Number Line Segment in a Given Ratio
• Finding a Point that Partitions a Segment in the Plane into a Given Ratio
Geometric Measurement and Dimension
Explain volume formulas and use them to solve problems.
G-GMD1 - Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
• How to Find the Circumference of a Circle
• How to Find the Circumference and Area of a Circle
• How to Find the Volume of a Cylinder
• Using Cross Sections to Identify Solids with the Same Volume
• How to Find the Volume of a Cone
• Using Cavalieri's Principle for 2D Figures
• Using Cavalieri's Principle for 3D Figures
G-GMD3 - Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
• How to Find the Volume of a Pyramid
• How to Find the Volume of a Cylinder
• How to Find the Volume of a Cylinder
• How to Find the Volume of a Sphere
• How to Find the Volume of a Cone
• How to Find the Volume of a Cone
• How to Solve a Word Problem Involving the Volume of a Cylinder
• How to Solve a Word Problem Involving the Volume of a Cone
• How to Solve a Word Problem Involving the Volume of a Sphere
G-GMD5 - Know that the effect of a scale factor k greater than zero on length, area, and volume is to multiply each by k, k^2, and k^3, respectively; determine length, area and volume measures using scale factors. CA
• Computing Ratios of Side Lengths for Similar Solids
• Computing Ratios of Surface Areas for Similar Solids
• Computing Ratios of Volumes for Similar Solids
• How to Find the Unknown Side Length Given Two Similar Solids
• How to Find the Surface Area of a Similar Solid
• How to Find the Volume of a Similar Solid
G-GMD6 - Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve real-world and mathematical problems. CA
• How to Use Triangle Inequality to Determine if Side Lengths Form a Triangle
• Using the Triangle Inequality to Determine Possible Lengths of a Third Side
• How to Determine if a Triangle is Possible Based on Given Angle Measures
• How to Determine if Given Measurements Define a Unique Triangle, More than One Triangle or No Triangle
• Finding the Relationship Between Angle Measures & Side Lengths in a Triangle
• Finding the Relationship Between Angle Measures & Side Lengths in 2 Triangles
Statistics and Probability
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Conditional Probability and Rules of Probability
Understand independence and conditional probability and use them to interpret data.
S-CP1 - 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").
• Determining a Sample Space For a Simple Event
• Determining Outcomes For a Simple Event
• Determining a Sample Space For a Compound Event
• Determining Outcomes for a Compound Event
• Determining Outcomes & Event Probability
• Determining Outcomes for Compound Events
• Determining Outcomes for Complements of Events
S-CP2 - Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
• Calculating Probability of Independent Events
• Using Probabilities to Identify Independent Events
S-CP3 - Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
• Computing Basic Conditional Probability
• Using Probabilities to Identify Independent Events
• Using Probabilities to Identify Independent Events
S-CP4 - Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
• How to Construct a Basic Two Way Frequency Table
• How to Construct an Advanced Two Way Frequency Table
• Making an Inference Using a Two Way Frequency Table
• Computing Conditional Probability Using a 2-Way Frequency Table
• Computing Conditional Probability to Make an Inference Using a 2-Way Frequency Table
• Computing Conditional Probability Using a Large 2-Way Frequency Table Conditional Probability
S-CP5 - Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
• Identifying Independent Events Given Descriptions of Experiments
Use the rules of probability to compute probabilities of compound events in a uniform probability model.
S-CP6 - Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.
• Calculating Probability of Dependent Events
• Computing the Conditional Probability of an Event
S-CP7 - Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.
• Calculating Probability Using a Venn Diagram
• Using a Venn Diagram to Understand the Addition Rule For Probability
• Calculating Event Probability Using the Addition Rule
S-CP8 - Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
• Calculating Probabilities of Draws with Replacement
• Calculating Probability of Dependent Events
• Calculating Probability of Dependent Events
• Using a Venn Diagram to Understand the Multiplication Rule for Probability
S-CP9 - Use permutations and combinations to compute probabilities of compound events and solve problems.
• Introduction to Permutations
• Computing Permutations
• Introduction to Combinations
• Computing Combinations
• How to Compute Permutations & Combinations
• How to Compute Missing Values Given an Equivalent Permutation or Combination Equation
• Solving Word Problems Involving Permutations
• Solving Word Problems Involving Combinations
• Finding Probabilities Using Permutations
• Finding Probabilities Using Combinations in One Step
• Using Permutations & Combinations to Calculate a Probability
• Calculating Probabilities of Draws Without Replacement
Using Probability to Make Decisions
Use probability to evaluate outcomes of decisions.
S-MD6 - Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
• Using a Random Number Table to Make a Fair Decision
S-MD7 - Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
• Computing Conditional Probability to Make an Inference Using a 2-Way Frequency Table

This collection of Integrated Pathway Mathematics 2 resources is designed to help students learn and master the fundamental Integrated Pathway Mathematics 2 skills. Our library includes thousands of Integrated Pathway Mathematics 2 practice problems, step-by-step explanations, and video walkthroughs. All materials align with California's common core standards for Integrated Pathway Mathematics 2.

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