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  • South Carolina Standards
  • Mathematics: Pre-Calculus

South Carolina - Mathematics: Pre-Calculus

SC--College- and Career-Ready Standards | Adopted: 2015

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.

PC.AAPR: : Arithmetic with Polynomials and Rational Expressions


PC.AAPR.2: : Know and apply the Division Theorem and the Remainder Theorem for polynomials.

Screenshot of Dividing Polynomials Using Synthetic Division

Dividing Polynomials Using Synthetic Division

Divide a polynomial by dragging the correct numbers into the correct positions for synthetic division. Compare the interpreted polynomial division to the synthetic division. 5 Minute Preview


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Screenshot of Polynomials and Linear Factors

Polynomials and Linear Factors

Create a polynomial as a product of linear factors. Vary the values in the linear factors to see how their connection to the roots of the function. 5 Minute Preview


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PC.AAPR.3: : Graph polynomials identifying zeros when suitable factorizations are available and indicating end behavior. Write a polynomial function of least degree corresponding to a given graph.

Screenshot of Graphs of Polynomial Functions

Graphs of Polynomial Functions

Study the graphs of polynomials up to the fourth degree. Vary the coefficients of the equation and investigate how the graph changes in response. Explore things like intercepts, end behavior, and even near-zero behavior. 5 Minute Preview


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Screenshot of Modeling the Factorization of <em>x</em><sup>2</sup>+<em>bx</em>+<em>c</em>

Modeling the Factorization of x2+bx+c

Factor a polynomial with a leading coefficient equal to 1 using an area model. Use step-by-step feedback to diagnose any mistakes. 5 Minute Preview


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Launch Gizmo
Screenshot of Polynomials and Linear Factors

Polynomials and Linear Factors

Create a polynomial as a product of linear factors. Vary the values in the linear factors to see how their connection to the roots of the function. 5 Minute Preview


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Launch Gizmo
Screenshot of Quadratics in Factored Form

Quadratics in Factored Form

Investigate the factors of a quadratic through its graph and through its equation. Vary the roots of the quadratic and examine how the graph and the equation change in response. 5 Minute Preview


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Screenshot of Quadratics in Vertex Form

Quadratics in Vertex Form

Compare the graph of a quadratic to its equation in vertex form. Vary the terms of the equation and explore how the graph changes in response. 5 Minute Preview


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Launch Gizmo
Screenshot of Roots of a Quadratic

Roots of a Quadratic

Find the root of a quadratic using its graph or the quadratic formula. Explore the graph of the roots and the point of symmetry in the complex plane. Compare the axis of symmetry and graph of the quadratic in the real plane. 5 Minute Preview


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Screenshot of Zap It! Game

Zap It! Game

Adjust the values in a quadratic function, in vertex form or in polynomial form, to "zap" as many data points as possible. 5 Minute Preview


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PC.AAPR.4: : Prove polynomial identities and use them to describe numerical relationships.

Screenshot of Factoring Special Products

Factoring Special Products

Choose the correct steps to factor a polynomial involving perfect-square binomials, differences of squares, or constant factors. Use the feedback to diagnose incorrect steps. 5 Minute Preview


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PC.AAPR.5: : Apply the Binomial Theorem to expand powers of binomials, including those with one and with two variables. Use the Binomial Theorem to factor squares, cubes, and fourth powers of binomials.

Screenshot of Binomial Probabilities

Binomial Probabilities

Find the probability of a number of successes or failures in a binomial experiment using a tree diagram, a bar graph, and direct calculation. 5 Minute Preview


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PC.AREI: : Reasoning with Equations and Inequalities


PC.AREI.7: : Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions.

Screenshot of Solving Equations by Graphing Each Side

Solving Equations by Graphing Each Side

Solve an equation by graphing each side and finding the intersection of the lines. Vary the coefficients in the equation and explore how the graph changes in response. 5 Minute Preview


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Screenshot of Solving Linear Systems (Matrices and Special Solutions)

Solving Linear Systems (Matrices and Special Solutions)

Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview


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Screenshot of Solving Linear Systems (Slope-Intercept Form)

Solving Linear Systems (Slope-Intercept Form)

Solve systems of linear equations, given in slope-intercept form, both graphically and algebraically. Use a draggable green point to examine what it means for an (x, y) point to be a solution of one equation, or of a system of two equations. 5 Minute Preview


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Screenshot of Solving Linear Systems (Standard Form)

Solving Linear Systems (Standard Form)

Solve systems of linear equations, written in standard form. Explore what it means to solve systems algebraically (with substitution or elimination) and graphically. Also, use a draggable green point to see what it means when (x, y) values are solutions of an equation, or of a system of equations. 5 Minute Preview


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PC.AREI.8: : Represent a system of linear equations as a single matrix equation in a vector variable.

Screenshot of Solving Linear Systems (Matrices and Special Solutions)

Solving Linear Systems (Matrices and Special Solutions)

Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview


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PC.AREI.9: : Using technology for matrices of dimension 3 × 3 or greater, find the inverse of a matrix if it exists and use it to solve systems of linear equations.

Screenshot of Solving Linear Systems (Matrices and Special Solutions)

Solving Linear Systems (Matrices and Special Solutions)

Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview


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PC.AREI.11: : Solve an equation of the form f(x)=g(x) graphically by identifying the 𝑥-coordinate(s) of the point(s) of intersection of the graphs of y = f(x) and y = g(x).

Screenshot of Absolute Value Equations and Inequalities

Absolute Value Equations and Inequalities

Solve an inequality involving absolute values using a graph of the absolute-value function. Vary the terms of the absolute-value function and vary the value that you are comparing it to. Then explore how the graph and solution set change in response. 5 Minute Preview


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Screenshot of Cat and Mouse (Modeling with Linear Systems)

Cat and Mouse (Modeling with Linear Systems)

Experiment with a system of two lines representing a cat-and-mouse chase. Adjust the speeds of the cat and mouse and the head start of the mouse, and immediately see the effects on the graph and on the chase. Connect real-world meaning to slope, y-intercept, and the intersection of lines. 5 Minute Preview


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Screenshot of Point-Slope Form of a Line

Point-Slope Form of a Line

Compare the point-slope form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response. 5 Minute Preview


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Screenshot of Solving Equations by Graphing Each Side

Solving Equations by Graphing Each Side

Solve an equation by graphing each side and finding the intersection of the lines. Vary the coefficients in the equation and explore how the graph changes in response. 5 Minute Preview


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Launch Gizmo
Screenshot of Solving Linear Systems (Matrices and Special Solutions)

Solving Linear Systems (Matrices and Special Solutions)

Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Solving Linear Systems (Slope-Intercept Form)

Solving Linear Systems (Slope-Intercept Form)

Solve systems of linear equations, given in slope-intercept form, both graphically and algebraically. Use a draggable green point to examine what it means for an (x, y) point to be a solution of one equation, or of a system of two equations. 5 Minute Preview


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Screenshot of Standard Form of a Line

Standard Form of a Line

Compare the standard form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response. 5 Minute Preview


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PC.ASE: : Structure and Expressions


PC.ASE.1: : Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions.

Screenshot of Compound Interest

Compound Interest

Explore compound interest in-depth, from compounded annually to compounded continuously. In addition, compare the END POINTS graph, with dots that fit an exponential curve, to the ALL TIME graph, which has a more step-like appearance. 5 Minute Preview


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Screenshot of Operations with Radical Expressions

Operations with Radical Expressions

Identify the correct steps to complete operations with a radical expression. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview


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Screenshot of Simplifying Algebraic Expressions I

Simplifying Algebraic Expressions I

Meet Spidro, a quirky critter with an appetite for algebraic expressions! As Spidro's adopted owner, it's your responsibility to feed him so that he grows into… whatever it is that a Spidro grows into. But be careful - Spidro is a picky eater who prefers his food to be as simple as possible. Use the commutative property, distributive property, and the other properties of addition and multiplication to put expressions in simplest (and tastiest) form. 5 Minute Preview


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Launch Gizmo
Screenshot of Simplifying Algebraic Expressions II

Simplifying Algebraic Expressions II

Will you adopt Spidro, Centeon, or Ping Bee? They're three very different critters with one thing in common: a hunger for simplified algebraic expressions! Learn how the distributive property can be used to combine variable terms, producing expressions that will help your pet grow up healthy and strong. You'll become a pro at identifying terms that can be combined – even terms with exponents and multiple variables. With enough practice, you and your pet will be ready for the competitive expression eating circuit. Good luck! 5 Minute Preview


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PC.ASE.2: : Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions.

Screenshot of Dividing Exponential Expressions

Dividing Exponential Expressions

Choose the correct steps to divide exponential expressions. Use the feedback to diagnose incorrect steps. 5 Minute Preview


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Launch Gizmo
Screenshot of Equivalent Algebraic Expressions II

Equivalent Algebraic Expressions II

Continue your meteoric rise in the undersea culinary world in this follow-up to Equivalent Algebraic Expressions I. Make equivalent expressions by using the distributive property forwards and backwards, sort expressions by equivalence, and personally assist Chef Grumpy himself with a project that will bring him (and maybe you) fame and fortune. 5 Minute Preview


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Screenshot of Exponents and Power Rules

Exponents and Power Rules

Choose the correct steps to simplify expressions with exponents using the rules of exponents and powers. Use feedback to diagnose incorrect steps. 5 Minute Preview


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Screenshot of Factoring Special Products

Factoring Special Products

Choose the correct steps to factor a polynomial involving perfect-square binomials, differences of squares, or constant factors. Use the feedback to diagnose incorrect steps. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Modeling the Factorization of <em>ax</em><sup>2</sup>+<em>bx</em>+<em>c</em>

Modeling the Factorization of ax2+bx+c

Factor a polynomial with a leading coefficient greater than 1 using an area model. Use step-by-step feedback to diagnose any mistakes. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Modeling the Factorization of <em>x</em><sup>2</sup>+<em>bx</em>+<em>c</em>

Modeling the Factorization of x2+bx+c

Factor a polynomial with a leading coefficient equal to 1 using an area model. Use step-by-step feedback to diagnose any mistakes. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Multiplying Exponential Expressions

Multiplying Exponential Expressions

Choose the correct steps to multiply exponential expressions. Use the feedback to diagnose incorrect steps. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Simplifying Algebraic Expressions I

Simplifying Algebraic Expressions I

Meet Spidro, a quirky critter with an appetite for algebraic expressions! As Spidro's adopted owner, it's your responsibility to feed him so that he grows into… whatever it is that a Spidro grows into. But be careful - Spidro is a picky eater who prefers his food to be as simple as possible. Use the commutative property, distributive property, and the other properties of addition and multiplication to put expressions in simplest (and tastiest) form. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Simplifying Algebraic Expressions II

Simplifying Algebraic Expressions II

Will you adopt Spidro, Centeon, or Ping Bee? They're three very different critters with one thing in common: a hunger for simplified algebraic expressions! Learn how the distributive property can be used to combine variable terms, producing expressions that will help your pet grow up healthy and strong. You'll become a pro at identifying terms that can be combined – even terms with exponents and multiple variables. With enough practice, you and your pet will be ready for the competitive expression eating circuit. Good luck! 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Simplifying Trigonometric Expressions

Simplifying Trigonometric Expressions

Choose the correct steps to simplify a trigonometric function. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview


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Launch Gizmo
Screenshot of Solving Algebraic Equations II

Solving Algebraic Equations II

Is solving equations tricky? If you know how to isolate a variable, you're halfway there. The other half? Don't do anything to upset the balance of an equation. Join our plucky variable friend as he encounters algebraic equations and a (sometimes grumpy) equal sign. With a little practice, you'll find that solving equations isn't tricky at all. 5 Minute Preview


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Screenshot of Using Algebraic Expressions

Using Algebraic Expressions

Translate algebraic expressions into English phrases, and translate English phrases into algebraic expressions. Read the expression or phrase and select word tiles or symbol tiles to form the corresponding phrase or expression. 5 Minute Preview


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PC.FBF: : Building Functions


PC.FBF.1: : Write a function that describes a relationship between two quantities.

PC.FBF.1.b: : Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations.

Screenshot of Addition and Subtraction of Functions

Addition and Subtraction of Functions

Explore the graphs of two polynomials and the graph of their sum or difference. Vary the coefficients in the polynomials and investigate how the graphs change in response. 5 Minute Preview


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PC.FBF.3: : Describe the effect of the transformations kf (x), f(x) +k, f(x + k), and combinations of such transformations on the graph of y = f(x) for any real number k. Find the value of k given the graphs and write the equation of a transformed parent function given its graph.

Screenshot of Absolute Value with Linear Functions

Absolute Value with Linear Functions

Compare the graph of a linear function, the graph of an absolute-value function, and the graphs of their translations. Vary the coefficients and constants in the functions and investigate how the graphs change in response. 5 Minute Preview


Lesson Info
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Screenshot of Exponential Functions

Exponential Functions

Explore the graph of an exponential function. Vary the coefficient and base of the function and investigate the changes to the graph of the function. 5 Minute Preview


Lesson Info
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Screenshot of Introduction to Exponential Functions

Introduction to Exponential Functions

Explore the graph of the exponential function. Vary the initial amount and base of the function. Investigate the changes to the graph. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Logarithmic Functions

Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Logarithmic Functions: Translating and Scaling

Logarithmic Functions: Translating and Scaling

Vary the values in the equation of a logarithmic function and examine how the graph is translated or scaled. Connect these transformations with the domain of the function, and the asymptote in the graph. 5 Minute Preview


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Launch Gizmo
Screenshot of Quadratics in Vertex Form

Quadratics in Vertex Form

Compare the graph of a quadratic to its equation in vertex form. Vary the terms of the equation and explore how the graph changes in response. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Radical Functions

Radical Functions

Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Rational Functions

Rational Functions

Compare the graph of a rational function to its equation. Vary the terms of the equation and explore how the graph is translated and stretched as a result. Examine the domain on a number line and compare it to the graph of the equation. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Translating and Scaling Functions

Translating and Scaling Functions

Vary the coefficients in the equation of a function and examine how the graph of the function is translated or scaled. Select different functions to translate and scale, and determine what they have in common. 5 Minute Preview


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Screenshot of Translating and Scaling Sine and Cosine Functions

Translating and Scaling Sine and Cosine Functions

Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview


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Screenshot of Translations

Translations

Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation. 5 Minute Preview


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Screenshot of Zap It! Game

Zap It! Game

Adjust the values in a quadratic function, in vertex form or in polynomial form, to "zap" as many data points as possible. 5 Minute Preview


Lesson Info
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PC.FBF.4: : Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, as f and g are inverse functions if and only if f(x)=y and g(y)=x, for all values of x in the domain of f and all values of y in the domain of g, and find inverse functions for one-to-one function or by restricting the domain.

PC.FBF.4.a: : Use composition to verify one function is an inverse of another.

Screenshot of Logarithmic Functions

Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


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PC.FBF.4.b: : If a function has an inverse, find values of the inverse function from a graph or table.

Screenshot of Function Machines 3 (Functions and Problem Solving)

Function Machines 3 (Functions and Problem Solving)

Drop a number into a function machine, and see what number comes out! You can use one of the six pre-set function machines, or program your own function rule into one of the blank machines. Stack up to three function machines together. Input and output can be recorded in a table and on a graph. 5 Minute Preview


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Screenshot of Logarithmic Functions

Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


Lesson Info
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PC.FBF.5: : Understand and verify through function composition that exponential and logarithmic functions are inverses of each other and use this relationship to solve problems involving logarithms and exponents.

Screenshot of Logarithmic Functions

Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


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PC.FIF: : Interpreting Functions


PC.FIF.4: : Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity.

Screenshot of Absolute Value with Linear Functions

Absolute Value with Linear Functions

Compare the graph of a linear function, the graph of an absolute-value function, and the graphs of their translations. Vary the coefficients and constants in the functions and investigate how the graphs change in response. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Compound Interest

Compound Interest

Explore compound interest in-depth, from compounded annually to compounded continuously. In addition, compare the END POINTS graph, with dots that fit an exponential curve, to the ALL TIME graph, which has a more step-like appearance. 5 Minute Preview


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Screenshot of Exponential Functions

Exponential Functions

Explore the graph of an exponential function. Vary the coefficient and base of the function and investigate the changes to the graph of the function. 5 Minute Preview


Lesson Info
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Screenshot of Function Machines 3 (Functions and Problem Solving)

Function Machines 3 (Functions and Problem Solving)

Drop a number into a function machine, and see what number comes out! You can use one of the six pre-set function machines, or program your own function rule into one of the blank machines. Stack up to three function machines together. Input and output can be recorded in a table and on a graph. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of General Form of a Rational Function

General Form of a Rational Function

Compare the equation of a rational function to its graph. Multiply or divide the numerator and denominator by linear factors and explore how the graph changes in response. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Graphs of Polynomial Functions

Graphs of Polynomial Functions

Study the graphs of polynomials up to the fourth degree. Vary the coefficients of the equation and investigate how the graph changes in response. Explore things like intercepts, end behavior, and even near-zero behavior. 5 Minute Preview


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Screenshot of Introduction to Exponential Functions

Introduction to Exponential Functions

Explore the graph of the exponential function. Vary the initial amount and base of the function. Investigate the changes to the graph. 5 Minute Preview


Lesson Info
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Screenshot of Introduction to Functions

Introduction to Functions

Determine if a relation is a function using the mapping diagram, ordered pairs, or the graph of the relation. Drag arrows from the domain to the range, type in ordered pairs, or drag points to the graph to add inputs and outputs to the relation. 5 Minute Preview


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Screenshot of Logarithmic Functions

Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


Lesson Info
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Screenshot of Points, Lines, and Equations

Points, Lines, and Equations

Compare the graph of a linear function to its rule and to a table of its values. Change the function by dragging two points on the line. Examine how the rule and table change. 5 Minute Preview


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Screenshot of Quadratics in Factored Form

Quadratics in Factored Form

Investigate the factors of a quadratic through its graph and through its equation. Vary the roots of the quadratic and examine how the graph and the equation change in response. 5 Minute Preview


Lesson Info
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Screenshot of Quadratics in Polynomial Form

Quadratics in Polynomial Form

Compare the graph of a quadratic to its equation in polynomial form. Vary the coefficients of the equation and explore how the graph changes in response. 5 Minute Preview


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Screenshot of Quadratics in Vertex Form

Quadratics in Vertex Form

Compare the graph of a quadratic to its equation in vertex form. Vary the terms of the equation and explore how the graph changes in response. 5 Minute Preview


Lesson Info
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Screenshot of Radical Functions

Radical Functions

Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation. 5 Minute Preview


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PC.FIF.5: : Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes.

Screenshot of General Form of a Rational Function

General Form of a Rational Function

Compare the equation of a rational function to its graph. Multiply or divide the numerator and denominator by linear factors and explore how the graph changes in response. 5 Minute Preview


Lesson Info
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Screenshot of Introduction to Functions

Introduction to Functions

Determine if a relation is a function using the mapping diagram, ordered pairs, or the graph of the relation. Drag arrows from the domain to the range, type in ordered pairs, or drag points to the graph to add inputs and outputs to the relation. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Logarithmic Functions

Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Radical Functions

Radical Functions

Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Rational Functions

Rational Functions

Compare the graph of a rational function to its equation. Vary the terms of the equation and explore how the graph is translated and stretched as a result. Examine the domain on a number line and compare it to the graph of the equation. 5 Minute Preview


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PC.FIF.6: : Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context.

Screenshot of Cat and Mouse (Modeling with Linear Systems)

Cat and Mouse (Modeling with Linear Systems)

Experiment with a system of two lines representing a cat-and-mouse chase. Adjust the speeds of the cat and mouse and the head start of the mouse, and immediately see the effects on the graph and on the chase. Connect real-world meaning to slope, y-intercept, and the intersection of lines. 5 Minute Preview


Lesson Info
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Screenshot of Slope

Slope

Explore the slope of a line, and learn how to calculate slope. Adjust the line by moving points that are on the line, and see how its slope changes. 5 Minute Preview


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PC.FIF.7: : Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases.

PC.FIF.7.a: : Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

Screenshot of General Form of a Rational Function

General Form of a Rational Function

Compare the equation of a rational function to its graph. Multiply or divide the numerator and denominator by linear factors and explore how the graph changes in response. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Rational Functions

Rational Functions

Compare the graph of a rational function to its equation. Vary the terms of the equation and explore how the graph is translated and stretched as a result. Examine the domain on a number line and compare it to the graph of the equation. 5 Minute Preview


Lesson Info
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PC.FIF.7.b: : Graph radical functions over their domain show end behavior.

Screenshot of Radical Functions

Radical Functions

Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.FIF.7.c: : Graph exponential and logarithmic functions, showing intercepts and end behavior.

Screenshot of Cosine Function

Cosine Function

Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
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Screenshot of Exponential Functions

Exponential Functions

Explore the graph of an exponential function. Vary the coefficient and base of the function and investigate the changes to the graph of the function. 5 Minute Preview


Lesson Info
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Screenshot of Introduction to Exponential Functions

Introduction to Exponential Functions

Explore the graph of the exponential function. Vary the initial amount and base of the function. Investigate the changes to the graph. 5 Minute Preview


Lesson Info
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Screenshot of Logarithmic Functions

Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Logarithmic Functions: Translating and Scaling

Logarithmic Functions: Translating and Scaling

Vary the values in the equation of a logarithmic function and examine how the graph is translated or scaled. Connect these transformations with the domain of the function, and the asymptote in the graph. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Sine Function

Sine Function

Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Tangent Function

Tangent Function

Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Translating and Scaling Sine and Cosine Functions

Translating and Scaling Sine and Cosine Functions

Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.FIF.7.d: : Graph trigonometric functions, showing period, midline, and amplitude.

Screenshot of Cosine Function

Cosine Function

Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Exponential Functions

Exponential Functions

Explore the graph of an exponential function. Vary the coefficient and base of the function and investigate the changes to the graph of the function. 5 Minute Preview


Lesson Info
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Screenshot of Introduction to Exponential Functions

Introduction to Exponential Functions

Explore the graph of the exponential function. Vary the initial amount and base of the function. Investigate the changes to the graph. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Logarithmic Functions

Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Logarithmic Functions: Translating and Scaling

Logarithmic Functions: Translating and Scaling

Vary the values in the equation of a logarithmic function and examine how the graph is translated or scaled. Connect these transformations with the domain of the function, and the asymptote in the graph. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Sine Function

Sine Function

Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Tangent Function

Tangent Function

Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Translating and Scaling Functions

Translating and Scaling Functions

Vary the coefficients in the equation of a function and examine how the graph of the function is translated or scaled. Select different functions to translate and scale, and determine what they have in common. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Translating and Scaling Sine and Cosine Functions

Translating and Scaling Sine and Cosine Functions

Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.FLQE: : Linear, Quadratic, and Exponential


PC. FLQE.4: : Express a logarithm as the solution to the exponential equation, ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Screenshot of Compound Interest

Compound Interest

Explore compound interest in-depth, from compounded annually to compounded continuously. In addition, compare the END POINTS graph, with dots that fit an exponential curve, to the ALL TIME graph, which has a more step-like appearance. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Logarithmic Functions

Logarithmic Functions

Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.FT: : Trigonometry


PC.FT.1: : Understand that the radian measure of an angle is the length of the arc on the unit circle subtended by the angle.

Screenshot of Sine Function

Sine Function

Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Tangent Function

Tangent Function

Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.FT.2: : Define sine and cosine as functions of the radian measure of an angle in terms of the x - and y - coordinates of the point on the unit circle corresponding to that angle and explain how these definitions are extensions of the right triangle definitions.

PC.FT.2.a: : Define the tangent, cotangent, secant, and cosecant functions as ratios involving sine and cosine.

Screenshot of Simplifying Trigonometric Expressions

Simplifying Trigonometric Expressions

Choose the correct steps to simplify a trigonometric function. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Sine, Cosine, and Tangent Ratios

Sine, Cosine, and Tangent Ratios

Reshape and resize a right triangle and examine how the sine of angle A, the cosine of angle A, and the tangent of angle A change. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.FT.2.b: : Write cotangent, secant, and cosecant functions as the reciprocals of tangent, cosine, and sine, respectively.

Screenshot of Simplifying Trigonometric Expressions

Simplifying Trigonometric Expressions

Choose the correct steps to simplify a trigonometric function. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Tangent Function

Tangent Function

Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.FT.3: : Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π − x, π+ x, and 2π − x in terms of their values for x, where x is any real number.

Screenshot of Cosine Function

Cosine Function

Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Sine Function

Sine Function

Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Sum and Difference Identities for Sine and Cosine

Sum and Difference Identities for Sine and Cosine

Choose the correct steps to evaluate a trigonometric expression using sum and difference identities. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Tangent Function

Tangent Function

Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Translating and Scaling Sine and Cosine Functions

Translating and Scaling Sine and Cosine Functions

Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.FT.4: : Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Screenshot of Cosine Function

Cosine Function

Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Sine Function

Sine Function

Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Tangent Function

Tangent Function

Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Translating and Scaling Sine and Cosine Functions

Translating and Scaling Sine and Cosine Functions

Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.FT.5: : Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Screenshot of Translating and Scaling Functions

Translating and Scaling Functions

Vary the coefficients in the equation of a function and examine how the graph of the function is translated or scaled. Select different functions to translate and scale, and determine what they have in common. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Translating and Scaling Sine and Cosine Functions

Translating and Scaling Sine and Cosine Functions

Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.FT.8: : Justify the Pythagorean, even/odd, and cofunction identities for sine and cosine using their unit circle definitions and symmetries of the unit circle and use the Pythagorean identity to find sinA, cosA, or tanA, given sinA, cosA, or tanA, and the quadrant of the angle.

Screenshot of Cosine Function

Cosine Function

Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Sine Function

Sine Function

Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.FT.9: : Justify the sum and difference formulas for sine, cosine, and tangent and use them to solve problems.

Screenshot of Sum and Difference Identities for Sine and Cosine

Sum and Difference Identities for Sine and Cosine

Choose the correct steps to evaluate a trigonometric expression using sum and difference identities. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.CI: : Circles


PC.GCI.5: : Derive the formulas for the length of an arc and the area of a sector in a circle, and apply these formulas to solve mathematical and real-world problems.

Screenshot of Chords and Arcs

Chords and Arcs

Explore the relationship between a central angle and the arcs it intercepts. Also explore the relationship between chords and their distance from the circle's center. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.GGPE: : Expressing Geometric Properties with Equations


PC.GGPE.2: : Use the geometric definition of a parabola to derive its equation given the focus and directrix.

Screenshot of Parabolas

Parabolas

Explore parabolas in a conic section context. Find the relationship among the vertex, focus, and directrix of a parabola, and how that relates to its equation. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.GGPE.3: : Use the geometric definition of an ellipse and of a hyperbola to derive the equation of each given the foci and points whose sum or difference of distance from the foci are constant.

Screenshot of Ellipses

Ellipses

Compare the equation of an ellipse to its graph. Vary the terms of the equation of the ellipse and examine how the graph changes in response. Drag the vertices and foci, explore their Pythagorean relationship, and discover the string property. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Hyperbolas

Hyperbolas

Compare the equation of a hyperbola to its graph. Vary the terms of the equation of the hyperbola. Examine how the graph of the hyperbola and its asymptotes changes in response. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NCNS: : Complex Number System


PC.NCNS.2: : Use the relation i²=−1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Screenshot of Points in the Complex Plane

Points in the Complex Plane

Identify the imaginary and real coordinates of a point in the complex plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NCNS.3: : Find the conjugate of a complex number in rectangular and polar forms and use conjugates to find moduli and quotients of complex numbers.

Screenshot of Points in the Complex Plane

Points in the Complex Plane

Identify the imaginary and real coordinates of a point in the complex plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Roots of a Quadratic

Roots of a Quadratic

Find the root of a quadratic using its graph or the quadratic formula. Explore the graph of the roots and the point of symmetry in the complex plane. Compare the axis of symmetry and graph of the quadratic in the real plane. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NCNS.4: : Graph complex numbers on the complex plane in rectangular and polar form and explain why the rectangular and polar forms of a given complex number represent the same number.

Screenshot of Points in the Complex Plane

Points in the Complex Plane

Identify the imaginary and real coordinates of a point in the complex plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NCNS.5: : Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

Screenshot of Points in the Complex Plane

Points in the Complex Plane

Identify the imaginary and real coordinates of a point in the complex plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NCNS.6: : Determine the modulus of a complex number by multiplying by its conjugate and determine the distance between two complex numbers by calculating the modulus of their difference.

Screenshot of Points in the Complex Plane

Points in the Complex Plane

Identify the imaginary and real coordinates of a point in the complex plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NCNS.7: : Solve quadratic equations in one variable that have complex solutions.

Screenshot of Points in the Complex Plane

Points in the Complex Plane

Identify the imaginary and real coordinates of a point in the complex plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Roots of a Quadratic

Roots of a Quadratic

Find the root of a quadratic using its graph or the quadratic formula. Explore the graph of the roots and the point of symmetry in the complex plane. Compare the axis of symmetry and graph of the quadratic in the real plane. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NVMQ: : Vector and Matrix Quantities


PC.NVMQ.1: : Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.

Screenshot of Adding Vectors

Adding Vectors

Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Vectors

Vectors

Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NVMQ.2: : Represent and model with vector quantities. Use the coordinates of an initial point and of a terminal point to find the components of a vector.

Screenshot of Adding Vectors

Adding Vectors

Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Vectors

Vectors

Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NVMQ.3: : Represent and model with vector quantities. Solve problems involving velocity and other quantities that can be represented by vectors.

Screenshot of Adding Vectors

Adding Vectors

Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NVMQ.4: : Perform operations on vectors.

PC.NVMQ.4.a: : Add and subtract vectors using components of the vectors and graphically.

Screenshot of Adding Vectors

Adding Vectors

Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Vectors

Vectors

Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NVMQ.4.b: : Given the magnitude and direction of two vectors, determine the magnitude of their sum and of their difference.

Screenshot of Vectors

Vectors

Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NVMQ.5: : Multiply a vector by a scalar, representing the multiplication graphically and computing the magnitude of the scalar multiple.

Screenshot of Dilations

Dilations

Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in (x, y) form and in matrix form. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Vectors

Vectors

Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NVMQ.7: : Perform operations with matrices of appropriate dimensions including addition, subtraction, and scalar multiplication.

Screenshot of Dilations

Dilations

Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in (x, y) form and in matrix form. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Translations

Translations

Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation. 5 Minute Preview


Lesson Info
Launch Gizmo

PC.NVMQ.11: : Apply 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Screenshot of Dilations

Dilations

Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in (x, y) form and in matrix form. 5 Minute Preview


Lesson Info
Launch Gizmo
Screenshot of Translations

Translations

Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation. 5 Minute Preview


Lesson Info
Launch Gizmo

Correlation last revised: 9/16/2020

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