# Florida - Mathematics:

## Benchmarks for Excellent Student Thinking Standards | Adopted: 2020

### MA.912.NSO: : Number Sense and Operations

MA.912.NSO.1: : Generate equivalent expressions and perform operations with expressions involving exponents, radicals or logarithms.

MA.912.NSO.1.2: : Generate equivalent monomial algebraic expressions using the properties of exponents.

Dividing Exponential Expressions

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

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

Multiplying Exponential Expressions

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

MA.912.NSO.1.3: : Generate equivalent algebraic expressions involving radicals or rational exponents using the properties of exponents. Radicands are limited to monomial algebraic expressions.

Simplifying Radical Expressions

Simplify a radical expression. Use step-by-step feedback to diagnose any incorrect steps. 5 Minute Preview

MA.912.NSO.1.4: : Apply previous understanding of operations with rational numbers to add, subtract, multiply and divide numerical radicals.

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

Simplifying Radical Expressions

Simplify a radical expression. Use step-by-step feedback to diagnose any incorrect steps. 5 Minute Preview

MA.912.NSO.1.5: : Add, subtract, multiply and divide algebraic expressions involving radicals. Radicands are limited to monomial algebraic expressions.

Simplifying Radical Expressions

Simplify a radical expression. Use step-by-step feedback to diagnose any incorrect steps. 5 Minute Preview

MA.912.NSO.1.6: : Given an algebraic logarithmic expression, generate an equivalent algebraic expression using the properties of logarithms or exponents.

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

MA.912.NSO.2: : Represent and perform operations with expressions within the complex number system.

MA.912.NSO.2.1: : Extend previous understanding of the real number system to include the complex number system. Add, subtract, multiply and divide complex numbers.

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

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

MA.912.NSO.2.2: : Represent addition, subtraction, multiplication and conjugation of complex numbers geometrically on 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

MA.912.NSO.2.3: : Calculate the distance and midpoint between two numbers on the complex coordinate 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

MA.912.NSO.2.4: : Solve mathematical and real-world problems involving complex numbers represented algebraically or on the coordinate plane.

Points in the Complex Plane

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

MA.912.NSO.2.5: : Represent complex numbers on the complex plane in rectangular and polar forms. Explain why the rectangular and polar forms of a given complex number represent the same number.

Points in the Complex Plane

MA.912.NSO.2.6: : Rewrite complex numbers to trigonometric form. Multiply complex numbers in trigonometric form.

Points in the Complex Plane

MA.912.NSO.3: : Represent and perform operations with vectors.

MA.912.NSO.3.1: : Apply appropriate notation and symbols to represent vectors in the plane as directed line segments. Determine the magnitude and direction of a vector in component form.

Adding Vectors

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

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

MA.912.NSO.3.2: : Represent vectors in component form, linear form or trigonometric form. Rewrite vectors from one form to another.

Adding Vectors

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

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

MA.912.NSO.3.3: : Solve mathematical and real-world problems involving velocity and other quantities that can be represented by vectors.

Adding Vectors

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

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

MA.912.NSO.3.4: : Solve mathematical and real-world problems involving vectors in two dimensions using the dot product and vector projections.

Vectors

MA.912.NSO.3.6: : Add and subtract vectors algebraically or graphically.

Adding Vectors

Vectors

MA.912.NSO.3.7: : Given the magnitude and direction of two or more vectors, determine the magnitude and direction of their sum.

Adding Vectors

Vectors

MA.912.NSO.3.8: : Multiply a vector by a scalar algebraically or graphically.

Vectors

MA.912.NSO.3.9: : Compute the magnitude and direction of a vector scalar multiple.

Vectors

MA.912.NSO.4: : Represent and perform operations with matrices.

MA.912.NSO.4.1: : Given a mathematical or real-world context, represent and manipulate data using matrices.

Dilations

Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in

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

Translations

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

MA.912.NSO.4.2: : Given a mathematical or real-world context, represent and solve a system of two- or three-variable linear equations using matrices.

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

MA.912.NSO.4.3: : Solve mathematical and real-world problems involving addition, subtraction and multiplication of matrices.

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

Translations

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

MA.912.NSO.4.4: : Solve mathematical and real-world problems using the inverse and determinant of matrices.

Solving Linear Systems (Matrices and Special Solutions)

*x*, *y*) point to be a solution of an equation, or of a system of equations.
5 Minute Preview

### MA.912.AR: : Algebraic Reasoning

MA.912.AR.1: : Interpret and rewrite algebraic expressions and equations in equivalent forms.

MA.912.AR.1.1: : Identify and interpret parts of an expression that represent a quantity in terms of a mathematical or real-world context, including viewing one or more of its parts as a single entity.

Arithmetic Sequences

Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response. 5 Minute Preview

Arithmetic and Geometric Sequences

Find the value of individual terms in an arithmetic or geometric sequence using graphs of the sequence and direct computation. Vary the common difference and common ratio and examine how the sequence changes in response. 5 Minute Preview

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

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

Geometric Sequences

Explore geometric sequences by varying the initial term and the common ratio and examining the graph. Compute specific terms in the sequence using the explicit and recursive formulas. 5 Minute Preview

Permutations and Combinations

Experiment with permutations and combinations of a number of letters represented by letter tiles selected at random from a box. Count the permutations and combinations using a dynamic tree diagram, a dynamic list of permutations, and a dynamic computation by the counting principle. 5 Minute Preview

MA.912.AR.1.2: : Rearrange equations or formulas to isolate a quantity of interest.

Solving Formulas for any Variable

Choose the correct steps to solve a formula for a given variable. Use the feedback to diagnose incorrect steps. 5 Minute Preview

MA.912.AR.1.3: : Add, subtract and multiply polynomial expressions with rational number coefficients.

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

Addition of Polynomials

Add polynomials using an area model. Use step-by-step feedback to diagnose any mistakes. 5 Minute Preview

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

Modeling the Factorization of *ax*^{2}+*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

Modeling the Factorization of *x*^{2}+*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

MA.912.AR.1.5: : Divide polynomial expressions using long division, synthetic division and algebraic manipulation.

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

MA.912.AR.1.6: : Solve mathematical and real-world problems involving addition, subtraction, multiplication or division of polynomials.

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

Addition of Polynomials

Add polynomials using an area model. Use step-by-step feedback to diagnose any mistakes. 5 Minute Preview

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

Modeling the Factorization of *ax*^{2}+*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

Modeling the Factorization of *x*^{2}+*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

MA.912.AR.1.7: : Rewrite a polynomial expression as a product of polynomials.

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

Modeling the Factorization of *ax*^{2}+*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

Modeling the Factorization of *x*^{2}+*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

MA.912.AR.2: : Write, solve and graph linear equations, functions and inequalities in one and two variables.

MA.912.AR.2.1: : Given a real-world context, write and solve one-variable multi-step linear equations.

Modeling and Solving Two-Step Equations

Solve a two-step equation using a cup-and-counter model. Use step-by-step feedback to diagnose and correct incorrect steps. 5 Minute Preview

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

Solving Equations on the Number Line

Solve an equation involving decimals using dynamic arrows on a number line. 5 Minute Preview

Solving Two-Step Equations

Choose the correct steps to solve a two-step equation. Use the feedback to diagnose incorrect steps. 5 Minute Preview

MA.912.AR.2.2: : Write a linear two-variable equation to represent relationships between quantities from a graph, a written description or a table of values within a mathematical or real-world context.

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

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

Slope-Intercept Form of a Line

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

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

MA.912.AR.2.3: : Write a linear two-variable equation for a line that is parallel or perpendicular to a given line and goes through a given point.

Solving Linear Systems (Matrices and Special Solutions)

*x*, *y*) point to be a solution of an equation, or of a system of equations.
5 Minute Preview

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*)

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

MA.912.AR.2.4: : Given a table, equation or written description of a linear function, graph that function, and determine and interpret its key features.

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

Slope-Intercept Form of a Line

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

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

MA.912.AR.2.5: : Solve and graph mathematical and real-world problems that are modeled with linear functions. Interpret key features and determine domain constraints in terms of the context.

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

Solving Linear Systems (Matrices and Special Solutions)

*x*, *y*) point to be a solution of an equation, or of a system of equations.
5 Minute Preview

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

MA.912.AR.2.6: : Given a mathematical or real-world context, write and solve one-variable linear inequalities, including compound inequalities. Represent solutions algebraically or graphically.

Compound Inequalities

Explore the graphs of two inequalities and find their union or intersection. Determine the relationship between the endpoints of the inequalities and the endpoints of the compound inequality. 5 Minute Preview

Exploring Linear Inequalities in One Variable

Solve inequalities in one variable. Examine the inequality on a number line and determine which points are solutions to the inequality. 5 Minute Preview

Solving Linear Inequalities in One Variable

Solve one-step inequalities in one variable. Graph the solution on a number line. 5 Minute Preview

MA.912.AR.2.7: : Write two-variable linear inequalities to represent relationships between quantities from a graph or a written description within a mathematical or real-world context.

Linear Inequalities in Two Variables

Find the solution set to a linear inequality in two variables using the graph of the linear inequality. Vary the terms of the inequality and vary the inequality symbol. Examine how the boundary line and shaded region change in response. 5 Minute Preview

MA.912.AR.2.8: : Given a mathematical or real-world context, graph the solution set to a two-variable linear inequality.

Linear Inequalities in Two Variables

Find the solution set to a linear inequality in two variables using the graph of the linear inequality. Vary the terms of the inequality and vary the inequality symbol. Examine how the boundary line and shaded region change in response. 5 Minute Preview

MA.912.AR.3: : Write, solve and graph quadratic equations, functions and inequalities in one and two variables.

MA.912.AR.3.1: : Given a mathematical or real-world context, write and solve one-variable quadratic equations over the real number system.

Points in the Complex Plane

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

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

MA.912.AR.3.2: : Given a mathematical or real-world context, write and solve one-variable quadratic equations over the real and complex number systems.

Points in the Complex Plane

Roots of a Quadratic

MA.912.AR.3.3: : Given a mathematical or real-world context, write and solve one-variable quadratic inequalities over the real number system. Represent solutions algebraically or graphically.

Quadratic Inequalities

Find the solution set to a quadratic inequality using its graph. Vary the terms of the inequality and the inequality symbol. Examine how the boundary curve and shaded region change in response. 5 Minute Preview

MA.912.AR.3.4: : Write a quadratic function to represent the relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context.

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

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

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

MA.912.AR.3.5: : Given the x-intercepts and another point on the graph of a quadratic function, write the equation for the function.

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

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

MA.912.AR.3.6: : Given an expression or equation representing a quadratic function, determine the vertex and zeros and interpret them in terms of a real-world context.

Quadratics in Factored 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

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

Roots of a Quadratic

MA.912.AR.3.7: : Given a table, equation or written description of a quadratic function, graph that function, and determine and interpret its key features.

Quadratics in Factored Form

Quadratics in Polynomial 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

Roots of a Quadratic

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

MA.912.AR.3.8: : Solve and graph mathematical and real-world problems that are modeled with quadratic functions. Interpret key features and determine constraints in terms of the context.

Quadratics in Polynomial Form

MA.912.AR.3.9: : Given a mathematical or real-world context, write two-variable quadratic inequalities to represent relationships between quantities from a graph or a written description.

Quadratic Inequalities

Find the solution set to a quadratic inequality using its graph. Vary the terms of the inequality and the inequality symbol. Examine how the boundary curve and shaded region change in response. 5 Minute Preview

MA.912.AR.3.10: : Given a mathematical or real-world context, graph the solution set to a two-variable quadratic inequality.

Quadratic Inequalities

Find the solution set to a quadratic inequality using its graph. Vary the terms of the inequality and the inequality symbol. Examine how the boundary curve and shaded region change in response. 5 Minute Preview

MA.912.AR.4: : Write, solve and graph absolute value equations, functions and inequalities in one and two variables.

MA.912.AR.4.1: : Given a mathematical or real-world context, write and solve one-variable absolute value equations.

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

MA.912.AR.4.2: : Given a mathematical or real-world context, write and solve one-variable absolute value inequalities. Represent solutions algebraically or graphically.

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

MA.912.AR.4.3: : Given a table, equation or written description of an absolute value function, graph that function and determine its key features.

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

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

MA.912.AR.4.4: : Solve and graph mathematical and real-world problems that are modeled with absolute value functions. Interpret key features and determine domain constraints in terms of the context.

Absolute Value Equations and Inequalities

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

MA.912.AR.5: : Write, solve and graph exponential and logarithmic equations and functions in one and two variables.

MA.912.AR.5.1: : Solve one-variable exponential equations using the properties of exponents.

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

MA.912.AR.5.2: : Solve equations involving one-variable logarithms or exponents. Interpret solutions as viable in terms of the context and identify any extraneous solutions.

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

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

MA.912.AR.5.3: : Given a mathematical or real-world context, classify an exponential function as representing growth or decay.

Exponential Growth and Decay

Explore the graph of the exponential growth or decay function. Vary the initial amount and the rate of growth or decay and investigate the changes to the graph. 5 Minute Preview

MA.912.AR.5.4: : Write an exponential function to represent a relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context.

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

Exponential Growth and Decay

Explore the graph of the exponential growth or decay function. Vary the initial amount and the rate of growth or decay and investigate the changes to the graph. 5 Minute Preview

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

MA.912.AR.5.5: : Given an expression or equation representing an exponential function, reveal the constant percent rate of change per unit interval using the properties of exponents. Interpret the constant percent rate of change in terms of a real-world context.

Exponential Growth and Decay

Explore the graph of the exponential growth or decay function. Vary the initial amount and the rate of growth or decay and investigate the changes to the graph. 5 Minute Preview

MA.912.AR.5.6: : Given a table, equation or written description of an exponential function, graph that function and determine its key features.

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

Exponential Growth and Decay

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

MA.912.AR.5.7: : Solve and graph mathematical and real-world problems that are modeled with exponential functions. Interpret key features and determine domain constraints in terms of the context.

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

Exponential Growth and Decay

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

MA.912.AR.5.8: : Given a table, equation or written description of a logarithmic function, graph that function and determine its key features.

Logarithmic Functions

*y* = *x* to compare the associated exponential function.
5 Minute Preview

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

MA.912.AR.5.9: : Solve and graph mathematical and real-world problems that are modeled with logarithmic functions. Interpret key features and determine constraints in terms of the context.

Logarithmic Functions

*y* = *x* to compare the associated exponential function.
5 Minute Preview

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

MA.912.AR.6: : Solve and graph polynomial equations and functions in one and two variables.

MA.912.AR.6.1: : Given a mathematical or real-world context, when suitable factorization is possible, solve one-variable polynomial equations of degree 3 or higher over the real and complex number systems.

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

MA.912.AR.6.2: : Explain and apply the Remainder Theorem.

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

MA.912.AR.6.3: : Given a table, equation or written description of a polynomial function of degree 3 or higher, graph that function and determine its key features.

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

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

MA.912.AR.6.4: : Sketch a rough graph of a polynomial function of degree 3 or higher using zeros, multiplicity and knowledge of end behavior.

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

MA.912.AR.6.5: : Solve and graph mathematical and real-world problems that are modeled with polynomial functions of degree 3 or higher. Interpret key features in terms of the context.

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

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

MA.912.AR.7: : Solve and graph radical equations and functions in one and two variables.

MA.912.AR.7.1: : Solve one-variable radical equations. Interpret solutions as viable in terms of context and identify any extraneous solutions.

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

MA.912.AR.7.2: : Given a table, equation or written description of a square root or cube root function, graph that function and determine its key features.

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

MA.912.AR.7.3: : Solve and graph mathematical and real-world problems that are modeled with square root or cube root functions. Interpret key features in context.

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

MA.912.AR.8: : Solve and graph rational equations and functions in one and two variables.

MA.912.AR.8.1: : Write and solve one-variable rational equations. Interpret solutions as viable in terms of the context and identify any extraneous solutions.

Direct and Inverse Variation

Adjust the constant of variation and explore how the graph of the direct or inverse variation function changes in response. Compare direct variation functions to inverse variation functions. 5 Minute Preview

MA.912.AR.8.2: : Given a table, equation or written description of a rational function, graph that function and determine its key features.

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

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

MA.912.AR.8.3: : Solve and graph mathematical and real-world problems that are modeled with rational functions. Interpret key features in terms of the context.

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

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

MA.912.AR.9: : Write and solve a system of two- and three-variable equations and inequalities that describe quantities or relationships.

MA.912.AR.9.1: : Given a mathematical or real-world context, write and solve a system of two-variable linear equations algebraically or graphically.

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

Solving Linear Systems (Matrices and Special Solutions)

*x*, *y*) point to be a solution of an equation, or of a system of equations.
5 Minute Preview

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*)

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

MA.912.AR.9.4: : Graph the solution set of a system of two-variable linear inequalities.

Systems of Linear Inequalities (Slope-intercept form)

Compare a system of linear inequalities to its graph. Vary the coefficients and inequality symbols in the system and explore how the boundary lines, shaded regions, and the intersection of the shaded regions change in response. 5 Minute Preview

MA.912.AR.9.5: : Given a real-world context, represent constraints as systems of linear equations or inequalities. Interpret solutions to problems as viable or nonviable options.

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

Linear Programming

Use the graph of the feasible region to find the maximum or minimum value of the objective function. Vary the coefficients of the objective function and vary the constraints. Explore how the graph of the feasible region changes in response. 5 Minute Preview

Solving Linear Systems (Matrices and Special Solutions)

*x*, *y*) point to be a solution of an equation, or of a system of equations.
5 Minute Preview

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*)

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

Systems of Linear Inequalities (Slope-intercept form)

Compare a system of linear inequalities to its graph. Vary the coefficients and inequality symbols in the system and explore how the boundary lines, shaded regions, and the intersection of the shaded regions change in response. 5 Minute Preview

MA.912.AR.9.7: : Solve real-world problems involving linear programming.

Linear Programming

Use the graph of the feasible region to find the maximum or minimum value of the objective function. Vary the coefficients of the objective function and vary the constraints. Explore how the graph of the feasible region changes in response. 5 Minute Preview

MA.912.AR.9.9: : Graph and solve mathematical and real-world problems that are modeled with piecewise functions. Interpret key features and determine constraints in terms of the context.

Absolute Value Equations and Inequalities

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

Distance-Time Graphs

Create a graph of a runner's position versus time and watch the runner complete a 40-yard dash based on the graph you made. Notice the connection between the slope of the line and the speed of the runner. What will the runner do if the slope of the line is zero? What if the slope is negative? Add a second runner (a second graph) and connect real-world meaning to the intersection of two graphs. 5 Minute Preview

Distance-Time and Velocity-Time Graphs

Create a graph of a runner's position versus time and watch the runner run a 40-yard dash based on the graph you made. Notice the connection between the slope of the line and the velocity of the runner. Add a second runner (a second graph) and connect real-world meaning to the intersection of two graphs. Also experiment with a graph of velocity versus time for the runners, and also distance traveled versus time. 5 Minute Preview

MA.912.AR.10: : Write and solve sequence and series equations, functions and inequalities in one and two variables.

MA.912.AR.10.1: : Given a mathematical or real-world context, write and solve problems involving arithmetic sequences.

Arithmetic Sequences

Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response. 5 Minute Preview

Arithmetic and Geometric Sequences

Find the value of individual terms in an arithmetic or geometric sequence using graphs of the sequence and direct computation. Vary the common difference and common ratio and examine how the sequence changes in response. 5 Minute Preview

MA.912.AR.10.2: : Given a mathematical or real-world context, write and solve problems involving geometric sequences.

Arithmetic and Geometric Sequences

Find the value of individual terms in an arithmetic or geometric sequence using graphs of the sequence and direct computation. Vary the common difference and common ratio and examine how the sequence changes in response. 5 Minute Preview

Geometric Sequences

Explore geometric sequences by varying the initial term and the common ratio and examining the graph. Compute specific terms in the sequence using the explicit and recursive formulas. 5 Minute Preview

MA.912.AR.10.5: : Given a mathematical or real-world context, write a sequence using function notation, defined explicitly or recursively, to represent relationships between quantities from a written description.

Arithmetic Sequences

Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response. 5 Minute Preview

Arithmetic and Geometric Sequences

Geometric Sequences

Explore geometric sequences by varying the initial term and the common ratio and examining the graph. Compute specific terms in the sequence using the explicit and recursive formulas. 5 Minute Preview

MA.912.AR.10.6: : Given a mathematical or real-world context, find the domain of a given sequence defined recursively or explicitly.

Arithmetic Sequences

### MA.912.F: : Functions

MA.912.F.1: : Understand, compare and analyze properties of functions.

MA.912.F.1.1: : Given an equation or graph that defines a function, determine the function type. Given an input-output table, determine a function type that could represent it.

Absolute Value with Linear Functions

Exponential Functions

Graphs of Polynomial Functions

Introduction to Exponential Functions

Point-Slope Form of a Line

Radical Functions

Slope-Intercept Form of a Line

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

Standard Form of a Line

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

MA.912.F.1.2: : Given a function represented in function notation, evaluate the function for an input in its domain. For a real-world context, interpret the output.

Absolute Value with Linear Functions

Exponential Functions

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

Quadratics in Polynomial Form

Radical Functions

MA.912.F.1.3: : Calculate and interpret the average rate of change of a real-world situation represented graphically, algebraically or in a table over a specified interval.

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

Distance-Time Graphs

Create a graph of a runner's position versus time and watch the runner complete a 40-yard dash based on the graph you made. Notice the connection between the slope of the line and the speed of the runner. What will the runner do if the slope of the line is zero? What if the slope is negative? Add a second runner (a second graph) and connect real-world meaning to the intersection of two graphs. 5 Minute Preview

Distance-Time and Velocity-Time Graphs

Create a graph of a runner's position versus time and watch the runner run a 40-yard dash based on the graph you made. Notice the connection between the slope of the line and the velocity of the runner. Add a second runner (a second graph) and connect real-world meaning to the intersection of two graphs. Also experiment with a graph of velocity versus time for the runners, and also distance traveled versus time. 5 Minute Preview

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

MA.912.F.1.4: : Demonstrate understanding of the concept of limit and estimate limits from graphs and tables of values, as related to the concept of the derivative of a function.

Graphs of Derivative Functions

What does the graph of a derivative function look like? What can a derivative function tell you about the original function? What can't it tell you? Explore these questions for five different types of functions: linear, quadratic, cubic, absolute value, and sine. 5 Minute Preview

MA.912.F.1.5: : Compare key features of linear and nonlinear functions each represented in the same way, such as algebraically, graphically, in tables or written descriptions.

Graphs of Polynomial Functions

MA.912.F.1.6: : Compare key features of two functions each represented in a different way such as algebraically, graphically, in tables or written descriptions.

Direct and Inverse Variation

Adjust the constant of variation and explore how the graph of the direct or inverse variation function changes in response. Compare direct variation functions to inverse variation functions. 5 Minute Preview

Logarithmic Functions

*y* = *x* to compare the associated exponential function.
5 Minute Preview

MA.912.F.1.7: : Determine whether a linear, quadratic or exponential function best models a given real-world situation.

Direct and Inverse Variation

Adjust the constant of variation and explore how the graph of the direct or inverse variation function changes in response. Compare direct variation functions to inverse variation functions. 5 Minute Preview

MA.912.F.1.8: : Determine whether a function is even, odd or neither when represented algebraically, graphically or in a table.

Graphs of Polynomial Functions

MA.912.F.2: : Identify and describe the effects of transformations on functions. Create new functions given transformations.

MA.912.F.2.1: : Identify the effect on the graph or table of a given function after replacing f(x) by f(x) + k, kf(x) and f(x + k) for specific values of k.

Absolute Value with Linear Functions

Point-Slope Form of a Line

Quadratics in Polynomial Form

Quadratics in Vertex Form

Slope-Intercept Form of a Line

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

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

MA.912.F.2.2: : Identify the effect on the graph of a given function of two or more transformations defined by adding a real number to the x- or y-values or multiplying the x- or y-values by a real number.

Absolute Value with Linear Functions

Point-Slope Form of a Line

Quadratics in Polynomial Form

Quadratics in Vertex Form

Slope-Intercept Form of a Line

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

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

MA.912.F.2.3: : Given the graph or table of f(x) and the graph or table of f(x) + k, kf(x), f(kx) and f(x + k), state the type of transformation and find the value of the real number k.

Absolute Value with Linear Functions

Point-Slope Form of a Line

Quadratics in Polynomial Form

Quadratics in Vertex Form

Slope-Intercept Form of a Line

Translating and Scaling Functions

Zap It! Game

MA.912.F.2.4: : Given the graph or table of values of two or more transformations of a function, state the type of transformation and find the values of the real number that defines the transformation.

Absolute Value with Linear Functions

Point-Slope Form of a Line

Quadratics in Polynomial Form

Quadratics in Vertex Form

Slope-Intercept Form of a Line

Translating and Scaling Functions

Zap It! Game

MA.912.F.2.5: : Given two or more transformations and a function, create the table or graph of the transformed function.

Absolute Value with Linear Functions

Point-Slope Form of a Line

Quadratics in Polynomial Form

Quadratics in Vertex Form

Slope-Intercept Form of a Line

Translating and Scaling Functions

MA.912.F.2.6: : Given a graph or table of values of two or more transformations of a function, write the equation of the transformed function.

Point-Slope Form of a Line

Translating and Scaling Functions

MA.912.F.3: : Create new functions from existing functions.

MA.912.F.3.1: : Given a mathematical or real-world context, combine two functions, limited to linear and quadratic, using arithmetic operations. When appropriate, include domain restrictions for the new function.

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

Solving Linear Systems (Standard Form)

*x*, *y*) values are solutions of an equation, or of a system of equations.
5 Minute Preview

MA.912.F.3.2: : Given a mathematical or real-world context, combine two or more functions, limited to linear, quadratic, exponential and polynomial, using arithmetic operations. When appropriate, include domain restrictions for the new function.

Addition and Subtraction of Functions

Solving Linear Systems (Standard Form)

*x*, *y*) values are solutions of an equation, or of a system of equations.
5 Minute Preview

MA.912.F.3.3: : Solve mathematical and real-world problems involving functions that have been combined using arithmetic operations.

Addition and Subtraction of Functions

Solving Linear Systems (Standard Form)

*x*, *y*) values are solutions of an equation, or of a system of equations.
5 Minute Preview

MA.912.F.3.6: : Determine whether an inverse function exists by analyzing tables, graphs and equations.

Logarithmic Functions

*y* = *x* to compare the associated exponential function.
5 Minute Preview

Radical Functions

MA.912.F.3.7: : Represent the inverse of a function algebraically, graphically or in a table. Use composition of functions to verify that one function is the inverse of the other.

Logarithmic Functions

*y* = *x* to compare the associated exponential function.
5 Minute Preview

Radical Functions

MA.912.F.3.9: : Solve mathematical and real-world problems involving inverse functions.

Logarithmic Functions

*y* = *x* to compare the associated exponential function.
5 Minute Preview

Radical Functions

### MA.912.FL: : Financial Literacy

MA.912.FL.1: : Determine simple and compound interest and demonstrate its relationship to functions. Calculate and use net present and net future values.

MA.912.FL.1.1: : Compare simple, compound and continuously compounded interest over time.

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

MA.912.FL.1.2: : Solve problems involving simple, compound and continuously compounded interest, including determining the present value and future value of money.

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

MA.912.FL.1.4: : Explain the relationship between compound interest and exponential growth and the relationship between continuously compounded interest and exponential growth.

Compound Interest

MA.912.FL.1.6: : Solve problems involving potential profit and actual cost.

Linear Programming

Use the graph of the feasible region to find the maximum or minimum value of the objective function. Vary the coefficients of the objective function and vary the constraints. Explore how the graph of the feasible region changes in response. 5 Minute Preview

MA.912.FL.2: : Describe the advantages and disadvantages of short-term and long-term purchases.

MA.912.FL.2.2: : Calculate the finance charges and total amount due on a bill using various forms of credit.

Compound Interest

MA.912.FL.2.7: : Solve problems involving student, personal and car loans, including finding the total amount to be paid, adjustable rates and refinancing options.

Compound Interest

MA.912.FL.2.12: : Compare interest rate calculations and annual percentage rate calculations, and distinguish between the two rates.

Compound Interest

### MA.912.GR: : Geometric Reasoning

MA.912.GR.1: : Prove and apply geometric theorems to solve problems.

MA.912.GR.1.1: : Prove relationships and theorems about lines and angles. Solve mathematical and real-world problems involving postulates, relationships and theorems of lines and angles.

Concurrent Lines, Medians, and Altitudes

Explore the relationships between perpendicular bisectors, the circumscribed circle, angle bisectors, the inscribed circle, altitudes, and medians using a triangle that can be resized and reshaped. 5 Minute Preview

Investigating Angle Theorems

Explore the properties of complementary, supplementary, vertical, and adjacent angles using a dynamic figure. 5 Minute Preview

Parallel, Intersecting, and Skew Lines

Explore the properties of intersecting, parallel, and skew lines as well as lines in the plane. Rotate the plane and lines in three-dimensional space to ensure a full understanding of these objects. 5 Minute Preview

MA.912.GR.1.2: : Prove triangle congruence or similarity using Side-Side-Side, Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, Angle-Angle and Hypotenuse-Leg.

Congruence in Right Triangles

Apply constraints to two right triangles. Then drag their vertices around under those conditions. Determine under what conditions the triangles are guaranteed to be congruent. 5 Minute Preview

Proving Triangles Congruent

Apply constraints to two triangles. Then drag the vertices of the triangles around and determine which constraints guarantee congruence. 5 Minute Preview

Similarity in Right Triangles

Divide a right triangle at the altitude to the hypotenuse to get two similar right triangles. Explore the relationship between the two triangles. 5 Minute Preview

MA.912.GR.1.3: : Prove relationships and theorems about triangles. Solve mathematical and real-world problems involving postulates, relationships and theorems of triangles.

Concurrent Lines, Medians, and Altitudes

Explore the relationships between perpendicular bisectors, the circumscribed circle, angle bisectors, the inscribed circle, altitudes, and medians using a triangle that can be resized and reshaped. 5 Minute Preview

Isosceles and Equilateral Triangles

Investigate the graph of a triangle under constraints. Determine which constraints guarantee isosceles or equilateral triangles. 5 Minute Preview

Segment and Angle Bisectors

Explore the special properties of a point that lies on the perpendicular bisector of a segment, and of a point that lies on an angle bisector. Manipulate the point, the segment, and the angle to see that these properties are always true. 5 Minute Preview

Triangle Angle Sum

Measure the interior angles of a triangle and find the sum. Examine whether that sum is the same for all triangles. Also, discover how the measure of an exterior angle relates to the interior angle measures. 5 Minute Preview

Triangle Inequalities

Discover the inequalities related to the side lengths and angle measures of a triangle. Reshape and resize the triangle to confirm that these properties are true for all triangles. 5 Minute Preview

MA.912.GR.1.4: : Prove relationships and theorems about parallelograms. Solve mathematical and real-world problems involving postulates, relationships and theorems of parallelograms.

Parallelogram Conditions

Apply constraints to a dynamic quadrilateral. Then drag its vertices around. Determine which constraints guarantee that the quadrilateral is always a parallelogram. 5 Minute Preview

Polygon Angle Sum

Derive the sum of the angles of a polygon by dividing the polygon into triangles and summing their angles. Vary the number of sides and determine how the sum of the angles changes. Dilate the polygon to see that the sum is unchanged. 5 Minute Preview

Special Parallelograms

Apply constraints to a parallelogram and experiment with the resulting figure. What type of shape can you be sure that you have under each condition? 5 Minute Preview

MA.912.GR.1.5: : Prove relationships and theorems about trapezoids. Solve mathematical and real-world problems involving postulates, relationships and theorems of trapezoids.

Classifying Quadrilaterals

Apply constraints to a quadrilateral, and then reshape and resize it. Classify the figure by its constraints. Explore the differences between the different kinds of quadrilaterals. 5 Minute Preview

MA.912.GR.1.6: : Solve mathematical and real-world problems involving congruence or similarity in two-dimensional figures.

Congruence in Right Triangles

Apply constraints to two right triangles. Then drag their vertices around under those conditions. Determine under what conditions the triangles are guaranteed to be congruent. 5 Minute Preview

Similar Figures

Vary the scale factor and rotation of an image and compare it to the preimage. Determine how the angle measures and side lengths of the two figures are related. 5 Minute Preview

Similarity in Right Triangles

Divide a right triangle at the altitude to the hypotenuse to get two similar right triangles. Explore the relationship between the two triangles. 5 Minute Preview

MA.912.GR.2: : Apply properties of transformations to describe congruence or similarity.

MA.912.GR.2.1: : Given a preimage and image, describe the transformation and represent the transformation algebraically using coordinates.

Dilations

Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in

Rotations, Reflections, and Translations

Rotate, reflect, and translate a figure in the plane. Compare the translated figure to the original figure. 5 Minute Preview

Translations

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

MA.912.GR.2.2: : Identify transformations that do or do not preserve distance.

Dilations

Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in

Rotations, Reflections, and Translations

Rotate, reflect, and translate a figure in the plane. Compare the translated figure to the original figure. 5 Minute Preview

Translations

MA.912.GR.2.3: : Specify a sequence of transformations that will map a given figure onto itself or onto another congruent or similar figure.

Rotations, Reflections, and Translations

Rotate, reflect, and translate a figure in the plane. Compare the translated figure to the original figure. 5 Minute Preview

MA.912.GR.2.5: : Apply rigid transformations to map one figure onto another to justify that the two figures are congruent.

Reflections

Reshape and resize a figure and examine how its reflection changes in response. Move the line of reflection and explore how the reflection is translated. 5 Minute Preview

Rotations, Reflections, and Translations

Translations

MA.912.GR.2.6: : Justify the criteria for triangle congruence using the definition of congruence in terms of rigid transformations.

Reflections

Reshape and resize a figure and examine how its reflection changes in response. Move the line of reflection and explore how the reflection is translated. 5 Minute Preview

Rotations, Reflections, and Translations

Translations

MA.912.GR.2.7: : Apply an appropriate transformation to map one figure onto another to justify that the two figures are similar.

Dilations

MA.912.GR.2.8: : Justify the criteria for triangle similarity using the definition of similarity in terms of non-rigid transformations.

Dilations

MA.912.GR.3: : Use coordinate geometry to solve problems or prove relationships.

MA.912.GR.3.1: : Given a mathematical or real-world context, use coordinate geometry to classify or justify definitions, properties and theorems involving circles, triangles or quadrilaterals.

Circles

Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview

MA.912.GR.3.2: : Solve geometric problems involving circles, triangles and quadrilaterals on the coordinate plane.

Circles

Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview

MA.912.GR.3.3: : Solve mathematical and real-world problems on the coordinate plane that involve finding the coordinates of a point on a line segment including the midpoint.

Circles

Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview

MA.912.GR.3.4: : Solve mathematical and real-world problems on the coordinate plane involving perimeter or area of polygons.

Distance Formula

Explore the distance formula as an application of the Pythagorean theorem. Learn to see any two points as the endpoints of the hypotenuse of a right triangle. Drag those points and examine changes to the triangle and the distance calculation. 5 Minute Preview

MA.912.GR.4: : Use geometric measurement and dimensions to solve problems.

MA.912.GR.4.1: : Identify the shapes of two-dimensional cross-sections of three-dimensional figures.

Pyramids and Cones

Vary the height and base-edge or radius length of a pyramid or cone and examine how its three-dimensional representation changes. Determine the area of the base and the volume of the solid. Compare the volume of a skew pyramid or cone to the volume of a right pyramid or cone. 5 Minute Preview

MA.912.GR.4.2: : Identify three-dimensional objects generated by rotations of two-dimensional figures.

Prisms and Cylinders

Vary the height and base-edge or radius length of a prism or cylinder and examine how its three-dimensional representation changes. Determine the area of the base and the volume of the solid. Compare the volume of an oblique prism or cylinder to the volume of a right prism or cylinder. 5 Minute Preview

Pyramids and Cones

Vary the height and base-edge or radius length of a pyramid or cone and examine how its three-dimensional representation changes. Determine the area of the base and the volume of the solid. Compare the volume of a skew pyramid or cone to the volume of a right pyramid or cone. 5 Minute Preview

MA.912.GR.4.3: : Determine how changes in dimensions affect the area of two-dimensional figures and the surface area or volume of three-dimensional figures.

Area of Parallelograms

Examine and manipulate a parallelogram and find its area. Explore the relationship between the area of a parallelogram and the area of a rectangle using an animation. 5 Minute Preview

Area of Triangles

Use a dynamic triangle to explore the area of a triangle. With the help of an animation, see that any triangle is always half of a parallelogram (with the same base and height). Likewise, a similar animation shows the connection between parallelograms and rectangles. 5 Minute Preview

Circumference and Area of Circles

Resize a circle and compare its radius, circumference, and area. 5 Minute Preview

Perimeter and Area of Rectangles

Discover how to find the perimeter and area of a rectangle, and of a square (which is really just a special case of a rectangle). 5 Minute Preview

Perimeters and Areas of Similar Figures

Manipulate two similar figures and vary the scale factor to see what changes are possible under similarity. Explore how the perimeters and areas of two similar figures compare. 5 Minute Preview

Prisms and Cylinders

Vary the height and base-edge or radius length of a prism or cylinder and examine how its three-dimensional representation changes. Determine the area of the base and the volume of the solid. Compare the volume of an oblique prism or cylinder to the volume of a right prism or cylinder. 5 Minute Preview

Pyramids and Cones

Vary the height and base-edge or radius length of a pyramid or cone and examine how its three-dimensional representation changes. Determine the area of the base and the volume of the solid. Compare the volume of a skew pyramid or cone to the volume of a right pyramid or cone. 5 Minute Preview

Surface and Lateral Areas of Prisms and Cylinders

Vary the dimensions of a prism or cylinder and investigate how the surface area changes. Use the dynamic net of the solid to compute the lateral area and the surface area of the solid. 5 Minute Preview

Surface and Lateral Areas of Pyramids and Cones

Vary the dimensions of a pyramid or cone and investigate how the surface area changes. Use the dynamic net of the solid to compute the lateral area and the surface area of the solid. 5 Minute Preview

MA.912.GR.4.4: : Solve mathematical and real-world problems involving the area of two-dimensional figures.

Area of Parallelograms

Examine and manipulate a parallelogram and find its area. Explore the relationship between the area of a parallelogram and the area of a rectangle using an animation. 5 Minute Preview

Area of Triangles

Use a dynamic triangle to explore the area of a triangle. With the help of an animation, see that any triangle is always half of a parallelogram (with the same base and height). Likewise, a similar animation shows the connection between parallelograms and rectangles. 5 Minute Preview

Circumference and Area of Circles

Resize a circle and compare its radius, circumference, and area. 5 Minute Preview

Perimeter and Area of Rectangles

Discover how to find the perimeter and area of a rectangle, and of a square (which is really just a special case of a rectangle). 5 Minute Preview

MA.912.GR.4.5: : Solve mathematical and real-world problems involving the volume of three-dimensional figures limited to cylinders, pyramids, prisms, cones and spheres.

Prisms and Cylinders

Vary the height and base-edge or radius length of a prism or cylinder and examine how its three-dimensional representation changes. Determine the area of the base and the volume of the solid. Compare the volume of an oblique prism or cylinder to the volume of a right prism or cylinder. 5 Minute Preview

Pyramids and Cones

MA.912.GR.4.6: : Solve mathematical and real-world problems involving the surface area of three-dimensional figures limited to cylinders, pyramids, prisms, cones and spheres.

Surface and Lateral Areas of Prisms and Cylinders

Vary the dimensions of a prism or cylinder and investigate how the surface area changes. Use the dynamic net of the solid to compute the lateral area and the surface area of the solid. 5 Minute Preview

Surface and Lateral Areas of Pyramids and Cones

Vary the dimensions of a pyramid or cone and investigate how the surface area changes. Use the dynamic net of the solid to compute the lateral area and the surface area of the solid. 5 Minute Preview

MA.912.GR.5: : Make formal geometric constructions with a variety of tools and methods.

MA.912.GR.5.1: : Construct a copy of a segment or an angle.

Constructing Congruent Segments and Angles

Construct congruent segments and angles using a straightedge and compass. Use step-by-step explanations and feedback to develop understanding of the construction. 5 Minute Preview

MA.912.GR.5.2: : Construct the bisector of a segment or an angle, including the perpendicular bisector of a line segment.