Mathematics Florida Standards

MAFS.912.A-APR: Arithmetic with Polynomials and Rational Expressions

MAFS.912.A-APR.1: Perform arithmetic operations on polynomials

MAFS.912.A-APR.1.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Addition of Polynomials - Activity A

MAFS.912.A-APR.2: Understand the relationship between zeros and factors of polynomials

MAFS.912.A-APR.2.2: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

Dividing Polynomials Using Synthetic Division

MAFS.912.A-APR.2.3: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Cubic Function Activity

Fourth-Degree Polynomials - Activity A

Polynomials and Linear Factors

MAFS.912.A-APR.3: Use polynomial identities to solve problems

MAFS.912.A-APR.3.4: Prove polynomial identities and use them to describe numerical relationships.

Factoring Special Products

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

MAFS.912.A-CED: Creating Equations

MAFS.912.A-CED.1: Create equations that describe numbers or relationships

MAFS.912.A-CED.1.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

MAFS.912.A-CED.1.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

Linear Programming - Activity A

System of Two Quadratic Inequalities

Systems of Linear Inequalities (Slope-intercept form) - Activity A

MAFS.912.A-CED.1.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Solving Formulas for any Variable

MAFS.912.A-REI: Reasoning with Equations and Inequalities

MAFS.912.A-REI.1: Understand solving equations as a process of reasoning and explain the reasoning

MAFS.912.A-REI.1.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Modeling and Solving Two-Step Equations

Solving Formulas for any Variable

Solving Two-Step Equations

MAFS.912.A-REI.2: Solve equations and inequalities in one variable

MAFS.912.A-REI.2.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Modeling and Solving Two-Step Equations

Solving Equations By Graphing Each Side

Solving Linear Inequalities using Addition and Subtraction

Solving Linear Inequalities using Multiplication and Division

Solving Two-Step Equations

System of Two Quadratic Inequalities

MAFS.912.A-REI.2.4: Solve quadratic equations in one variable.

MAFS.912.A-REI.2.4.a: Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form.

MAFS.912.A-REI.2.4.b: Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Factoring Special Products

Roots of a Quadratic

Square Roots

MAFS.912.A-REI.3: Solve systems of equations

MAFS.912.A-REI.3.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Solving Linear Systems by Graphing

Special Types of Solutions to Linear Systems

Systems of Linear Equations - Activity A

MAFS.912.A-REI.3.7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Solving Linear Systems by Graphing

Special Types of Solutions to Linear Systems

Systems of Linear Equations - Activity A

MAFS.912.A-REI.3.9: Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

Systems of Linear Equations - Activity A

MAFS.912.A-REI.4: Represent and solve equations and inequalities graphically

MAFS.912.A-REI.4.10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Modeling and Solving Two-Step Equations

Solving Two-Step Equations

MAFS.912.A-REI.4.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Cubic Function Activity

Fourth-Degree Polynomials - Activity A

Function Machines 2 (Functions, Tables, and Graphs)

General Form of a Rational Function

Logarithmic Functions - Activity A

Logarithmic Functions: Translating and Scaling

Quadratic and Absolute Value Functions

Rational Functions

Slope-Intercept Form of a Line - Activity B

MAFS.912.A-REI.4.12: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Inequalities Involving Absolute Values

Linear Inequalities in Two Variables - Activity A

Linear Programming - Activity A

Solving Linear Inequalities using Addition and Subtraction

Solving Linear Inequalities using Multiplication and Division

System of Two Quadratic Inequalities

Systems of Linear Inequalities (Slope-intercept form) - Activity A

MAFS.912.A-SSE: Seeing Structure in Expressions

MAFS.912.A-SSE.1: Interpret the structure of expressions

MAFS.912.A-SSE.1.1: Interpret expressions that represent a quantity in terms of its context.

MAFS.912.A-SSE.1.1.a: Interpret parts of an expression, such as terms, factors, and coefficients.

Finding Factors with Area Models

MAFS.912.A-SSE.2: Write expressions in equivalent forms to solve problems

MAFS.912.A-SSE.2.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

MAFS.912.A-SSE.2.3.a: Factor a quadratic expression to reveal the zeros of the function it defines.

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

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

Quadratics in Factored Form

Roots of a Quadratic

MAFS.912.A-SSE.2.3.b: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Parabolas - Activity A

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

Roots of a Quadratic

MAFS.912.A-SSE.2.3.c: Use the properties of exponents to transform expressions for exponential functions.

Multiplying Exponential Expressions

MAFS.912.F-BF: Building Functions

MAFS.912.F-BF.1: Build a function that models a relationship between two quantities

MAFS.912.F-BF.1.1: Write a function that describes a relationship between two quantities.

MAFS.912.F-BF.1.1.a: Determine an explicit expression, a recursive process, or steps for calculation from a context.

Function Machines 1 (Functions and Tables)

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Modeling and Solving Two-Step Equations

Solving Two-Step Equations

MAFS.912.F-BF.1.1.b: Combine standard function types using arithmetic operations.

Addition and Subtraction of Polynomials

Function Machines 1 (Functions and Tables)

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Linear Functions

MAFS.912.F-BF.1.1.c: Compose functions.

Function Machines 1 (Functions and Tables)

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Linear Functions

MAFS.912.F-BF.1.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Arithmetic Sequences

Arithmetic and Geometric Sequences

Geometric Sequences

MAFS.912.F-BF.2: Build new functions from existing functions

MAFS.912.F-BF.2.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Translating and Scaling Functions

MAFS.912.F-BF.2.4: Find inverse functions.

MAFS.912.F-BF.2.4.a: Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

Function Machines 3 (Functions and Problem Solving)

Modeling and Solving Two-Step Equations

Solving Two-Step Equations

MAFS.912.F-BF.2.4.c: Read values of an inverse function from a graph or a table, given that the function has an inverse.

Cosine Function

Fourth-Degree Polynomials - Activity A

Function Machines 1 (Functions and Tables)

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Logarithmic Functions - Activity A

Points, Lines, and Equations

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

MAFS.912.F-BF.2.4.d: Produce an invertible function from a non-invertible function by restricting the domain.

Function Machines 3 (Functions and Problem Solving)

MAFS.912.F-BF.2.5: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Function Machines 3 (Functions and Problem Solving)

Logarithmic Functions - Activity A

MAFS.912.F-IF: Interpreting Functions

MAFS.912.F-IF.1: Understand the concept of a function and use function notation

MAFS.912.F-IF.1.1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Cosine Function

Function Machines 1 (Functions and Tables)

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Functions Involving Square Roots

Introduction to Functions

Rational Functions

Sine Function

Tangent Function

MAFS.912.F-IF.1.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Linear Functions

Points, Lines, and Equations

MAFS.912.F-IF.1.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Arithmetic and Geometric Sequences

Function Machines 1 (Functions and Tables)

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Linear Functions

MAFS.912.F-IF.2: Interpret functions that arise in applications in terms of the context

MAFS.912.F-IF.2.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Cosine Function

Fourth-Degree Polynomials - Activity A

Function Machines 1 (Functions and Tables)

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Functions Involving Square Roots

Linear Functions

Points, Lines, and Equations

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

Rational Functions

Sine Function

Tangent Function

MAFS.912.F-IF.2.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Functions Involving Square Roots

MAFS.912.F-IF.2.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Direct and Inverse Variation

Function Machines 1 (Functions and Tables)

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Introduction to Functions

Linear Functions

Points, Lines, and Equations

MAFS.912.F-IF.3: Analyze functions using different representations

MAFS.912.F-IF.3.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

MAFS.912.F-IF.3.7.a: Graph linear and quadratic functions and show intercepts, maxima, and minima.

Function Machines 2 (Functions, Tables, and Graphs)

Linear Functions

Point-Slope Form of a Line - Activity A

Quadratic and Absolute Value Functions

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

Roots of a Quadratic

Slope-Intercept Form of a Line - Activity B

MAFS.912.F-IF.3.7.b: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Function Machines 2 (Functions, Tables, and Graphs)

Functions Involving Square Roots

Quadratic and Absolute Value Functions

Radical Functions

Square Roots

MAFS.912.F-IF.3.7.c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Cubic Function Activity

Fourth-Degree Polynomials - Activity A

Function Machines 2 (Functions, Tables, and Graphs)

Polynomials and Linear Factors

MAFS.912.F-IF.3.7.d: Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

Function Machines 2 (Functions, Tables, and Graphs)

General Form of a Rational Function

Polynomials and Linear Factors

Rational Functions

MAFS.912.F-IF.3.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

MAFS.912.F-IF.3.8.a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Function Machines 2 (Functions, Tables, and Graphs)

Parabolas - Activity A

Polynomials and Linear Factors

Quadratic and Absolute Value Functions

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

Roots of a Quadratic

MAFS.912.F-IF.3.8.b: Use the properties of exponents to interpret expressions for exponential functions.

Multiplying Exponential Expressions

MAFS.912.F-IF.3.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Cosine Function

Fourth-Degree Polynomials - Activity A

Function Machines 1 (Functions and Tables)

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Functions Involving Square Roots

Introduction to Functions

Linear Functions

Logarithmic Functions - Activity A

Points, Lines, and Equations

Polynomials and Linear Factors

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

Rational Functions

Sine Function

Tangent Function

Using Algebraic Equations

Using Algebraic Expressions

MAFS.912.F-LE: Linear, Quadratic, and Exponential Models

MAFS.912.F-LE.1: Construct and compare linear, quadratic, and exponential models and solve problems

MAFS.912.F-LE.1.1: Distinguish between situations that can be modeled with linear functions and with exponential functions.

MAFS.912.F-LE.1.1.a: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

Function Machines 2 (Functions, Tables, and Graphs)

Linear Functions

Point-Slope Form of a Line - Activity A

Points, Lines, and Equations

Slope-Intercept Form of a Line - Activity B

MAFS.912.F-LE.1.2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Arithmetic Sequences

Arithmetic and Geometric Sequences

Function Machines 1 (Functions and Tables)

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Geometric Sequences

Linear Functions

Points, Lines, and Equations

Slope-Intercept Form of a Line - Activity B

MAFS.912.F-LE.1.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Function Machines 2 (Functions, Tables, and Graphs)

Introduction to Exponential Functions

MAFS.912.F-LE.2: Interpret expressions for functions in terms of the situation they model

MAFS.912.F-LE.2.5: Interpret the parameters in a linear or exponential function in terms of a context.

Function Machines 2 (Functions, Tables, and Graphs)

Slope-Intercept Form of a Line - Activity B

MAFS.912.F-TF: Trigonometric Functions

MAFS.912.F-TF.1: Extend the domain of trigonometric functions using the unit circle

MAFS.912.F-TF.1.2: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Cosine Function

Sine Function

Tangent Function

Unit Circle

MAFS.912.F-TF.1.3: Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi–x, pi+x, and 2pi–x in terms of their values for x, where x is any real number.

Cosine Function

Sine Function

Sine and Cosine Ratios - Activity A

Sine, Cosine and Tangent

Tangent Function

Tangent Ratio

Unit Circle

MAFS.912.F-TF.1.4: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Cosine Function

Sine Function

Tangent Function

Unit Circle

MAFS.912.F-TF.2: Model periodic phenomena with trigonometric functions

MAFS.912.F-TF.2.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Cosine Function

Sine Function

Tangent Function

MAFS.912.F-TF.2.6: Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Cosine Function

Sine Function

Tangent Function

MAFS.912.F-TF.3: Prove and apply trigonometric identities

MAFS.912.F-TF.3.8: Prove the Pythagorean identity sin²(theta) + cos²(theta) = 1 and use it to calculate trigonometric ratios.

Simplifying Trigonometric Expressions

Sine and Cosine Ratios - Activity A

Sine, Cosine and Tangent

MAFS.912.G-C: Circles

MAFS.912.G-C.1: Understand and apply theorems about circles

MAFS.912.G-C.1.1: Prove that all circles are similar.

Perimeters and Areas of Similar Figures

MAFS.912.G-C.1.2: Identify and describe relationships among inscribed angles, radii, and chords.

Chords and Arcs

Inscribed Angles

MAFS.912.G-C.1.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Classifying Quadrilaterals - Activity B

Parallelogram Conditions

Segment and Angle Bisectors

MAFS.912.G-C.2: Find arc lengths and areas of sectors of circles

MAFS.912.G-C.2.5: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Area of Triangles

Chords and Arcs

Inscribed Angles

MAFS.912.G-CO: Congruence

MAFS.912.G-CO.1: Experiment with transformations in the plane

MAFS.912.G-CO.1.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Function Machines 1 (Functions and Tables)

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

MAFS.912.G-CO.1.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Reflections

Rotations, Reflections and Translations

Segment and Angle Bisectors

MAFS.912.G-CO.1.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Reflections

Rotations, Reflections and Translations

Translations

MAFS.912.G-CO.1.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Reflections

Rotations, Reflections and Translations

Translations

MAFS.912.G-CO.2: Understand congruence in terms of rigid motions

MAFS.912.G-CO.2.6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Dilations

Reflections

Rotations, Reflections and Translations

MAFS.912.G-CO.2.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Congruence in Right Triangles

Dilations

Proving Triangles Congruent

Reflections

Rotations, Reflections and Translations

MAFS.912.G-CO.4: Make geometric constructions

MAFS.912.G-CO.4.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

Constructing Congruent Segments and Angles

Constructing Parallel and Perpendicular Lines

MAFS.912.G-CO.4.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Classifying Triangles

Isosceles and Equilateral Triangles

MAFS.912.G-GPE: Expressing Geometric Properties with Equations

MAFS.912.G-GPE.1: Translate between the geometric description and the equation for a conic section

MAFS.912.G-GPE.1.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

MAFS.912.G-GPE.1.2: Derive the equation of a parabola given a focus and directrix.

MAFS.912.G-GPE.1.3: Derive the equations of ellipses and hyperbolas given the foci and directrices.

Ellipse - Activity A

Hyperbola - Activity A

MAFS.912.G-GPE.2: Use coordinates to prove simple geometric theorems algebraically

MAFS.912.G-GPE.2.4: Use coordinates to prove simple geometric theorems algebraically.

Simplifying Trigonometric Expressions

MAFS.912.G-GPE.2.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Constructing Parallel and Perpendicular Lines

Slope - Activity B

MAFS.912.G-GPE.2.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Perimeters and Areas of Similar Figures

Similar Figures - Activity A

MAFS.912.G-GPE.2.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Area of Parallelograms

Distance Formula - Activity A

Perimeter and Area of Rectangles

Pythagorean Theorem - Activity B

Pythagorean Theorem with a Geoboard

Segment and Angle Bisectors

MAFS.912.G-GMD: Geometric Measurement and Dimension

MAFS.912.G-GMD.1: Explain volume formulas and use them to solve problems

MAFS.912.G-GMD.1.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

Circles: Circumference and Area

Prisms and Cylinders - Activity A

MAFS.912.G-GMD.1.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Prisms and Cylinders - Activity A

MAFS.912.G-MG: Modeling with Geometry

MAFS.912.G-MG.1: Apply geometric concepts in modeling situations

MAFS.912.G-MG.1.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Classifying Quadrilaterals - Activity B

Classifying Triangles

Parallelogram Conditions

Special Parallelograms

MAFS.912.G-MG.1.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Area of Triangles

Prisms and Cylinders - Activity A

MAFS.912.G-SRT: Similarity, Right Triangles, and Trigonometry

MAFS.912.G-SRT.1: Understand similarity in terms of similarity transformations

MAFS.912.G-SRT.1.1: Verify experimentally the properties of dilations given by a center and a scale factor:

MAFS.912.G-SRT.1.1.a: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

MAFS.912.G-SRT.1.1.b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Dilations

Perimeters and Areas of Similar Figures

Similar Figures - Activity A

MAFS.912.G-SRT.1.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Dilations

Perimeters and Areas of Similar Figures

Reflections

Rotations, Reflections and Translations

Similar Figures - Activity A

MAFS.912.G-SRT.1.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Perimeters and Areas of Similar Figures

Similar Figures - Activity A

MAFS.912.G-SRT.2: Prove theorems involving similarity

MAFS.912.G-SRT.2.4: Prove theorems about triangles.

Perimeters and Areas of Similar Figures

Segment and Angle Bisectors

MAFS.912.G-SRT.2.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Congruence in Right Triangles

Perimeters and Areas of Similar Figures

Proving Triangles Congruent

Similar Figures - Activity A

MAFS.912.G-SRT.3: Define trigonometric ratios and solve problems involving right triangles

MAFS.912.G-SRT.3.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Sine and Cosine Ratios - Activity A

Sine, Cosine and Tangent

Tangent Ratio

MAFS.912.G-SRT.3.7: Explain and use the relationship between the sine and cosine of complementary angles.

Cosine Function

Sine Function

Sine and Cosine Ratios - Activity A

Sine, Cosine and Tangent

Unit Circle

MAFS.912.G-SRT.3.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Cosine Function

Distance Formula - Activity A

Pythagorean Theorem - Activity B

Pythagorean Theorem with a Geoboard

Sine Function

Sine and Cosine Ratios - Activity A

Tangent Function

MAFS.912.G-SRT.4: Apply trigonometry to general triangles

MAFS.912.G-SRT.4.11: Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

MAFS.912.N-CN: The Complex Number System

MAFS.912.N-CN.1: Perform arithmetic operations with complex numbers.

MAFS.912.N-CN.1.1: Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.

Points in the Complex Plane - Activity A

MAFS.912.N-CN.1.3: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Absolute Value of a Complex Number

Points in the Complex Plane - Activity A

MAFS.912.N-CN.2: Represent complex numbers and their operations on the complex plane.

MAFS.912.N-CN.2.4: Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

Complex Numbers in Polar Form

Points in the Complex Plane - Activity A

MAFS.912.N-CN.2.5: Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

Points in the Complex Plane - Activity A

MAFS.912.N-CN.2.6: Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Absolute Value of a Complex Number

Distance Formula - Activity A

Points in the Complex Plane - Activity A

Pythagorean Theorem with a Geoboard

MAFS.912.N-CN.3: Use complex numbers in polynomial identities and equations.

MAFS.912.N-CN.3.8: Extend polynomial identities to the complex numbers.

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

Points in the Complex Plane - Activity A

MAFS.912.N-VM: Vector and Matrix Quantities

MAFS.912.N-VM.1: Represent and model with vector quantities.

MAFS.912.N-VM.1.1: 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 (e.g., v, |v|, ||v||, v).

MAFS.912.N-VM.1.2: Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

MAFS.912.N-VM.2: Perform operations on vectors.

MAFS.912.N-VM.2.4: Add and subtract vectors.

MAFS.912.N-VM.2.4.a: Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

MAFS.912.N-VM.2.4.b: Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

MAFS.912.N-VM.2.4.c: Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

MAFS.912.N-VM.3: Perform operations on matrices and use matrices in applications.

MAFS.912.N-VM.3.11: Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

Dilations

Reflections

Rotations, Reflections and Translations

MAFS.912.N-VM.3.12: Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Area of Triangles

Dilations

Real Number Line - Activity B

Reflections

Rotations, Reflections and Translations

MAFS.912.S-CP: Conditional Probability and the Rules of Probability

MAFS.912.S-CP.1: Understand independence and conditional probability and use them to interpret data

MAFS.912.S-CP.1.2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

Compound Independent Events

Compound Independent and Dependent Events

MAFS.912.S-CP.1.3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

Probability Simulations

Theoretical and Experimental Probability

MAFS.912.S-CP.1.4: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

Box-and-Whisker Plots

Compound Independent Events

Compound Independent and Dependent Events

Describing Data Using Statistics

Histograms

Scatter Plots - Activity A

Stem-and-Leaf Plots

MAFS.912.S-CP.1.5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

Compound Independent Events

Compound Independent and Dependent Events

MAFS.912.S-CP.2: Use the rules of probability to compute probabilities of compound events in a uniform probability model

MAFS.912.S-CP.2.6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

MAFS.912.S-CP.2.9: Use permutations and combinations to compute probabilities of compound events and solve problems.

Compound Independent Events

Compound Independent and Dependent Events

Permutations

Permutations and Combinations

MAFS.912.S-ID: Interpreting Categorical and Quantitative Data

MAFS.912.S-ID.1: Summarize, represent, and interpret data on a single count or measurement variable

MAFS.912.S-ID.1.1: Represent data with plots on the real number line (dot plots, histograms, and box plots).

Box-and-Whisker Plots

Describing Data Using Statistics

Histograms

Populations and Samples

MAFS.912.S-ID.1.2: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Box-and-Whisker Plots

Mean, Median and Mode

Populations and Samples

Real-Time Histogram

MAFS.912.S-ID.1.3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Describing Data Using Statistics

Mean, Median and Mode

MAFS.912.S-ID.1.4: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

MAFS.912.S-ID.2: Summarize, represent, and interpret data on two categorical and quantitative variables

MAFS.912.S-ID.2.5: Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Correlation

Solving Using Trend Lines

MAFS.912.S-ID.2.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

MAFS.912.S-ID.2.6.a: Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

Correlation

Scatter Plots - Activity A

Solving Using Trend Lines

MAFS.912.S-ID.2.6.b: Informally assess the fit of a function by plotting and analyzing residuals.

Correlation

Scatter Plots - Activity A

Solving Using Trend Lines

MAFS.912.S-ID.2.6.c: Fit a linear function for a scatter plot that suggests a linear association.

Correlation

Lines of Best Fit Using Least Squares - Activity A

Scatter Plots - Activity A

Solving Using Trend Lines

MAFS.912.S-ID.3: Interpret linear models

MAFS.912.S-ID.3.7: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Points, Lines, and Equations

Slope-Intercept Form of a Line - Activity B

MAFS.912.S-ID.3.8: Compute (using technology) and interpret the correlation coefficient of a linear fit.

MAFS.912.S-ID.3.9: Distinguish between correlation and causation.

MAFS.912.S-IC: Making Inferences and Justifying Conclusions

MAFS.912.S-IC.1: Understand and evaluate random processes underlying statistical experiments

MAFS.912.S-IC.1.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Polling: City

Polling: Neighborhood

MAFS.912.S-IC.2: Make inferences and justify conclusions from sample surveys, experiments, and observational studies

MAFS.912.S-IC.2.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

MAFS.912.S-IC.2.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Polling: Neighborhood

Probability Simulations

MAFS.912.S-IC.2.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

Populations and Samples

Probability Simulations

MAFS.912.S-IC.2.6: Evaluate reports based on data.

MAFS.912.S-MD: Using Probability to Make Decisions

MAFS.912.S-MD.1: Calculate expected values and use them to solve problems

MAFS.912.S-MD.1.1: Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

Box-and-Whisker Plots

Describing Data Using Statistics

Histograms

Scatter Plots - Activity A

Stem-and-Leaf Plots

MAFS.912.S-MD.1.2: Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

MAFS.912.S-MD.1.3: Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

Probability Simulations

Theoretical and Experimental Probability

MAFS.912.S-MD.1.4: Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

Geometric Probability - Activity A

Probability Simulations

Theoretical and Experimental Probability

MAFS.912.S-MD.2: Use probability to evaluate outcomes of decisions

MAFS.912.S-MD.2.5: Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

MAFS.912.S-MD.2.5.a: Find the expected payoff for a game of chance.

Binomial Probabilities

Geometric Probability - Activity A

MAFS.912.S-MD.2.6: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

Probability Simulations

Theoretical and Experimental Probability

MAFS.912.S-MD.2.7: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Correlation last revised: 8/7/2014

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