Grade Level and High School Content Expectations

L1.1.1: Know the different properties that hold in different number systems and recognize that the applicable properties change in the transition from the positive integers to all integers, to the rational numbers, and to the real numbers.

Addition of Polynomials

Rational Numbers, Opposites, and Absolute Values

L1.1.3: Explain how the properties of associativity, commutativity, and distributivity, as well as identity and inverse elements, are used in arithmetic and algebraic calculations.

Equivalent Algebraic Expressions I

Equivalent Algebraic Expressions II

Operations with Radical Expressions

Simplifying Algebraic Expressions I

Simplifying Algebraic Expressions II

Solving Algebraic Equations I

Square Roots

L1.1.6: Explain the importance of the irrational numbers the square root of 2 and the square root of 3 in basic right triangle trigonometry, and the importance of pi because of its role in circle relationships.

Circumference and Area of Circles

L1.2.1: Use mathematical symbols to represent quantitative relationships and situations.

Square Roots

Using Algebraic Expressions

L1.2.3: Use vectors to represent quantities that have magnitude and direction, interpret direction and magnitude of a vector numerically, and calculate the sum and difference of two vectors.

L1.2.4: Organize and summarize a data set in a table, plot, chart, or spreadsheet; find patterns in a display of data; understand and critique data displays in the media.

Box-and-Whisker Plots

Correlation

Describing Data Using Statistics

Stem-and-Leaf Plots

L1.3.1: Describe, explain, and apply various counting techniques; relate combinations to Pascal?s triangle; know when to use each technique.

L1.3.2: Define and interpret commonly used expressions of probability.

Probability Simulations

Theoretical and Experimental Probability

L2.1.2: Calculate fluently with numerical expressions involving exponents; use the rules of exponents; evaluate numerical expressions involving rational and negative exponents; transition easily between roots and exponents.

L2.1.4: Know that the complex number i is one of two solutions to x² = -1.

Points in the Complex Plane

Roots of a Quadratic

L2.1.5: Add, subtract, and multiply complex numbers; use conjugates to simplify quotients of complex numbers.

Points in the Complex Plane

Roots of a Quadratic

L2.1.7: Understand the mathematical bases for the differences among voting procedures.

L2.2.1: Find the nth term in arithmetic, geometric, or other simple sequences.

Arithmetic Sequences

Geometric Sequences

L2.3.1: Convert units of measurement within and between systems; explain how arithmetic operations on measurements affect units, and carry units through calculations correctly.

L2.4.3: Know the meaning of and interpret statistical significance, margin of error, and confidence level.

Polling: City

Polling: Neighborhood

L3.1.3: Define and explain the roles of axioms (postulates), definitions, theorems, counterexamples, and proofs in the logical structure of mathematics. Identify and give examples of each.

Biconditional Statements

Investigating Angle Theorems

Isosceles and Equilateral Triangles

L3.2.1: Know and use the terms of basic logic.

Biconditional Statements

Conditional Statements

L3.2.2: Use the connectives ?not,? ?and,? ?or,? and ?if?, then,? in mathematical and everyday settings. Know the truth table of each connective and how to logically negate statements involving these connectives.

L3.2.4: Write the converse, inverse, and contrapositive of an ?if?, then?? statement. Use the fact, in mathematical and everyday settings, that the contrapositive is logically equivalent to the original, while the inverse and converse are not.

Biconditional Statements

Conditional Statements

L3.3.1: Know the basic structure for the proof of an ?if?, then?? statement (assuming the hypothesis and ending with the conclusion) and that proving the contrapositive is equivalent.

Biconditional Statements

Conditional Statements

L3.3.3: Explain the difference between a necessary and a sufficient condition within the statement of a theorem. Determine the correct conclusions based on interpreting a theorem in which necessary or sufficient conditions in the theorem or hypothesis are satisfied.

Biconditional Statements

Conditional Statements

A1.1.1: Give a verbal description of an expression that is presented in symbolic form, write an algebraic expression from a verbal description, and evaluate expressions given values of the variables.

Solving Equations on the Number Line

Using Algebraic Equations

Using Algebraic Expressions

A1.1.3: Factor algebraic expressions using, for example, greatest common factor, grouping, and the special product identities.

Factoring Special Products

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

Simplifying Algebraic Expressions II

A1.1.4: Add, subtract, multiply, and simplify polynomials and rational expressions.

Addition and Subtraction of Functions

Addition of Polynomials

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

Simplifying Algebraic Expressions II

A1.1.7: Transform trigonometric expressions into equivalent forms using basic identities such as sin² theta + cos² theta = 1 and tan² theta + 1 = sec² theta

Simplifying Trigonometric Expressions

Sum and Difference Identities for Sine and Cosine

A1.2.1: Write equations and inequalities with one or two variables to represent mathematical or applied situations, and solve.

Compound Inequalities

Modeling and Solving Two-Step Equations

Solving Algebraic Equations II

Solving Equations on the Number Line

Solving Linear Inequalities in One Variable

A1.2.3: Solve linear and quadratic equations and inequalities including systems of up to three linear equations with three unknowns. Justify steps in the solution, and apply the quadratic formula appropriately.

Cat and Mouse (Modeling with Linear Systems)

Exploring Linear Inequalities in One Variable

Linear Inequalities in Two Variables

Modeling One-Step Equations

Modeling and Solving Two-Step Equations

Point-Slope Form of a Line

Quadratic Inequalities

Roots of a Quadratic

Solving Algebraic Equations II

Solving Equations by Graphing Each Side

Solving Linear Inequalities in One Variable

Solving Linear Systems (Matrices and Special Solutions)

Solving Linear Systems (Slope-Intercept Form)

Solving Linear Systems (Standard Form)

Solving Two-Step Equations

Standard Form of a Line

Systems of Linear Inequalities (Slope-intercept form)

A1.2.4: Solve absolute value equations and inequalities, and justify steps in the solution.

Absolute Value Equations and Inequalities

Absolute Value with Linear Functions

Compound Inequalities

A1.2.6: Solve power equations and equations including radical expressions, justify steps in the solution, and explain how extraneous solutions may arise.

Operations with Radical Expressions

Radical Functions

A1.2.7: Solve exponential and logarithmic equations, and justify steps in the solution.

A1.2.9: Know common formulas and apply appropriately in contextual situations.

Cat and Mouse (Modeling with Linear Systems)

Circles

Compound Interest

A2.1.1: Determine whether a relationship (given in contextual, symbolic, tabular, or graphical form) is a function and identify its domain and range.

A2.1.3: Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations.

Function Machines 1 (Functions and Tables)

Function Machines 3 (Functions and Problem Solving)

Introduction to Functions

Points, Lines, and Equations

A2.1.7: Identify and interpret the key features of a function from its graph or its formula(e).

Absolute Value with Linear Functions

Exponential Functions

Linear Functions

Radical Functions

A2.2.1: Combine functions by addition, subtraction, multiplication, and division.

Addition and Subtraction of Functions

Function Machines 1 (Functions and Tables)

A2.2.3: Recognize whether a function (given in tabular or graphical form) has an inverse and recognize simple inverse pairs.

Function Machines 3 (Functions and Problem Solving)

Logarithmic Functions

A2.2.6: Know and interpret the function notation for inverses and verify that two functions are inverses using composition.

Function Machines 1 (Functions and Tables)

Function Machines 3 (Functions and Problem Solving)

Logarithmic Functions

A2.3.1: Identify a function as a member of a family of functions based on its symbolic or graphical representation; recognize that different families of functions have different asymptotic behavior.

Absolute Value with Linear Functions

Exponential Functions

Linear Functions

Logarithmic Functions

Radical Functions

A2.3.2: Describe the tabular pattern associated with functions having constant rate of change (linear); or variable rates of change.

Compound Interest

Direct and Inverse Variation

Function Machines 1 (Functions and Tables)

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Points, Lines, and Equations

Slope-Intercept Form of a Line

A2.4.4: Use methods of linear programming to represent and solve simple real-life problems.

A3.1.1: Write the symbolic forms of linear functions (standard, point-slope, and slope-intercept) given appropriate information, and convert between forms.

Point-Slope Form of a Line

Points, Lines, and Equations

Slope-Intercept Form of a Line

Standard Form of a Line

A3.1.2: Graph lines (including those of the form x = h and y = k) given appropriate information.

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Point-Slope Form of a Line

Slope-Intercept Form of a Line

Standard Form of a Line

A3.1.4: Find an equation of the line parallel or perpendicular to given line, through a given point; understand and use the facts that non-vertical parallel lines have equal slopes, and that non-vertical perpendicular lines have slopes that multiply to give -1.

Cat and Mouse (Modeling with Linear Systems)

A3.2.1: Write the symbolic form and sketch the graph of an exponential function given appropriate information.

Compound Interest

Exponential Functions

Introduction to Exponential Functions

Logarithmic Functions

A3.2.2: Interpret the symbolic forms and recognize the graphs of exponential and logarithmic functions; recognize the logarithmic function as the inverse of the exponential function.

Compound Interest

Exponential Functions

Introduction to Exponential Functions

Logarithmic Functions

A3.2.3: Apply properties of exponential and logarithmic functions.

Exponential Functions

Logarithmic Functions

A3.2.4: Understand and use the fact that the base of an exponential function determines whether the function increases or decreases and understand how the base affects the rate of growth or decay.

A.3.2.5: Relate exponential and logarithmic functions to real phenomena, including half-life and doubling time.

A3.3.1: Write the symbolic form and sketch the graph of a quadratic function given appropriate information.

Addition and Subtraction of Functions

Exponential Functions

Quadratics in Factored Form

Quadratics in Polynomial Form

Roots of a Quadratic

Translating and Scaling Functions

Zap It! Game

A3.3.2: Identify the elements of a parabola (vertex, axis of symmetry, direction of opening) given its symbolic form or its graph, and relate these elements to the coefficient(s) of the symbolic form of the function.

Addition and Subtraction of Functions

Ellipses

Holiday Snowflake Designer

Hyperbolas

Parabolas

Translating and Scaling Functions

Zap It! Game

A3.3.3: Convert quadratic functions from standard to vertex form by completing the square.

A3.3.4: Relate the number of real solutions of a quadratic equation to the graph of the associated quadratic function.

Quadratics in Factored Form

Quadratics in Polynomial Form

Roots of a Quadratic

A3.3.5: Express quadratic functions in vertex form to identify their maxima or minima, and in factored form to identify their zeros.

Quadratics in Factored Form

Quadratics in Polynomial Form

A3.4.2: Express direct and inverse relationships as functions and recognize their characteristics.

A3.5.1: Write the symbolic form and sketch the graph of simple polynomial functions.

Graphs of Polynomial Functions

Polynomials and Linear Factors

Quadratics in Factored Form

A3.5.2: Understand the effects of degree, leading coefficient, and number of real zeros on the graphs of polynomial functions of degree greater than 2.

Graphs of Polynomial Functions

Polynomials and Linear Factors

Quadratics in Factored Form

Zap It! Game

A3.5.3: Determine the maximum possible number of zeros of a polynomial function, and understand the relationship between the x-intercepts of the graph and the factored form of the function.

Graphs of Polynomial Functions

Polynomials and Linear Factors

Quadratics in Factored Form

A3.6.1: Write the symbolic form and sketch the graph of simple rational functions.

General Form of a Rational Function

Rational Functions

A3.6.2: Analyze graphs of simple rational functions and understand the relationship between the zeros of the numerator and denominator and the function?s intercepts, asymptotes, and domain.

General Form of a Rational Function

Rational Functions

A3.7.1: Use the unit circle to define sine and cosine; approximate values of sine and cosine; use sine and cosine to define the remaining trigonometric functions; explain why the trigonometric functions are periodic.

Cosine Function

Simplifying Trigonometric Expressions

Sine Function

Sine, Cosine, and Tangent Ratios

Sum and Difference Identities for Sine and Cosine

A3.7.2: Use the relationship between degree and radian measures to solve problems.

Cosine Function

Sine Function

Tangent Function

A3.7.4: Graph the sine and cosine functions; analyze graphs by noting domain, range, period, amplitude, and location of maxima and minima.

Cosine Function

Sine Function

Translating and Scaling Sine and Cosine Functions

A3.7.5: Graph transformations of basic trigonometric functions (involving changes in period, amplitude, and midline) and understand the relationship between constants in the formula and the transformed graph.

Translating and Scaling Functions

Translating and Scaling Sine and Cosine Functions

G1.1.1: Solve multi-step problems and construct proofs involving vertical angles, linear pairs of angles supplementary angles, complementary angles, and right angles.

Investigating Angle Theorems

Triangle Angle Sum

G1.1.2: Solve multi-step problems and construct proofs involving corresponding angles, alternate interior angles, alternate exterior angles, and same-side (consecutive) interior angles.

Congruence in Right Triangles

Proving Triangles Congruent

Similar Figures

Similarity in Right Triangles

Triangle Angle Sum

G1.1.3: Perform and justify constructions, including midpoint of a line segment and bisector of an angle, using straightedge and compass.

Constructing Parallel and Perpendicular Lines

Segment and Angle Bisectors

G1.1.4: Given a line and a point, construct a line through the point that is parallel to the original line using straightedge and compass. Given a line and a point, construct a line through the point that is perpendicular to the original line. Justify the steps of the constructions.

Constructing Congruent Segments and Angles

Constructing Parallel and Perpendicular Lines

G1.1.6: Recognize Euclidean geometry as an axiom system. Know the key axioms and understand the meaning of and distinguish between undefined terms, axioms, definitions, and theorems.

G1.2.1: Prove that the angle sum of a triangle is 180° and that an exterior angle of a triangle is the sum of the two remote interior angles.

G1.2.3: Know a proof of the Pythagorean Theorem, and use the Pythagorean Theorem and its converse to solve multistep problems.

Circles

Cosine Function

Distance Formula

Pythagorean Theorem

Pythagorean Theorem with a Geoboard

Sine Function

Surface and Lateral Areas of Pyramids and Cones

Tangent Function

G1.2.4: Prove and use the relationships among the side lengths and the angles of 30º- 60º- 90º triangles and 45º- 45º- 90º triangles.

Cosine Function

Sine Function

Tangent Function

G1.3.1: Define the sine, cosine, and tangent of acute angles in a right triangle as ratios of sides. Solve problems about angles, side lengths, or areas using trigonometric ratios in right triangles.

Cosine Function

Sine Function

Sine, Cosine, and Tangent Ratios

Tangent Function

G1.3.3: Determine the exact values of sine, cosine, and tangent for 0°, 30°, 45°, 60°, and their integer multiples and apply in various contexts.

Cosine Function

Sine Function

Tangent Function

G1.4.1: Solve multistep problems and construct proofs involving angle measure, side length, diagonal length, perimeter, and area of squares, rectangles, parallelograms, kites, and trapezoids.

Area of Parallelograms

Area of Triangles

Classifying Quadrilaterals

Parallelogram Conditions

Perimeter and Area of Rectangles

Perimeters and Areas of Similar Figures

Similar Figures

Square Roots

G1.4.3: Describe and justify hierarchical relationships among quadrilaterals.

Classifying Quadrilaterals

Parallelogram Conditions

Special Parallelograms

G1.4.4: Prove theorems about the interior and exterior angle sums of a quadrilateral.

G1.4.5: Understand the definition of a cyclic quadrillateral and know and use the basic properties of cyclic quadrilaterals.

G1.5.1: Know and use subdivision or circumscription methods to find areas of polygons.

Area of Parallelograms

Area of Triangles

Perimeter and Area of Rectangles

Perimeters and Areas of Similar Figures

G1.5.2: Know, justify, and use formulas for the perimeter and area of a regular n-gon and formulas to find interior and exterior angles of a regular n-gon and their sums.

Area of Triangles

Polygon Angle Sum

G1.6.2: Solve problems and justify arguments about chords and lines tangent to circles.

G1.6.3: Solve problems and justify arguments about central angles, inscribed angles, and triangles in circles.

Chords and Arcs

Inscribed Angles

G1.6.4: Know and use properties of arcs and sectors and find lengths of arcs and areas of sectors.

G.1.7.1: Find an equation of a circle given its center and radius; given the equation of a circle, find its center and radius.

G1.7.2: Identify and distinguish among geometric representations of parabolas, circles, ellipses, and hyperbolas; describe their symmetries, and explain how they are related to cones.

Addition and Subtraction of Functions

Circles

Ellipses

Hyperbolas

Parabolas

Zap It! Game

G1.7.3: Graph ellipses and hyperbolas with axes parallel to the x- and y-axes, given equations.

G1.7.4: Know and use the relationship between the vertices and foci in and ellipse, the vertices and foci in a hyperboia, and the directrix and focus in a parabola, interpret these relationships in applied contexts.

G2.1.1: Know and demonstrate the relationships between the area formula of a triangle, the area formula of a parallelogram, and the area formula of a trapezoid.

Area of Parallelograms

Area of Triangles

Perimeter and Area of Rectangles

G2.1.2: Know and demonstrate the relationships between the area formulas of various quadrilaterals.

Area of Parallelograms

Area of Triangles

Perimeter and Area of Rectangles

G2.1.3: Know and use the relationship between the volumes of pyramids and prisms (of equal base and height) and cones and cylinders (of equal base and height).

Prisms and Cylinders

Pyramids and Cones

G2.2.1: Identify or sketch a possible three-dimensional figure, given two-dimensional views. Create a two-dimensional representation of a three-dimensional figure.

Surface and Lateral Areas of Prisms and Cylinders

G2.3.1: Prove that triangles are congruent using the SSS, SAS, ASA, and AAS criteria, and that right triangles, are congruent using the hypotenuse-leg criterion.

Congruence in Right Triangles

Proving Triangles Congruent

G2.3.2: Use theorems about congruent triangles to prove additional theorems and solve problems, with and without use of coordinates.

Congruence in Right Triangles

Proving Triangles Congruent

G2.3.3: Prove that triangles are similar by using SSS, SAS, and AA conditions for similarity.

G2.3.4: Use theorems about similar triangles to solve problems with and without use of coordinates.

G3.1.1: Define reflection, rotation, translation, and glide reflection and find the image of a figure under a given isometry.

Rotations, Reflections, and Translations

Similar Figures

Translations

G3.1.2: Given two figures that are images of each other under an isometry, find the isometry and describe it completely.

Dilations

Reflections

Rotations, Reflections, and Translations

Translations

G3.2.1: Know the definition of dilation and find the image of a figure under a given dilation.

Circles

Dilations

Rotations, Reflections, and Translations

Translations

G3.2.2: Given two figures that are images of each other under some dilation, identify the center and magniture of the dilation.

S1.1.1: Construct and interpret dot plots, histograms, relative frequency histograms, bar graphs, basic control charts, and box plots with appropriate labels and scales; determine which kinds of plots are appropriate for different types of data; compare data sets and interpret differences based on graphs and summary statistics.

Box-and-Whisker Plots

Correlation

Histograms

Mean, Median, and Mode

Reaction Time 1 (Graphs and Statistics)

Real-Time Histogram

Sight vs. Sound Reactions

Stem-and-Leaf Plots

S1.1.2: Given a distribution of a variable in a data set, describe its shape, including symmetry or skewness, and state how the shape is related to measures of center (mean and median) and measures of variation (range and standard deviation) with particular attention to the effects of outliers on these measures.

Box-and-Whisker Plots

Describing Data Using Statistics

Mean, Median, and Mode

Populations and Samples

Reaction Time 1 (Graphs and Statistics)

Real-Time Histogram

Sight vs. Sound Reactions

Stem-and-Leaf Plots

S1.2.1: Calculate and interpret measures of center including: mean, median, and mode; explain uses, advantages and disadvantages of each measure given a particular set of data and its context.

Box-and-Whisker Plots

Describing Data Using Statistics

Mean, Median, and Mode

Populations and Samples

Reaction Time 1 (Graphs and Statistics)

Sight vs. Sound Reactions

Stem-and-Leaf Plots

S1.2.2: Estimate the position of the mean, median, and mode in both symmetrical and skewed distributions, and from a frequency distribution or histogram.

S1.2.3: Compute and interpret measures of variation, including percentiles, quartiles, interquartile range, variance, and standard deviation.

Box-and-Whisker Plots

Polling: City

Real-Time Histogram

Sight vs. Sound Reactions

S1.3.1: Explain the concept of distribution and the relationship between summary statistics for a data set and parameters of a distribution.

Box-and-Whisker Plots

Describing Data Using Statistics

Mean, Median, and Mode

Populations and Samples

Reaction Time 1 (Graphs and Statistics)

Real-Time Histogram

Stem-and-Leaf Plots

S1.3.2: Describe characteristics of the normal distribution, including its shape and the relationships among its mean, median, and mode.

Polling: City

Populations and Samples

Real-Time Histogram

Sight vs. Sound Reactions

S1.3.3: Know and use the fact that about 68%, 95%, and 99.7% of the data lie within one, two, and three standard deviations of the mean, respectively in a normal distribution.

S2.1.2: Given a scatterplot, identify patterns, clusters, and outliers. Recognize no correlation, weak correlation, and strong correlation.

Correlation

Least-Squares Best Fit Lines

Solving Using Trend Lines

Trends in Scatter Plots

S2.1.3: Estimate and interpret Pearson?s correlation coefficient for a scatterplot of a bivariate data set. Recognize that correlation measures the strength of linear association.

Correlation

Least-Squares Best Fit Lines

Solving Using Trend Lines

Trends in Scatter Plots

S2.1.4: Differentiate between correlation and causation. Know that a strong correlation does not imply a cause-and-effect relationship. Recognize the role of lurking variables in correlation.

S2.2.1: For bivariate data that appear to form a linear pattern, find the least squares regression line by estimating visually and by calculating the equation of the regression line. Interpret the slope of the equation for a regression line.

Correlation

Least-Squares Best Fit Lines

Solving Using Trend Lines

S2.2.2: Use the equation of the least squares regression line to make appropriate predictions.

Correlation

Least-Squares Best Fit Lines

Solving Using Trend Lines

S3.1.1: Know the meanings of a sample from a population and a census of a population, and distinguish between sample statistics and population parameters.

S3.1.2: Identify possible sources of bias in data collection and sampling methods and simple experiments; describe how such bias can be reduced and controlled by random sampling; explain the impact of such bias on conclusions made from analysis of the data; and know the effect of replication on the precision of estimates.

Polling: Neighborhood

Populations and Samples

S3.1.4: Design simple experiments or investigations to collect data to answer questions of interest; interpret and present results.

Describing Data Using Statistics

Polling: City

Real-Time Histogram

S3.1.5: Understand methods of sampling, including random sampling, stratified sampling, and convenience samples, and be able to determine, in context, the advantages and disadvantages of each.

Polling: City

Polling: Neighborhood

Populations and Samples

S3.1.6: Explain the importance of randomization, double-blind protocols, replication, and the placebo effect in designing experiments and interpreting the results of studies.

Box-and-Whisker Plots

Polling: City

Polling: Neighborhood

Populations and Samples

S4.1.1: Understand and construct sample spaces in simple situations.

Independent and Dependent Events

Theoretical and Experimental Probability

S4.1.2: Define mutually exclusive events, independent events, dependent events, compound events, complementary events and conditional probabilities; and use the definitions to compute probabilities.

Binomial Probabilities

Independent and Dependent Events

Probability Simulations

Theoretical and Experimental Probability

S4.1.3: Design and carry out an appropriate simulation using random digits to estimate answers to questions about probability; estimate probabilities using results of a simulation; compare results of simulations to theoretical probabilities.

Geometric Probability

Independent and Dependent Events

Probability Simulations

Theoretical and Experimental Probability

S4.2.1: Compute probabilities of events using tree diagrams, formulas for combinations and permutations, Venn diagrams, or other counting techniques.

Binomial Probabilities

Permutations and Combinations

S4.2.2: Apply probability concepts to practical situations, in such settings as finance, health, ecology, or epidemiology, to make informed decisions.

Estimating Population Size

Probability Simulations

Theoretical and Experimental Probability

Correlation last revised: 1/20/2017

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