Correlation of the ALEKS High School Courses to …Word problem involving permutations Word problem involving combinations Permutations, combinations, and the multiplication principle

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Correlation of the ALEKS High School Courses to the Michigan High

School Mathematics Content Expectations

Strand 1 : Quantitative Literacy and Logic

= ALEKS course topic that addresses the standardTD = Teacher Directed

L1: Based on their knowledge of the properties of arithmetic, students understand and

reason about numbers, number systems, and the relationships between them. They

represent quantitative relationships using mathematical symbols, and interpret

relationships from those representations.

Number Systems and Number Sense

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.

Distributive property: Whole number coefficients Distributive property: Integer coefficients Properties of addition Properties of real numbers

Explain why the multiplicative inverse of a number has the same sign as the number, while the additive inverse of a number has the opposite sign.

TD

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

Properties of addition Properties of real numbers Distributive property: Whole number coefficients Distributive property: Integer coefficients

Describe the reasons for the different effects of multiplication by, or exponentiation of, a positive number by a number less than 0, a number between 0 and 1, and a number greater than 1.

TD

Justify numerical relationships

Ordering fractions Ordering decimals Ordering fractions and decimals Ordering integers Ordering fractions with variables

Explain the importance of the irrational numbers √2 and √3 in basic right triangle trigonometry, and theimportance of π because of its role in circle relationships.

Circumference and area of a circle Special right triangles

Representations and Relationships

Use mathematical symbols to represent quantitative relationships and situations.


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Ordering fractions Ordering decimals Ordering fractions and decimals Ordering integers Writing a one-step variable expression for a real-world situation Translating a sentence into a one-step equation Translating a sentence into a simple inequality Writing a multi-step inequality for a real-world situation Translating a sentence into a two-step expression Writing a multi-step equation for a real-world situation Writing a simple inequality for a real-world situation

Interpret representations that reflect absolute value relationships in such contexts as error tolerance.TD

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.

Scalar multiplication of a vector: Geometric Approach Multiplication of a vector by a scalar: Geometric approach Vector addition and scalar multiplication Vector addition: Geometric approach Vector subtraction: Geometric approach Magnitude of a vector Calculating the magnitude and direction of a vector Linear combination of vectors: Algebraic approach

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.

Histograms for numerical data Interpreting bar graphs Interpreting line graphs Double bar charts Histograms for grouped data Frequency polygons for grouped data Interpreting relative frequency histograms Interpreting a stem-and-leaf plot Using back-to-back stem-and-leaf plots to compare data sets Box-and-whisker plots

Read and interpret representations from various technological sources, such as contour or isobar diagrams.TD

Counting and Probabilistic Reasoning

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

Introduction to permutations and combinations Permutations and combinations: Problem type 1 Permutations and combinations: Problem type 2 Permutations and combinations: Problem type 3 Counting principle Word problem involving permutations Word problem involving combinations Permutations, combinations, and the multiplication principle for counting

Define and interpret commonly used expressions of probability.

Probabilities of draws with replacement Probabilities of draws without replacement


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Introduction to probability of an event Experimental and theoretical probability Probability of an event Area as probability Probability of independent events Probability of dependent events Outcomes and event probability Die rolling Probability of intersection or union: Word problems Independent events: Basic Probability of union: Basic Probability of the union of two events Probability of independent events Probability of dependent events

Recognize and explain common probability misconceptions such as "hot streaks" and "being due."TD

L2: Students calculate fluently, estimate proficiently, and describe and use algorithms

in appropriate situations (e.g., approximating solutions to equations). They understand

the basic ideas of iteration and algorithms.

Calculation Using Real and Complex Numbers

Explain the meaning and uses of weighted averages.

Weighted mean Estimating the mean of grouped data

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.

Evaluating expressions with exponents: Problem type 1 Evaluating expressions with exponents: Problem type 2 Exponents and order of operations Introduction to exponents Writing a positive number without a negative exponent Writing a negative number without a negative exponent Introduction to the product rule of exponents Product rule with negative exponents Quotients of expressions involving exponents Introduction to the power rule of exponents Power rule with positive exponents Power rule with negative exponents: Problem type 1 Power rule with negative exponents: Problem type 2 Converting between radical form and exponent form Rational exponents: Basic Rational exponents: Negative exponents and fractional bases Rational exponents: Products and quotients Rational exponents: Powers of powers Exponents and integers: Problem type 1 Exponents and integers: Problem type 2 Exponents and signed fractions Evaluating expressions with exponents of zero Writing a simple algebraic expression without negative exponents Understanding the product rule of exponents Product rule with positive exponents Quotient rule with negative exponents: Problem type 1


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Understanding the power rule of exponents Using the power and product rules to simplify expressions with positive exponents Using the power, product, and quotient rules to simplify expressions with negative exponents

Explain the exponential relationship between a number and its base 10 logarithm and use it to relate rules of logarithms to those of exponents in expressions involving numbers.

TD

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

Using i to rewrite square roots of negative numbers

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

Using i to rewrite square roots of negative numbers Simplifying a product or quotient involving roots of negative numbers Adding and subtracting complex numbers Multiplying complex numbers Dividing complex numbers

Understand the mathematical bases for the differences among voting procedures.TD

Sequences and Iteration

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

Finding the first terms of a sequence Finding a specified term of an arithmetic sequence given the common difference and first term Finding a specified term of a geometric sequence given the common ratio and first term Arithmetic and geometric sequences: Identifying and writing in standard form Arithmetic sequences Geometric sequences

Compute sums of finite arithmetic and geometric sequences.

Sum of the first n terms of an arithmetic sequence Sum of the first n terms of a geometric sequence

Use iterative processes in such examples as computing compound interest or applying approximation procedures.

Solving a word problem using an exponential equation: Problem type 1 Compound interest

Compute sums of infinite geometric sequences.

Sum of a geometric series

Measurement Units, Calculations, and Scales

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

Customary unit conversion with whole number values Metric distance conversion with whole number values Converting between metric and customary unit systems Converting between compound units: Basic Converting between compound units: Advanced

Describe and interpret logarithmic relationships in such contexts as the Richter scale, the pH scale, or decibel measurements; solve applied problems.

TD

Understanding Error

Determine what degree of accuracy is reasonable for measurements in a given situation; express accuracy


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through use of significant digits, error tolerance, or percent of error; describe how errors in measurements are magnified by computation; recognize accumulated error in applied situations.

TD

Describe and explain round-off error, rounding, and truncating.TD

Know the meaning of and interpret statistical significance, margin of error, and confidence level.TD

L3: Students understand mathematical reasoning as being grounded in logic and proof

and can distinguish mathematical arguments from other types of arguments. They can

interpret arguments made about quantitative situations in the popular media. Students

know the language and laws of logic and can apply them in both mathematical and

everyday settings. They write proofs using direct and indirect methods and use

counterexamples appropriately to show that statements are false.

Mathematical Reasoning

Distinguish between inductive and deductive reasoning, identifying and providing examples of each.TD

Differentiate between statistical arguments (statements verified empirically using examples or data) and logical arguments based on the rules of logic.

TD

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.

TD

Language and Laws of Logic

Know and use the terms of basic logic.

Conditional statements and negations The converse, inverse, and contrapositive of a conditional statement Conditional statements and deductive reasoning

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.

Conditional statements and negations The converse, inverse, and contrapositive of a conditional statement Conditional statements and deductive reasoning Venn diagrams with two sets Venn diagrams with three sets

Use the quantifiers "there exists" and "all" in mathematical and everyday settings and know how to logically negate statements involving them.

TD

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.

The converse, inverse, and contrapositive of a conditional statement

Proof

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.

Conditional statements and deductive reasoning


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Construct proofs by contradiction. Use counterexamples, when appropriate, to disprove a statement.

Indirect proof (proof by contradiction)

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.

TD

Strand 2 : Algebra and Functions


A1: Students recognize, construct, interpret, and evaluate expressions. They fluently

transform symbolic expressions into equivalent forms. They determine appropriate

techniques for solving each type of equation, inequality, or system of equations, apply

the techniques correctly to solve, justify the steps in the solutions, and draw

conclusions from the solutions. They know and apply common formulas.

Construction, Interpretation, and Manipulation of Expressions

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.

Introduction to evaluating an algebraic expression Evaluating a linear expression in two variables Evaluating a quadratic expression in one variable Writing a one-step variable expression for a real-world situation Translating a sentence into a one-step equation Translating a sentence into a simple inequality Writing a multi-step inequality for a real-world situation Translating a sentence into a two-step expression Writing a multi-step equation for a real-world situation Writing a simple inequality for a real-world situation

Know the definitions and properties of exponents and roots, transition fluently between them, and apply them in algebraic expressions.

Introduction to the product rule of exponents Product rule with negative exponents Quotients of expressions involving exponents Introduction to the power rule of exponents Power rule with positive exponents Power rule with negative exponents: Problem type 1 Power rule with negative exponents: Problem type 2 Square root of a perfect square monomial Simplifying a radical expression: Problem type 1 Simplifying a radical expression: Problem type 2 Square root addition Simplifying a sum of radical expressions Square root multiplication Simplifying a product of radical expressions Simplifying a product of radical expressions: Advanced Converting between radical form and exponent form Rational exponents: Basic Rational exponents: Products and quotients


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Rational exponents: Powers of powers Writing a simple algebraic expression without negative exponents Product rule with positive exponents Quotient rule with negative exponents: Problem type 1 Using the power and product rules to simplify expressions with positive exponents Using the power, product, and quotient rules to simplify expressions with negative exponents Rational exponents: Negative exponents and fractional bases Square root of a rational perfect square Simplifying products or quotients of higher index radicals with different indices Simplifying a higher radical: Problem type 2

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

Factoring a quadratic with leading coefficient 1 Factoring a quadratic with leading coefficient greater than 1 Factoring a perfect square trinomial Factoring a quadratic polynomial in two variables with leading coefficient greater than 1 Factoring a product of a quadratic trinomial and a monomial Factoring a difference of squares Factoring a sum or difference of two cubes Factoring out a binomial from a polynomial Greatest common factor of two multivariate monomials Factoring out a monomial from a polynomial: Univariate Factoring a polynomial by grouping: Problem type 1 Factoring a polynomial by grouping: Problem type 2

Add, subtract, multiply, and simplify polynomials and rational expressions.

Product rule with positive exponents Combining like terms: Advanced Simplifying a sum or difference of two univariate polynomials Simplifying a sum or difference of three univariate polynomials Multiplying a monomial and a polynomial: Univariate with positive leading coefficients Multiplying binomials with leading coefficients of 1 Multiplying conjugate binomials: Univariate Multiplying binomials in two variables Squaring a binomial: Univariate Multiplication involving binomials and trinomials in two variables Multiplying rational expressions: Problem type 1 Multiplying rational expressions: Problem type 2 Adding rational expressions with common denominators Adding rational expressions with different denominators: ax, bx Adding rational expressions with different denominators: Multivariate Adding rational expressions with different denominators: x+a, x+b Adding rational expressions with different denominators: Quadratic Simplifying a ratio of polynomials: Problem type 1 Simplifying a ratio of polynomials: Problem type 2 Ratio of multivariate polynomials Combining like terms in a quadratic expression

Divide a polynomial by a monomial.

Dividing a polynomial by a monomial: Univariate Dividing a polynomial by a monomial: Multivariate

Transform exponential and logarithmic expressions into equivalent forms using the properties of exponents and logarithms, including the inverse relationship between exponents and logarithms.

Converting between logarithmic and exponential equations


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Converting between natural logarithmic and exponential equations Evaluating a logarithmic expression Basic properties of logarithms Writing expressions as a single logarithm Expanding a logarithmic expression: Problem type 1 Expanding a logarithmic expression: Problem type 2

Transform trigonometric expressions into equivalent forms using basic identities such as sin² θ + cos² θ = 1and tan² θ + 1 = sec² θ

Simplifying trigonometric expressions Verifying a trigonometric identity Proving trigonometric identities: Problem type 1

Solutions of Equations and Inequalities

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

Solving a fraction word problem using a linear equation of the form Ax = B Translating a sentence into a one-step equation Solving a word problem with two unknowns using a linear equation Solving a decimal word problem using a linear equation with the variable on both sides Solving a decimal word problem using a linear equation of the form Ax + B = C Solving a fraction word problem using a linear equation with the variable on both sides Translating a sentence into a simple inequality Writing a compound inequality Writing a multi-step inequality for a real-world situation Word problem with linear inequalities Solving a word problem using a system of linear inequalities Solving a word problem using linear programming Solving a word problem involving a sum and another simple relationship using a system of linearequations Solving a value mixture problem using a system of linear equations Solving a distance, rate, time problem using a system of linear equations Solving a percent mixture problem using a system of linear equations Solving a tax rate or interest rate problem using a system of linear equations Solving a word problem using a quadratic equation with rational roots Finding the perimeter or area of a rectangle given one of these values Translating a sentence into a two-step expression Writing a multi-step equation for a real-world situation Writing a simple inequality for a real-world situation Solving a value mixture problem using a linear equation Solving a percent mixture problem using a linear equation Solving a distance, rate, time problem using a linear equation Word problem with linear inequalities: Problem type 1 Word problem with linear inequalities: Problem type 2 Writing an equation and drawing its graph to model a real-world situation

Associate a given equation with a function whose zeros are the solutions of the equation.

Finding zeros of a polynomial function written in factored form Finding the x-intercept(s) and the vertex of a parabola

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.

Additive property of equality with whole numbers Additive property of equality with decimals Additive property of equality with integers Additive property of equality with a negative coefficient


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Multiplicative property of equality with whole numbers Multiplicative property of equality with signed fractions Solving a two-step equation with integers Solving a two-step equation with signed fractions Solving a linear equation with several occurrences of the variable: Variables on the same side anddistribution Solving a linear equation with several occurrences of the variable: Variables on both sides andfractional coefficients Solving a linear equation with several occurrences of the variable: Variables on both sides anddistribution Solving a linear equation with several occurrences of the variable: Variables on both sides and twodistributions Solving a linear equation with several occurrences of the variable: Fractional forms with binomialnumerators Solving a linear inequality: Problem type 1 Solving a linear inequality: Problem type 2 Solving a linear inequality: Problem type 3 Solving a linear inequality: Problem type 4 Graphically solving a system of linear equations Solving a simple system using substitution Solving a system of linear equations using elimination with multiplication and addition Solving a system that is inconsistent or consistent dependent Solving a system of 3 equations in 3 unknowns Graphing a system of linear inequalities Finding the roots of a quadratic equation with leading coefficient 1 Finding the roots of a quadratic equation with leading coefficient greater than 1 Solving a quadratic equation needing simplification Solving a quadratic equation using the square root property: Problem type 1 Solving a quadratic equation using the square root property: Problem type 2 Applying the quadratic formula: Exact answers Solving a quadratic inequality written in factored form Solving a quadratic inequality Solving a quadratic equation with complex roots Multiplicative property of equality with integers Using two steps to solve an equation with signed decimals Solving equations with zero, one, or infinitely many solutions

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

Simple absolute value equation Solving an equation involving absolute value: Basic Solving an equation involving absolute value: Advanced Solving an inequality involving absolute value Solving an inequality involving absolute value: Basic

Solve polynomial equations and equations involving rational expressions, and justify steps in the solution.

Roots of a product of polynomials Solving a rational equation that simplifies to a linear equation: Problem type 1 Solving a rational equation that simplifies to a linear equation: Problem type 2 Solving a rational equation that simplifies to a linear equation: Problem type 3 Solving a rational equation that simplifies to a linear equation: Problem type 4 Solving a rational equation that simplifies to a quadratic equation: Problem type 1 Solving a rational equation that simplifies to a quadratic equation: Problem type 2 Solving a rational equation that simplifies to a quadratic equation: Problem type 3 Solving equations written in factored form

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


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Solving a radical equation that simplifies to a linear equation: One radical, basic Solving a radical equation that simplifies to a linear equation: Two radicals Solving a radical equation that simplifies to a quadratic equation: One radical Solving a radical equation that simplifies to a quadratic equation: Two radicals Solving a quadratic equation using the square root property: Problem type 1 Odd root property Solving a quadratic equation using the square root property: Problem type 2 Solving an equation with exponent using the odd-root property Solving an equation with positive rational exponent Solving an equation with negative rational exponent Solving an equation with a root index greater than 2

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

Solving a logarithmic equation: Problem type 1 Solving a logarithmic equation: Problem type 2 Solving a logarithmic equation: Problem type 3 Solving a logarithmic equation: Problem type 4 Solving an exponential equation: Problem type 1 Solving an exponential equation: Problem type 2 Solving an exponential equation: Problem type 3 Solving a logarithmic equation: Problem type 5

Solve an equation involving several variables (with numerical or letter coefficients) for a designated variable. Justify steps in the solution.

Introduction to algebraic symbol manipulation Algebraic symbol manipulation Algebraic symbol manipulation: Problem type 2 Algebraic symbol manipulation: Problem type 1

Know common formulas and apply appropriately in contextual situations.

Simple interest Compound interest Application problem with a linear function: Problem type 1 Application problem with a linear function: Problem type 2 Solving a word problem using an exponential equation: Problem type 1 Solving a word problem using an exponential equation: Problem type 2 Solving a word problem using an exponential equation: Problem type 3 Solving a word problem using a quadratic equation with irrational roots

Use special values of the inverse trigonometric functions to solve trigonometric equations over specific intervals.

Finding solutions in an interval for a basic equation involving sine or cosine Finding solutions in an interval for a basic tangent, cotangent, secant, or cosecant equation Finding solutions in an interval for a trigonometric equation with a squared function: Problem type 1 Finding solutions in an interval for a trigonometric equation with a squared function: Problem type 2 Finding solutions in an interval for a trigonometric equation in factored form

A2: Students understand functions, their representations, and their attributes. They

perform transformations, combine and compose functions, and find inverses. Students

classify functions and know the characteristics of each family. They work with

functions with real coefficients fluently. Students construct or select a function to

model a real-world situation in order to solve applied problems. They draw on their

knowledge of families of functions to do so.

Definitions, Representations, and Attributes of Functions


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

Identifying functions from relations Determining whether an equation defines a function Vertical line test Domain and range from ordered pairs Domain of a square root function Domain and range from the graph of a continuous function Domain and range from the graph of a piecewise function Domain of a rational function Range of a quadratic function Domain of a square root function

Read, interpret, and use function notation and evaluate a function at a value in its domain.

Function tables Evaluating functions: Problem type 1 Evaluating functions: Problem type 2 Evaluating a piecewise-defined function Graphing integer functions Finding inputs and outputs of a function from its graph

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

Graphing a line given its equation in slope-intercept form Graphing a line given its equation in standard form Writing an equation and drawing its graph to model a real-world situation Writing an equation of a line given the y-intercept and another point Writing the equation of a line given the slope and a point on the line Writing the equation of the line through two given points Graphing an equation involving absolute value in the plane: Advanced Function tables Evaluating functions: Problem type 1 Graphing integer functions Writing a function rule given a table of ordered pairs: One-step rules Writing a function rule given a table of ordered pairs: Two-step rules Finding inputs and outputs of a function from its graph Graphing a simple cubic function Graphing a piecewise-defined function Writing a quadratic equation given the roots and the leading coefficient Matching graphs with polynomial functions Sketching the graph of a rational function: Problem type 1 Sketching the graph of a rational function: Problem type 2 Graphing rational functions with holes Writing the equation of a rational function given its graph

Graphing a parabola of the form y = ax2

Graphing a parabola of the form y = (x-a)2 + c

Graphing a parabola of the form y = ax2 + bx + c: Integer coefficients Writing the equation of a quadratic function given its graph Sketching the graph of an exponential function: Basic Choosing a graph to fit a narrative

Graphing a parabola of the form y = ax2 + bx + c Graphing a function involving a square root Sketching the graph of a logarithmic function: Basic The graph, domain, and range of a logarithmic function The graph, domain, and range of an exponential function


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Recognize that functions may be defined by different expressions over different intervals of their domains; such functions are piecewise-defined.

Evaluating a piecewise-defined function Graphing a piecewise-defined function

Recognize that functions may be defined recursively. Compute values of and graph simple recursively defined functions.

Composition of two functions: Basic Sum, difference, and product of two functions Quotient of two functions

Identify the zeros of a function, the intervals where the values of a function are positive or negative, and describe the behavior of a function as x approaches positive or negative infinity, given the symbolic and graphical representations.

Finding x- and y-intercepts of a line given the equation: Advanced Finding zeros of a polynomial function written in factored form Finding x- and y-intercepts given a polynomial function Determining the end behavior of the graph of a polynomial function Finding the x-intercept(s) and the vertex of a parabola Finding intercepts and zeros of a function given the graph Finding where a function is increasing, decreasing, or constant given the graph Using a graphing calculator to find zeros of a polynomial function

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

Finding x- and y-intercepts of a line given the equation: Advanced Domain and range from ordered pairs Domain of a square root function Domain of a square root function Finding intercepts and zeros of a function given the graph Finding where a function is increasing, decreasing, or constant given the graph Finding local maxima and minima of a function given the graph Domain and range from the graph of a continuous function Domain and range from the graph of a piecewise function Finding zeros of a polynomial function written in factored form Finding x- and y-intercepts given a polynomial function Determining the end behavior of the graph of a polynomial function Using a graphing calculator to find zeros of a polynomial function Using a graphing calculator to find local extrema of a polynomial function Inferring properties of a polynomial function from its graph Domain of a rational function Finding the asymptotes of a rational function: Problem type 1 Finding the asymptotes of a rational function: Problem type 2 Range of a quadratic function Word problem using the maximum or minimum of a quadratic function Finding the x-intercept(s) and the vertex of a parabola Y-intercept of a line Rewriting a quadratic function to find the vertex of its graph Using a graphing calculator to find the vertex and x-intercepts of a quadratic function Finding the maximum or minimum of a quadratic function Classifying the graph of a function How the leading coefficient affects the shape of a parabola Matching graphs with polynomial functions The graph, domain, and range of an exponential function The graph, domain, and range of a logarithmic function

Operations and Transformations


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Combine functions by addition, subtraction, multiplication, and division.

Sum, difference, and product of two functions Quotient of two functions Combining functions: Advanced

Apply given transformations to basic functions and represent symbolically.

Writing an equation for a function after a vertical translation Writing an equation for a function after a vertical and horizontal translation Translating the graph of a function: One step Translating the graph of a function: Two steps Transforming the graph of a function by reflecting over an axis Transforming the graph of a function by shrinking or stretching Transforming the graph of a function using more than one transformation Translating the graph of a logarithmic or exponential function

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

Horizontal line test

If a function has an inverse, find the expression(s) for the inverse.

Inverse functions: Problem type 1 Inverse functions: Problem type 2 Inverse functions: Problem type 3

Write an expression for the composition of one function with another; recognize component functions when a function is a composition of other functions.

Composition of two functions: Advanced Expressing a function as a composition of two functions

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

Determining whether two functions are inverses of each other Inverse functions: Problem type 1

Representations of Functions

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.

Finding the asymptotes of a rational function: Problem type 1 Finding the asymptotes of a rational function: Problem type 2 Matching graphs with polynomial functions Classifying the graph of a function Determining the end behavior of the graph of a polynomial function

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

Writing a function rule given a table of ordered pairs: One-step rules Writing a function rule given a table of ordered pairs: Two-step rules

Write the general symbolic forms that characterize each family of functions.TD

Models of Real-world Situations Using Families of Functions

Identify the family of function best suited for modeling a given real-world situation.TD

Adapt the general symbolic form of a function to one that fits the specification of a given situation by using the information to replace arbitrary constants with numbers.


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Writing an equation and drawing its graph to model a real-world situation Writing a quadratic equation given the roots and the leading coefficient Writing an equation of a line given the y-intercept and another point Writing the equation of a line given the slope and a point on the line Writing the equation of a quadratic function given its graph

Using the adapted general symbolic form, draw reasonable conclusions about the situation being modeled.

Solving a word problem using a quadratic equation with irrational roots Word problem using the maximum or minimum of a quadratic function Evaluating an exponential function that models a real-world situation

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

Linear programming Solving a word problem using linear programming

A3: Students study the symbolic and graphical forms of each function family. By

recognizing the unique characteristics of each family, they can use them as tools for

solving problems or for modeling real-world situations.

Lines and Linear Functions

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

Writing an equation and drawing its graph to model a real-world situation Writing an equation of a line given the y-intercept and another point Writing the equation of a line given the slope and a point on the line Writing the equation of the line through two given points Writing the equations of vertical and horizontal lines through a given point

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

Graphing a line given the x- and y-intercepts Graphing a line given its equation in slope-intercept form Graphing a line given its equation in standard form Graphing a line through a given point with a given slope Graphing a vertical or horizontal line

Relate the coefficients in a linear function to the slope and x- and y-intercepts of its graph.

Y-intercept of a line Finding the slope of a line given its equation

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.

Slopes of parallel and perpendicular lines: Problem type 1 Slopes of parallel and perpendicular lines: Problem type 2

Exponential and Logarithmic Functions

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

Sketching the graph of an exponential function: Basic The graph, domain, and range of an exponential function Sketching the graph of an exponential function: Advanced

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

Evaluating an exponential function that models a real-world situation Sketching the graph of an exponential function: Basic The graph, domain, and range of an exponential function


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Sketching the graph of an exponential function: Advanced Sketching the graph of a logarithmic function: Basic The graph, domain, and range of a logarithmic function Translating the graph of a logarithmic or exponential function

Apply properties of exponential and logarithmic functions.

Converting between logarithmic and exponential equations Converting between natural logarithmic and exponential equations Evaluating a logarithmic expression Basic properties of logarithms Writing expressions as a single logarithm Expanding a logarithmic expression: Problem type 1 Expanding a logarithmic expression: Problem type 2 Change of base for logarithms: Problem type 1 Solving a logarithmic equation: Problem type 1 Solving a logarithmic equation: Problem type 2 Solving a logarithmic equation: Problem type 3 Solving a logarithmic equation: Problem type 4 Solving an exponential equation: Problem type 1 Solving an exponential equation: Problem type 2 Solving an exponential equation: Problem type 3 Solving a logarithmic equation: Problem type 5

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.

Sketching the graph of an exponential function: Basic The graph, domain, and range of an exponential function

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

Evaluating an exponential function that models a real-world situation Solving a word problem using an exponential equation: Problem type 1 Solving a word problem using an exponential equation: Problem type 2 Solving a word problem using an exponential equation: Problem type 3 Solving a word problem using an exponential equation: Problem type 4 Compound interest

Quadratic Functions

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

Writing a quadratic equation given the roots and the leading coefficient


Graphing a parabola of the form y = (x-a)2 + c

Graphing a parabola of the form y = ax2 + bx + c: Integer coefficients Writing the equation of a quadratic function given its graph

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.

Finding the x-intercept(s) and the vertex of a parabola Rewriting a quadratic function to find the vertex of its graph How the leading coefficient affects the shape of a parabola

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

Rewriting a quadratic function to find the vertex of its graph

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

Finding the x-intercept(s) and the vertex of a parabola


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

Finding the x-intercept(s) and the vertex of a parabola Rewriting a quadratic function to find the vertex of its graph

Power Functions

Write the symbolic form and sketch the graph of power functions.

Graphing a simple cubic function


Express direct and inverse relationships as functions and recognize their characteristics.

Word problem on direct variation Word problem on inverse variation

Analyze the graphs of power functions, noting reflectional or rotational symmetry.TD

Polynomial Functions

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

Graphing a simple cubic function Matching graphs with polynomial functions

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

Inferring properties of a polynomial function from its graph Matching graphs with polynomial functions

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.

Finding zeros of a polynomial function written in factored form Linear factors theorem and conjugate zeros theorem

Rational Functions

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

Sketching the graph of a rational function: Problem type 1 Sketching the graph of a rational function: Problem type 2 Graphing rational functions with holes Writing the equation of a rational function given its graph

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.

Domain of a rational function Finding the asymptotes of a rational function: Problem type 1 Finding the asymptotes of a rational function: Problem type 2 Matching graphs with rational functions: Two vertical asymptotes

Trigonometric Functions

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.

Finding trigonometric ratios from a point on the unit circle

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

Converting between degree and radian measure: Problem type 1 Converting between degree and radian measure: Problem type 2

Use the unit circle to determine the exact values of sine and cosine, for integer multiples of π/6 and π/4.


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Trigonometric functions and special angles: Problem type 1 Trigonometric functions and special angles: Problem type 2 Trigonometric functions and special angles: Problem type 3

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

Sketching the graph of a sine or cosine function: Problem type 1 Sketching the graph of a sine or cosine function: Problem type 2 Sketching the graph of a sine or cosine function: Problem type 3

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.

Amplitude and period of sine and cosine functions Amplitude, period, and phase shift of sine and cosine functions Writing the equation of a sine or cosine function given its graph: Problem type 1 Writing the equation of a sine or cosine function given its graph: Problem type 2 Sketching the graph of a sine or cosine function: Problem type 1 Sketching the graph of a sine or cosine function: Problem type 2 Sketching the graph of a sine or cosine function: Problem type 3 Sketching the graph of a secant or cosecant function: Problem type 1 Sketching the graph of a secant or cosecant function: Problem type 2 Sketching the graph of a tangent or cotangent function: Problem type 1 Sketching the graph of a tangent or cotangent function: Problem type 2 Matching graphs and equations for secant, cosecant, tangent, and cotangent functions

Strand 3 : Geometry and Trigonometry


G1: Students represent basic geometric figures, polygons, and conic sections and

apply their definitions and properties in solving problems and justifying arguments,

including constructions and representations in the coordinate plane. Students

represent three-dimensional figures, understand the concepts of volume and surface

area, and use them to solve problems. They know and apply properties of common

three-dimensional figures.

Lines and Angles; Basic Euclidean and Coordinate Geometry

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

Supplementary and complementary angles Identifying linear pairs and vertical angles Solving equations involving vertical angles Vertical angles and linear pairs

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

Corresponding and alternate angles Introduction to proofs involving parallel lines Proofs involving parallel lines

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


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Constructing congruent angles Constructing an angle bisector Constructing the perpendicular bisector of a line segment Constructing a pair of perpendicular lines Constructing a pair of parallel lines

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 a pair of perpendicular lines Constructing a pair of parallel lines

Given a line segment in terms of its endpoints in the coordinate plane, determine its length and midpoint.

Midpoint of a line segment in the plane Distance between two points in the plane

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.

TD

Triangles and Their Properties

Prove that the angle sum of a triangle is 180° and that an exterior angle of a triangle is the sum of the tworemote interior angles.

TD

Construct and justify arguments and solve multistep problems involving angle measure, side length, perimeter, and area of all types of triangles.

Scalene, isosceles, and equilateral triangles Triangle inequality: Problem type 1 Triangle inequality: Problem type 2 Relationship between angle measures and side lengths in a triangle: Problem type 1 Relationship between angle measures and side lengths in a triangle: Problem type 2 Sum of the angle measures of a triangle Finding angle measures in extended triangles Angle measures with triangles and parallel lines Angle measures of right or isosceles triangles with variables Finding an angle measure for a triangle with an extended side Finding an angle measure for a triangle sharing a side with another triangle Sides of polygons having the same perimeter Area involving rectangles and triangles Area between two concentric circles

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

Pythagorean Theorem Computing an area using the Pythagorean Theorem

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

Special right triangles Circles inscribed in and circumscribed about regular polygons

Solve multistep problems and construct proofs about the properties of medians, altitudes, perpendicular bisectors to the sides of a triangle, and the angle bisectors of a triangle. Using a straightedge and compass, construct these lines.

Proofs involving congruent triangles: Problem type 2 Proofs involving congruent triangles: Problem type 4


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Triangles and Trigonometry

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.

Sine, cosine, and tangent ratios Solving a right triangle Using a trigonometric ratio to find a side length in a right triangle Using trigonometry to find distances Using a trigonometric ratio to find an angle measure in a right triangle Using trigonometry to find angles of elevation or depression Finding trigonometric ratios given a right triangle

Know and use the Law of Sines and the Law of Cosines and use them to solve problems. Find the area of atriangle with sides a and b and included angle θ using the formula Area = (1/2) absinθ.

Solving a triangle with the law of sines: Problem type 1 Solving a triangle with the law of sines: Problem type 2 Solving a word problem using the law of sines Solving a triangle with the law of cosines Solving a word problem using the law of cosines Finding the area of a triangle using trigonometry

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

Trigonometric functions and special angles: Problem type 1 Trigonometric functions and special angles: Problem type 2 Trigonometric functions and special angles: Problem type 3

Quadrilaterals and Their Properties

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

Classifying quadrilaterals Classifying quadrilaterals: Advanced problem Properties of parallelograms: Problem type 1 Properties of parallelograms: Problem type 2 Properties of rectangles Properties of rhombi Finding a side length given the perimeter and side lengths with variables Finding the side length of a rectangle given its perimeter or area Areas of rectangles with the same perimeter Finding the perimeter or area of a rectangle given one of these values Area between two rectangles Area involving rectangles and triangles Perimeter involving rectangles and circles Area involving rectangles and circles Area involving rectangles and circles: Advanced problem

Solve multistep problems and construct proofs involving quadrilaterals using Euclidean methods or coordinate geometry.

Finding coordinates of vertices of polygons Area of quadrilaterals in the coordinate plane Finding the coordinates of a point to make a parallelogram

Describe and justify hierarchical relationships among quadrilaterals.

Classifying quadrilaterals Classifying parallelograms Classifying quadrilaterals: Advanced problem


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Prove theorems about the interior and exterior angle sums of a quadrilateral.TD

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

TD

Other Polygons and Their Properties

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

Area of a piecewise rectangular figure Area of a regular polygon

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.

The sum of interior angle measures in a convex polygon Interior and exterior angle measures in a regular polygon Area of a regular polygon

Circles and Their Properties

Solve multistep problems involving circumference and area of circles.

Finding the radius or the diameter of a circle given its circumference Perimeter involving rectangles and circles Circumference ratios Area between two concentric circles Area involving rectangles and circles Area involving rectangles and circles: Advanced problem

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

Introduction to circle: diameter, radius, and chord Identifying central angles, inscribed angles, arcs, chords, and tangents of a circle Tangents of a circle: Problem type 1 Tangents of a circle: Problem type 2 Lengths of chords, secants, and tangents Inscribed angles of a circle Central angles and inscribed angles of a circle Angles of intersecting secants and tangents

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

Identifying central angles, inscribed angles, arcs, chords, and tangents of a circle Inscribed angles of a circle Central angles and inscribed angles of a circle Angle measures in circle graphs

Know and use properties of arcs and sectors and find lengths of arcs and areas of sectors.

Arc length and area of a sector of a circle Area of a sector of a circle

Conic Sections and Their Properties

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

Writing an equation of a circle given its center and a point on the circle

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

TD

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


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Graphing an ellipse centered at the origin Graphing an ellipse given its equation in standard form Graphing an ellipse given its equation in general form Graphing a hyperbola centered at the origin Graphing a hyperbola given its equation in standard form Graphing a hyperbola given its equation in general form

Know and use the relationship between the vertices and foci in an ellipse, the vertices and foci in a hyperbola, and the directrix and focus in a parabola, interpret these relationships in applied contexts.

Finding the foci of an ellipse Writing an equation of an ellipse given the foci and the major axis length Finding the foci of a hyperbola Writing an equation of a hyperbola given the foci and the vertices Writing an equation of a hyperbola given the foci and the asymptotes Finding the focus of a parabola

Three- Dimensional Figures

Solve multi-step problems involving surface area and volume of pyramids, prisms, cones, cylinders, hemispheres, and spheres.

Volume of a solid made of unit cubes Volume of a rectangular prism Volume of a piecewise rectangular prism Volume of a triangular prism Volume of a pyramid Volume of a cylinder Volume of a cone Rate of filling of a solid Ratio of volumes Volume of a sphere Surface area of a cube or a rectangular prism Surface area of a solid made of unit cubes Surface area of a triangular prism Surface area of a cylinder Surface area involving prisms or cylinders Surface area of a sphere

Identify symmetries of pyramids, prisms, cones, cylinders, hemispheres, and spheres.TD

G2: Students use and justify relationships between lines, angles, area and volume

formulas, and 2- and 3-dimensional representations. They solve problems and provide

proofs about congruence and similarity.

Relationships Between Area and Volume Formulas

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.

TD

Know and demonstrate the relationships between the area formulas of various quadrilaterals.TD

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).

TD

Relationships Between Two-dimensional and Three-dimensional Representations

Identify or sketch a possible three-dimensional figure, given two-dimensional views. Create a


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two-dimensional representation of a three-dimensional figure.

Nets of solids Side views of a solid made of cubes

Identify or sketch cross sections of three-dimensional figures. Identify or sketch solids formed by revolving two-dimensional figures around lines.

TD

Congruence and Similarity

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

Identifying and naming congruent triangles Proofs involving congruent triangles: Problem type 1 Proofs involving congruent triangles: Problem type 2 Proofs involving congruent triangles: Problem type 3 Proofs involving congruent triangles: Problem type 5

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

Identifying and naming congruent triangles Proofs involving congruent triangles: Problem type 1 Proofs involving congruent triangles: Problem type 2 Proofs involving congruent triangles: Problem type 3 Proofs involving congruent triangles: Problem type 4 Proofs involving congruent triangles: Problem type 5

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

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

Right triangles and geometric mean Triangles and parallel lines Similar right triangles Indirect measurement

Know and apply the theorem stating that the effect of a scale factor of k relating one two-dimensional figure to another or one three-dimensional figure to another, on the length, area, and volume of the figures is to

multiply each by k, k2, and k3, respectively.

Similar solids: Problem type 1 Similar solids: Problem type 2

G3: Students will solve problems about distance-preserving transformations and

shape-preserving transformations. The transformations will be described synthetically

and, in simple cases, by analytic expressions in coordinates.

Distance-preserving Transformations: Isometries

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

Translation of a polygon Reflection of a polygon over a vertical or horizontal line Rotation of a figure about the origin

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

Identifying transformations


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Find the image of a figure under the composition of two or more isometries and determine whether the resulting figure is a reflection, rotation, translation, or glide reflection image of the original figure.

TD

Shape-preserving Transformations: Dilations and Isometries

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

Dilation

Given two figures that are images of each other under some dilation, identify the center and magnitude of the dilation.

TD

Find the image of a figure under the composition of a dilation and an isometry.TD

Strand 4 : Statistics and Probability


S1: Students plot and analyze univariate data by considering the shape of distributions

and analyzing outliers; they find and interpret commonly-used measures of center and

variation; and they explain and use properties of the normal distribution.

Producing and Interpreting Plots

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.

Histograms for numerical data Interpreting bar graphs Double bar charts Histograms for grouped data Frequency polygons for grouped data Interpreting relative frequency histograms Box-and-whisker plots

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.

Mean, median, and mode: Comparisons

Measures of Center and Variation

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.

Comparing means without calculation Mode of a data set Mean and median of a data set Choosing the best measure to describe data How changing a value affects the mean and median Mean, median, and mode: Computations Rejecting unreasonable claims based on average statistics Estimating the mean of grouped data Mean, median, and mode: Comparisons


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Weighted mean

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

Comparing means without calculation Estimating the mean of grouped data

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

Comparing standard deviations without calculation Percentiles Population standard deviation Sample standard deviation Estimating the standard deviation of grouped data Transforming the mean and standard deviation of a data set

The Normal Distribution

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

TD

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

Normal versus standard normal density curves

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.

Chebyshev's theorem and the empirical rule Word problem involving calculations from a normal distribution

Calculate z-scores, use z-scores to recognize outliers, and use z-scores to make informed decisions.

Standard normal probabilities Standard normal values: Basic Standard normal values: Advanced Normal distribution raw scores Normal distribution: Word problems

S2: Students plot and interpret bivariate data by constructing scatterplots, recognizing

linear and nonlinear patterns, and interpreting correlation coefficients; they fit and

interpret regression models, using technology as appropriate.

Scatterplots and Correlation

Construct a scatterplot for a bivariate data set with appropriate labels and scales.TD

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

Linear relationship and the sample correlation coefficient

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

Linear relationship and the sample correlation coefficient

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.

TD

Linear Regression


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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.

Sketching the least-squares regression line Computing the sample correlation coefficient and the coefficients for the least-squares regression line

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

Predictions from the least-squares regression line

S3: Students understand and apply sampling and various sampling methods, examine

surveys and experiments, identify bias in methods of conducting surveys, and learn

strategies to minimize bias. They understand basic principles of good experimental

design.

Data Collection and Analysis

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

TD

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.

TD

Distinguish between an observational study and an experimental study, and identify, in context, the conclusions that can be drawn from each.

TD

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

TD

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.

TD

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

TD

Explain the basic ideas of statistical process control, including recording data from a process over time.TD

Read and interpret basic control charts; detect patterns and departures from patterns.TD

S4: Students understand probability and find probabilities in various situations,

including those involving compound events, using diagrams, tables, geometric models

and counting strategies; they apply the concepts of probability to make decisions.

Probability

Understand and construct sample spaces in simple situations.

Outcomes and event probability Die rolling

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


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Probabilities of draws with replacement Introduction to probability of an event Probability of an event Probability of independent events Probability of dependent events Independent events: Basic Probability of union: Basic Probability of the union of two events Mutually exclusive events: Two events Probability of independent events Independent events: Two events Conditional probability: Basic Probability of dependent events Intersection and conditional probability Conditional probability: Mutually exclusive events Conditional probability: Independent events Probability of intersection or union: Word problems

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.

Experimental and theoretical probability

Application and Representation

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

Probabilities of draws without replacement Tree diagrams for conditional probabilities

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

Introduction to expectation Calculating relative frequencies in a contingency table Conditional probability: Basic Conditional probability: Independent events Probability of intersection or union: Word problems

Documents

Correlation of the ALEKS High School Courses to …Word problem involving permutations Word problem involving combinations Permutations, combinations, and the multiplication principle