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Click on any standard to search for aligned resources. This data may be subject to copyright. You may download a CSV of the Georgia Mathematics Learning Standards if your intention constitutes fair use.

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Students will solve problems using concepts in graph theory including directed and undirected graphs, the Handshaking Theorem, isomorphism, paths, connectedness, and Euler and Hamilton Paths.

Students will generate and interpret equivalent numeric and algebraic expressions.

Apply properties of operations emphasizing when the commutative property applies. (MGSE7.EE.1)

Use area models to represent the distributive property and develop understandings of addition and multiplication (all positive rational numbers should be included in the models). (MGSE3.MD.7)

Model numerical expressions (arrays) leading to the modeling of algebraic expressions. (MGSE7.EE.1,2; MGSE9-12.A.SSE.1,3)

Add, subtract, and multiply algebraic expressions. (MGSE6.EE.3, MGSE6.EE.4, MC7.EE.1, MGSE9-12.A.SSE.3)

Generate equivalent expressions using properties of operations and understand various representations within context. For example, distinguish multiplicative comparison from additive comparison. Students should be able to explain the difference between 3 more and 3 times. (MGSE4.0A.2; MGSE6.EE.3, MGSE7.EE.1,2;MGSE9-12.A.SSE.3)

Evaluate formulas at specific values for variables. For example, use formulas such as A = l x w and find the area given the values for the length and width. (MGSE6.EE.2)

Students will create and solve equations and inequalities in one variable.

Use variables to represent an unknown number in a specified set. (MGSE.6.EE2,5,6)

Explain each step in solving simple equations and inequalities using the equality properties of numbers. (MGSE9-12.A.REI.1)

Construct viable arguments to justify the solutions and methods of solving equations and inequalities. (MGSE9-12.A.REI.1)

Use variables to solve real-world and mathematical problems. (MGSE6.EE.7,MGSE7.EE.4)

Solve multi-step real world problems, analyzing the relationships between all four operations. For example, understand division as an unknown-factor problem in order to solve problems. Knowing that 50 x 40 = 2000 helps students determine how many boxes of cupcakes they will need in order to ship 2000 cupcakes in boxes that hold 40 cupcakes each. (MGSE3.OA.6, MGSE4.OA.3)

Explain patterns in the placement of decimal points when multiplying or dividing by powers of ten. (MGSE5.NBT.2)

Compare fractions and decimals to the thousandths place. For fractions, use strategies other than cross multiplication. For example, locating the fractions on a number line or using benchmark fractions to reason about relative size. For decimals, use place value. (MGSE4.NF.2;MGSE5.NBT.3,4)

Students will explain equivalent ratios by using a variety of models. For example, tables of values, tape diagrams, bar models, double number line diagrams, and equations.

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range.(MGSE9-12.F.IF.1)

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. (MGSE9-12.F.IF.5)

Graph functions using sets of ordered pairs consisting of an input and the corresponding output. (MGSE8.F.1, 2)

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x2 + y2 = 3.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression

Write a function that describes a relationship between two quantities.

Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. Sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Graph functions expressed algebraically and show key features of the graph both by hand and by using technology.

Distinguish between situations that can be modeled with linear functions and with exponential functions

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

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

Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, ?3) lies on the circle centered at the origin and containing the point (0,2). (Focus on quadrilaterals, right triangles, and circles.)

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Understand there is a complex number i such that i2 = ?1, and every complex number has the form a + bi where a and b are real numbers.

Explain how the meaning of rational exponents follows from extending the properties of integer exponents to rational numbers, allowing for a notation for radicals in terms of rational exponents. For example, we define 5(1/3) to be the cube root of 5 because we want [5(1/3)] 3 = 5[(1/3) x 3] to hold, so [5(1/3)] 3 must equal 5.

Explain why the sum or product of rational numbers is rational; why the sum of a rational number and an irrational number is irrational; and why the product of a nonzero rational number and an irrational number is irrational.

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

Students will use, apply, and solve linear first-order differential equations.

Solve linear first-order differential equations of the form y' + p(x)y = q(x) with an integrating factor.

Solve homogeneous linear first-order differential equations using a variable substitution.

Write linear first-order differential equations that represent real-world phenomena and solve them, such as those arising from Kirchhoff's Law and mixing problems.

Students will solve linear second-order differential equations of the form y" + p(x)y' + q(x)y = c using the characteristic equation where the characteristic equation has two real roots, one real root, or no real roots.

Apply and adapt a variety of appropriate strategies to solve problems

Monitor and reflect on the process of mathematical problem solving

Students will use vectors and matrices to organize and describe problem situations.

Represent situations and solve problems using vectors in areas such as transportation, computer graphics, and the physics of force and motion.

Represent geometric transformations and solve problems using matrices in fields such as computer animations.

Students will use a variety of network models to organize data in quantitative situations, make informed decisions, and solve problems

Solve problems represented by a vertex-edge graph, and find critical paths, Euler paths, and minimal spanning trees.

Construct, analyze, and interpret flow charts to develop an algorithm to describe processes such as quality control procedures.

Students will create and analyze mathematical models to make decisions related to earning, investing, spending, and borrowing money.

Use exponential functions to model change in a variety of financial situations

Determine, represent, and analyze mathematical models for income, expenditures, and various types of loans and investments.

Students will analyze and evaluate the mathematics behind various methods of voting and selection

Evaluate various voting and selection processes to determine an appropriate method for a given situation

Apply various ranking algorithms to determine an appropriate method for a given situation.

Students will determine probability and expected value to inform everyday decision making

Determine conditional probabilities and probabilities of compound events to make decisions in problem situations.

Use probabilities to make and justify decisions about risks in everyday life

Calculate expected value to analyze mathematical fairness, payoff, and risk.

Students will build the skills and vocabulary necessary to analyze and critique reported statistical information, summaries, and graphical displays.

Students will apply statistical methods to design, conduct, and analyze statistical studies.

Students will use functions to model problem situations in both discrete and continuous relationships.

Determine whether a problem situation involving two quantities is best modeled by a discrete (pattern identification, population growth, compound interest) or continuous (medication dosage, climate change, bone decay) relationship.

Use linear, exponential, logistic, piecewise and sine functions to construct a model.

Students will create and use two- and three-dimensional representations of authentic situations.

Students will solve geometric problems involving inaccessible distances using basic trigonometric principles, including the Law of Sines and the Law of Cosines

Students will extend the understanding of proportional reasoning, ratios, rates, and percents by applying them to various settings to include business, media, and consumerism.

Solve problems involving large quantities that are not easily measured.

Understand how identification numbers, such as UPCs, are created and verified.

Students will classify differential equations in terms of ordinary/partial, order, and linearity.

Students will solve first and second order linear homogeneous equations by finding integrating factors and auxiliary equations.

Students will determine whether a first or second order initial value problem has a unique solution over a given interval.

Students will partially differentiate functions of multiple variables.

Students will explore counting principles such as: recurrence relations, Polyas Enumeration Theorem, inclusion-exclusion, and the Pigeonhole principle.

Students will explore game theory including Nash Equilibrium and two-player zerosum games.

Students will determine if a relation is an equivalence relation over a given set.

Students will describe sets using set builder notation. Students will define, use notation of, and pictorially represent set theory concepts, including union, intersection, element of, cardinality, complement, subset, and proper subset.

Students will apply quantifiers, conditionals, negations, contrapositives, converses, and inverses to determine the validity of logic statements.

Students will represent sentences in symbolic form and use the tools of truth tables and symbolic logic in assessing equivalence and validity.

Students will determine equivalence between sentences involving conjunctions, disjunctions, negations, and or/conditions.

Students will determine truth tables for sentences and use Venn diagrams to illustrate the relationships represented by these truth tables.

Students will use Karnaugh Maps to simplify Boolean algebra expressions.

Students will utilize appropriate methods of proof such as: direct proof, proof by mathematical induction, proof by transposition, and proof by contradiction

Students will differentiate between mathematical theorems and axioms, using graph theory, counting principles, and game theory.

Students will write theorems containing a hypothesis and conclusion, using graph theory, counting principles and game theory.

Students will prove previously recognized mathematical theorems, such as but not limited to the Pythagorean Theorem, the Minimax Theorem, the Binomial Theorem, and Cantors Theorem.

Students will demonstrate knowledge of both the definition and the graphical interpretation of limit of values of functions

Use theorems and algebraic concepts in evaluating the limits of sums, products, quotients, and composition of functions

Students will demonstrate knowledge of both the definition and graphical interpretation of continuity of a function.

Evaluate limits of functions and apply properties of limits, including one-sided limits.

Describe asymptotic behavior in terms of limits involving infinity.

Apply the definition of continuity to a function at a point and determine if a function is continuous over an interval.

Students will demonstrate knowledge of differentiation using algebraic functions.

Use differentiation and algebraic manipulations to sketch, by hand, graphs of functions.

Identify maxima, minima, inflection points, and intervals where the function is increasing and decreasing

Use differentiation and algebraic manipulations to solve optimization (maximum - minimum problems) in a variety of pure and applied contexts.

Students will demonstrate an understanding of the definition of the derivative of a function at a point, and the notion of differentiability.

Demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function

Demonstrate an understanding of the interpretation of the derivative as instantaneous rate of change

Use derivatives to solve a variety of problems coming from physics, chemistry, economics, etc. that involve the rate of change of a function

Demonstrate an understanding of the relationship between differentiability and continuity.

Use derivative formulas to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.

Use the Chain Rule and applications to the calculation of the derivative of a variety of composite functions

Find the derivatives of relations and use implicit differentiation in a wide variety of problems from physics, chemistry, economics, etc.

Demonstrate an understanding of and apply Rolle's Theorem, the Mean Value Theorem.

Apply the definition of the integral to model problems in physics, economics, etc, obtaining results in terms of integrals

Demonstrate knowledge of the Fundamental Theorem of Calculus, and use it to interpret integrals as anti-derivatives.

Use definite integrals in problems involving area, velocity, acceleration, and the volume of a solid

Compute, by hand, the integrals of a wide variety of functions using substitution.

Students will apply knowledge of mathematics, science, and engineering design to solve problems.

Determine the equations of lines and surfaces using vectors and 3D graphing

Apply dot and cross products of vectors to express equations of planes, parallelism, perpendicularity, angles

Students will learn to evaluate matrices and apply their properties to solve engineering problems.

Use Gaussian elimination to compute solution sets of linear systems.

Students will investigate functions of two and three independent variables to model engineering systems.

. Students will use visual and written communication to express basic design elements in the appropriate mathematics notation.

Demonstrate fundamentals of technical sketching using computer-generated visuals by using the appropriate mathematics scale.

Present a technical design, using computer-generated model, for an assigned design project utilizing the appropriate scientific units (US standards and SI units)

Students will evaluate and apply partial differentiation of multivariable functions with two or more independent variables.

Use the general chain rule to determine the partial derivatives of composite functions

Solve engineering optimization problems by applying partial differentiation or Lagrange multipliers

Utilize partial derivatives in developing the appropriate system balances (ex: mass balance) in engineering problems.

Students will apply the techniques of double and triple integration to multivariable scalar- and vector-valued functions.

Manipulate integrals by changing the order of integration, introducing variable substitutions, or changing to curvilinear coordinates.

Apply properties of integrals to calculate and represent area, volume, or mass.

Use integrals of vectors to define and apply the gradient, divergence, or the curl

Interpret the theorems of Green, Stokes, or Gauss and apply them to the study of real-world phenomena.

Students will solve engineering-based calculus problems (using appropriate technology)

Apply and adapt a variety of appropriate strategies to solve problems, such as considering realistic constraints relevant to the design of a system, component, or process.

Monitor and reflect on the process of mathematical problem solving and interpret problem solutions.

Students will use visual and written communication to express basic design elements and will communicate mathematically.

Organize and consolidate their mathematical thinking through communication

Communicate and use the language of mathematics to articulate their mathematical thinking coherently.

Present a technical design, using computer-generated model, for an assigned design project.

Students will describe the history of technological advancement and make connections among mathematical ideas and to other disciplines.

Recognize and use connections among mathematical and engineering ideas

Recognize and apply mathematics in contexts outside of mathematics.

Explain the impact of key persons and historical events and their impact on engineering and society

Describe the issues of necessity that have influenced innovation and technological development.

Create and use representations to organize, record, and communicate mathematical ideas

Select, apply, and translate among mathematical representations to solve problems.

Use representations to model and interpret physical, and mathematical phenomena.

Students will develop vocabulary and communication skills by reading text material and problem descriptions associated with engineering and technology education.

Students will explore and summarize the educational requirements and professional expectations associated with engineering career paths.

Substitute numeric values into formulas containing exponents, interpreting units consistently. (MGSE6.EE.2, MGSE9-12.N.Q.1, MGSE9-12.A.SSE.1, MGSE9-12.N.RN.2)

Use properties of integer exponents to find equivalent numerical expressions. For example, 32 x 3 -5 = 3 -3 = 1 33 = 1 27. (MGSE8.EE.1)

Evaluate square roots of perfect squares and cube roots of perfect cubes (MGSE8.EE.2)

Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x 3 = p, where p is a positive rational number. (MGSE8.EE.2)

Use the Pythagorean Theorem to solve triangles based on real-world contexts (Limit to finding the hypotenuse given two legs). (MGSE8.G.7)

Students will use units as a way to understand problems and guide the solutions of multi-step problems.

Choose and interpret graphs and data displays, including the scale and comparisons of data. (MGSE3.MD.3, MGSE9-12.N.Q.1)

Graph points in all four quadrants of the coordinate plane. (MGSE6.NS.8)

Create an algebraic model from a context using two-variable equations. (MGSE6.EE.6, MGSE8.EE.8, MGSE9-12.A.CED.2)

Find approximate solutions using technology to graph, construct tables of values, and find successive approximations. (MGSE9-12.A.REI.10,11)

Represent solutions to systems of equations graphically or by using a table of values. (MGSE6.EE.5; MGSE7.EE3; MGSE8.EE.8; MGSE9-12.A.CED.2)

Analyze the reasonableness of the solutions of systems of equations within a given context. (MGSE6.EE.5,6,MGSE7.EE4)

Solve for any variable in a multi-variable equation. (MGSE6.EE.9,MGSE9-12.A.REI.3)

Rearrange formulas to highlight a particular variable using the same reasoning as in solving equations. For example, solve for the base in A = bh. (MGSE9-12.A.CED.4)

Students will conceptualize positive and negative numbers (including decimals and fractions).

Explain meanings of real numbers in a real world context. (MGSE6.NS.5)

Students will recognize that there are numbers that are not rational, and approximate them with rational numbers

Numbers. a. Find an estimated decimal expansion of an irrational number locating the approximations on a number line. For example, for ?2, show that ?2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue this pattern in order to obtain better approximations. (MGSE8.NS.1,2)

Explain the results of adding and multiplying with rational and irrational numbers. (MGSE9- 12.N.RN.3)

Students will apply and extend previous understanding of addition, subtraction, multiplication, and division

Find sums, differences, products, and quotients of multi-digit decimals using strategies based on place value, the properties of operations, and/or relationships between operations. (MGSE5.NBT.7; MGSE6.NS.3)

Find sums, differences, products, and quotients of all forms of rational numbers, stressing the conceptual understanding of these operations. (MGSE7.NS.1,2)

Interpret and solve contextual problems involving division of fractions by fractions. For example, how many 3/4-cup servings are in 2/3 of a cup of yogurt? (MGSE6.NS.1)

. Illustrate and explain calculations using models and line diagrams. ( MGSE7.NS.1,2)

Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using estimation strategies and graphing technology. (MGSE7.NS.3, MGSE7.EE.3, MGSE9-12.N.Q.3)

Students will recognize and represent proportional relationships between quantities

Relate proportionality to fraction equivalence and division. For example, 3 6 is equal to 4 8 because both yield a quotient of and, in both cases, the denominator is double the value of the numerator. (MGSE4.NF.1)

Understand real-world rate/ratio/percent problems by finding the whole given a part and find a part given the whole. (MGSE6.RP.1,2,3;MGSE7.RP.1,2)

Use proportional relationships to solve multistep ratio and percent problems. (MGSE7.RP.2,3)

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. (MGSE8.EE.6)

Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. (MGSE8.EE.5)

Calculate rates of change of functions, comparing when rates increase, decrease, or stay constant. For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. (MGSE6.RP.2;MGSE7.RP.1,2,3;MGSE8.F.2,5; MGSE9-12.F.IF.6)

Graph by hand simple functions expressed symbolically (use all four quadrants). (MGSE9- 12.F.IF.7)

Interpret the equation y = mx + b as defining a linear function whose graph is a straight line. (MGSE8.F.3)

Use technology to graph non-linear functions. (MGSE8.F.3, MGSE9-12.F.IF.7)

Analyze graphs of functions for key features (intercepts, intervals of increase/decrease, maximums/minimums, symmetries, and end behavior) based on context. (MGSE9-12.F.IF.4,7)

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the great rate of change. (MGSE8.F.2)

Write a function that describes a relationship between two quantities. (MGSE8.F.4, MGSE9- 12.F.BF.1)

Use variables to represent two quantities in a real-world problem that change in relationship to one another (conceptual understanding of a variable). (MGSE6.EE.9)

Add, subtract, and multiply polynomials; understand that polynomials form a system analogous to the integers in that they are closed under these operations.

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

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2 ) 2 = (x2 y 2 ) 2 + (2xy)2 can be used to generate Pythagorean triples.

Know and apply that the Binomial Theorem gives the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined using Pascals Triangle.

Rewrite simple rational expressions in different forms using inspection, long division, or a computer algebra system; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x).

Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, simple rational, and exponential functions (integer inputs only).

Create linear, quadratic, and exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (The phrase in two or more variables refers to formulas like the compound interest formula, in which A = P(1 + r/n)nt has multiple variables.)

Represent constraints by equations or inequalities, and by systems of equation and/or inequalities, and interpret data points as possible (i.e. a solution) or not possible (i.e. a non-solution) under the established constraints.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Examples: Rearrange Ohms law V = IR to highlight resistance R; Rearrange area of a circle formula A = ?r2 to highlight the radius r.

Using algebraic properties and the properties of real numbers, justify the steps of a simple, one-solution equation. Students should justify their own steps, or if given two or more steps of an equation, explain the progression from one step to the next using properties.

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.

1 Using graphs, tables, or successive approximations, show that the solution to the equation f(x) = g(x) is the x-value where the y-values of f(x) and g(x) are the same.

Graph the solution set to a linear inequality in two variables.

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. For example, given ax + 3 = 7, solve for x.

Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p)2 = q that has the same solutions. Derive the quadratic formula from ax2 + bx + c = 0.

Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions).

Show and explain why the elimination method works to solve a system of two-variable equations.

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x2 + y2 = 3.

Represent a system of linear equations as a single matrix equation in a vector variable

Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 3 or greater).

Interpret expressions that represent a quantity in terms of its context

Interpret parts of an expression, such as terms, factors, and coefficients.

b Given situations which utilize formulas or expressions with multiple terms and/or factors, interpret the meaning (in context) of individual terms or factors

Use the structure of an expression to rewrite it in different equivalent forms. For example, see x4 y 4 as (x2 ) 2 - (y2 ) 2 , thus recognizing it as a difference of squares that can be factored as (x2 y 2 ) (x2 + y2 ).

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Factor any quadratic expression to reveal the zeros of the function defined by the expression.

Complete the square in a quadratic expression to reveal the maximum and minimum value of the function defined by the expression.

Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15t , where t is in years, can be rewritten as [1.15(1/12)] (12t) ? 1.012(12t) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments

Write a function that describes a relationship between two quantities.

Determine an explicit expression, a recursive process (steps for calculation) from a context. For example, if Jimmy starts out with $15 and earns $2 a day, the explicit expression 2x + 15 can be described recursively (either in writing or verbally) as to find out how much money Jimmy will have tomorrow, you add $2 to his total today. Jn = J n 1 + 2, J0 = 15.

Combine standard function types using arithmetic operations in contextual situations (Adding, subtracting, and multiplying functions of different types).

Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

Write arithmetic and geometric sequences both recursively and explicitly, use them to model situations, and translate between the two forms. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions.

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2(x3 ) or f(x) = (x+1)/(x-1) for x ? 1.

Verify by composition that one function is the inverse of another.

Read values of an inverse function from a graph or a table, given that the function has an inverse.

Produce an invertible function from a non-invertible function by restricting the domain.

Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range, i.e. each input value maps to exactly one output value. If f is a function, and x is the input (an element of the domain), then f(x) is the output (an element of the range). Graphically, the graph is y = f(x).

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (Generally, the scope of high school math defines this subset as the set of natural numbers 1, 2, 3, 4, ) By graphing or calculating terms, students should be able to show how the recursive sequence a1 = 7, an = an 1 + 2; the sequence sn = 2(n 1) + 7; and the function f(x) = 2x + 5 (when x is a natural number) all define the same sequence.

Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. Sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Graph functions expressed algebraically and show key features of the graph both by hand and by using technology

Graph linear and quadratic functions and show intercepts, maxima, and minima (as determined by the function or by context).

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior

Graph exponential and logarithmic functions, showing intercepts and end behavior

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function

Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. For example, compare and contrast quadratic functions in standard, vertex, and intercept forms.

Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t , y = (0.97)t , y = (1.01)(12t), y = (1.2)(t/10), and classify them as representing exponential growth and decay.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Distinguish between situations that can be modeled with linear functions and with exponential functions

Show that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals. (This can be shown by algebraic proof, with a table showing differences, or by calculating average rates of change over equal intervals).

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

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

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

For exponential models, express as a logarithm the solution to ab(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology

Interpret the parameters in a linear (f(x) = mx + b) and exponential (f(x) = a ? dx ) function in terms of a context. (In the functions above, m and b are the parameters of the linear function, and a and d are the parameters of the exponential function.) In context, students should describe what these parameters mean in terms of change and starting value.

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

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

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

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Prove the Pythagorean identity (sin A)2 + (cos A)2 = 1 and use it to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle.

Prove addition, subtraction, double, and half-angle formulas for sine, cosine, and tangent and use them to solve problems.

Identify and describe relationships among inscribed angles, radii, chords, tangents, and secants. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Construct a tangent line from a point outside a given circle to the circle.

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

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle.

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

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

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

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

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) fo

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segments endpoints

Give informal arguments for geometric formulas. a. Give informal arguments for the formulas of the circumference of a circle and area of a circle using dissection arguments and informal limit arguments. b. Give informal arguments for the formula of the volume of a cylinder, pyramid, and cone using Cavalieris principle.

Give informal arguments for the formulas of the circumference of a circle and area of a circle using dissection arguments and informal limit arguments.

Give informal arguments for the formula of the volume of a cylinder, pyramid, and cone using Cavalieris principle.

Give an informal argument using Cavalieris principle for the formulas for the volume of a sphere and other solid figures.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Identify the shapes of two-dimensional cross-sections of threedimensional objects, and identify three-dimensional objects generated by rotations of twodimensional objects.

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Derive the equation of a parabola given a focus and directrix.

Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, ?3) lies on the circle centered at the origin and containing the point (0, 2). (Focus on quadrilaterals, right triangles, and circles.)

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

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Verify experimentally the properties of dilations given by a center and a scale factor.

Prove the Laws of Sines and Cosines and use them to solve problems.

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

The dilation of a line not passing through the center of the dilation results in a parallel line and leaves a line passing through the center unchanged

The dilation of a line segment is longer or shorter according to the ratio given by the scale factor.

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

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar

Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, (and its converse); the Pythagorean Theorem using triangle similarity.

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Explain and use the relationship between the sine and cosine of complementary angles.

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Derive the formula A = (1/2)ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Understand there is a complex number i such that i2 = ?1, and every complex number has the form a + bi where a and b are real numbers.

Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Find the conjugate of a complex number; use the conjugate to find the absolute value (modulus) and quotient of complex numbers.

Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + ?3i)3 = 8 because (-1 + ?3i) has modulus 2 and argument 120.

Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Solve quadratic equations with real coefficients that have complex solutions by (but not limited to) square roots, completing the square, and the quadratic formula.

Extend polynomial identities to include factoring with complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x 2i).

Use the Fundamental Theorem of Algebra to find all roots of a polynomial equation

Use units within multi-step problems and formulas; interpret units of input and resulting units of output.

Identify, use, and record appropriate units of measure within context, within data displays, and on graphs;

Convert units and rates using dimensional analysis (English-to-English and Metric-to-Metric without conversion factor provided and between English and Metric with conversion factor);

Use units within multi-step problems and formulas; interpret units of input and resulting units of output.

Define appropriate quantities for the purpose of descriptive modeling. Given a situation, context, or problem, students will determine, identify, and use appropriate quantities for representing the situation.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. For example, money situations are generally reported to the nearest cent (hundredth). Also, an answers precision is limited to the precision of the data given.

Explain how the meaning of rational exponents follows from extending the properties of integer exponents to rational numbers, allowing for a notation for radicals in terms of rational exponents. For example, we define 5(1/3) to be the cube root of 5 because we want [5(1/3)] 3 = 5[(1/3) x 3] to hold, so [5(1/3)] 3 must equal 5.

Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Explain why the sum or product of rational numbers is rational; why the sum of a rational number and an irrational number is irrational; and why the product of a nonzero rational number and an irrational number is irrational.

Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

Work with 2 X 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Solve problems involving velocity and other quantities that can be represented by vectors.

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

Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum

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

Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ? 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

Use matrices to represent and manipulate data, e.g., transformations of vectors.

Add, subtract, and multiply matrices of appropriate dimensions.

Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (or, and, not).

Understand that if two events A and B are independent, the probability of A and B occurring together is the product of their probabilities, and that if the probability of two events A and B occurring together is the product of their probabilities, the two events are independent.

Understand the conditional probability of A given B as P (A and B)/P(B). Interpret independence of A and B in terms of conditional probability; that is, the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, use collected data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

Find the conditional probability of A given B as the fraction of Bs outcomes that also belong to A, and interpret the answer in context.

Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answers in context.

Apply the general Multiplication Rule in a uniform probability model, P(A and B) = [P(A)]x[P(B|A)] =[P(B)]x[P(A|B)], and interpret the answer in terms of the model.

Use permutations and combinations to compute probabilities of compound events and solve problems.

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0. 5. Would a result of 5 tails in a row cause you to question the model?

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant

Evaluate reports based on data. For example, determining quantitative or categorical data; collection methods; biases or flaws in data.

Represent data with plots on the real number line (dot plots, histograms, and box plots)

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation, mean absolute deviation) of two or more different data sets.

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

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

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

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

Decide which type of function is most appropriate by observing graphed data, charted data, or by analysis of context to generate a viable (rough) function of best fit. Use this function to solve problems in context. Emphasize linear, quadratic and exponential models

Using given or collected bivariate data, fit a linear function for a scatter plot that suggests a linear association.

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Compute (using technology) and interpret the correlation coefficient r of a linear fit. (For instance, by looking at a scatterplot, students should be able to tell if the correlation coefficient is positive or negative and give a reasonable estimate of the r value.) After calculating the line of best fit using technology, students should be able to describe how strong the goodness of fit of the regression is, using r.

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

Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Students will explore and use historical methods for expressing and solving equations.

Express the geometrical algebra found in historical works (such as the Elements ofEuclid) in modern algebraic notation.

Solve systems of linear and nonlinear equations using Diophantus' method.

Translate into modern notation problems appearing in ancient and medieval texts that involve linear, quadratic, or cubic equations and solve them.

Use Cardanos cubic formula and Khayyams geometric construction to find a solution to a cubic equation.

Students will explore abstract algebra and group-theoretic concepts.

Explore matrix products other than the Cayley product (including Lie and Jordan) by determining whether these products are associative or commutative.

Prove statements concerning figurate numbers using both graphical (as in the manner of the Greeks) and algebraic methods.

Use Fermats Little Theorem and Eulers Theorem to simplify expressions of the form a k mod m.

Use Gauss Law of Quadratic Reciprocity to determine quadratic residues of two odd primes; i.e., solve quadratic congruences of the form x 2 = p mod q.

Discover that the real primes that can be expressed as the sum of two squares are no longer prime in the field of Gaussian integers.

Students will use the algebraic techniques of Fermat, Barrow, and Newton to determine tangents to quadratic curves.

Students will compute the ratio of winnings in an interrupted game.

Students will understand and recognize the use of definitions, postulates, and axioms in defining a deductive system such as Euclidean geometry.

Prove the first five propositions in Book I of Euclids Elements. c. Construct a regular pentagon with a straight-edge and compass.

Students will compute lengths, areas, and volumes according to historical formulas.

Find the volume of a truncated pyramid using the Babylonian, Chinese, and Egyptian formulas

Identify cyclic quadrilaterals and find associated lengths by Ptolemys Theorem.

Students will explore and prove statements in non-Euclidean geometry

Prove that the summit angles of an isosceles birectangle are congruent, but that it is impossible to prove they are right without referring to the parallel postulate or one of its consequences

Describe the hypothesis of the acute angle (Hyperbolic), the hypothesis of the right angle (Euclidean), and the hypothesis of the obtuse angle (Spherical).

Prove that under the hypothesis of the acute angle, similarity implies congruence

Students will identify Hindu-Arabic numerals as a prime scientific advancement

Describe the limitations of the Babylonian, Roman, Egyptian (hieratic and hieroglyphic), Chinese, and Greek number systems as compared to Hindu-Arabic numerals

Describe the transition of Hindu-Arabic numerals from regional use in the 10th century to wide-spread use in the 15th (including the influence of Fibonacci for the use of the numerals and the Italian abascists against their use).

Identify the number system and notation used by a society as an influence on the types of mathematics developed by that society.

Students will describe factors involved in the rise and fall of ancient Greek society. a. D

Describe the theories for the rise of intellectual thought in ancient Greece and the factors involved in its collapse.

Describe the cultural aspects of Greek society that influenced the way mathematics developed in ancient Greece.

Explain the distinction made between number and magnitude, commensurable and incommensurable, and arithmetic and logistic, the cultural factors inherent in this distinction, and the logical crisis that occurred concerning incommensurable (irrational) magnitudes.

Students will trace the centers of development of mathematical ideas from the 5th century to the 18th century

Describe the transmission of ideas from the Greeks, through the Islamic peoples, to medieval Europe

Describe the influence of the Catholic Church and Charlemagne on the establishment of mathematics as one of the central pillars of education.

Explain the cultural factors that encouraged the development of algebra in 15th century Italy, and how this development influenced mathematical thought throughout Europe.

Identify the works of Galileo, Copernicus, and Kepler as a landmark in scientific thought, describe the conflict between their explanation of the workings of the solar system and then-current perspectives, and contrast their works to those of Aristotle

Describe the contributions of Fermat, Pascal, Descartes, Newton, and Gauss to mathematics.

Identify Euler as the first modern mathematician and a motivating force behind all aspects of mathematics for the 18th century.

Describe the influence the French Revolution had on education (establishment of the Ecole Normale and the Ecole Polytechnique, Monge, Lagrange, Legendre, Laplace).

Students will identify the 19th and 20th centuries as the time when mathematics became more specialized and more rigorous.

Describe the societal factors that inhibited the developement of non-Euclidean geometry.

Explain how the ancient Greek pattern of material axiomatics evolved into abstract axiomatics (non-Euclidean geometry, non-commutative algebra)

Identify Cantor as the most original mathematician since the ancient Greeks.

Describe the implications of Kleins Erlanger Programme and Godel's Incompleteness Theorem on the nature of mathematical discovery and proof.

Use Babylonian, Roman, Egyptian (hieratic and hieroglyphic), Chinese, and Greek number systems to represent quantities.

Use historical multiplication and division algorithms (including the Egyptian method of duplation and mediation, the medieval method of gelosia, and Napiers rods).

Students will explore the implications of infinite sets of real numbers.

Describe denumerable and nondenumerable sets and provide examples of each.

Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving.

Recognize reasoning and proof as fundamental aspects of mathematics

Organize and consolidate their mathematical thinking through communication.

Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

Analyze and evaluate the mathematical thinking and strategies of others.

Use the language of mathematics to express mathematical ideas precisely.

Students will make connections among mathematical ideas and to other disciplines.

Understand how mathematical ideas interconnect and build on one another to produce a coherent whole

Recognize and apply mathematics in contexts outside of mathematics.

Create and use representations to organize, record, and communicate mathematical ideas

Select, apply, and translate among mathematical representations to solve problems.

Use representations to model and interpret physical, social, and mathematical phenomena.

Apply and adapt a variety of appropriate strategies to solve problems.

Monitor and reflect on the process of mathematical problem solving.

Recognize reasoning and proof as fundamental aspects of mathematics.

Organize and consolidate their mathematical thinking through communication

Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

Analyze and evaluate the mathematical thinking and strategies of others.

Use the language of mathematics to express mathematical ideas precisely

Students will make connections among mathematical ideas and to other disciplines.

Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

Recognize and apply mathematics in contexts outside of mathematics.

Create and use representations to organize, record, and communicate mathematical ideas

Select, apply, and translate among mathematical representations to solve problems.

Use representations to model and interpret physical, social, and mathematical phenomena.

Students will investigate the relationship between points, lines, and planes in three-dimensions.

Express analytic geometry of three dimensions (equations of planes, parallelism, perpendicularity, angles) in terms of the dot product and cross product of vectors.

Represent a 3-by-3 system of linear equations as a matrix and solve the system in multiple ways: the inverse matrix, row operations, and Cramers Rule

Apply properties of similar and orthogonal matrices to prove statements about matrices.

Find and apply the eigenvectors and eigenvalues of a 3-by-3 matrix.

Students will explore functions of two independent variables of the form z = f(x, y) and implicit functions of the form f(x, y, z) = 0.

Students will explore the continuity of functions of two independent variables in terms of the limits of such functions as (x, y) approaches a given point in the plane.

Students will explore, find, use, and apply partial differentiation of functions of two independent variables of the form z = f(x, y) and implicit functions of the form f(x, y, z) = 0.

Approximate the partial derivatives at a point of a function defined by a table of data

Find expressions for the first and second partial derivatives of a function

Define and apply the total differential to approximate real-world phenomena

Represent the partial derivatives of a system of two functions in two variables using the Jacobian

Find the partial derivatives of the composition of functions using the general chain rule.

Apply partial differentiation to problems of optimization, including problems requiring the use of the Lagrange multiplier.

Students will define and apply the gradient, the divergence, and curl in terms of differential vector operations.

Students will integrate functions of the form z = f(x, y) or w = f(x, y, z)

Define, use, and interpret double and triple integrals in terms of volume and mass.

Integrate functions through various techniques such as changing the order of integration, substituting variables, or changing to polar coordinates.

Students will apply and interpret the theorems of Green, Stokes, and Gauss.

Apply line and surface integrals to functions representing real-world phenomena

Recognize, understand, and use line integrals that are independence of path.

Define and apply the gradient, the divergence, and the curl in terms of integrals of vectors.

Apply and adapt a variety of appropriate strategies to solve problems.

Monitor and reflect on the process of mathematical problem solving

Recognize reasoning and proof as fundamental aspects of mathematics

Organize and consolidate their mathematical thinking through communication.

Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

Analyze and evaluate the mathematical thinking and strategies of others.

Use the language of mathematics to express mathematical ideas precisely.

Students will make connections among mathematical ideas and to other disciplines.

Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

Recognize and apply mathematics in contexts outside of mathematics.

Create and use representations to organize, record, and communicate mathematical ideas.

Select, apply, and translate among mathematical representations to solve problems.

Use representations to model and interpret physical, social, and mathematical phenomena.

Students will use basic functions to solve and model problems related to stock transactions, banking and credit, employment and taxes, rent and mortgages, retirement planning, and other related finance applications.

Students will understand the characteristics of these functions as they relate to financial situations.

Understand and apply limits as end behavior of modeling functions.

Students will use formulas to investigate investments in banking and retirement planning.

Students will understand and use matrices to represent data and solve banking and retirement planning problems.

Students will use measures of central tendency to investigate data found in the stock market, retirement planning, transportation, budgeting, and home rental or ownership.

Students will use data displays including bar graphs, line graphs, stock bar charts, candlestick charts, box and whisker plots, stem and leaf plots, circle graphs, and scatterplots to recognize and interpret trends related to the stock market, retirement planning, insurance, car purchasing, and home rental or ownership

Students will use linear, quadratic, and cubic regressions as well as the correlation coefficient to move supply and demand, revenue, profit, and other financial problem situations.

Students will use probability, the Monte Carlo method, and expected value to model and predict outcomes related to the stock market, retirement planning, insurance, and investing.

Students will apply the concepts of area, volume, scale factors, and scale drawings to planning for housing.

Students will apply the properties of angles and segments in circles.

Students will use fractions, percents, and ratios to solve problems related to stock transactions, credit cards, taxes, budgets, automobile purchases, fuel economy, Social Security, Medicare, retirement planning, checking and saving accounts and other related finance applications.

Apply and adapt a variety of appropriate strategies to solve problems

Monitor and reflect on the process of mathematical problem solving.

Recognize reasoning and proof as fundamental aspects of mathematics.

Organize and consolidate their mathematical thinking through communication.

Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

Analyze and evaluate the mathematical thinking and strategies of others.

Use the language of mathematics to express mathematical ideas precisely.

Students will make connections among mathematical ideas and to other disciplines.

Understand how mathematical ideas interconnect and build on one another to produce a coherent whole

Recognize and apply mathematics in contexts outside of mathematics.

Create and use representations to organize, record, and communicate mathematical ideas.

Select, apply, and translate among mathematical representations to solve problems

Use representations to model and interpret physical, social, and mathematical phenomena.

Students will determine optimal locations and use them to make appropriate decisions.

Find the optimal location given three equally weighted, non-collinear points.

Students will determine optimal paths and use them to make appropriate decisions.

Apply appropriate recursive algorithms for minimum spanning tree, shortest path, and critical path management.

Recognize reasoning and proof as fundamental aspects of mathematics

Organize and consolidate their mathematical thinking through communication

Communicate their mathematical thinking coherently and clearly to peers, teachers, and others

Analyze and evaluate the mathematical thinking and strategies of others

Use the language of mathematics to express mathematical ideas precisely.

Students will make connections among mathematical ideas and to other disciplines.

Understand how mathematical ideas interconnect and build on one another to produce a coherent whole

Recognize and apply mathematics in contexts outside of mathematics.

Create and use representations to organize, record, and communicate mathematical ideas

Select, apply, and translate among mathematical representations to solve problem

Use representations to model and interpret physical, social, and mathematical phenomena

Students will use properties of normal distributions to make decisions about optimization and efficiency.

Calculate theoretical and empirical probabilities using standardized and nonstandardized data.

Consider contextual factors and investigate issues within the decision-making process.

Students will use properties of other distributions (e.g. binomial, geometric, Poisson) to make decisions about optimization and efficiency

Calculate theoretical and empirical probabilities using standardized and nonstandardized data.

Consider contextual factors and investigate issues within the decisionmaking process.

Use program evaluation review technique (PERT) to investigate completion times of a project

Develop and apply transition matrices to make predictions using Markov Chains

Consider contextual factors and investigate issues within the decision making process.

Students will use distributions to identify the key features of the data collected. Students will describe the distribution for quantitative and categorical data.

Describe and interpret the measures of center for the distribution.

Describe and interpret the patterns in variability for the distribution.

Describe and interpret patterns that exist for the distribution.

Describe and interpret any outliers or gaps in the distribution.

Students will use distributions to compare two or more groups. Students will compare two or more groups by analyzing distributions.

Use graphical and numerical attributes of distributions to make comparisons between distributions.

Students will determine if an association exists between two variables (pattern or trend in bivariate data) and use values of one variable to predict values of another variable. Students will analyze associations between variables and make predictions from one variable to another

Make predictions and draw conclusions from two-variable data based on data displays

Students will distinguish between a population distribution, a sample data distribution, and a sampling distribution

Recognize a population distribution has fixed values of its parameters that are usually unknown.

Recognize a sample data distribution is taken from a population distribution and the data distribution is what is seen in practice hoping it approximates the population distribution.

Recognize a sampling distribution is the distribution of a sample statistic (such as a sample mean or a sample proportion) obtained from repeated samples. The sampling distribution provides the key for determining how close to expect a sample statistic approximates the population parameter.

Students will create sample data distributions and a sampling distribution.

Create a sampling distribution of a statistic by taking repeated samples from a population (either hands-on or by simulation with technology).

Students will understand that randomness should be incorporated into a sampling or experimental procedure. Students will be able to implement a reasonable random method for selecting a sample or for assigning treatments in an experiment

Students will distinguish between the three types of study designs for collecting data (sample survey, experiment, and observational study) and will know the scope of the interpretation for each design type. Students will be able to distinguish between the three types of study designs for collecting data (sample survey, experiment, and observational study) and know the scope of the interpretation for each design type.

Determine the type of study design appropriate for answering a statistical question.

Determine the appropriate scope of inference for the study design used

Students will distinguish between the role of randomness and the role of sample size with respect to using a statistic from a sample to estimate a population parameter. Students will be able to distinguish the roles of randomization and sample size with designing studies

Recognize that randomization reduces bias where bias occurs when certain outcomes are systematically more likely to appear.

Recognize that random selection from a population plays a different role than random assignment in an experiment.

Recognize that sample size impacts the precision with which estimates of the population parameters can be made (larger the sample size the more precision).

Students will apply the statistical method to real-world situations

Formulate questions to clarify the problem at hand and formulate one (or more) questions that can be answered with data.

Collect data by designing a plan to collect appropriate data and employ the plan to collect the data

Analyze data by selecting appropriate graphical and numerical methods and using these methods to analyze the data.

Interpret results by interpreting the analysis and relating the interpretation to the original question.

Students will identify whether the data are categorical or quantitative (numerical). Students will be able to identify the difference between categorical and quantitative (numerical) data

Determine the appropriate graphical display for each type of data

Determine the type of data used to produce a given graphical display.

Students will ask if the difference between two sample proportions or two sample means is due to random variation or if the difference is significant. Students will be able to determine if there are differences between two population parameters or treatment effects.

Using simulation, determine the appropriate model to decide if there is a difference between two population parameters

Using simulation, determine the appropriate model to decide if there is a difference between two treatment effects.

Students will understand that when randomness is incorporated into a sampling or experimental procedure, probability provides a way to describe the long-run behavior of a statistic as described by its sampling distribution. Students will be able to create simulated sampling distributions and understand how to use the sampling distribution to make predictions about a population parameter(s) or the difference in treatment effects.

Create an appropriate simulated sampling distribution (using technology) and develop a margin of error

Create an appropriate simulated sampling distribution (using technology) and develop a p-value.

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