NUMBER AND OPERATION



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|NUMBER & OPERATION |

|Whole Number: Counting & Representation |

|Grades K-3 |

|K |1 |2 |3 |

| Understand the relationship between quantities and |Count, compare and represent whole numbers up to 120,|Compare and represent whole numbers up to 1000 with |Compare and represent whole numbers up to 100,000 |

|whole numbers up to 31. |with an emphasis on groups of tens and ones. |an emphasis on place value and equality. |with an emphasis on place value and equality. |

| | | | |

|K.1.1.1 Recognize that a number can be used to |1.1.1.1 Use place value to describe whole numbers |2.1.1.1 Read, write and represent whole numbers up |3.1.1.1 Read, write and represent whole numbers up |

|represent how many objects are in a set or to |between 10 and 100 in terms of tens and ones. |to 1000. Representations may include numerals, |to 100,000. Representations may include numerals, |

|represent the position of an object in a sequence. |1.1.1.2 Read, write and represent whole numbers up |addition, subtraction, multiplication, words, |expressions with operations, words, pictures, number |

|K.1.1.2 Read, write, and represent whole numbers |to 120. Representations may include numerals, |pictures, tally marks, number lines and |lines, and manipulatives such as bundles of sticks |

|from 0 to at least 31. Representations may include |addition and subtraction, pictures, tally marks, |manipulatives, such as bundles of sticks and base 10 |and base 10 blocks. |

|numerals, pictures, real objects and picture graphs, |number lines and manipulatives, such as bundles of |blocks. |3.1.1.2 Use place value to describe whole numbers |

|spoken words, and manipulatives such as connecting |sticks and base 10 blocks. |2.1.1.2 Use place value to describe whole numbers |between 1000 and 100,000 in terms of ten thousands, |

|cubes. |1.1.1.3 Count, with and without objects, forward and|between 10 and 1000 in terms of hundreds, tens and |thousands, hundreds, tens and ones. |

|K.1.1.3 Count, with and without objects, forward and|backward from any given number up to 120. |ones. Know that 100 is 10 tens, and 1000 is 10 |3.1.1.3 Find 10,000 more or 10,000 less than a given|

|backward to at least 20. |1.1.1.4 Find a number that is 10 more or 10 less |hundreds. |five-digit number. Find 1000 more or 1000 less than a|

|K.1.1.4 Find a number that is 1 more or 1 less than |than a given number. |2.1.1.3 Find 10 more or 10 less than a given |given four- or five-digit. Find 100 more or 100 less |

|a given number. |1.1.1.5 Compare and order whole numbers up to 120. |three-digit number. Find 100 more or 100 less than a |than a given four- or five-digit number. |

|K.1.1.5 Compare and order whole numbers, with and |1.1.1.6 Use words to describe the relative size of |given three-digit number. |3.1.1.4 Round numbers to the nearest 10,000, 1000, |

|without objects, from 0 to 20. |numbers. |2.1.1.4 Round numbers up to the nearest 10 and 100 |100 and 10. Round up and round down to estimate sums |

| |1.1.1.7 Use counting and comparison skills to create|and round numbers down to the nearest 10 and 100. |and differences. |

| |and analyze bar graphs and tally charts. |2.1.1.5 Compare and order whole numbers up to 1000. |3.1.1.5 Compare and order whole numbers up to |

| | | |100,000. |

|NUMBER & OPERATION |

|Whole Number: Operations |

|Grades K-5 |

|K |1 |2 |3 |4 |5 |

|NUMBER & OPERATION |

|Fractions & Decimals: Representations and Relationships |

|Grades 3-7 |

|3 |4 |5 |6 |7 |

|Understand meanings and uses of |Represent and compare fractions and decimals in |Read, write, represent and compare |Read, write, represent and compare | Read, write, represent and compare |

|fractions in real-world and mathematical|real-world and mathematical situations; use place |fractions and decimals; recognize and |positive rational numbers expressed as |positive and negative rational numbers, |

|situations. |value to understand how decimals represent |write equivalent fractions; convert |fractions, decimals, percents and |expressed as integers, fractions and |

| |quantities. |between fractions and decimals; use |ratios; write positive integers as |decimals. |

|3.1.3.1 Read and write fractions with | |fractions and decimals in real-world |products of factors; use these | |

|words and symbols. Recognize that |4.1.2.1 Represent equivalent fractions using |and mathematical situations. |representations in real-world and |7.1.1.1 Know that every rational number |

|fractions can be used to represent parts|fraction models such as parts of a set, fraction | |mathematical situations. |can be written as the ratio of two |

|of a whole, parts of a set, points on a |circles, fraction strips, number lines and other |5.1.2.1 Read and write decimals using | |integers or as a terminating or repeating|

|number line, or distances on a number |manipulatives. Use the models to determine |place value to describe decimals in |6.1.1.1 Locate positive rational |decimal. Recognize that π is not |

|line. |equivalent fractions. |terms of groups from millionths to |numbers on a number line and plot pairs|rational, but that it can be approximated|

|3.1.3.2 Understand that the size of a |4.1.2.2 Locate fractions on a number line. Use |millions. |of positive rational numbers on a |by rational numbers such as [pic] and |

|fractional part is relative to the size |models to order and compare whole numbers and |5.1.2.2 Find 0.1 more than a number |coordinate grid. |3.14. |

|of the whole. |fractions, including mixed numbers and improper |and 0.1 less than a number. Find 0.01 |6.1.1.2 Compare positive rational |7.1.1.2 Understand that division of two |

|3.1.3.3 Order and compare unit |fractions. |more than a number and 0.01 less than a|numbers represented in various forms. |integers will always result in a rational|

|fractions and fractions with like |4.1.2.3 Use fraction models to add and subtract |number. Find 0.001 more than a number |Use the symbols < , = and >. |number. Use this information to interpret|

|denominators by using models and an |fractions with like denominators in real-world and|and 0.001 less than a number. |6.1.1.3 Understand that percent |the decimal result of a division problem |

|understanding of the concept of |mathematical situations. Develop a rule for |5.1.2.3 Order fractions and decimals, |represents parts out of 100 and ratios |when using a calculator. |

|numerator and denominator. |addition and subtraction of fractions with like |including mixed numbers and improper |to 100. |7.1.1.3 Locate positive and negative |

| |denominators. |fractions, and locate on a number line.|6.1.1.4 Determine equivalences among |rational numbers on a number line, |

| |4.1.2.4 Read and write decimals with words and | |fractions, decimals and percents; |understand the concept of opposites, and |

| |symbols; use place value to describe decimals in |5.1.2.4 Recognize and generate |select among these representations to |plot pairs of positive and negative |

| |terms of thousands, hundreds, tens, ones, tenths, |equivalent decimals, fractions, mixed |solve problems. |rational numbers on a coordinate grid. |

| |hundredths and thousandths. |numbers and improper fractions in |6.1.1.5 Factor whole numbers; express a|7.1.1.4 Compare positive and negative |

| |4.1.2.5 Compare and order decimals and whole |various contexts. |whole number as a product of prime |rational numbers expressed in various |

| |numbers using place value, a number line and |5.1.2.5 Round numbers to the nearest |factors with exponents. |forms using the symbols < , > , = , ≤ , |

| |models such as grids and base 10 blocks. |0.1, 0.01 and 0.001. |6.1.1.6 Determine greatest common |≥ . |

| |4.1.2.6 Read and write tenths and hundredths in | |factors and least common multiples. Use|7.1.1.5 Recognize and generate equivalent|

| |decimal and fraction notations using words and | |common factors and common multiples to |representations of positive and negative |

| |symbols; know the fraction and decimal equivalents| |calculate with fractions and find |rational numbers, including equivalent |

| |for halves and fourths. | |equivalent fractions. |fractions. |

| |4.1.2.7 Round decimals to the nearest tenth. | |6.1.1.7 Convert between equivalent | |

| | | |representations of positive rational | |

| | | |numbers. | |

|NUMBER & OPERATION |

|Fractions & Decimals: Operations |

|Grades 5-7 |

|5 |6 |7 |

|Add and subtract fractions, mixed numbers and decimals to solve |Multiply and divide decimals, fractions and mixed numbers; solve | Calculate with positive and negative rational numbers, and rational |

|real-world and mathematical problems. |real-world and mathematical problems using arithmetic with positive|numbers with whole number exponents, to solve real-world and mathematical |

| |rational numbers. |problems. |

|5.1.3.1 Add and subtract decimals and fractions, using efficient and| | |

|generalizable procedures, including standard algorithms. |6.1.3.1 Multiply and divide decimals and fractions, using efficient|7.1.2.1 Add, subtract, multiply and divide positive and negative rational |

|5.1.3.2 Model addition and subtraction of fractions and decimals |and generalizable procedures, including standard algorithms. |numbers that are integers, fractions and terminating decimals; use |

|using a variety of representations. |6.1.3.2 Use the meanings of fractions, multiplication, division and|efficient and generalizable procedures, including standard algorithms; |

|5.1.3.3 Estimate sums and differences of decimals and fractions to |the inverse relationship between multiplication and division to |raise positive rational numbers to whole-number exponents. |

|assess the reasonableness of results. |make sense of procedures for multiplying and dividing fractions. |7.1.2.2 Use real-world contexts and the inverse relationship between |

|5.1.3.4 Solve real-world and mathematical problems requiring |6.1.3.3 Calculate the percent of a number and determine what |addition and subtraction to explain why the procedures of arithmetic with |

|addition and subtraction of decimals, fractions and mixed numbers, |percent one number is of another number to solve problems in |negative rational numbers make sense. |

|including those involving measurement, geometry and data. |various contexts. |7.1.2.3 Understand that calculators and other computing technologies often|

| |6.1.3.4 Solve real-world and mathematical problems requiring |truncate or round numbers. |

| |arithmetic with decimals, fractions and mixed numbers. |7.1.2.4 Solve problems in various contexts involving calculations with |

| |6.1.3.5 Estimate solutions to problems with whole numbers, |positive and negative rational numbers and positive integer exponents, |

| |fractions and decimals and use the estimates to assess the |including computing simple and compound interest. |

| |reasonableness of results in the context of the problem. |7.1.2.5 Use proportional reasoning to solve problems involving ratios in |

| | |various contexts. |

| | |7.1.2.6 Demonstrate an understanding of the relationship between the |

| | |absolute value of a rational number and distance on a number line. Use the|

| | |symbol for absolute value. |

|ALGEBRA |

|Algebra Concepts: Ratios to Proportions to Functions |

|Grades 6-11 |

|6 |7 |8 |9-11 |

| Understand the concept of ratio and its |Understand the concept of proportionality |Understand the concept of function in real-world |Understand the concept of function, and identify important features of |

|relationship to fractions and to the |in real-world and mathematical situations, |and mathematical situations, and distinguish |functions and other relations using symbolic and graphical methods where |

|multiplication and division of whole |and distinguish between proportional and |between linear and nonlinear functions. |appropriate. |

|numbers. Use ratios to solve real-world and |other relationships. | | |

|mathematical problems. | |8.2.1.1 Understand that a function is a |9.2.1.1 Understand the definition of a function. Use functional notation|

| |7.2.1.1 Understand that a relationship |relationship between an independent variable and a |and evaluate a function at a given point in its domain. |

|6.1.2.1 Identify and use ratios to compare |between two variables, x and y, is |dependent variable in which the value of the |9.2.1.2 Distinguish between functions and other relations defined |

|quantities; understand that comparing |proportional if it can be expressed in the |independent variable determines the value of the |symbolically, graphically or in tabular form. |

|quantities using ratios is not the same as |form [pic]or[pic]. Distinguish proportional|dependent variable. Use functional notation, such |9.2.1.3 Find the domain of a function defined symbolically, graphically |

|comparing quantities using subtraction. |relationships from other relationships, |as f(x), to represent such relationships. |or in a real-world context. |

|6.1.2.2 Apply the relationship between |including inversely proportional |8.2.1.2 Use linear functions to represent |9.2.1.4 Obtain information and draw conclusions from graphs of functions|

|ratios, equivalent fractions and percents to|relationships ([pic]or[pic]). |relationships in which changing the input variable |and other relations. |

|solve problems in various contexts, |7.2.1.2 Understand that the graph of a |by some amount leads to a change in the output |9.2.1.5 Identify the vertex, line of symmetry and intercepts of the |

|including those involving mixtures and |proportional relationship is a line through|variable that is a constant times that amount. |parabola corresponding to a quadratic function, using symbolic and |

|concentrations. |the origin whose slope is the unit rate |8.2.1.3 Understand that a function is linear if it|graphical methods, when the function is expressed in the form f (x) = |

|6.1.2.3 Determine the rate for ratios of |(constant of proportionality). Know how to |can be expressed in the form [pic] or if its graph |ax2 + bx + c, in the form f (x) = a(x – h)2 + k , or in factored form. |

|quantities with different units. |use graphing technology to examine what |is a straight line. |9.2.1.6 Identify intercepts, zeros, maxima, minima and intervals of |

|6.1.2.4 Use reasoning about multiplication |happens to a line when the unit rate is |8.2.1.4 Understand that an arithmetic sequence is |increase and decrease from the graph of a function. |

|and division to solve ratio and rate |changed. |a linear function that can be expressed in the |9.2.1.7 Understand the concept of an asymptote and identify asymptotes |

|problems. | |form[pic], where x = 0, 1, 2, 3,…. |for exponential functions and reciprocals of linear functions, using |

| | |8.2.1.5 Understand that a geometric sequence is a |symbolic and graphical methods. |

| | |non-linear function that can be expressed in the |9.2.1.8 Make qualitative statements about the rate of change of a |

| | |form [pic], where |function, based on its graph or table of values. |

| | |x = 0, 1, 2, 3,…. |9.2.1.9 Determine how translations affect the symbolic and graphical |

| | | |forms of a function. Know how to use graphing technology to examine |

| | | |translations. |

|ALGEBRA |

|Algebra: Relationships and Functions |

|Grades K-11 |

|K |1 |2 |3 |4 |5 |

|ALGEBRA |

|Algebra: Relationships and Functions |

|Grades K-11 |

|6 |7 |8 |9-11 |

|Recognize and represent |Recognize proportional relationships in |Recognize linear functions in real-world and mathematical|Recognize linear, quadratic, exponential and other common functions in |

|relationships between varying |real-world and mathematical situations; |situations; represent linear functions and other |real-world and mathematical situations; represent these functions with |

|quantities; translate from one |represent these and other relationships with|functions with tables, verbal descriptions, symbols and |tables, verbal descriptions, symbols and graphs; solve problems involving |

|representation to another; use |tables, verbal descriptions, symbols and |graphs; solve problems involving these functions and |these functions, and explain results in the original context. |

|patterns, tables, graphs and rules |graphs; solve problems involving |explain results in the original context. | |

|to solve real-world and |proportional relationships and explain | |9.2.2.1 Represent and solve problems in various contexts using linear and|

|mathematical problems. |results in the original context. |8.2.2.1 Represent linear functions with tables, verbal |quadratic functions. |

| | |descriptions, symbols, equations and graphs; translate |9.2.2.2 Represent and solve problems in various contexts using |

|6.2.1.1 Understand that a variable|7.2.2.1 Represent proportional |from one representation to another. |exponential functions, such as investment growth, depreciation and |

|can be used to represent a quantity|relationships with tables, verbal |8.2.2.2 Identify graphical properties of linear |population growth. |

|that can change, often in |descriptions, symbols, equations and graphs;|functions including slopes and intercepts. Know that the |9.2.2.3 Sketch graphs of linear, quadratic and exponential functions, and|

|relationship to another changing |translate from one representation to |slope equals the rate of change, and that the y-intercept|translate between graphs, tables and symbolic representations. Know how to|

|quantity. Use variables in various |another. Determine the unit rate (constant |is zero when the function represents a proportional |use graphing technology to graph these functions. |

|contexts. |of proportionality or slope) given any of |relationship. |9.2.2.4 Express the terms in a geometric sequence recursively and by |

|6.2.1.2 Represent the relationship|these representations. |8.2.2.3 Identify how coefficient changes in the equation|giving an explicit (closed form) formula, and express the partial sums of |

|between two varying quantities with|7.2.2.2 Solve multi-step problems involving|f (x) = mx + b affect the graphs of linear functions. |a geometric series recursively. |

|function rules, graphs and tables; |proportional relationships in numerous |Know how to use graphing technology to examine these |9.2.2.5 Recognize and solve problems that can be modeled using finite |

|translate between any two of these |contexts. |effects. |geometric sequences and series, such as home mortgage and other compound |

|representations. |7.2.2.3 Use knowledge of proportions to |8.2.2.4 Represent arithmetic sequences using equations, |interest examples. Know how to use spreadsheets and calculators to explore|

| |assess the reasonableness of solutions. |tables, graphs and verbal descriptions, and use them to |geometric sequences and series in various contexts. |

| |7.2.2.4 Represent real-world or |solve problems. |9.2.2.6 Sketch the graphs of common non-linear functions such as [pic], |

| |mathematical situations using equations and |8.2.2.5 Represent geometric sequences using equations, |[pic], [pic], f (x) = x3, and translations of these functions, such as |

| |inequalities involving variables and |tables, graphs and verbal descriptions, and use them to |[pic]. Know how to use graphing technology to graph these functions. |

| |positive and negative rational numbers. |solve problems. | |

|ALGEBRA |

|Algebra: Expressions |

|Grades 5-11 |

|5 |6 |7 |8 |9-11 |

|Use properties of arithmetic to|Use properties of arithmetic to|Apply understanding of order of operations |Generate equivalent numerical and |Generate equivalent algebraic expressions involving polynomials and |

|generate equivalent numerical |generate equivalent numerical |and algebraic properties to generate |algebraic expressions and use |radicals; use algebraic properties to evaluate expressions. |

|expressions and evaluate |expressions and evaluate |equivalent numerical and algebraic |algebraic properties to evaluate | |

|expressions involving whole |expressions involving positive |expressions containing positive and |expressions. |9.2.3.1 Evaluate polynomial and rational expressions and expressions|

|numbers. |rational numbers. |negative rational numbers and grouping | |containing radicals and absolute values at specified points in their |

| | |symbols; evaluate such expressions. |8.2.3.1 Evaluate algebraic |domains. |

|5.2.2.1 Apply the commutative,|6.2.2.1 Apply the associative,| |expressions, including expressions|9.2.3.2 Add, subtract and multiply polynomials; divide a polynomial |

|associative and distributive |commutative and distributive |7.2.3.1 Use properties of algebra to |containing radicals and absolute |by a polynomial of equal or lower degree. |

|properties and order of |properties and order of |generate equivalent numerical and algebraic|values, at specified values of |9.2.3.3 Factor common monomial factors from polynomials, factor |

|operations to generate |operations to generate |expressions containing rational numbers, |their variables. |quadratic polynomials, and factor the difference of two squares. |

|equivalent numerical |equivalent expressions and to |grouping symbols and whole number |8.2.3.2 Justify steps in |9.2.3.4 Add, subtract, multiply, divide and simplify algebraic |

|expressions and to solve |solve problems involving |exponents. Properties of algebra include |generating equivalent expressions |fractions. |

|problems involving whole |positive rational numbers. |associative, commutative and distributive |by identifying the properties |9.2.3.5 Check whether a given complex number is a solution of a |

|numbers. | |laws. |used, including the properties of |quadratic equation by substituting it for the variable and evaluating|

| | |7.2.3.2 Evaluate algebraic expressions |algebra. Properties include the |the expression, using arithmetic with complex numbers. |

| | |containing rational numbers and whole |associative, commutative and |9.2.3.6 Apply the properties of positive and negative rational |

| | |number exponents at specified values of |distributive laws, and the order |exponents to generate equivalent algebraic expressions, including |

| | |their variables. |of operations, including grouping |those involving nth roots. |

| | |7.2.3.3 Apply understanding of order of |symbols. |9.2.3.7 Justify steps in generating equivalent expressions by |

| | |operations and grouping symbols when using | |identifying the properties used. Use substitution to check the |

| | |calculators and other technologies. | |equality of expressions for some particular values of the variables; |

| | | | |recognize that checking with substitution does not guarantee equality|

| | | | |of expressions for all values of the variables. |

|ALGEBRA |

|Algebra: Equations and Inequalities |

|Grades 1-11 |

|1 |2 |3 |4 |5 |

|Use number sentences involving addition |Use number sentences involving addition, |Use number sentences involving |Use number sentences involving |Understand and interpret equations and |

|and subtraction basic facts to represent |subtraction and unknowns to represent and|multiplication and division basic facts and|multiplication, division and unknowns to |inequalities involving variables and |

|and solve real-world and mathematical |solve real-world and mathematical |unknowns to represent and solve real-world |represent and solve real-world and |whole numbers, and use them to represent |

|problems; create real-world situations |problems; create real-world situations |and mathematical problems; create |mathematical problems; create real-world |and solve real-world and mathematical |

|corresponding to number sentences. |corresponding to number sentences. |real-world situations corresponding to |situations corresponding to number |problems. |

| | |number sentences. |sentences. | |

|1.2.2.1 Represent real-world situations |2.2.2.1 Understand how to interpret | | |5.2.3.1 Determine whether an equation or|

|involving addition and subtraction basic |number sentences involving addition, |3.2.2.1 Understand how to interpret number|4.2.2.1 Understand how to interpret |inequality involving a variable is true |

|facts, using objects and number sentences.|subtraction and unknowns represented by |sentences involving multiplication and |number sentences involving multiplication,|or false for a given value of the |

|1.2.2.2 Determine if equations involving |letters. Use objects and number lines and|division basic facts and unknowns. Create |division and unknowns. Use real-world |variable. |

|addition and subtraction are true. |create real-world situations to represent|real-world situations to represent number |situations involving multiplication or |5.2.3.2 Represent real-world situations |

|1.2.2.3 Use number sense and models of |number sentences. |sentences. |division to represent number sentences. |using equations and inequalities |

|addition and subtraction, such as objects |2.2.2.2 Use number sentences involving |3.2.2.2 Use multiplication and division |4.2.2.2 Use multiplication, division and |involving variables. Create real-world |

|and number lines, to identify the missing |addition, subtraction, and unknowns to |basic facts to represent a given problem |unknowns to represent a given problem |situations corresponding to equations and|

|number in an equation such as: |represent given problem situations. Use |situation using a number sentence. Use |situation using a number sentence. Use |inequalities. |

|2 + 4 = ♦; 3 + ♦ = 7; |number sense and properties of addition |number sense and multiplication and |number sense, properties of |5.2.3.3 Evaluate expressions and solve |

|5 = ♦ – 3. |and subtraction to find values for the |division basic facts to find values for the|multiplication, and the relationship |equations involving variables when values|

|1.2.2.4 Use addition or subtraction basic|unknowns that make the number sentences |unknowns that make the number sentences |between multiplication and division to |for the variables are given. |

|facts to represent a given problem |true. |true. |find values for the unknowns that make the| |

|situation using a number sentence. | | |number sentences true. | |

|ALGEBRA |

|Algebra: Equations and Inequalities |

|Grades 1-11 |

|Grade 6 |Grade 7 |Grade 8 |Grades 9-11 |

|Understand and interpret equations and|Represent real-world and mathematical |Represent real-world and mathematical situations using |Represent real-world and mathematical situations using equations |

|inequalities involving variables and |situations using equations with variables.|equations and inequalities involving linear expressions. Solve |and inequalities involving linear, quadratic, exponential and nth |

|positive rational numbers. Use |Solve equations symbolically, using the |equations and inequalities symbolically and graphically. |root functions. Solve equations and inequalities symbolically and |

|equations and inequalities to |properties of equality. Also solve |Interpret solutions in the original context. |graphically. Interpret solutions in the original context. |

|represent real-world and mathematical |equations graphically and numerically. | | |

|problems; use the idea of maintaining |Interpret solutions in the original |8.2.4.1 Use linear equations to represent situations involving|9.2.4.1 Represent relationships in various contexts using |

|equality to solve equations. Interpret|context. |a constant rate of change, including proportional and |quadratic equations and inequalities. Solve quadratic equations |

|solutions in the original context. | |non-proportional relationships. |and inequalities by appropriate methods including factoring, |

| |7.2.4.1 Represent relationships in |8.2.4.2 Solve multi-step equations in one variable. Solve for |completing the square, graphing and the quadratic formula. Find |

|6.2.3.1 Represent real-world or |various contexts with equations involving |one variable in a multi-variable equation in terms of the other|non-real complex roots when they exist. Recognize that a |

|mathematical situations using |variables and positive and negative |variables. Justify the steps by identifying the properties of |particular solution may not be applicable in the original context.|

|equations and inequalities involving |rational numbers. Use the properties of |equalities used. |Know how to use calculators, graphing utilities or other |

|variables and positive rational |equality to solve for the value of a |8.2.4.3 Express linear equations in slope-intercept, |technology to solve quadratic equations and inequalities. |

|numbers. |variable. Interpret the solution in the |point-slope and standard forms, and convert between these |9.2.4.2 Represent relationships in various contexts using |

|6.2.3.2 Solve equations involving |original context. |forms. Given sufficient information, find an equation of a |equations involving exponential functions; solve these equations |

|positive rational numbers using number|7.2.4.2 Solve equations resulting from |line. |graphically or numerically. Know how to use calculators, graphing |

|sense, properties of arithmetic and |proportional relationships in various |8.2.4.4 Use linear inequalities to represent relationships in |utilities or other technology to solve these equations. |

|the idea of maintaining equality on |contexts. |various contexts. |9.2.4.3 Recognize that to solve certain equations, number systems|

|both sides of the equation. Interpret | |8.2.4.5 Solve linear inequalities using properties of |need to be extended from whole numbers to integers, from integers |

|a solution in the original context and| |inequalities. Graph the solutions on a number line. |to rational numbers, from rational numbers to real numbers, and |

|assess the reasonableness of results. | |8.2.4.6 Represent relationships in various contexts with |from real numbers to complex numbers. In particular, non-real |

| | |equations and inequalities involving the absolute value of a |complex numbers are needed to solve some quadratic equations with |

| | |linear expression. Solve such equations and inequalities and |real coefficients. |

| | |graph the solutions on a number line. |9.2.4.4 Represent relationships in various contexts using systems|

| | |8.2.4.7 Represent relationships in various contexts using |of linear inequalities; solve them graphically. Indicate which |

| | |systems of linear equations. Solve systems of linear equations |parts of the boundary are included in and excluded from the |

| | |in two variables symbolically, graphically and numerically. |solution set using solid and dotted lines. |

| | |8.2.4.8 Understand that a system of linear equations may have |9.2.4.5 Solve linear programming problems in two variables using |

| | |no solution, one solution, or an infinite number of solutions. |graphical methods. |

| | |Relate the number of solutions to pairs of lines that are |9.2.4.6 Represent relationships in various contexts using |

| | |intersecting, parallel or identical. Check whether a pair of |absolute value inequalities in two variables; solve them |

| | |numbers satisfies a system of two linear equations in two |graphically. |

| | |unknowns by substituting the numbers into both equations. |9.2.4.7 Solve equations that contain radical expressions. |

| | |8.2.4.9 Use the relationship between square roots and squares |Recognize that extraneous solutions may arise when using symbolic |

| | |of a number to solve problems. |methods. |

| | | |9.2.4.8 Assess the reasonableness of a solution in its given |

| | | |context and compare the solution to appropriate graphical or |

| | | |numerical estimates; interpret a solution in the original context.|

|GEOMETRY & MEASUREMENT |

|Geometry: Shapes |

|Grades K-5 |

|K |1 |2 |3 |4 |5 |

|GEOMETRY & MEASUREMENT |

|Geometry: Measurement |

|Grades K-11 |

|K |1 |2 |3 |4 |5 |

|GEOMETRY & MEASUREMENT |

|Geometry: Measurement |

|Grades K-11 |

|Grade 6 |Grade 7 |Grade 8 |Grades 9-11 |

|Calculate perimeter, area, surface area and volume of two- and |Use reasoning with proportions and ratios to |Solve problems involving right triangles using|Calculate measurements of plane and solid geometric |

|three-dimensional figures to solve real-world and mathematical |determine measurements, justify formulas and |the Pythagorean Theorem and its converse. |figures; know that physical measurements depend on |

|problems. |solve real-world and mathematical problems | |the choice of a unit and that they are |

| |involving circles and related geometric |8.3.1.1 Use the Pythagorean Theorem to solve |approximations. |

|6.3.1.1 Calculate the surface area and volume of prisms and use|figures. |problems involving right triangles. | |

|appropriate units, such as cm2 and cm3. Justify the formulas | |8.3.1.2 Determine the distance between two |9.3.1.1 Determine the surface area and volume of |

|used. Justification may involve decomposition, nets or other |7.3.1.1 Demonstrate an understanding of the |points on a horizontal or vertical line in a |pyramids, cones and spheres. Use measuring devices |

|models. |proportional relationship between the diameter |coordinate system. Use the Pythagorean Theorem|or formulas as appropriate. |

|6.3.1.2 Calculate the area of quadrilaterals. Quadrilaterals |and circumference of a circle and that the unit|to find the distance between any two points in|9.3.1.2 Compose and decompose two- and |

|include squares, rectangles, rhombuses, parallelograms, |rate (constant of proportionality) is [pic]. |a coordinate system. |three-dimensional figures; use decomposition to |

|trapezoids and kites. When formulas are used, be able to explain|Calculate the circumference and area of circles| |determine the perimeter, area, surface area and |

|why they are valid. |and sectors of circles to solve problems in | |volume of various figures. |

|6.3.1.3 Estimate the perimeter and area of irregular figures on|various contexts. | |9.3.1.3 Understand that quantities associated with |

|a grid when they cannot be decomposed into common figures and |7.3.1.2 Calculate the volume and surface area | |physical measurements must be assigned units; apply |

|use correct units, such as cm and cm2. |of cylinders and justify the formulas used. | |such units correctly in expressions, equations and |

| | | |problem solutions that involve measurements; and |

| | | |convert between measurement systems. |

| | | |9.3.1.4 Understand and apply the fact that the |

| | | |effect of a scale factor k on length, area and |

| | | |volume is to multiply each by k, k2 and k3, |

| | | |respectively. |

| | | |9.3.1.5 Make reasonable estimates and judgments |

| | | |about the accuracy of values resulting from |

| | | |calculations involving measurements. |

|GEOMETRY & MEASUREMENT |

|Algebra in Geometry |

|Grades 6-11 |

|Grade 6 |Grade 7 |Grade 8 |Grades 9-11 |

|Choose appropriate units of |Analyze the effect of change of |Solve problems involving parallel and |Solve real-world and mathematical geometric problems using algebraic methods. |

|measurement and use ratios to |scale, translations and |perpendicular lines on a coordinate | |

|convert within measurement systems|reflections on the attributes of |system. |9.3.4.1 Understand how the properties of similar right triangles allow the trigonometric ratios to |

|to solve real-world and |two-dimensional figures. | |be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. |

|mathematical problems. | |8.3.2.1 Understand and apply the | |

| |7.3.2.1 Describe the properties |relationships between the slopes of |9.3.4.2 Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as |

|6.3.3.1 Solve problems in various|of similarity, compare geometric |parallel lines and between the slopes of |determining lengths and areas in right triangles and in figures that can be decomposed into right |

|contexts involving conversion of |figures for similarity, and |perpendicular lines. Dynamic graphing |triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios.|

|weights, capacities, geometric |determine scale factors. |software may be used to examine these | |

|measurements and times within |7.3.2.2 Apply scale factors, |relationships. |9.3.4.3 Use calculators, tables or other technologies in connection with the trigonometric ratios |

|measurement systems using |length ratios and area ratios to |8.3.2.2 Analyze polygons on a coordinate |to find angle measures in right triangles in various contexts. |

|appropriate units. |determine side lengths and areas |system by determining the slopes of their |9.3.4.4 Use coordinate geometry to represent and analyze line segments and polygons, including |

|6.3.3.2 Estimate weights, |of similar geometric figures. |sides. |determining lengths, midpoints and slopes of line segments. |

|capacities and geometric |7.3.2.3 Use proportions and |8.3.2.3 Given a line on a coordinate |9.3.4.5 Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y |

|measurements using benchmarks in |ratios to solve problems involving|system and the coordinates of a point not |– k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. |

|measurement systems with |scale drawings and conversions of |on the line, find lines through that point|9.3.4.6 Use numeric, graphic and symbolic representations of transformations in two dimensions, |

|appropriate units. |measurement units. |that are parallel and perpendicular to the|such as reflections, translations, scale changes and rotations about the origin by multiples of 90˚,|

| | |given line, symbolically and graphically. |to solve problems involving figures on a coordinate grid. |

| | | |9.3.4.7 Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving |

| | | |for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to |

| | | |obtain a quadratic equation for a length in a geometric figure. |

|GEOMETRY & MEASUREMENT |

|Euclidean Geometry and Reasoning |

|Grades 4-11 |

|Grade 4 |Grade 6 |Grade 7 |Grade 8 |Grades 9-11 |

|Use translations, |Understand and use |Analyze the effect of |Solve problems |Construct logical arguments, based on axioms, definitions and theorems, to prove theorems and other results in |

|reflections and rotations|relationships between |change of scale, |involving right |geometry. |

|to establish congruency |angles in geometric |translations and |triangles using the | |

|and understand |figures. |reflections on the |Pythagorean Theorem and|9.3.2.1 Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. |

|symmetries. | |attributes of |its converse. |9.3.2.2 Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." |

| |6.3.2.1 Solve problems|two-dimensional | |Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. |

|4.3.3.1 Apply |using the relationships|figures. |8.3.1.3 Informally |9.3.2.3 Assess the validity of a logical argument and give counterexamples to disprove a statement. |

|translations (slides) to |between the angles | |justify the Pythagorean|9.3.2.4 Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by|

|figures. |formed by intersecting |7.3.2.4 Graph and |Theorem by using |contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph |

|4.3.3.2 Apply |lines. |describe translations |measurements, diagrams |proofs, flow charts or illustrations. |

|reflections (flips) to |6.3.2.2 Determine |and reflections of |and computer software. |9.3.2.5 Use technology tools to examine theorems, make and test conjectures, perform constructions and develop |

|figures by reflecting |missing angle measures |figures on a | |mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic |

|over vertical or |in a triangle using the|coordinate grid and | |geometry software, design software or Internet applets. |

|horizontal lines and |fact that the sum of |determine the | | |

|relate reflections to |the interior angles of |coordinates of the | |Know and apply properties of geometric figures to solve real-world and mathematical problems and to logically |

|lines of symmetry. |a triangle is 180˚. Use|vertices of the figure| |justify results in geometry. |

|4.3.3.3 Apply rotations |models of triangles to |after the | | |

|(turns) of 90˚ clockwise |illustrate this fact. |transformation. | |9.3.3.1 Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a |

|or counterclockwise. |6.3.2.3 Develop and | | |transversal, to solve problems and logically justify results. |

|4.3.3.4 Recognize that |use formulas for the | | |9.3.3.2 Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary |

|translations, reflections|sums of the interior | | |and supplementary angles, to solve problems and logically justify results. |

|and rotations preserve |angles of polygons by | | |9.3.3.3 Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically |

|congruency and use them |decomposing them into | | |justify results. |

|to show that two figures |triangles. | | |9.3.3.4 Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. |

|are congruent. | | | |9.3.3.5 Know and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to |

| | | | |solve problems and logically justify results. |

| | | | |9.3.3.6 Know and apply properties of congruent and similar figures to solve problems and logically justify results.|

| | | | | |

| | | | |9.3.3.7 Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, |

| | | | |solve problems and logically justify results. |

| | | | |9.3.3.8 Know and apply properties of a circle to solve problems and logically justify results. |

|DATA ANALYSIS |

|Grades 3-11 |

|3 |4 |5 |7 |8 |9-11 |

|PROBABILITY |

|Grades 6-11 |

|Grade 6 |Grade 7 |Grades 9-11 |

|Use probabilities to solve real-world and | | |

|mathematical problems; represent probabilities using|Calculate probabilities and reason about |Calculate probabilities and apply probability concepts to solve real-world and mathematical problems. |

|fractions, decimals and percents. |probabilities using proportions to solve real-world | |

| |and mathematical problems. |9.4.3.1 Select and apply counting procedures, such as the multiplication and addition principles and tree|

|6.4.1.1 Determine the sample space (set of possible| |diagrams, to determine the size of a sample space (the number of possible outcomes) and to calculate |

|outcomes) for a given experiment and determine which|7.4.3.1 Use random numbers generated by a |probabilities. |

|members of the sample space are related to certain |calculator or a spreadsheet or taken from a table to|9.4.3.2 Calculate experimental probabilities by performing simulations or experiments involving a |

|events. Sample space may be determined by the use of|simulate situations involving randomness, make a |probability model and using relative frequencies of outcomes. |

|tree diagrams, tables or pictorial representations. |histogram to display the results, and compare the |9.4.3.3 Understand that the Law of Large Numbers expresses a relationship between the probabilities in a |

|6.4.1.2 Determine the probability of an event using|results to known probabilities. |probability model and the experimental probabilities found by performing simulations or experiments |

|the ratio between the size of the event and the size|7.4.3.2 Calculate probability as a fraction of |involving the model. |

|of the sample space; represent probabilities as |sample space or as a fraction of area. Express |9.4.3.4 Use random numbers generated by a calculator or a spreadsheet, or taken from a table, to perform |

|percents, fractions and decimals between 0 and 1 |probabilities as percents, decimals and fractions. |probability simulations and to introduce fairness into decision making. |

|inclusive. Understand that probabilities measure |7.4.3.3 Use proportional reasoning to draw |9.4.3.5 Apply probability concepts such as intersections, unions and complements of events, and |

|likelihood. |conclusions about and predict relative frequencies |conditional probability and independence, to calculate probabilities and solve problems. |

|6.4.1.3 Perform experiments for situations in which|of outcomes based on probabilities. |9.4.3.6 Describe the concepts of intersections, unions and complements using Venn diagrams. Understand |

|the probabilities are known, compare the resulting | |the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and |

|relative frequencies with the known probabilities; | |spreadsheets. |

|know that there may be differences. | |9.4.3.7 Understand and use simple probability formulas involving intersections, unions and complements of|

|6.4.1.4 Calculate experimental probabilities from | |events. |

|experiments; represent them as percents, fractions | |9.4.3.8 Apply probability concepts to real-world situations to make informed decisions. |

|and decimals between 0 and 1 inclusive. Use | |9.4.3.9 Use the relationship between conditional probabilities and relative frequencies in contingency |

|experimental probabilities to make predictions when | |tables. |

|actual probabilities are unknown. | | |

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