# Performance Level Descriptors Grade 6 Mathematics

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1 Performance Level Descriptors Grade 6 Mathematics Multiplying and Dividing with Fractions 6.NS.1-2 Grade 6 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Level 4: Meets Solves word problems involving division of fractions by fractions. Divides fractions with unlike denominators and solves word problems with prompting embedded within the problem. Divides fractions with common denominators and solves word problems with prompting embedded within the problem. Divides fractions with common denominators. Ratios 6.RP.1 6.RP.2 6.RP.3a 6.RP.3b 6.RP.3c-1 6.RP.3c-2 6.RP.3d Uses ratio and rate reasoning to solve real-world and mathematical problems, including ratio, unit rate, percent and unit conversion problems. Uses and connects a variety of representations and strategies to solve these problems. Uses ratio and rate reasoning to solve real-world and mathematical problems, including ratio, unit rate, percent and unit conversion problems using a limited variety of representations and strategies. Uses ratio and rate reasoning to solve mathematical problems, including ratio, unit rate, percent and unit conversion problems using a limited variety of representations and strategies. Solves problems including ratio, unit rate, percent and unit conversion problems using a limited variety of representations and strategies. Finds missing values in tables and plots values on the coordinate plane. Finds missing values in tables and locates and plots values on the coordinate plane. Finds missing values in tables and locates or plots values on the coordinate plane. Revised November 2015 Page 1 of 11

2 6.NS.5 6.NS.6a 6.NS.6b-1 6.NS.6b-2 6.NS.6c-1 6.NS.6c-2 6.NS.7a 6.NS.7b 6.NS.7c-1 6.NS.7c-2 6.NS.7d 6.NS.8 Performance Level Descriptors Grade 6 Mathematics Grade 6 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Level 4: Meets Rational Numbers Understands that positive and negative numbers describe mathematical or real-world quantities which have opposite values or directions and can be represented on a number line and compared with or without the use of a number line. Understands and interprets the absolute value of a rational number. Understands (or recognizes) that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. Distinguishes comparisons of absolute value from statements about order. Understands that positive and negative numbers describe mathematical or real-world quantities which have opposite values or directions and can be represented on a number line and compared with or without the use of a number line. Understands the absolute value of a rational number. Plots ordered pairs on a Plots ordered pairs on a coordinate coordinate plane to solve realworld and mathematical problems. mathematical plane to solve real-world and problems. Understands that positive and negative numbers describe mathematical or real-world quantities which have opposite values or directions and can be represented on a number line. Determines the absolute value of a rational number. Locates or plots ordered pairs on a coordinate plane to solve mathematical problems. Understands that positive and negative numbers describe mathematical or real-world quantities which have opposite values or directions and can be represented on a number line. Determines the absolute value of a rational number. Revised November 2015 Page 2 of 11

3 Expressions and Inequalities 6.EE EE EE.2a 6.EE.2b 6.EE.2c-1 6.EE.2c-2 6.EE.4 Performance Level Descriptors Grade 6 Mathematics Grade 6 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Level 4: Meets Writes, reads and evaluates numerical and algebraic expressions, including those that contain whole number exponents. Reads and evaluates numerical and algebraic expressions, including those that contain whole number exponents. Writes numerical expressions and some algebraic expressions, including those that contain whole number exponents. Reads numerical and algebraic expressions including those that contain whole number exponents. Identifies parts of algebraic and numerical expressions using mathematical terms and views one or more parts of an expression as a single entity. Identifies parts of algebraic and numerical expressions using mathematical terms. Identifies parts of algebraic and numerical expressions using mathematical terms. Identifies parts of an algebraic or numerical expression using mathematical terms. Identifies equivalent expressions using properties of operations. Identifies equivalent expressions using properties of operations. Equations and Inequalities 6.EE EE EE.6 6.EE.7 6.EE.8 6.EE.9 Uses variables to represent numbers and writes expressions and single-step equations to solve real-world and mathematical problems and understand their solutions. Expresses a relationship between dependent and independent variables and Uses variables to represent numbers and writes expressions and single-step equations to solve real-world or mathematical problems. Relates tables and graphs to the equations. Uses variables to represent numbers and writes expressions without exponents, and singlestep equations to solve mathematical problems. Relates tables and graphs to the equations. Uses variables to represent numbers and writes expressions without exponents, and singlestep equations to solve mathematical problems Revised November 2015 Page 3 of 11

4 Performance Level Descriptors Grade 6 Mathematics Grade 6 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds relates tables and graphs to equations. Level 4: Meets Writes and graphs inequalities to represent a constraint or condition in a real-world or mathematical problem. Understands that there are an infinite number of solutions for an inequality. Writes and graphs inequalities to represent a constraint or condition in a real-world or mathematical problem. Graphs inequalities to represent a constraint or condition in a mathematical problem. Revised November 2015 Page 4 of 11

5 Factors and Multiples 6.NS NS.4-2 Performance Level Descriptors Grade 6 Mathematics Grade 6 Math: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Finds greatest common factors and least common multiples. Uses the distributive property to express a sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor. Level 4: Meets Finds greatest common factors and least common multiples. Uses the distributive property to rewrite a sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor. Identifies greatest common factors and least common multiples. Identifies greatest common factors or least common multiples. Geometry 6.G.1 6.G G G.3 6.G.4 Solves real-world and mathematical problems involving area of polygons by composing into rectangles or decomposing into triangles and other shapes. Determines measurements of polygons in the coordinate plane. Solves real-world and mathematical problems involving area of polygons by either composing into rectangles or decomposing into triangles and other shapes. Determines measurements of polygons in the coordinate plane. Solves mathematical problems involving area of polygons by either composing into rectangles or decomposing into triangles and other shapes. Determines measurements of polygons in the coordinate plane. Solves mathematical problems involving area of polygons by composing into rectangles. Determines and uses nets of three-dimensional figures to find surface area. Determines volume of right rectangular prisms with fractional edge lengths by packing them with unit cubes and using formulas. Determines and uses nets of three-dimensional figures to find surface area. Determines volume of right rectangular prisms with fractional edge lengths by packing them with unit cubes and using formulas. Uses nets of three-dimensional figures to find surface area. Determines volume of right rectangular prisms with fractional edge lengths by packing them with unit cubes and using formulas. Revised November 2015 Page 5 of 11

6 Performance Level Descriptors Grade 6 Mathematics Grade 6 Math: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Uses volume formulas to find unknown measurements. Level 4: Meets Statistics and Probability 6.SP.1 6.SP.2 6.SP.3 6.SP.4 6.SP.5 Understands the concepts of area and volume to solve unscaffolded problems. Recognizes a statistical question and understands that a set of collected data has a distribution which can be described by its center, spread and overall shape. Understands the purpose of center and variability and that it can be summarized with a single number. Recognizes a statistical question and understands that a set of collected data has a distribution which can be described by its center, spread and overall shape. Understands the purpose of center and that it can be summarized with a single number. Recognizes a statistical question and understands that a set of collected data has a distribution which can be described by its center, spread and overall shape. Understands the purpose of center and that it can be summarized with a single number. Understands that a set of collected data has a distribution which can be described by its center, spread and overall shape. Understands that the center of a set of data can be summarized with a single number. Displays numerical data in plots on a number line, including dot plots, histograms and box plots, and determines which display is the most appropriate. Summarizes numerical data sets in relation to their context, such as by reporting the number of observations, describing the nature of the attributes under investigation and using measures of center and variability. Revised November 2015 Page 6 of 11

7 Performance Level Descriptors Grade 6 Mathematics Grade 6 Math: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Determines which measures of center and variability are the most appropriate for a set of data. Level 4: Meets Operations with Multi-Digit Numbers 6.NS.2 6.NS NS NS NS Int.1 Solves two-step word problems and other problems by dividing multi-digit numbers and adding, subtracting, multiplying and dividing multi-digit decimals and assesses reasonableness of the result using different methods. Solves one-step word problems and other problems with some level of accuracy by dividing multi-digit numbers and adding, subtracting, multiplying and dividing multi-digit decimals. Solves one-step problems by dividing multi-digit numbers and adding, subtracting, multiplying and dividing multi-digit decimals. Solves one-step problems with limited accuracy by dividing multi-digit numbers and adding, subtracting, multiplying and dividing multi-digit decimals. Revised November 2015 Page 7 of 11

8 Properties of Operations 6.C C.2 Performance Level Descriptors Grade 6 Mathematics Grade 6: Sub-Claim C In connection with content, the student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Exceeds Level 4: Meets student clearly constructs and communicates a complete response based on the properties of operations and the relationship between addition and subtraction or between multiplication and division, including: a logical and complete progression of steps generalization of an argument or evaluating, interpreting, and critiquing the validity and efficiency of other s responses, approaches and reasoning, and providing counter-examples where applicable. the student clearly constructs and communicates a complete response based on the properties of operations and the relationship between addition and subtraction or between multiplication and division, including: a logical and complete vocabulary, symbols and labels evaluating, interpreting and critiquing the validity of other s responses, approaches and reasoning. the student constructs and communicates a complete response based on the properties of operations and the relationship between addition and subtraction or between multiplication and division, including: a logical, but incomplete, minor calculation errors some use of grade-level partial justification of a evaluating the validity of other s approaches and s. student constructs and communicates an incomplete response based on the properties of operations and the relationship between addition and subtraction or between multiplication and division, which may include: a faulty approach based on a an incomplete or illogical major calculation errors limited use of grade-level partial justification of a Revised November 2015 Page 8 of 11

9 Concrete Referents and Diagrams 6.C.3 6.C.4 6.C.5 Performance Level Descriptors Grade 6 Mathematics Grade 6: Sub-Claim C In connection with content, the student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Exceeds Level 4: Meets student clearly constructs and communicates a complete response based on concrete referents provided in the prompt or constructed by the student such as: diagrams that are connected to a written (symbolic) method, number line diagrams or coordinate plane diagrams, including: a logical and complete progression of steps generalization of an argument or evaluating, interpreting and critiquing the validity and efficiency of other s responses, approaches and reasoning, and provides a counter-example where applicable. the student clearly constructs and communicates a complete response based on concrete referents provided in the prompt or constructed by the student such as: diagrams that are connected to a written (symbolic) method, number line diagrams or coordinate plane diagrams, including: a logical and complete evaluating, interpreting and critiquing the validity of other s responses, approaches and reasoning student constructs and communicates a complete response based on concrete referents provided in the prompt or in simple cases, constructed by the student such as: diagrams that are connected to a written (symbolic) method, number line diagrams or coordinate plane diagrams, including: a logical, but incomplete, minor calculation errors some use of grade-level partial justification of a evaluating the validity of other s approaches and s. student constructs and communicates an incomplete response based on concrete referents provided in the prompt such as: diagrams, number line diagrams or coordinate plane diagrams, which may include: a faulty approach based on a or faulty an incomplete or illogical major calculation errors limited use of grade-level partial justification of a Revised November 2015 Page 9 of 11

10 Distinguish Correct Explanation/ Reasoning from that which is Flawed 6.C.6 6.C.7 6.C C C.9 Performance Level Descriptors Grade 6 Mathematics Grade 6: Sub-Claim C In connection with content, the student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Exceeds Level 4: Meets student clearly constructs and communicates a complete response to a given equation, multi-step problem, proposition or conjecture, including: a logical approach based on a a logical and complete progression of steps precision of calculation correct use of grade-level complete justification of a generalization of an argument or evaluating, interpreting and critiquing the validity and efficiency of other s responses, approaches and reasoning, and providing a counter-example where applicable. identifying and describing errors in solutions and presents correct solutions. distinguishing correct explanation/reasoning from that which is flawed. If there is a flaw, presents correct reasoning. the student clearly constructs and communicates a complete response to a given equation, multi-step problem, proposition or conjecture, including: a logical and complete evaluating, interpreting and critiquing the validity of other s responses, approaches and reasoning. identifying and describing error in solutions and presents correct solutions. student constructs and communicates a complete response to a given equation, multi-step problem, proposition or conjecture, including: a logical, but incomplete, minor calculation errors some use of grade-level partial justification of a evaluating the validity of other s approaches and. identifying and describing errors in solutions. student constructs and communicates an incomplete response to a given equation, multi-step problem, proposition or conjecture, including: an approach based on a conjecture and/or stated or faulty an incomplete or illogical major calculation errors limited use of grade-level partial justification of a. Revised November 2015 Page 10 of 11

11 Modeling 6.D.1 6.D.2 6.D.3 Performance Level Descriptors Grade 6 Mathematics Grade 6: Sub-Claim D In connection with content, the student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems, knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure and/or looking for and expressing regularity in repeated reasoning. Level 5: Exceeds Level 4: Meets knowledge, skills, and abilities described in Sub-claims A and B, the student devises a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: using stated and making and approximations to simplify a real-world situation mapping relationships between important quantities by selecting appropriate tools to create models analyzing relationships mathematically between important quantities to draw s writing a complete, clear and correct algebraic expression or equation to describe a situation applying proportional reasoning writing/using functions to describe how one quantity of interest depends on another using reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity reflecting on whether the results make sense improving the model if it has not served its purpose interpreting mathematical results in the context of the situation analyzing and/or creating limitations, relationships and interpreting goals within the model analyzing, justifying and defending models which lead to a described in Sub-claims A and B, the student devises a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: using stated and making and approximations to simplify a real-world situation mapping relationships between important quantities by selecting appropriate tools to create models analyzing relationships mathematically between important quantities to draw s writing a complete, clear, and correct algebraic expression or equation to describe a situation applying proportional reasoning writing/using functions to describe how one quantity of interest depends on another using reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity reflecting on whether the results make sense improving the model if it has not served its purpose interpreting mathematical results in the context of the situation student devises a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: using stated and approximations to simplify a realworld situation illustrating relationships between important quantities by using provided tools to create models analyzing relationships mathematically between important quantities to draw s writing an incomplete algebraic expression or equation to describe a situation applying proportional reasoning writing/using functions to describe how one quantity of interest depends on another using reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity reflecting on whether the results make sense modifying the model if it has not served its purpose interpreting mathematical results in a simplified context the student devises a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: using stated and approximations to simplify a realworld situation identifying important quantities by using provided tools to create models analyzing relationships mathematically to draw s writing an incomplete algebraic expression or equation to describe a situation applying proportional reasoning using functions to describe how one quantity of interest depends on another using unreasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity Revised November 2015 Page 11 of 11

12 Proportional Relationships 7.RP.1 7.RP.2a 7.RP.2b 7.RP.2c 7.RP.2d 7.RP RP.3-2 Performance Level Descriptors Grade 7 Mathematics Grade 7 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Level 4: Meets Analyzes and uses proportional relationships to solve real-world and mathematical problems, including multi-step ratio/percent problems. Computes unit rates of quantities associated with ratios of fractions. Decides whether two quantities are in a proportional relationship and identifies the constant of proportionality (unit rate) in tables, equations, diagrams, verbal descriptions and graphs. Interprets a point (x, y) on the graph Uses equations representing a Interprets a point (x, y) on the graph of a proportional relationship in terms proportional relationship to solve of a proportional relationship in terms of the situation, with special of the situation, with special attention to the points (0, 0) and (1, mathematical and real-world problems, including ratio and attention to the points (0, 0) and (1, r) where r is the unit rate. percent problems. r) where r is the unit rate. Represents proportional relationships by equations and uses them to solve mathematical and real-world problems, including multi-step ratio and percent problems. Determines when it is appropriate to use unit rates and understands its limitations. Analyzes and uses proportional relationships to solve real-world and mathematical problems, including simple ratio/percent problems. Computes unit rates of quantities associated with ratios of fractions. Decides whether two quantities are in a proportional relationship and identifies the constant of proportionality (unit rate) in tables, equations, diagrams, verbal descriptions and graphs. Represents proportional relationships by equations and uses them to solve mathematical and realworld problems, including simple ratio and percent problems. Uses proportional relationships to solve real-world and mathematical problems, including simple ratio/percent problems. Computes unit rates of quantities associated with ratios of fractions. Decides whether two quantities are in a proportional relationship and identifies the constant of proportionality (unit rate) in tables, equations, diagrams, verbal descriptions and graphs. Identifies proportional relationships to solve mathematical problems, including ratio/percent problems. Identifies whether two quantities are in a proportional relationship. Revised November 2015 Page 1 of 11

13 Operations with Fractions 7.NS.1a 7.NS.1b-1 7.NS.1b-2 7.NS.1c-1 7.NS.1d 7.NS.2a-1 7.NS.2a-2 7.NS.2b-1 7.NS.2b-2 7.NS.2c 7.NS.3 7.EE.3 Expressions, Equations and Inequalities 7.EE.1 7.EE.2 7.EE.4a-1 7.EE.4a-2 7.EE.4b Performance Level Descriptors Grade 7 Mathematics Grade 7 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Level 4: Meets Performs operations on positive and negative rational numbers in multi-step mathematical and realworld problems. Represents addition and subtraction on a horizontal or vertical number line and recognizes situations in which opposite quantities combine to make zero. Determines reasonableness of a solution and interprets solutions in real-world contexts. Using the properties of operations, justifies the steps taken to solve multi-step mathematical and real-world problems involving rational numbers. Applies properties of operations as strategies to add, subtract, factor and expand linear expressions. Solves multi-step linear equations with rational coefficients. In mathematical or real-world contexts, uses variables to represent quantities, construct and solve equations and Performs operations on positive and negative rational numbers in multistep mathematical and real-world problems. Represents addition and subtraction on a horizontal or vertical number line and recognizes situations in which opposite quantities combine to make zero. Determines reasonableness of a solution. Applies properties of operations as strategies to add, subtract, factor and expand linear expressions. Solves two-step linear equations with rational coefficients. In a mathematical or real-world context, uses variables to represent quantities, construct and solve equations and inequalities, and graph solution sets. Performs operations on positive and negative rational numbers in mathematical and real-world problems. Represents addition and subtraction on a horizontal or vertical number line and recognizes situations in which opposite quantities combine to make zero. Applies properties of operations as strategies to add, subtract and expand linear expressions. Solves two-step linear equations with rational coefficients. In a mathematical context, uses variables to represent quantities, construct and solve equations and inequalities, and graph solution sets. Performs operations on positive and negative rational numbers in mathematical problems. Represents addition and subtraction on a horizontal or vertical number line. Applies properties of operations as strategies to add and subtract linear expressions. Solves one-step linear equations with rational coefficients. Revised November 2015 Page 2 of 11

14 Performance Level Descriptors Grade 7 Mathematics Grade 7 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds inequalities, and graph and interpret solution sets. Rewrites an expression in different forms. Describes the relationship between equivalent quantities that are expressed algebraically in different forms in a problem context and explains their equivalence in light of the context of the problem. Level 4: Meets Revised November 2015 Page 3 of 11

15 Representing Geometric Figures 7.G.2 7.G.3 Performance Level Descriptors Grade 7 Mathematics Grade 7 Math: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Draws geometric figures freehand, with a ruler and protractor or with technology and describes their attributes. Level 4: Meets Draws geometric figures freehand, with a ruler and protractor or with technology and describes their attributes. Draws geometric figures freehand, with a ruler and protractor, or with technology and describes some of their attributes. Draws geometric figures freehand, with a ruler and protractor, or with technology and describes some of their attributes. Constructs triangles with given angle and side conditions and notices when those conditions determine a unique triangle, more than one triangle or no triangle. Constructs triangles with given angle and side conditions. Constructs triangles with given angle and side conditions. Drawings and Measurement 7.G.1 7.G G G.5 7.G.6 Describes two-dimensional figures that result from slicing threedimensional figures by a plane which may or may not be parallel or perpendicular to a base or face. Solves mathematical and realworld problems involving circumference, area, surface area and volume of two-and threedimensional objects, including composite objects. Solves problems involving scale drawings of geometric figures, including reproducing a scale drawing at a different scale. Describes the two-dimensional figures that result from slicing three-dimensional figures by a plane parallel or perpendicular to a base or face. Solves mathematical and realworld problems involving circumference, area, surface area and volume of two-and threedimensional objects. Solves problems involving scale drawings of geometric figures, including reproducing a scale drawing at a different scale. Solves mathematical problems involving circumference, area, surface area and volume of twoand three-dimensional objects. Solves problems involving scale drawings of geometric figures. Solves mathematical problems involving circumference and area of two-dimensional objects. Solves problems involving scale drawings of geometric figures. Represents angle relationships using equations to solve for unknown angles. Represents angle relationships using equations to solve for unknown angles. Uses facts about angle relationships to determine the measure of unknown angles. Revised November 2015 Page 4 of 11

16 Performance Level Descriptors Grade 7 Mathematics Grade 7 Math: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Level 4: Meets Produces a logical about the relationship between the circumference and area of a circle. Random Sampling and Comparative Inferences Understands and uses random sampling to draw inferences about a population. Understands and uses random sampling to draw inferences about a population. Draws inferences about a population from a table or graph of random samples. Compares two populations based on measures of center and measures of variability. 7.SP.1 7.SP.2 7.SP.3 7.SP.4 Draws relevant informal comparative inferences about two populations, including assessing the degree of visual overlap of two numerical data distributions with similar variabilities. Draws relevant informal comparative inferences about two populations. Draws informal comparative inferences about two populations. Generates multiple samples of the same size to gauge the variation in estimates or predictions. Analyzes whether a sample is representative of a population. Revised November 2015 Page 5 of 11

17 Chance Processes and Probability Models 7.SP.5 7.SP.6 7.SP.7a 7.SP.7b 7.SP.8a 7.SP.8b 7.SP.8c Performance Level Descriptors Grade 7 Mathematics Grade 7 Math: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Understands that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Generates a sample space to determine the probability of simple or compound events using methods such as organized lists, tables, tree diagrams or simulations. Approximates the probability of a chance event by collecting data. Develops probability models to determine the probabilities of events. Level 4: Meets Understands that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Finds probabilities when given sample spaces for simple and compound events using methods such as organized lists, tables and tree diagrams. Develops a model to approximate the probability of a chance event and predicts approximate frequencies when given the probability or by observing frequencies in data generated from the process. Understands that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Finds probabilities when given sample spaces for simple events using methods such as organized lists and tables. Understands that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Designs and uses a simulation to generate frequencies for compound events. Designs and uses a simulation to estimate the probability of a compound event. Revised November 2015 Page 6 of 11

18 Properties of Operations 7.C C C.2 Performance Level Descriptors Grade 7 Mathematics Grade 7 Math: Sub-Claim C In connection with content, the student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Exceeds Level 4: Meets student clearly constructs and communicates a complete response based on the properties of operations and the relationship between addition and subtraction or between multiplication and division, including: a logical and complete progression of steps generalization of an argument or evaluating, interpreting, and critiquing the validity of other s responses, approaches, s and reasoning, and correcting and providing counter-examples where applicable. student clearly constructs and communicates a complete response based on the properties of operations and the relationship between addition and subtraction or between multiplication and division, including: a logical and complete evaluating, interpreting and critiquing the validity of other s responses, approaches, s, and reasoning. the student constructs and communicates a complete response based on the properties of operations and the relationship between addition and subtraction or between multiplication and division, including: the student constructs and communicates an incomplete response based on the properties of operations and the relationship between addition and subtraction or between multiplication and division, including: a faulty approach based on a a logical, but incomplete, an incomplete or illogical minor calculation errors major calculation errors some use of grade-level limited use of grade-level partial justification of a partial justification of a evaluating the validity of other s approaches and s Revised November 2015 Page 7 of 11

19 Concrete Referents and Diagrams 7.C.3 7.C.4 Performance Level Descriptors Grade 7 Mathematics Grade 7 Math: Sub-Claim C In connection with content, the student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Exceeds Level 4: Meets student clearly constructs and communicates a complete response based on concrete referents provided in the prompt or constructed by the student such as diagrams that are connected to a written (symbolic) method, number line diagrams or coordinate plane diagrams, including: a logical and complete progression of steps generalization of an argument or evaluating, interpreting and critiquing the validity and efficiency of other s responses, approaches, s and reasoning, and providing a counterexample where applicable. student clearly constructs and communicates a complete response based on concrete referents provided in the prompt or constructed by the student such as: diagrams that are connected to a written (symbolic) method, number line diagrams or coordinate plane diagrams, including: a logical and complete evaluating, interpreting and critiquing the validity of other s responses, approaches, s and reasoning. the student constructs and communicates an incomplete response based on concrete referents provided in the prompt or in simple cases, constructed by the student such as: diagrams that are connected to a written (symbolic) method, number line diagrams or coordinate plane diagrams, including: a logical, but incomplete, minor calculation errors some use of grade-level partial justification of a evaluation the validity of other s approaches and s. the student constructs and communicates an incomplete response based on concrete referents provided in the prompt such as: diagrams, number line diagrams or coordinate plane diagrams, which may include: a faulty approach based on a an illogical and incomplete major calculation errors limited use of grade-level partial justification of a Revised November 2015 Page 8 of 11

20 Distinguish Correct Explanation / Reasoning from that which is Flawed 7.C.5 7.C C C C C C.8 Performance Level Descriptors Grade 7 Mathematics Grade 7 Math: Sub-Claim C In connection with content, the student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Exceeds Level 4: Meets student clearly constructs and communicates a complete response to a given equation, multi-step problem, proposition or conjecture, including: a logical and complete progression of steps generalization of an argument or evaluating, interpreting and critiquing the validity and efficiency of other s responses, approaches, s and reasoning, and provides a counterexample where applicable. identifying and describing errors in solutions and presents correct solutions distinguishing correct explanation/reasoning from that which is flawed. If there is a flaw, presents correct reasoning. student clearly constructs and communicates a complete response to a given equation, multi-step problem, proposition or conjecture, including: a logical and complete evaluating, interpreting and critiquing the validity of other s responses, approaches, s and reasoning. identifying and describing errors in solutions and presents correct solutions. the student constructs and communicates a complete response to a given equation, multi-step problem, proposition or conjecture, including: a logical, but incomplete, minor calculation errors some use of grade-level partial justification of a evaluating the validity of other s approaches and s. identifying and describing errors in solutions. the student constructs and communicates an incomplete response to a given equation, multi-step problem, proposition or conjecture, including: a faulty approach based on a an illogical and incomplete major calculation errors limited use of grade-level partial justification of a Revised November 2015 Page 9 of 11

21 Performance Level Descriptors Grade 7 Mathematics Grade 7 Math: Sub-Claim D In connection with content, the student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems, knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure and/or looking for and expressing regularity in repeated reasoning Level 5: Exceeds Level 4: Meets Modeling 7.D.1 7.D.2 7.D.3 7.D.4 described in Sub-claims A and B, the student devises a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: using stated and making and approximations to simplify a real-world situation mapping relationships between important quantities by selecting appropriate tools to create models analyzing relationships mathematically between important quantities to draw s writing a complete, clear and correct algebraic expression or equation to describe a situation applying proportional reasoning writing/using functions to describe how one quantity of interest depends on another using reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity reflecting on whether the results make sense improving the model if it has not served its purpose student devises a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: using stated and making and approximations to simplify a realworld situation mapping relationships between important quantities by selecting appropriate tools to create models analyzing relationships mathematically between important quantities to draw s writing a complete, clear and correct algebraic expression or equation to describe a situation applying proportional reasoning writing/using functions to describe how one quantity of interest depends on another using reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity reflecting on whether the results make sense student devises a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: using stated and approximations to simplify a realworld situation illustrating relationships between important quantities by using provided tools to create models analyzing relationships mathematically between important quantities to draw s writing an incomplete algebraic expression or equation to describe a situation applying proportional reasoning writing/using functions to describe how one quantity of interest depends on another using reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity reflecting on whether the results make sense student devises a plan to apply mathematics in solving problems arising in everyday life, society and the workplace by: using stated and approximations to simplify a realworld situation identifying important quantities using provided tools to create models analyzing relationships mathematically to draw s writing an incomplete algebraic expression or equation to describe a situation applying proportional reasoning using functions to describe how one quantity of interest depends on another using unreasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity Revised November 2015 Page 10 of 11

22 Performance Level Descriptors Grade 7 Mathematics Grade 7 Math: Sub-Claim D In connection with content, the student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems, knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure and/or looking for and expressing regularity in repeated reasoning Level 5: Exceeds Level 4: Meets interpreting mathematical results in the context of the situation analyzing and/or creating constraints, relationships and goals analyzing, justifying and defending models which lead to a improving the model if it has not served its purpose interpreting mathematical results in the context of the situation modifying the model if it has not served its purpose interpreting mathematical results in a simplified context Revised November 2015 Page 11 of 11

23 Expressions and Equations 8 EE.1 8 EE.2 Performance Level Descriptors Grade 8 Mathematics Grade 8 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Level 4: Meets Evaluates and generates equivalent numerical expressions using and applying properties of integer exponents. Evaluates and generates equivalent numerical expressions using and applying properties of integer exponents. Evaluates numerical expressions using properties of integer exponents. Evaluates numerical expressions using properties of integer exponents. Scientific Notation 8.EE.3 8.EE EE.4-2 Solves equations of the form x 2 = p and x 3 = p, representing solutions using or symbols. Using scientific notation, estimates very large and very small quantities and determines how many times as large one number is in relation to another. Performs operations with numbers expressed in scientific notation. Interprets scientific notation that has been generated by technology. Solves equations of the form x 2 = p, where p is a perfect square, and solves equations of the form x 3 = p, where p is a a perfect cube. Using scientific notation, estimates very large and very small quantities. Performs operations with numbers expressed in scientific notation. Partially solves equations of the form x 2 = p, where p is a positive rational number and a perfect square less than or equal to 100, by representing only the positive solution of the equation. Using scientific notation, estimates very large quantities. Performs operations with numbers expressed in scientific notation. Using scientific notation, estimates very large quantities. Chooses appropriate units for measuring very large or very small quantities. Interprets scientific notation in context. Revised November 2015 Page 1 of 11

24 Proportional Relationships and Linear Equations Performance Level Descriptors Grade 8 Mathematics Grade 8 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Level 4: Meets Graphs linear relationships in the form y=mx+b, including proportional relationships. Graphs linear relationships, in the form y=mx+b, including proportional relationships. Graphs linear relationships, in the form y=mx+b, including proportional relationships. Graphs linear relationships, in the form y=mx+b. 8.EE EE EE F.3-1 Interprets the unit rate as the slope of the graph of a proportional relationship and applies these concepts to solve real-world problems. Interprets the unit rate as the slope of the graph of a proportional relationship and applies these concepts to solve real-world problems. Interprets the unit rate as the slope of the graph of a proportional relationship. Compares two different proportional relationships represented in different ways. Interprets y=mx+b as defining a linear function. Compares two different proportional relationships represented in different ways. Makes some comparisons between two different proportional relationships represented in different ways. Uses similar triangles to show that the slope is the same between any two distinct points on a non-vertical line in the coordinate plane. Solving Linear Equations 8.EE.7b 8.EE.C.Int.1 Solves mathematical and realworld problems linear equations in one variable, with rational number coefficients, including those that require use of the distributive property and combining like terms. Solves linear equations in one variable, with rational number coefficients, including those that require use of the distributive property and combining like terms. Solves linear equations in one variable, with rational number coefficients, including those that require use of the distributive property or combining like terms. Solves linear equations in one variable, with rational number coefficients. Revised November 2015 Page 2 of 11

25 Simultaneous Linear Equations 8.EE.8a 8.EE.8b-1 8.EE.8b-2 8.EE.8b-3 8.EE.8c Performance Level Descriptors Grade 8 Mathematics Grade 8 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards for Mathematical Practice. Level 5: Exceeds Level 4: Meets Analyzes and solves mathematical and real-world problems leading to pairs of simultaneous linear equations graphically, algebraically and by inspection. Understands the relationship between the graphic representation and the algebraic solution to the system. Analyzes and solves mathematical problems leading to pairs of simultaneous linear equations graphically and algebraically. Solves mathematical problems leading to pairs of simultaneous linear equations graphically and by inspection. Solves mathematical problems leading to pairs of simultaneous linear equations graphically, where the graph is provided. Verifies a solution utilizing multiple methods to prove accuracy. Functions 8.F F F.2 8.F.3-2 Understands that a function is a rule assigning to each input exactly one output, which can be graphed as a set of ordered pairs. Compares properties of two functions represented in different ways. Understands that a function is a rule that assigns to each input exactly one output and can be graphed as a set of ordered pairs. Compares properties of two functions represented in different ways. Understands that a function is a rule that assigns to each input exactly one output and can be graphed as a set of ordered pairs. Understands that a function is a rule that assigns to each input exactly one output. Identifies and proves functions that are non-linear. Revised November 2015 Page 3 of 11

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