School District of South Orange and Maplewood

School District of South Orange and Maplewood Mathematics Department

Appendix C: Level Distinctions in Grade 6

Leveling is not a function of intelligence or mathematical talent or the ability to learn. Leveling in math begins with a consideration of the mathematical content that has to be developed, takes a measure of students’ prior learning, and enacts a plan to maximize learning across a spectrum of student achievement.

This appendix addresses two categorical distinctions regarding the variation in students’ preparation to do mathematics in the various levels. The first are the parameters of content by level for each learning objective based on the content outline. At each level, students will address every learning objective prescribed by State Standards. Variations and modifications in the content outline are based on evidence of foundational knowledge, the instructional time required for students to obtain mastery of essential aspects of content, and the opportunity to make mathematical decisions as the content is developed.

In a broader sense, the second distinction addresses the Mathematical Practices, that is, the processes and strategic competencies students need to use in order to think mathematically and develop mathematical concepts and skills. This section describes the distinctions in instructional approach within and across levels. The Mathematical Practices are defined and required by national and state standards.

May 2013 Page 1 Mathematics Curriculum: Grade 6 Objective College Prep Honors Concepts Concepts 1. Expand the All work with integers values and negative No restrictions on the range of integers and understanding of the rational numbers limited to common negative rational numbers. number system and the fractions (8ths, 10ths, 25ths) sense of magnitudes of Relations Relations number to include integers, 1. Represent equivalent forms of 1. Represent equivalent forms of and negative rational integers and negative rational integers and negative rational numbers numbers. numbers(values that are reasonable to with no restrictions. model; symbolic extensions limited to 2. Order and sequence integers and reasonable patterns: -125, -200, -50.5, -37 negative rational numbers. ½ , and negative rational numbers 3. Write, interpret, and explain NJ CCSS 6 NS 5, 6 (restricted to fractions listed above). statements of order for any rational 2. Order and sequence integers and numbers in real world contexts. negative rational numbers (restricted to fractions listed above). 3. Write, interpret, and explain statements of order for common rational numbers in real world contexts.

2. Represent, order, and 1.Represent the absolute value of restricted 1. Represent the absolute value of any interpret in real situations to common rational numbers on a number rational number on a number line. involving the absolute value line. 2. Interpret or determine absolute value as of rational numbers. 2.Interpret or determine absolute value as magnitude for a positive or negative magnitude for a positive or negative quantity in any real-world situation. quantity in a familiar real-world situation. 3. Recognize and create unrestricted NJCCSS 6 NS 7 3.Recognize and create select equivalent equivalent values in an absolute value values in an absolute value expression. expression. 4.Explain the meaning of zero in context. 4. Explain the meaning of zero in context. 5. Recognize that the opposite of the opposite of a number is the number itself.

May 2013 Page 2 Mathematics Curriculum: Grade 6 Objective College Prep Honors

3. Solve real world and 1. Represent points in the plane with 1. Represent points in the plane with any mathematical problems by integers and limited negative rational integers and negative rational number graphing points in all four number coordinates. (common fractions coordinates. quadrants of the coordinate only) 2. Find and position integers and any plane. Use coordinates 2. Find and position integers and select rational numbers on the coordinate and absolute value to find rational numbers on the coordinate plane. plane. the distance between points 3. Investigate and express that signs of 3. Describe from applications signs of with the same first or the numbers in ordered pairs indicate numbers in ordered pairs indicate same second coordinate. locations in quadrants. locations in quadrants. 4. Solve real world and select mathematical 4. Solve real world and unrestricted problems by graphing mathematical problems by graphing 5. Draw polygons in the coordinate plane 5. Draw polygons in the coordinate given selected rational numbers plane given any rational coordinates for coordinates for the vertices. the vertices. Use coordinates to find NJCCSS 6 NS 8, 6 G 3 6. Use coordinates to find distance between distance between points with the same points with the same first coordinate or first coordinate or the same second the same second coordinate-integer coordinate. Any rational number values values only 6. Generalize that when 2 ordered pairs 7. Investigate that when 2 ordered pairs differ only by signs, the locations of the differ only by signs, the locations of the points are related by reflections across points are related by reflections across one or both axes. one or both axes. 7. Enrich with other transformations

May 2013 Page 3 Mathematics Curriculum: Grade 6 Objective College Prep Honors 1. Refresh and extend GCF, LCM, prime 1.Briefly review GCF, LCM, prime factorization and divisibility. factorization, and divisibility. 4. Apply number theory concepts to real-world and 2. Find the GCF of 2 numbers less < 100 & 2.Find the GCF of 2 numbers less < 100 & mathematical problem the LCM of 2 numbers < 12. the LCM of 2 numbers < 12. situations. 3. Use the distributive property to express a 3.Use the distributive property to express a sum of 2 selected whole numbers 1-100 sum of 2 whole numbers 1-100 with a with a common factor. common factor. NJCCSS NS 4 4.Represent prime factorization with exponents.

Concepts Review and Expansion Concept Review 5. Extend the 1.part/whole representations a.Primarily a review understanding of the a.linear, area, and discrete models 1. Focus review on positive and negative number system by 2. for all models: emphasize the connection non-unit fractions and values exceeding 1 constructing meanings for between unit and non-unit fractions and 2.density and magnitude fractions. values exceeding 1 3.density and magnitude Relations 4.negative number coordinates Lesson: review through challenging Relations problems (e.g., SAT samples) and pre- Lessons: guided discovery Algebra extensions 1.equivalence 1. equivalence 2.relation to half 2.relation to half 3.order and sequence 3.order and sequence

May 2013 Page 4 Mathematics Curriculum: Grade 6 Objective College Prep Honors Addition & subtraction - Review at a brisk pace: 6. Extend the 1. Estimation w/ benchmarking Addition & subtraction - understanding and 2. focus on subtraction w/regrouping 1. Estimation w/ benchmarking use of arithmetic 3. use contexts to review and extend 2. focus on subtraction w/regrouping operations to 3. use contexts to review and extend fractions; develop, Multiplication & division - Multiplication & division - apply and explain 1. Recognize logical distinctions 4. Recognize logical distinctions procedures for 2. Estimate 5. 5. Estimate, compute and interpret computation and Review/ extend through familiar contexts and quotients for estimation with 3. Interpret and compute quotients for division of c. mult/div by whole no. fractions. fraction problems d. mult./div by fractions a. mult/div by whole no. Develop Prealgebra examples with b. mult./div by fractions mixed operations 4. Represent/interpret stories and visual models, and 6. Investigate properties in NJCCSS 6 NS 1 compute operations on fractions with and without variables. 1.Id. the number of places in solutions 1. Demonstrate a flexible approach to 2. Use computational patterns to reason about and computation. E.g., when possible, 7. Develop the reinforce place value divide out common factors; use number sense 3. Represent division as a fraction. If possible, divide out metal math strategies, such as necessary for common factors in dividend and divisor before dividing decomposition to find solutions. computation. 4. Students informally model select division situations w/ 2. Fluently divide multi-digit numbers Base 10 to make sense of the standard algorithm. using the standard algorithm. Compute fluently with 5. Fluently divide multi-digit numbers w/ standard 3. Fluently add, subtract, multiply, and multi-digit numbers, algorithm-prompt w/B-10s divide multi-digit decimals using the including decimals. Extended time required for this obj. standard algorithm for each 6. Fluently add, subtract, multiply, and divide multi-digit operation. decimals. 4. Select and apply properties to solve NJCCSS 6 NS 2,3 7. Recognize the advantage of using properties: problems: commutative, commutative, associative, distributive, identity associative, distributive, identity

May 2013 Page 5 Mathematics Curriculum: Grade 6 Objective College Prep Honors 1. Ratios 1. Ratios 8. Develop meaning a. Review the concept of ratio. Distinguish a. 1-1 ½ period review of part to part and for ratios and part to part and part to whole part to whole, the concept of a unit rate proportions; develop, b. Limited exploration in the a/b associated with a ratio a:b with b ≠ 0, apply, and explain development of the concept of a unit and use ratio language in the context of methods for solving rate a/b associated with a ratio a:b with a ratio relationship and determine equal problems involving b ≠ 0, and use real examples to show ratios and rates proportions in a variety relationships, such as 21/0 (girls to 2. Proportions (with the focus on proportional of situations. boys) reasoning) c. Determine equal ratios and rates a. Develop conceptual meaning by 9. Develop meaning Learn unit rates and chunking reasoning through qualitative for percents; develop, strategies comparisons apply, and explain 2. Proportions (with the focus on proportional b. Use proportional language in describing, methods for solving reasoning) explaining, justifying or evaluating problems involving a. Develop conceptual meaning challenging (SAT) examples and percents in a variety of b. Make representations situations situations. c. Reason through select qualitative 3. Applications: comparisons a. Make tables with equivalent ratios d. Use proportional language in describing, relating quantities with integer and explaining, justifying or evaluating rational number measurements; use examples and situations the tables to compare ratios. 3. Applications: b. Use proportional reasoning to convert a. Make tables with equivalent ratios measurement units NJCCSS 6 RP 1,2,3 relating quantities with whole number c. Plot values on the coordinate plane to measurements; use the tables to compare equal ratios and lay a compare ratios. foundation for slope. b. Use proportional reasoning to convert d. Generate the relationships between measurement units, emphasizing the coordinates of dilated or expanded multiplicative nature of this activity. shapes. c. Plot pairs of values on the coordinate e. Solve unit rate problems, including those plane; compare to find equal ratios on involving unit pricing and constant

May 2013 Page 6 Continuation of… a line. speed. d. Solve unit rate problems, including f. Solve SAT-like problems involving scale 8. Develop meaning those involving unit pricing and limited drawings for ratios and examples of constant speed. g. Solve SAT-like problems involving proportions; develop, e. Solve problems involving scale similar figures apply, and explain drawings 4. Percents methods for solving f. Solve problems involving similar figures a. Conceptual meaning using percent bars problems involving 4. Percents b. Find the percent of a quantity as a rate proportions in a variety a. Conceptual meaning using percent bars per 100. of situations. b. Representations- area and discrete c. Solve problems involving finding the models whole, given the part and percent. 9. Develop meaning c. Find the percent of a quantity as a rate d. Applications for percents; develop, per 100. 5. Memorize equivalence of fractions, apply, and explain d. Estimate the percent of a given whole. decimals, and percents ½ through 1/12, methods for solving e. Solve problems involving finding the and 1/20, 1/25, 1/50, and 1/100. problems involving whole, given the part and percent. percents in a variety of f. Applications situations. 5. Build patterns so that students NJCCSS 6 RP 1,2,3 recognize relationships using, eg, 1/10 &1/20, 1/25, 1/50, & 1/100. 6. Cue with percent bars. 7. Memorize equivalence of fractions, decimals, and percents ½ - 1/10 L3- Focus on ratios and proportions L4- Focus on proportions (everyday math (Groundworks better buy, SF examples, some activities for unit rates, Groundworks better everyday math activities). Percent focus on buy, comparing quantities activities). Percent application to real life situations (shopping, focus on mark down and mark up, various gratuity, mark up, using easy percents to strategies for solving using proportional estimate) reasoning.

May 2013 Page 7 Mathematics Curriculum: Grade 6 Objective College Prep Honors 1.Using visual and concrete models, identify 1. Find the area of right triangles, other triangles, 10. Develop, explain, and the relationship between triangles and special quadrilaterals, and other polygons by apply a variety of strategies rectangles. Use this to find the area of composing into rectangles or decomposing for determining areas of right right triangles and other triangles. into triangles and other shapes. triangles, other triangles, 2.Relate and justify formula for area of 2. Apply all of the above in the context of solving special quadrilaterals, and triangles using models of right, acute and real world/math problems. other polygons. Solve real obtuse triangles. 3. Transform formulas to solve in terms of a world problems involving 3.Apply all of the above in the context of specified dimension or variable. area and surface area. solving real world/math problems. 4. Use algebraic expressions to specify 4.Informally determine the area of special dimensions of given shapes (e.g., Given the quadrilaterals and other polygons by area of a rectangle with sides of x and ½ x, NJCCSS 6 G 8 decomposing these shapes, rearranging find x.) or removing pieces, and relating the 5. Use algebraic expressions to define a shapes to triangles and rectangles. dimension or relationship of dimensions in a 5.Relate and apply formulas for these. given problem context (e.g., Given the area, 6.Apply all of the above in the context of determine the base of an isosceles trapezoid solving select applications of real with sides two-thirds the length of the larger world/math problems. base.) 1. Represent 3D figures using nets; use the 1. Use nets to find surface area of 3D figures. 11. Develop, explain, and nets to find surface area of the figures. Apply in real/math contexts. apply a variety of strategies Apply in limited contexts. 2. Generalize formulas for solving problems related for determining surface area, 2. Develop selected formulas for vol. Relate to volume. and volume of prisms and vol. formula for rectangular prisms to 3. Solve real world/ math problems by finding the pyramids. Demonstrate and other volume formulas. volume of a rectangular prism with any fractional explain the impact of the 3. Solve real world/ math problems by finding edge lengths change of an object’s linear the volume of a right rectangular prism 4. Model & explain the impact of change of linear dimensions on its perimeter, with selected fractional edge lengths. dimensions on volume. area, surface area, or 4. Distinguish the effect of change of 5. Transform formulas to solve in terms of a volume. dimension on volume. specified dimension. NJCCSS 6 G 9 6. Use variables in dimensions.

May 2013 Page 8 Mathematics Curriculum: Grade 6 Objective College Prep Honors Variables Variables 12. Understand 1. Represent and solve problems generated 1. Represent and solve problems generated from and use variables from pictorial and symbolic representations pictorial and symbolic representations and expressions. Expressions Expressions A. Develop an intuitive/logical meaning, with A. Develop an intuitive/logical meaning, with reference to time and position. reference to time and position. B. Apply and extend prior knowledge of B. Apply and extend prior knowledge of arithmetic to algebraic expressions arithmetic to algebraic expressions NJCCSS 6 EE 1. Write and evaluate numerical expressions 1. Write and evaluate numerical expressions 1,2,3,4 involving whole numbers & some exponents. involving rational number and exponents. 2. Write and evaluate simple expressions in 2. Write and evaluate complex expressions in which a variable is a placeholder. which a variable is a placeholder. a. Translate between simple & select real world a. Translate between real world expressions expressions written in words&symbols written in words& symbols b. Identify parts of an expression using b. Identify parts of an expression using mathematical terms mathematical terms c. Evaluate expressions at specific values of c. Evaluate expressions at specific values of their variables, including expressions that their variables, including expressions that arise from formulas used in selected real- arise from formulas used in a wide variety world problems. of real-world problems. d. Perform arithmetic operations, including d. Perform arithmetic operations, including those involving exponents of 2, in the those involving any whole number conventional order with and without exponents, in the conventional order with parentheses. (3-step maximum) and without parentheses. More than 3steps 3. Apply the properties of operations to 3. Apply the properties of operations to generate generate equivalent expressions. equivalent expressions.(incl.Distributive prop) 4. Identify when 2 expressions are equivalent, 4. Identify when 2 expressions are equivalent, regardless of which value is substituted into regardless of which value is substituted into the variable. Simple expressions the variable. Complex expressions

May 2013 Page 9 Mathematics Curriculum: Grade 6 Objective College Prep Honors Equation Equation 13. Solve simple 1. Build structure and logic through balance 1. Build structure and logic through balance linear equations representations w/ problems representations w/ problems using concrete, a. Focus on and emphasize equivalence as a. Focus on and emphasize equivalence as informal, and key across selected tasks. key across varied tasks. graphical methods, b. Recognize substitution as a logical b. Recognize substitution as a logical as well as approach to finding simple solutions. approach to finding any solutions. appropriate paper- 2. Solve real world and other math problems by 2. Solve real world and other math problems and-pencil writing and solving equations in the form of x by writing and solving equations in the form techniques. + p = q, and px = q for cases in which p, q, of x + p = q, and px = q for cases in which p, and x are all whole numbers. q, and x are all non-negative rational numbers. Ineq ualities Ineq ualities 1. Approach solving an inequality as a process 1. Approach solving an inequality as a NJCCSS 6 EE 5, 6, of answering the question: which values process of answering the question: which 7, 8 from a specific set, if any, make an values from a specific set, if any, make an inequality true? inequality true? 2. Represent solutions of inequalities on 2. Represent solutions of inequalities on number line diagrams. Coefficient =1 number line diagrams. Any coefficient 3. Substitute to determine if a given number in 3. Substitute to determine if a given number in a set makes an inequality true. a set makes an inequality true. 4. Recognize that the inequalities ( x>c or xc or xc or xc or x

May 2013 Page 10 Mathematics Curriculum: Grade 6 Objective College Prep Honors 1. Use variables to represent two whole 1. Use variables to represent two quantities 14. Understand and number quantities in a real world problem (Rational numbers) in a real world problem describe the that change in relationship to one another; that change in relationship to one another; relationships among write an equation to express one quantity write an equation to express one quantity various representations (dependent variable), in terms of the other (dependent variable), in terms of the other of patterns and quantity (independent variable). quantity (independent variable). functions. 2. Analyze the relationship between the 2. Analyze the relationship between the dependent & independent variables using dependent & independent variables using NJCCSS 6 EE 9 a. Tables and graphs a. Tables and graphs b. Relating selected examples to one b. Relating any examples of these to one another another 1. Recognize that a measure of center for a 1. Recognize that a measure of center for a 15. Select and use numerical data set summarizes all of its numerical data set summarizes all of its appropriate graphical values with a single number, while a values with a single number, while a representations and measure of variation describes how its measure of variation describes how its measures of central values vary with a single number. values vary with a single number. tendency for sets of 2. Distinguish measures of center (w/selected 2. Distinguish measures of center (any rational data. rational numbers) number) Mean mode median Mean mode median 3. Examine range, spread, gaps, and (effect 3. Examine range, spread, gaps, and (effect of) outliers (simple examples) of) outliers (wide variety of examples) NJCCSS 6 SP 3 4.Graphic representations 4.Graphic representations 5. Limited and selected real applications 5. All real world applications

May 2013 Page 11 Mathematics Curriculum: Grade 6 Objective College Prep Honors Summarize and describe distributions Summarize and describe distributions 16. Make 1. Display limited numerical data in plots on a 1. Display any numerical data in plots on a number inferences and number line, dot plots, histograms, and box plots. line, including dot plots, histograms, & box plots. formulate and 2. Summarize limited, selected numerical data sets 2. Summarize any real numerical data sets in evaluate in relation to their contexts; e.g. : relation to their contexts; e.g. : arguments a. Reporting a small number of observations a. Reporting the number of observations based on data b. Describing the nature of the attribute under b. Describing the nature of the attribute under analysis and investigation, including how it was measured investigation, including how it was measured data displays. and its unit of measurement. Select examples and its unit of measurement. Wide variety Develop c. Given quantitative measures of center (median c. Given quantitative measures of center understanding and/or mean) and variability (interquartile (median and/or mean) and variability of statistical range), as well as describing an overall pattern (interquartile range and /or mean absolute variability. and any striking deviations from the overall deviation), as well as describing an overall pattern with reference to the context in which pattern and any striking deviations from the NJCCSS 6 SP the data were gathered. overall pattern with reference to the context in 1, 2, 4, 5 d. Relating the choice of measures of center and which the data were gathered. variability to the shape of the data distribution d. Relating the choice of measures of center and and the context in which the data were variability to the shape of the data distribution gathered. Limited range and examples and the context in which data were gathered. Variability – primarily exposure, simple examples Variability – including wide range of real examples 1. Recognize that a measure of variation describes 1. Recognize that a measure of variation des- how its values vary with a single number. cribes how its values vary w/a single number 2. Recognize a statistical question as one that 2. Recognize a statistical question as one that anticipates variability in the data related to the anticipates variability in the data related to the question and accounts for it in the answer. question and accounts for it in the answer. 3. Understand that a set of data collected to answer 3. Understand that a set of data collected to answer a statistical question has a distribution that can be a statistical question has a distribution that can described by its center, spread and overall shape. be described by its center, spread and overall shape.

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