DRAFT Curriculum Map

DRAFT Curriculum Map Grade 6 Mathematics

Vision for Assessment and Instruction As a community of learners, we strive to implement a rigorous thinking curriculum that utilizes an inquiry- based, formative assessment process in order to provide opportunities for students to develop academic maturity as disciplinary thinkers.

Updated May 4, 2018 ______This document was created to support instructional design and delivery of Grade 6 mathematics using enVisionMATH from Pearson as a resource.

User’s Guide information…  Critical Focus Areas  Major, Supporting/Additional  Framework  Comment on resources…envision, FAP

SCUSD Curriculum Map Implementation 2014-15 The curriculum maps are designed as a guide to assist teachers with planning and implementing the Common Core Content and Practice Standards with fidelity throughout the school year. Included with the maps is a “Year-at-a-Glance”, with content aligned to interim Benchmark assessments (Dates TBD). These curriculum maps are intended to be “living documents” that will be revised and updated as teachers use them. The maps include support for the Social-Emotional Learning (SEL) Competencies, and the new English Language Development (ELD) standards. The standards have been separated into cohesive units of study, and information on each unit is divided into the following areas:

 Essential Questions

Thought provoking, open-ended questions to be used within daily lessons that:

 cause genuine and relevant inquiry into the big ideas and core content;

 provoke deep thought, lively discussion, sustained inquiry, and new understanding as well as more questions;

 require students to consider alternatives, weigh evidence, support their ideas, and justify their answers;

 stimulate vital, on-going rethinking of big ideas, assumptions, and prior lessons;

 spark meaningful connections with prior learning and personal experiences;

 naturally recur, creating opportunities for transfer to other situations and subjects.

 Assessments for Learning

Diagnostic, Formative, and Summative assessments used throughout the unit to inform instruction.

 Assessment tasks are aligned to Learning Outcomes

 Note: Assessments are suggested, not required.

 Sequence of Learning Outcomes

Sequence of learning is intentionally organized for student success. Each outcome is not necessarily intended to be taught within one class session.

 Strategies for Teaching and Learning

Instructional strategies that lead to desired learning outcomes.

 Differentiation, e.g. EL, SpEd, GATE

An illustrative list of high-leverage strategies to meet the needs of all students Strategies for differentiation for special populations – To be completed by district experts)

 Resources

Grade 6 Mathematics References to suggested curricular and professional resources that support strategies for teaching and learning as well as the development of content knowledge.

GRADE 6 MATHEMATICS: Year-at-a-Glance Content Standards Month Units (Bold =Major Standards) District Bench-marks Unit 1: Expressions and Equations

Students apply and extend previous understandings of arithmetic to algebraic expressions. Students will reason about and solve one-variable equations BM #1 Standards to be 6.EE.1 and inequalities. Students assessed: 6.EE.1-9 6.EE.2a will represent and analyze (BM #1 to be given between 6.EE.2b quantitative relationships Nov. 9 and Dec. 4) 6.EE.2c between dependent and 6.EE.3 Sept/ independent variables. 6.EE.4 Oct 6.EE.5 NOTE: The standards in 6.EE.6 6.EE call for students to 6.EE.7 work with exponential 6.EE.8 expressions in which the BM #2 Standards to be 6.EE.9 addressed base can be a whole 6.NS.4 6.NS.1-8 number, positive decimal, (BM#2 to be given between Feb or positive fraction. The 22 and Mar. 18) enVisionMATH materials in Topics 1-3 include only expressions with a whole- number as the base.

enVisionMATH, Topics 1-3

Nov/ Unit 2: Computation with 6.NS.1 Dec/ Multi-digit Numbers, 6.NS.2 Jan Decimals, and Fractions 6.NS.3 6.NS.4 Students apply and extend previous understandings of multiplication and division to divide fractions by fractions. They will

Grade 6 Mathematics GRADE 6 MATHEMATICS: Year-at-a-Glance Content Standards Month Units (Bold =Major Standards) District Bench-marks develop fluency with multi-digit numbers including multi-digit decimals using the standard algorithm for each operation. Students will find common factors and multiples.

enVisionMATH, Topics 4-6

Unit 3: The System of Rational Numbers

Students will apply and extend previous understandings of numbers to the system of 6.NS.5 rational numbers and 6.NS.6a,b,c begin their formal study of 6.NS.7a,b Jan/ negative numbers. 6.NS.7c,d Feb Students will graph points 6.NS.8 in the coordinate plane. 6.G.3 They will represent shapes 6.EE.9 in the coordinate plane and use these techniques to solve problems.

enVisionMATH, Topics 7 and 8

Feb/ Unit 4: Ratios and 6.RP.1 March Proportional 6.RP.2 Relationships 6.RP.3a 6.RP.3b Students will develop 6.RP.3c understandings of ratios 6.RP.3d and unit rates. They will 6.EE.9 connect ratio, rate, and percentage to whole- number multiplication and division and will use concepts of ratios and Page 4 Grade 6 Mathematics GRADE 6 MATHEMATICS: Year-at-a-Glance Content Standards Month Units (Bold =Major Standards) District Bench-marks rates to solve real world and mathematical problems. Students will create and reason about tables of equivalent ratios, tape diagrams, double number lines and equations.

enVisionMATH, Topics 9- 11 Unit 5: Geometry BM #3 (Required March Students will solve real-world and mathematical problems 6.G.1 1st grade, /April involving area, surface area, and volume. 6.G.2 Optional 6.G.4 2nd – 6th) enVisionMATH, Topics 12, 13 Standards to be assessed 6.RP.1-3 6.G.1-4 Unit 6: Statistics and Probability BM#3 to be 6.SP.1 given Students will develop understanding of statistical thinking. They will work with between 6.SP.2 May 9 – statistical variability and will represent and analyze data distributions. 6.SP.3 June 3) May/ Students will focus on the characterization of data distributions by measures of 6.SP.4 June center and spread. They will learn to describe and summarize data sets. 6.SP.5a,b,c, Students will develop the ability to use statistical reasoning. d

enVisionMATH, Topic 14

Grade 6 Mathematics OVERVIEW Unit 1: Expressions and Equations (Approx. 8 weeks) Students apply and extend previous understandings of arithmetic to algebraic expressions. Students will reason about and solve one-variable equations and inequalities. Students will represent and analyze quantitative relationships between dependent and independent variables.

NOTE: The standards in 6.EE call for students to work with exponential expressions in which the base can be a whole number, positive decimal, or positive fraction. The enVisionMATH materials in Topics 1-3 include only expressions with a whole-number as the base.

Over-Arching Essential Questions for Unit 1:  How does our understanding of arithmetic apply to algebraic expressions?  What purposes can be served by the use of variables?  How does understanding the structure of algebraic expressions help in modeling real-world situations?  How do different models and representations help us to understand real-world and mathematical problems?

In this unit students will:  Write and evaluate numerical expressions involving whole-number exponents including expressions where the base is a whole number, positive decimal, or a positive fraction.  Write, read and evaluate expressions in which letters stand for numbers.  Evaluate expressions using the conventional order of operations.  Apply the properties of operations to generate equivalent algebraic expressions; they will develop the ability to use the distributive property flexibly.  Identify equivalent expressions and justify their determination using substitution.  Understand that solving an equation or inequality is a process that answers the question, “Which values, if any, from a specified set make the equation/inequality true?”  Use algebraic expressions and equations of the forms x + p = q and px = q to solve real-world and mathematical problems.  Understand that variables can represent unknown numbers or any number in a specified set.

 Write inequalities of the forms x › c or x ‹ c to represent constraints or conditions in real-world or

mathematical problems. They will recognize that inequalities of these forms have infinitely many solutions and will represent solutions on number line diagrams.  Represent and analyze quantitative relationships between dependent and independent variables in the context of real-world problems using graphs, tables, and equations.

Culminating Task:

Grade 6 Mathematics OVERVIEW Unit 1: Expressions and Equations (Approx. 8 weeks)

Documents to Better Understand the Standards in this Unit CA Mathematics Framework  “ Instructional Strategies” chapter provides research-based strategies for teaching math, K-12

 “ Supporting High Quality Common Core Instruction” chapter addresses the development, implementation, and maintenance of high-quality, standards-based mathematics instructional programs

CA Mathematics Frameworks “Grade 6”  pages 30 – 37, Expressions and Equations

Kansas Association of Teachers of Mathematics (KATM) 6th Flipbook Provides illustrated examples, instructional strategies, additional resources/tools and misconceptions by standard.  pages 48 - 67, Expressions and Equations

Progressions Document for the CCSM Narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children’s cognitive development and by the logical structure of mathematics.  6–8 Expressions and Equations

North Carolina Unpacked Standards 6 th Grade Provides illustrated examples, instructional strategies, additional resources/tools and misconceptions by standard. Page 7 Grade 6 Mathematics OVERVIEW Unit 1: Expressions and Equations (Approx. 8 weeks)  pages 29 – 42, Expressions and Equations Content Standards Common Core State Standards–Mathematics Expressions and Equations 6.EE Apply and extend previous understandings of arithmetic to algebraic expressions. (Major) 1. Write and evaluate numerical expressions involving whole-number exponents. 2. Write, read, and evaluate expressions in which letters stand for numbers. a. Write expressions that record operations with numbers and letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order

when there are no parentheses to specify a order (Order of Operations). For example, use the formulas V = s³ and A = s² to

find the volume and surface area of a cube with side lengths of s = ½. 3.Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression3(2 + x) to produce the equivalent expressions 6 + 3x; apply the distr4ibutive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. 4. Identify when two expressions ae equivalent (i.e. when the two expressions name the same number regardless of the which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the sane number regardless of what number y stands for. Reason about and solve one-variable equations and inequalities. (Major) 5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. 7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers.

8. Write an inequality of the form x › c or x ‹ c to represent a constraint or condition in a real-world or mathematical problem.

Recognize that inequalities on the form x › c or x ‹ c have infinitely many solutions; represent solutions of such inequalities on

number line diagrams. Represent and analyze quantitative relationships between dependent and independent variables. (Major)

Grade 6 Mathematics OVERVIEW Unit 1: Expressions and Equations (Approx. 8 weeks) 9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables and relate these to the equation. For example, in a problem involving motion at a constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. The Number System 6.NS Compute fluently with multi-digit numbers and find common factors and multiples. (Supporting) 4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4(9 + 2). Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. Students Students seek the meaning of a problem and look for efficient represent a wide variety of real-world contexts by using rational ways to represent and solve it using algebraic expressions and numbers and variables in mathematical expressions, equations, equations. Particularly in their work with 6.EE.9 when students and inequalities. Students contextualize to understand the make sense of real-world problems and the relationships meaning of the number or variable as related to the problem between verbal descriptions, graphs, tables, and equations. and decontextualize to operate with symbolic representations by applying properties of operations or other meaningful moves. This is particularly true of students’ work with the standards 6.EE.2c, 6.EE.6, 6.EE.7, 6.EE.8, and 6.EE.9.To reinforce students’ reasoning and understanding, teachers might ask, “How do you know?” or “What is the relationship of the quantities?” SMP 3 Construct viable arguments and critique the reasoning SMP 4 Model with mathematics. Students represent problem of others. Students construct arguments with verbal or written situations symbolically, graphically, in tables, contextually, and explanations accompanied by expressions, equations, with drawings (e.g. tape diagrams) as needed. They form inequalities, models, graphs, and tables. expressions, equations, or inequalities from real-world contexts and connect symbolic and graphical representations. This is specifically true of their work with the standards 6.EE.2a, 6.EE.3, 6.EE.6, 6.EE.7, 6.EE.8, and 6.EE.9. In 6.EE.9 they begin to explore covariance and represent two quantities simultaneously. Students use number lines to represent inequalities in their work with 6.EE.8. SMP 6 Attend to precision. Students continue to refine their SMP 7 Look for and make use of structure. Students apply mathematical communication skills by using clear and precise properties to generate equivalent expressions and to solve language in their discussions with others and in their own equations. Students develop an understanding of the structure reasoning. In this unit specifically students focus on use of of expressions and use this structure to simplify and evaluate vocabulary to describe parts of algebraic expressions – see expressions. 6.EE.2b. Students also learn to be precise in their definition of a variable – for example, writing “n equals John’s age in years” as opposed to “n is John”. ELD Standards to Support Unit Part I: Interacting in Meaningful Ways A. Collaborative 1. Exchanging information and ideas with others through oral collaborative discussions on a range of social and academic Page 9 Grade 6 Mathematics OVERVIEW Unit 1: Expressions and Equations (Approx. 8 weeks) topics) 2. Interacting with others in written English in various communicative forms (print, communicative technology, and multimedia) ------4. Adapting language choices to various contexts (based on task, purpose, audience, and text type). B. Interpretive: 5. Listening actively to spoken English in a range of social and academic contexts. C. Productive: 11. Justifying own arguments and evaluating others’ arguments in writing. Part II: Learning About How English Works B. Expanding and Enriching Ideas 5. Modifying to add details. C. Connecting and Condensing Ideas 6. Connecting Ideas 7. Condensing Ideas Social and Emotional Learning Competencies (SEL) Self-awareness Self-management Social awareness Relationship skills Responsible decision making

Differentiation support Universal Design for Learning (UDL) including EL, Special Needs, GATE

Grade 6 Mathematics OVERVIEW Unit 1: Expressions and Equations (Approx. 8 weeks) “ Universal Design for Learning ” from CAST, the Anchor Activities: Center for Applied Special Technology 1. Content-related 2. Tasks for early finishers Use of Formative Assessment Process including 1. Game actionable feedback. 2. Investigation 3. Partner Activity Use of math journals for differentiation and 4. Stations formative assessment (use link below) https://www.teachingchannel.org/videos/math- Depth and Complexity Prompts/Icons: journals 1. Depth o Language of the Discipline Flexible grouping: o Patterns 1. Content o Unanswered Questions 2. Interest o Rules 3. Project/product Trends 4. Level (Heterogeneous/ Homogeneous) o o Big Ideas Tiered: o Complexity  Independent Management Plan (Must Do/May Do)  Grouping o Content o Rigor w/in the concept o Project-based learning o Homework o Grouping

Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

As a result of the learning experience, students will be able to…in order to… Intended to be used in/during class for teaching and learning

Strategies for Teaching and Learning

1. Write and evaluate numerical expressions involving whole-number exponents in order to understand the meaning of exponents. enVisionMATH:  Lesson 1-1 6.EE.1 Georgia Department of Education:  Exponents This task has students use dice to create exponential expressions. In order to meet the full extent of 6.EE.1 students could draw cards to determine their base and use a die for the exponent. Initially cards might include only whole numbers; cards with fractions and decimals might be added later. Illustrative Mathematics:  The Djinni's Offer NOTE: 6.EE.1 calls for students to work with expressions in which the base can be a whole number, positive decimal, or positive fraction. The enVisionMATH  Seven to the What ?!?  Sierpinski's Carpet This task also integrates 6.G.1. Inside Mathematics  Double Down

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated enVisionMATH:  Lesson 1-6 6.EE.2a 6.EE.6 Math Worksheets Land:  write expressions from phrases - matching worksheet with 7 pairs to match  write expressions from phrases worksheet with 10 phrases  Using Variables to Represent Numbers - matching 5 real-world scenarios  Using Variables to Represent Numbers 10 real-world scenarios Illustrative Mathematics:  Rectangle Perimeter  Distance to School Math Worksheets Land:  Translating between words and algebraic expressions 8 questions: 3- write expressions, 3-write words, 2-true/false

As students move from numerical to algebraic work, the multiplication and division symbols x and ÷ are replaced by the conventions of algebraic notation. Students learn to use either a dot for multiplication or simple juxtaposition for example, 3

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated

Visual representations and concrete models such as algebra tiles, counters, and cubes should be used to support students’ development of conceptual understanding as they move towards use of abstract symbolic representations. Provide opportunities for students to write expressions for numerical and real-world situations and to write multiple statements that represent a given algebraic expression. Translating between mathematical phrases and algebraic expressions will help students to develop the ability to write expressions (and later equations

3. Identify parts of an expression using mathematical terms and to view one or more parts of a single expression as a single entity in order to understand the enVisionMATH:  Lesson 1-7 66.EE.2b 6.EE.6

Mathematical vocabulary used by students to describe expressions should include the following: term, constant, coefficient, variable, sum, difference, factor, product, and quotient. view (8+7) as

4. Perform arithmetic operations, including those involving whole number exponents, in the conventional order. (Order of Operations) enVisionMATH:  Lesson 1- 3 6.EE.2c Georgia Department of Education:  Rules for Exponents How do we determine the order of operations to be used when simplifying or evaluating expressions? This task incorporates the use of technology ( basic and scientific calculators) to facilitate students’ exploration of the rules regarding Page 14 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated NOTE: The standards in 6.EE call for students to work with exponential expressions in which the base can be a whole number, positive decimal, or positive operations with exponents. fraction. The enVisionMATH materials in Topic 1 include only expressions with a whole-number as the base. National Council of Teachers of Mathematics:  Exploring Krypto Lesson plan and game.  Order of Operations - Krypto Challenges Worksheet from “Exploring Krypto” lesson. Illustrative Mathematics:  Watch Out for Parentheses It is important to help students distinguish between conventions regarding the order in which arithmetic operations are performed and properties of operations. Students should understand that the order of operations tells us how to interpret expressions but does not necessarily dictate how to calculate them. (Adapted from

5. Perform arithmetic operations in order to evaluate expressions at specific values of their variables, including expressions that arise from real-world problems. enVisionMATH:  Lessons: 1-5 and 1-8 6.EE.2c 6.EE.6 Georgia Department of Education:  Writing Expressions & Writing and Evaluating Expressions

Howard County Public School System My First Fish Tank Students should focus on what terms are to be simplified first rather than invoking the “PEMDAS” rule. (adapted from Kansas Flip Book)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated Provide a variety of expressions and formulas, along with values for the variables. Expressions should progress in complexity. Given values for variables should also follow a progression beginning with whole numbers and moving to fractions and decimals. Students should have opportunities to evaluate the same expression at

enVisionMATH, Lessons: 1-10 and 1-11

6.EE.3 Georgia Department of Education:  Conjectures About Properties  The Magic of Algebra parts 1, 2, & 3 Series of three lessons explores patterns and Page 16 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated “tricks” algebraically.

National Council of Teachers of Mathematics:  Distributing and Factoring Using Area

Common Core Sheets:  Applying Properties of Operations Students use the distributive property to generate equivalent expressions.

Illustrative Mathematics:  Equivalent Expressions  Rectangle Perimeter 2 This task also integrates 6.G.1.

Mathematics Assessment Project:  Representing the Laws of Arithmetic Formative assessment lesson - integrates 6.G.1.

LearnZillion (4 minute video) Page 17 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated  Visual Model of Distributive Property

Students should generate equivalent expressions using the associative, commutative, and distributive properties. + 8 does not simplify to 11n or 11. This could be addressed through the use of manipulatives or by providing real-world contexts. Students can also be encouraged to substitute values into the expression and evaluate to demonstrate the

Algebra tiles and other area models can be used to illustrate and make sense of the distributive property.

7. Use substitution as well as the properties of operations in order to identify when two expressions are equivalent. enVisionMATH, Lessons: 1-12

6.EE.4 Georgia Department of Education:  Are We Equal? This task incorporates the use of manipulatives (Algebra Tiles or Algeblocks) and also integrates 6.NS.4.

National Council of Teachers of Mathematics:

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated  Pan Balance - Shapes Lesson plan. Students use interactive pan balance tool to build algebraic thinking using shapes of unknown weight. They are challenged to find the weight of each shape in one of six built-in sets or a random set.  Pan Balance - Expressions This interactive pan balance allows numeric or algebraic expressions to be entered and compared. You can "weigh" the expressions you want to compare by entering them on either side of the balance. Using this interactive tool, you can practice arithmetic and algebraic skills, and investigate the important concept of equivalence.

Students may have the idea that the equal sign is a signal to perform indicated computations and produce an “answer”. It is critical that students understand the equal sign as indication of equivalence. Students can prove equivalence or non-equivalence using substitution and technology or paper-and-pencil strategies to evaluate expressions. Page 19 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated Students should also prove equivalence by simplifying each expression into the same form; they should justify each step, naming properties (e.g. associative, commutative, distributive) where appropriate.

enVisionMATH, Lessons: 1-9 and 1-13

6.EE.2a Georgia Department of Education: 6.EE.2c  Visual Patterns 6.EE.6 “Students need multiple experiences working with and explaining patterns. Giving students a visual pattern to build concretely, allows students to experience the growth of the pattern and explain it based on that experience.” –GA DOE

Illustrative Mathematics:  Triangular Tables

Mathematics Assessment Project:  Evaluating Conjectures - Consecutive Sums Formative assessment lesson. Connecting written expressions with word problems and/or drawing visual models can be used to provide context. It is important for students to read algebraic expressions in a manner that reinforces the understanding that the variable represents a number.

9. Understand that solving an equation is a process of answering the question, “which values from a specified set, if any, make the equation true?” Use substitution enVisionMATH, Lesson: 2-1

6.EE.5 Math Worksheets Land:  Solve real-world problems – matching Worksheet with 5 scenarios. Answer choices to match.

Illustrative Mathematics:  Log Ride  Busy Day Understanding of solving should be developed conceptually before students work on procedural skills. Students should be given opportunities to think about possible solutions prior to solving the equation. For example, in the equation n + 4) and/or true-false sentences (e.g. 3.45 + 5.2 = 3.35 + 5.3) can be used to encourage students to employ relational thinking in developing their understanding of solving. Developmentally Appropriate Instruction for Grades 6 – 8; pages 229 through 232 provide more on true-false and open sentences and activities and questions to support development of relational thinking. Understanding of 6.EE.5 can be reinforced by comparing arithmetic and algebraic solutions to simple word problems. (See pages 6 and 7 of Progressions document.) x + p = q and px = q for cases in which p, q, and x are all non-negative enVisionMATH, Lessons: 2-3, 2-4, 2-5, 2-6, and 2-9

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated 6.EE.7 6.EE.6 Math Worksheets Land:  Solve real-world problems How can we differentiate between situations that can be modeled by equations of the form x + p = q and those that can be modeled using the form px = q? Worksheet with 10 scenarios.  Solving equations with fractions and decimals – worksheet 1  Solving equations with fractions and decimals - worksheet 2 Illustrative Mathematics:  Morning Walk  Firefighter Allocation LearnZillion videos- Solving equations using tape diagrams/bar modeling:  Addition: Subtraction: Multiplication: Division Each link opens a separate video Manipulatives and drawings such as tape diagrams and bar models should be used to build conceptual understanding of the procedural strategies involved in solving equations using inverse operations. Emphasize understanding of the results of using additive inverse (0) versus multiplicative inverse (1) and incorporate use of appropriate vocabulary (e.g. coefficient and constant) in explanations. and multiplication by the reciprocal ( ). Students should have multiple opportunities to distinguish between situations which can be modeled with equations of the form x + p = q and those which can be modeled using the form px = q. and should progress to include positive fraction and decimal values. Solutions should not require the use of operations with negative numbers.

Understanding of 6.EE.6 can be reinforced by comparing arithmetic and algebraic solutions to simple word problems. (See pages 6 and 7 of Progressions document.) Page 22 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated Students should be expected to attend to precision in defining variables used in equations that model real-world problems. It is important for students to understand that a variable does not represent a word e.g. “t is time”

enVisionMATH, Lesson: 2-7

to represent a constraint or condition in a real-world or mathematical problem. NOTE: The 6th grade standard 6.EE.8 is to work with the forms

Common Core Sheets: 6.EE.8 6.EE.6  Writing Inequalities Writing from mathematical statements. Opus: . It is not inappropriate to introduce and include inequalities of the  6.EE.8: Write an inequality of the form x > c or x < c to represent... These forms can be used to support students’ development of conceptual understanding but should not assessed for grading purposes. Mostly real-world contexts.

Illustrative Mathematics:  Height Requirements NOTE: This task also includes the possibility of notation that involves ≤ and ≥.

Provide examples and items that support understanding of the distinction between the various relationships represented by < , ≤ , > , and ≥ based on both mathematical language (e.g. less than 57 versus 57 or less) and real-world contexts (e.g. a family bought more than $100 worth of groceries versus a family bought at least $100 worth of groceries). Discussions regarding this distinction provide an opportunity for students to create viable arguments and critique the reasoning of others (SMP 3).

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated Students should be encouraged to think flexibly when writing inequality statements. For example they should see the relationship between x > c and c < x. Examples may be extended to include situations modeled by a compound inequality of the form

Students can develop conceptual understanding of equality and inequality through the use of a balance. See lessons included in National Council of Teachers of Mathematics unit Everything Balances Out in the End . These lessons were also referenced in learning outcome 7. 12. Recognize that inequalities on the form x › c or x ‹ c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. enVisionMATH, Lesson: 2-8 Understand that solving an inequality is a process of answering the question, “which values from a specified set, if any, make the inequality true?” and use

6.EE.8 Math Worksheets Land: 6.EE.5  Solving Equations and Inequalities - Matching

 Solving Equations and Inequalities . It is not inappropriate to introduce and include inequalities of the These forms can be used to support students’ development of conceptual understanding but should not assessed for grading purposes. Common Core Sheets:  Expressing Inequalities on a Number line  Matching Inequalities to Number lines  Writing Inequalities from a Number line

Illustrative Mathematics:  Fishing Adventures Page 24 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated NOTE: This task introduces the notation of ≤ and ≥.

Provide multiple opportunities for students to consider situations and to determine whether there must be a single solution or if the scenario allows for multiple solutions. have infinitely many solutions. Provide situations in which the solution is not limited to the set of positive numbers. Students should have experience with solution sets that include other rational numbers (i.e. fractions and/or decimals) as well as negative values. Students should be aware that numbers less than zero could be part of a solution set. Students should have multiple opportunities to develop the understanding that shading on a number line indicates that possible values of solutions include fractions and decimals. Examples may be extended to include situations modeled by a compound inequality of the form

As students represent solutions on number lines they should make sense of the use of an open circle to represent boundary points. Provide examples and items that support understanding of the distinction between open and closed circles based on both mathematical language (e.g. less than 57 versus 57 or less) and real- world contexts (e.g. a family bought more than $100 worth of groceries versus a family bought at least $100 worth of groceries). Discussions regarding this distinction provide an opportunity for students to create viable arguments and critique the reasoning of others (SMP 3). The following examples are from the CA Frameworks page 35. Note the use of both open and closed circles in Example 2.

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated enVisionMATH, Lesson: 3-1

6.EE.9

Provide multiple opportunities for students to analyze situations and determine what unknown is dependent on other components. Include situations in which variables could be designated differently depending on the conditions. For example, distance traveled is dependent on the elapsed time and a constant rate of speed versus time being dependent on the rate of speed and the distance to be traveled. 14. Use variables to represent two quantities in a real-world problem that change in relation to one another; write an equation to represent the relationship; analyze enVisionMATH, Lessons: 3-2, 3-3, and 3-4

NOTE: The objectives of the enVisionMATH lessons in Topic 3 focus on the use of tables and equations. These lessons do not address the full content of the standard 6.EE.9 6.EE.9.

Math Worksheets Land:  Using Tables and Data Charts  Using Tables and Data Charts - Matching

National Council of Teachers of Mathematics:  Bouncing Tennis Balls Lesson plan.  Building Bridges Page 27 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated Lesson plan.

Illustrative Mathematics:  Chocolate Bar Sales

Mathematics Assessment Project:  Interpreting Equations Formative assessment lesson.  Modeling Relationships- Car Skid Marks Formative assessment lesson.

Inside Mathematics:  Gym.pdf - Performance Task  Truffles.pdf - Performance Task

Students should have multiple opportunities to show relationships between quantities with multiple representations, using language, a table, an equation, and/or a graph. Students should be able to start with any one of these representations and produce the others. Translating between multiple representations helps students understand that each form represents the same relationship and provides a different perspective. Students should have multiple opportunities to graph these relationships on the coordinate plane. They will need to recognize that the independent variable is graphed on the x-axis and the dependent variable is graphed on the y-axis. They should also develop an understanding of the difference between discrete data and data that can be graphed with a line. Relationships should be proportional with the line passing through the origin. The use of technology can facilitate collection of real-time data and the use of actual data to create tables and charts. Page 28 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

This task provides an opportunity for students to reason about equivalence and equations 6.EE.4 and 6.EE.5.

This mini-assessment is designed to illustrate the cluster 6.EE.1 through 6.EE.4.

OVERVIEW Unit 2: Computation with Multi-digit Numbers, Decimals, and Fractions (Approx. 7-8 weeks)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated Students apply and extend previous understandings of multiplication and division to divide fractions by fractions. They will develop fluency with multi-digit numbers including multi-digit decimals using the standard algorithm for each operation. Students will find common factors and multiples. .

Decompose whole numbers with fractions and decimals into their place value components to be able to add, subtract, multiply, and divide two or more numbers. Use estimation and rounding throughout problems to test possible answers before arriving at final solution.

chapter addresses the development, implementation, and maintenance of high-quality, standards-based mathematics instructional programs

Provides illustrated examples, instructional strategies, additional resources/tools and misconceptions by standard.

Narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children’s cognitive development and by the logical structure of mathematics.

Provides illustrated examples, instructional strategies, additional resources/tools and misconceptions by standard.

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction (In general, (a/b) ÷ (c/d) = ad/bc) How much chocolate will each person get if 3 people share 1/2 lb. of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of

Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. 4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.

Students seek the meaning of a problem and look for efficient ways to represent and solve it using fractions and decimals. Particularly SMP 2 Reason abstractly and quantitatively. Students represent a wide variety of real-world contexts by using rational numbers in mathematical expressions, equations, and inequalities. To reinforce students’ reasoning and understanding, teachers might ask, “How do you know what the place value is for each part of the fraction/decimal?” Page 36 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated in their work with 6.NS.1 and 6.NS.3, students will use place value to make sense of and reinforce their understanding of computing using numbers with fractions/decimals.

Students construct arguments with verbal or written explanations accompanied by numbers including fractions/decimals SMP 4 Model with mathematics. Students represent problem situations symbolically, graphically, in tables, contextually, and with drawings (e.g. tape diagrams) as needed. They form expressions, equations, or inequalities from real-world contexts and connect symbolic and graphical representations. This is specifically true of their work with the standards 6.NS.1 & 6.NS.3. In 6.NS.3 they begin to explore generalizations of rules for multiplying and dividing with decimals.

Students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. In this SMP 7 Look for and make use of structure. Students apply properties of fractions/decimals to look closer at answers for reasonableness. Students develop an understanding of the structure of unit specifically students focus on use of vocabulary to describe parts of division using both whole numbers and numbers including fractions/decimals – see 6.NS.1 & 6.NS.3. Students also learn to be fractions and generate the “multiply by the reciprocal” when dividing by a fraction. precise in their definition of place value in computation and precise about correct place value when multiplying and dividing with decimals/fractions by using estimation.

Exchanging information and ideas with others through oral collaborative discussions on a range of social and academic topics) Interacting with others in written English in various communicative forms (print, communicative technology, and multimedia)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated Anchor Activities: 3. Content-related 4. Tasks for early finishers 5. Game https://www.teachingchannel.org/videos/math-journals 6. Investigation 7. Partner Activity 8. Stations

Depth and Complexity Prompts/Icons: 2. Depth o Language of the Discipline o Patterns o Unanswered Questions o Rules o Trends o Big Ideas o Complexity

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated SEQUENCE OF INSTRUCTION Unit 2: Computation with Multi-digit Numbers, Decimals, and Fractions (Approx. 7-8 weeks)

Analyze a decimal, representing it numerically and pictorially, as both a single fraction and as a sum of the place value pieces of the fraction (e.g. 6.32 enVisionMATH: ”Adding, Subtracting and Multiplying Decimals” in order to recognize that ten of any place value can be re-written in a place value representing the next smaller power of ten.  Lesson 4-1, 4-2, 4-3

6.NS.3 Math Goodies  Adding and Subtracting Decimals Math Work Sheets Land  Adding Decimals EngageNY  Use topics 6-15 for 6.NS.3 SCUSD Resource Development Page 41 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated  Writing Decimals as Fractions Inside Mathematics:  “ Sewing”

This outcome can be used as a guided warmup, review of prior knowledge, or an independent challenge problem.

2. Add and subtract multi-digit decimals by decomposing the quantities into terms of whole numbers and fractions to understand place value. enVisionMATH: ”Adding, Subtracting and Multiplying Decimals”  Lesson 4-1, 4-2, 4-3 6.NS.3 Illustrative Mathematics:  “ Jayden’s Snacks”

When adding, why does “twelve- hundredths” get re-grouped as “one-tenth” and “two-hundredths”? What similarities are there between whole digit addition/subtraction and decimal addition/subtraction?

Same terminating place value decimal without re-grouping (e.g. 6.32 + 3.15) or borrowing,

Same and different terminating place value with regrouping (6.79 + 3.54 and 6.3 – 4.83)

When performing all operations with decimals and fractions, incorporate estimation into the experience throughout the problem.

Changing decimals and fraction addition and subtraction into multi-digit whole number addition and subtraction

3. Estimate products of decimal numbers using front-end estimation and by rounding to the largest place value. Compare the results of estimations without finding enVisionMATH: ”Adding, Subtracting and Multiplying Decimals”  Lesson 4-4

6.NS.3 Illustrative Mathematics:  Reasoning about Multiplication and Division and Place Value, Part I What would be a reasonable estimate for the product of 3.8 and 5.12? (extend this question to addition, subtraction, and division of decimals) Use rounding as a tool to estimate reasonableness for decimal placement throughout the problem. When performing all operations with decimals and fractions, incorporate estimation into the experience throughout the problem.

4. Write decimals as fractions and multiply, using the denominator of the product to determine place value. enVisionMATH: ”Adding, Subtracting and Multiplying Decimals” 6.NS.3  Lesson 4-5 Page 43 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

When performing all operations with decimals and fractions, incorporate estimation into the experience throughout the problem.

enVisionMATH: “Dividing Whole Numbers and Decimals”  Lesson 5-1, 5-2, 5-3 6.NS.2 Illustrative Mathematics  Baking Cookies Which method for dividing multi-digit whole numbers do you prefer – “scaffolded” or “stacking” –and why?  Price per Pound and Pound per Dollar  Reasoning about Multiplication and Division and Place Value  The Florist Shop SCUSD Resource Development  Scaffold Division Show Me.com  Long Division Scaffolded YouTube  Partial Quotients (Stacked for Division)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated Stacked long division – stacked division is a method of using repeated subtraction for division which connects place value to the standards algorithm.

6. Divide a decimal number by a whole number where the quotient is a decimal (e.g. . Divide decimal by a decimal number where the quotient is a decimal (e.g. enVisionMATH: “Dividing Whole Numbers and Decimals”  Lessons: 5-4 6.NS.3 Illustrative Mathematics  Grandma’s Gifts EngageNY  Dividing Decimals (with adding zeroes) SCUSD - student examples of:  Dividing decimals

Understand that when dividing a decimal by a whole number, the integrity of the place value is maintained by first multiplying by the appropriate power of 10, enVisionMATH: “Dividing Whole Numbers and Decimals” Example:  Lessons: 5-5 Illustrative Mathematics 6.NS.3  Reasoning Around Multiplication and Division and Place Value Part II How would you explain the reason we can “move the decimal” to create whole numbers and perform long division? What is really happening when we “bring up the decimal” when doing the traditional division algorithm?

Analyze the process used for dividing decimals to generalize and create short-cuts of moving the decimal and dividing whole numbers. This outcome provides the justification for why we can “bring up the decimal” when using the traditional division algorithm.

Divide multi-digit decimals with different terminating place values by enVisionMATH: “Dividing Whole Numbers and Decimals” writing as two fractions, finding a common denominator, then dividing straight across in order to perform long division with whole numbers.  Lessons: 5-5 Illustrative Mathematics Page 46 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated 6.NS.3  Buying Gas

Why do we “move the decimal” (Get rid of the decimal) in the divisor when we divide decimals?

9. Factor composite numbers up to 100 and use the prime factors to list all factor pairs. Use prime factorization to create list of factor pairs to find the greatest enVisionMATH: “Dividing Fractions”  Lesson 6-1 Illustrative Mathematics: 6.NS.4  Adding Multiples SCUSD Resource Development How does finding all the prime factors of two numbers help to find all common factors including the GCF?  Find Common Factors Common Core Sheets  Greatest Common Factor

*Nick baked 32 cupcakes and Gillian baked 48 cupcakes. They wanted to put the same number of cupcakes in each box. What is the greatest number of cupcakes that can fit in a box? How many boxes will they have altogether?

enVisionMATH, “Dividing Fractions”  Lessons: 6-2 6.NS.4 Illustrative Mathematics: How can you use prime factorization to find the greatest common factor (GCF) and least common multiple (LCM) of two numbers at the same time?  Bake Sale How can you use prime factorization to find the greatest common factor (GCF) and least common multiple (LCM) of two numbers at the same time? Mathematics Assessment Resource Service (MARS)  Factors and Multiples Common Core Sheets  Least Common Multiple

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated Students should make the connection to the fact that the GCF of two numbers is the product of all the common prime factors.

Study of GCF and LCM may provide opportunity for review of fraction operations learned in grade 4 and 5.

In grade four, students identified prime numbers, composite numbers, and factor pairs. In grade six, students build on prior knowledge and find the greatest common factor (GCF) of two whole numbers less than or equal to 100 and find the least common multiple (LCM) of two whole numbers less than or equal to 12 (6.NS.4). Teachers might employ compact methods for finding the LCM and GCF of two numbers, such as the ladder method discussed below and other methods.

As an extension, students should make the connection to the fact that the GCF of two numbers is the product of all common prime factors. Study of GCF and LCM may provide opportunity for review of fraction operations learned in grades 4 and 5.

enVisionMATH, “Dividing Fractions” 6.NS.1  Lessons: 6-3, 6-4, and 6-5 Illustrative Mathematics  Baking Cookies  Price per Pound and Pound per Dollar  Reasoning about Multiplication and Division and Place Value  The Florist Shop

Students may divide straight across (i.e. the first numerator divided by second numerator and first denominator divided by the second denominator) when both the numerators and denominators are (whole number) divisible:

fractions without common denominators with models to find a common denominator and to show that division means how many of one quantity goes enVisionMATH, “Dividing Fractions”  Lessons: 6-8 Illustrative Mathematics 6.NS.1  Baking Cookies  Price per Pound and Pound per Dollar  Reasoning about Multiplication and Division and Place Value  The Florist Shop

Students may create one equivalent fraction in order to divide straight across (by multiplying the first fraction by the equivalent form of the denominator of the second fraction, i.e. 7/7= 1) :

enVisionMATH, “Dividing Fractions”  Lessons: 6-5 Page 50 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 1: Expressions and Equations (Approx. 8 weeks)

Students should be given opportunities to work with a variety of situations and problem types. They should begin with whole-number bases and progress to exponential expressions that include bases that are positive fractions or positive decimals. product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated 6.NS.1

14. Analyze different cases of dividing fractions by fractions to generalize that one can multiply by the reciprocal. enVisionMATH, “Dividing Fractions”  Lessons: 6-6 6.NS.1

Teaching the invert and multiply model for dividing fractions without developing an understanding of why it works can confuse students and interfere with their ability to apply division of fractions to solve word problems.” (CA Math

Chapters 8 & 9, “Fraction Concepts and Computation” & Decimal Concepts and Computation“ , pages 104 – 167.

Grade 6 Mathematics OVERVIEW Unit 3: The System of Rational Numbers (Approx. 3-4 weeks) Students will apply and extend previous understandings of numbers to the system of rational numbers and begin their formal study of negative numbers. Students will graph points in the coordinate plane. They will represent shapes in the coordinate plane and use these techniques to solve problems.

Note: Teachers should attend to precision in using the word “minus” only when referring to the operation of subtraction. Teachers should refer to numbers such as “-1/2” as negative one-half or the opposite of one-half.

Over-Arching Essential Questions for Unit 3:  What is the meaning of a positive and/or negative number in real-life situations?  How and why are rational numbers ordered?  What is an absolute value and how is it used in the real world?  How do you locate points in the coordinate plane?  How can you find distances between points on the same vertical or horizontal line?

In this unit students will:  Students will be able explain why positive and negative numbers are used together to describe quantities having opposite directions or values.  Students will be able to use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.  Students will be able to locate numbers and their opposites on a number line.  Students will be able to identify that the opposite of the opposite of a number is the number itself.  Students will be able to explain that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.  Students will be able to find and position integers and other rational numbers on a horizontal or vertical number line diagram.  Students will be able to find and position coordinates on a coordinate plane.  Students will be able to define the absolute value of a rational number as its distance from 0 on the number line and interpret absolute value as magnitude for a positive or negative quantity in a real- world situation.  Students will be able to explain that the order of positive numbers is the same as the order of their absolute values. Students will be able to explain that the order of negative numbers is the opposite order of their absolute values.  Students will be able to interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Students will be able to write, interpret, and explain statements of order for rational numbers in real- world contexts.

Grade 6 Mathematics OVERVIEW Unit 3: The System of Rational Numbers (Approx. 3-4 weeks)  Students will be able to graph points in any quadrant of the coordinate plane to solve real-world problems.  Students will be able to use absolute value to find distances between two points with the same x- coordinate or the same y-coordinate.

Culminating Task:

Grade 6 Mathematics OVERVIEW Unit 3: The System of Rational Numbers (Approx. 3-4 weeks)

Documents to Better Understand the Standards in this Unit CA Mathematics Framework  “ Instructional Strategies” chapter provides research-based strategies for teaching math, K-12

 “ Supporting High Quality Common Core Instruction” chapter addresses the development, implementation, and maintenance of high-quality, standards-based mathematics instructional programs

CA Mathematics Frameworks “Grade 6”  pages 27-31, The Number System

Kansas Association of Teachers of Mathematics (KATM) 6th Flipbook Provides illustrated examples, instructional strategies, additional resources/tools and misconceptions by standard.  pages 39 - 47, Expressions and Equations

Progressions Document for the CCSM Narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children’s cognitive development and by the logical structure of mathematics.  6–8 The Number System Page 56 Grade 6 Mathematics OVERVIEW Unit 3: The System of Rational Numbers (Approx. 3-4 weeks)

North Carolina Unpacked Standards 6 th Grade Provides illustrated examples, instructional strategies, additional resources/tools and misconceptions by standard.  pages 23 – 28, The Number System Content Standards Common Core State Standards-Mathematics: Number System 6.NS Apply and extend previous understandings of numbers to the system of rational numbers. 5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. 6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite. b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. 7. Understand ordering and absolute value of rational numbers. a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right. b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3°C > –7°C to express the fact that –3°C is warmer than –7°C. c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars. d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars. 8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. Students Students seek the meaning of a problem and look for efficient represent a wide variety of real-world contexts by using rational ways to represent and solve problems using both vertical and numbers and variables in mathematical expressions, equations, Page 57 Grade 6 Mathematics OVERVIEW Unit 3: The System of Rational Numbers (Approx. 3-4 weeks) horizontal number lines. They will understand the meaning of and inequalities. Students contextualize to understand the zero and opposites on a number line(s). Particularly in their meaning of the number or variable as related to the problem work with 6.EE.9 when students make sense of real-world and decontextualize to operate with symbolic representations. problems and the relationships between verbal descriptions, This is particularly true of students’ work with the standards graphs, tables, and equations. 6.NS.7a, 6.NS.7b, 6.NS.7c, 6.NS.8, 6.G.3, & 6.EE.9.To reinforce students’ reasoning and understanding, teachers might ask, “How do you know?” or “What is the relationship of the quantities?” SMP 3 Construct viable arguments and critique the reasoning SMP 4 Model with mathematics. Students represent problem of others. Students construct arguments with verbal or written situations symbolically, graphically, in tables, contextually, and explanations accompanied by expressions, equations, with drawings (e.g. tape diagrams) as needed. They form inequalities, models, graphs, and tables. expressions, equations, or inequalities from real-world contexts and connect symbolic and graphical representations. This is specifically true of their work with the standards 6.EE.2a, 6.EE.3, 6.EE.6, 6.EE.7, 6.EE.8, and 6.EE.9. In 6.EE.9 they begin to explore covariance and represent two quantities simultaneously. Students use number lines to represent inequalities in their work with 6.EE.8. SMP 6 Attend to precision. Students continue to refine their SMP 7 Look for and make use of structure. Students apply mathematical communication skills by using clear and precise properties to generate number lines and graphs to solve language in their discussions with others and in their own equations. Students develop an understanding of vertical and reasoning. In this unit specifically students focus on use of horizontal number lines and subsequently, the coordinate vocabulary (positive, negative, zero, absolute value, and plane. Students will use this for graphing, finding distance, and magnitude) to describe numbers on a number line. locating objects.

ELD Standards to Support Unit Part I: Interacting in Meaningful Ways C. Collaborative 1. Exchanging information and ideas with others through oral collaborative discussions on a range of social and academic topics) 2. Interacting with others in written English in various communicative forms (print, communicative technology, and multimedia) ------4. Adapting language choices to various contexts (based on task, purpose, audience, and text type). B. Interpretive: 5. Listening actively to spoken English in a range of social and academic contexts. C. Productive: 11. Justifying own arguments and evaluating others’ arguments in writing. Part II: Learning About How English Works B. Expanding and Enriching Ideas 5. Modifying to add details. C. Connecting and Condensing Ideas 6. Connecting Ideas 7. Condensing Ideas Social and Emotional Learning Competencies (SEL)

Grade 6 Mathematics OVERVIEW Unit 3: The System of Rational Numbers (Approx. 3-4 weeks) Self-awareness Self-management Social awareness Relationship skills Responsible decision making

Differentiation support Universal Design for Learning (UDL) including EL, Special Needs, GATE

Grade 6 Mathematics OVERVIEW Unit 3: The System of Rational Numbers (Approx. 3-4 weeks) “ Universal Design for Learning ” from CAST, the Anchor Activities: Center for Applied Special Technology 5. Content-related 6. Tasks for early finishers Use of Formative Assessment Process including 9. Game actionable feedback. 10. Investigation 11. Partner Activity Use of math journals for differentiation and 12. Stations formative assessment (use link below) https://www.teachingchannel.org/videos/math- Depth and Complexity Prompts/Icons: journals 3. Depth o Language of the Discipline Flexible grouping: o Patterns 9. Content o Unanswered Questions 10. Interest o Rules 11. Project/product Trends 12. Level (Heterogeneous/ Homogeneous) o o Big Ideas Tiered: o Complexity  Independent Management Plan (Must Do/May Do)  Grouping o Content o Rigor w/in the concept o Project-based learning o Homework o Grouping

Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 3: The System of Rational Numbers (Approx. 3-4 weeks) Sequence of Learning Outcomes Resources As a result of the learning experience, students will be able to… Intended to be used in/during class for teaching and learning in order to…

1) Make sense of positive and negative integers in enVisionMATH: ”Integers and Other Rational context of a real-world situation, explaining the Numbers” meaning of zero in each situation. Students  Lesson 7-1 should understand that positive and negative  Lesson 7-6 numbers as numbers on opposite sides of 0 on EngageNY the number line, e.g. –(-3) as the opposite of (-3)  Temperature - Exit Ticket pg. 41 otherwise known as 3. Graph opposites on a LearnZillion number line.  Understanding Negative Numbers on a Number Line 6.NS.5  Opposites 6.NS.6a SCUSD Resource Development:  What is the relationship between positive and  Contest Winner negative numbers?  Is zero positive or negative? Explain.  Where do we see negative numbers in the real world?

 Teachers should use real-world contexts for integers such as: elevation, temperature, banking, and electric charges.  Students should think of the opposite as the number that is the same distance away from zero, but on the other side of zero. Use the diagram below to help illustrate the math involved.

Image from the Kansas Flipbook 6th grade

2. Compare and order integers on number lines and enVisionMATH: “Integers and Other Rational by using inequality symbols. Numbers”  Lesson 7-2 6.NS.7a Illustrative Mathematics: 6.NS.7b  Mile High  Warmer in Miami  Why are bigger negative numbers really  Integers on the Number Line I smaller? Page 61 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 3: The System of Rational Numbers (Approx. 3-4 weeks)  Integers on the Number Line II SCUSD Resource Development  Positive and negative events story LearnZillion  Compare using inequalities

 Ensure that number lines have equal spaces between segments. Some students are under the impression that if you break up a whole into five parts, that the four tick marks do not need to be equal. The Common Core State Standards do not specifically mention the set of integers (consisting of the whole numbers and their opposites) as a distinct set of numbers. Rather, the standards are focused on student understanding of the set of rational numbers in general (consisting of whole numbers, fractions, and their opposites). CA Framework p. 27

 Working with number line models helps students internalize the order of the numbers—larger numbers on the right or top of the number line and smaller numbers to the left or bottom of the number line.

3. Students identify absolute value and its relation to enVisionMATH: “Integers and Other Rational zero (absolute value is the distance of a number from Numbers” zero – and distance is always a positive number).  Lesson 7-3 Understand absolute value as magnitude for a  Lesson 7-1 positive or negative quantity in a real-world EngageNY situation, e.g. owing 30 dollars represents the  Absolute Value - Begins on p. 98 magnitude of. From Illustrative Mathematics: 6.NS.7  Comparing Temperatures  Jumping Flea  What is absolute value?  Above and Below Sea Level  When is absolute value used in the real world?  What does it mean for numbers to be opposites?

CA Framework 6th Grade p. 31

4. Graph all different types of rational numbers on a enVisionMATH: “Integers and Other Rational number line, paying attention to the order of the Numbers” numbers. Plot rational numbers on number lines,  Lesson 7-4 Illustrative Mathematics: Page 62 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 3: The System of Rational Numbers (Approx. 3-4 weeks) both horizontal and vertical, in context of a real  Plotting Points in the Coordinate Plane world situation such as temperature or elevation.  Extending the Number Line

6.NS.6c

 What are rational numbers?  How do you use a number line to represent a magnitude of a number?

 Use number lines to make sense of relative size based on location, both in and out of real-world context, recognizing that a number to the right is always greater than a number to the left.

5. Write, interpret, and explain inequality statements enVisionMATH: “Integers and Other Rational with rational numbers, using a number line to justify Numbers” reasoning.  Lessons: 7-2 LearnZillion 6.NS.7  Comparing Rationals using inequalities  Comparing Part II  Why are bigger negative numbers From Common Core Toolbox: really smaller?  Cake Weighing IXL Math Online:  Put Rational Numbers In Order Illustrative Mathematics  Fractions on a Number Line  Remember to include benchmark fractions/decimals to assist students in problem solving and their thinking. Remind them that 0, 1/4, 1/2, 3/4, 1, and whole numbers both on the positive and negative scales are (hopefully) easy numbers to use to compare. These numbers can help students eliminate possible incorrect thinking. If you are comparing two numbers (16/15 and 11/12) and one number is to the right of positive one and the other one is just below positive one, this kind of thinking makes the two numbers easier to compare instead of having to find a common denominator, and find out which one is greater. 6. Use reasoning (working backwards) to solve enVisionMATH, “Integers and Other Rational problems using both positive and negative integers Numbers” and rational numbers.  Lessons: 7-6 LearnZillion 6.NS.5  Understand Relationship Between Positive and Negative Numbers  When and why is it appropriate to work

Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 3: The System of Rational Numbers (Approx. 3-4 weeks) backwards in word problems?

 Sometimes a student may need to work backward to solve a problem. Remind students to use the opposite operations during the backwards journey. Once they arrive at a solution, have them check their answer by using the operations forwards again. 7. Interpret absolute value inequality statements in LearnZillion real world contexts. E.g. may represent a debt  Understand positive and negative numbers in greater than 10 dollars. real life situations

6.NS.7

 How can we represent inequalities that have absolute value in them?

 Remind students that just because a negative number is inside the absolute value symbol, they must still understand the context of where the negative comes from, i.e. “the debt is greater than 10 dollars” means that the distance from zero to the amount is more than 10 dollars in the negative direction. 8. Graph ordered pairs of integers on all four enVisionMATH, “Coordinate Geometry” quadrants of the coordinate plane.  Lessons: 8-1 From SCUSD Resource Development: 6.NS.6  Amusement Park 6.NS.8 From Internet for Classrooms:  How do we represent the four quadrants of  Coordinate Planes the coordinate plane? I recommend using:  What is the origin? Billy Bug (quadrant I only)  How do we use graphing integers in real- Billy Bug 2 (all four Qs) world contexts? Catch the Fly (all four Qs) From IXL.com:  Coordinate Graphs Review  Graph Points on a Coordinate Plane  Coordinate Graphs as Maps  Make sure to remind your students of what the signs are for each quadrant (I +/+, II -/+, III -/-, IV +/-). This will help them make sense of the placement of objects on the coordinate plane.  Remind them about absolute value and the effect it has on adding/subtracting to find the distance between two ordered pairs. 9. Graph ordered pairs of rational numbers on all enVisionMATH, “Coordinate Geometry” four quadrants of the coordinate plane.  Lesson 8-2 6.NS.6 Inside Mathematics: 6.NS.8  Percent Cards Page 64 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 3: The System of Rational Numbers (Approx. 3-4 weeks)  How does graphing rational numbers differ LearnZillion: from graphing integers?  Planning a Town  Howard County Public School System  Pre-Post Assessment 6.NS.8 From Internet for Classrooms:  Coordinate Planes I recommend using: Graphing Applet Graphing Ordered Pairs  Recognize characteristics of coordinate pairs in each of the four quadrants, e.g. (-2, 0.75) must be in quadrant II because the x-coordinate is negative and the y-coordinate is positive.  Recognize coordinate points with the same x-values and opposite y-values are reflections across the x- axis. Recognize that coordinate points with the same y-coordinate values and opposite x-values are reflections across the y-axis. 10. Find distance on the coordinate plane using enVisionMATH, “Coordinate Geometry” absolute value.  Lessons: 8-3 6.NS.8 IXL.com  Why is distance always positive?  Distance Between Two Points (no conceptual  Why do we use absolute value to find understanding - only practice problems) distance?  Coordinates could also be in two quadrants. For example, the distance between (3, –5) and (3, 7) would be 12 units. This would be a vertical line since the x-coordinates are the same. The distance can be found by using a number line to count from –5 to 7 or by recognizing that the distance (absolute value) from –5 to 0 is 5 units and the distance (absolute value) from 0 to 7 is 7 units so the total distance would be 5 + 7 or 12 units.

11. Find the perimeter of a polygon located on a enVisionMATH, “Coordinate Geometry” coordinate plane.  Lessons: 8-4 6.NS.8 Illustrative Mathematics: 6.G.3  Distance Between Points  Polygons in the Coordinate Plane  What other polygons can you make from GeoGebra triangles?  Geogebra  How can you combine (recomposing, as opposed 1. Start using - Geometry to decomposing) polygons to figure out the areas 2. Use the toggle (triangle-circle) bar on the of more complex shapes? right to bring up the coordinate plane and grid 3. Start creating Internet for Classrooms:  Coordinate Planes Spy Guys Interactive Page 65 Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 3: The System of Rational Numbers (Approx. 3-4 weeks) LearnZillion:  Create polygons using coordinate plane

 Use the distance formula multiple times to find the lengths of each of the sides of the polygon and add them together.

12. Graph a linear equation using variables for both enVisionMATH, “Coordinate Geometry” dependent and independent variables.  Lessons: 8-5 6.EE.9 From Illustrative Mathematics:  How can we represent and interpret  Chocolate Bar Sales independent and dependent points on the From Inside Mathematics: coordinate plane?  Gym  What kinds of representations (models) can we make for the relationship between independent and dependent variables? NOTE: Remember that the objectives of the enVisionMATH lessons in Topic 3 focused on the use of tables and equations. These lessons did not address the full content of the standard 6.EE.9. These lessons is Topic 8 do fully address 6.EE.9  The use of multiple representations simultaneously is key here. Translating between multiple representations helps students understand that each form represents the same relationship and provides different perspective on the relationship.

 Outcomes 12 & 13 may be taught concurrently rather than as distinct learning experiences in order to relate the situation, table, graph, and equation of a given real-world problem.  Create a table of values to represent a real-world situation with independent and dependent variables and represent the relationship with a list of values for the independent variable and corresponding values for the dependent variable.

 Analyze a real-world situation for the purpose of identifying the two quantities that change in relationship to one another, defining them with variables, and determining which variable is dependent upon the other variable in the relationship (dependent and independent variables).

13. Graph a linear relationship involving two enVisionMATH, “Coordinate Geometry” relationships between two variables.  Lessons: 8-6 NOTE: Remember that the objectives of the

Grade 6 Mathematics SEQUENCE OF INSTRUCTION Unit 3: The System of Rational Numbers (Approx. 3-4 weeks) 6.EE.9 enVisionMATH lessons in Topic 3 focused on the use  What does it mean for one quantity to of tables and equations. Those lessons did not depend on another? address the full content of the standard 6.EE.9.  Why can we have two relationships between These lessons is Topic 8 do fully address 6.EE.9 two variables? Can we have more than two? (lead toward 7th grade standards)

 Represent a series of values for independent (x axis) and corresponding dependent variables (y axis) on quadrant 1 of a coordinate plane and determine whether the points should be discrete or continuous based on the context of the problem.

 Model the relationship between independent and dependent variables by creating a table, graphing the coordinates, and analyzing the relationship between the two variables in the table and graph in order to write the associated equation.

Additional Resources for Unit 3

Teaching Student-Centered Mathematics Developmentally Appropriate Instruction for Grades 6 – 8  Chapter 10 “The Number System”, pages 170-197.

Inside Mathematics  percent cards.pdf - Performance Task This task provides an opportunity for students to reason about equivalence and equations 6.NS.6 and 6.NS.6c

Georgia Department of Education  6th-Math-Unit-7.pdf Addresses 6.NS.5 through 6.NS.8

EngageNY  Grade 6 Mathematics Module 3 | EngageNY

Erie 1 BOCES- Deep Curriculum Alignment Project for Mathematics  The Number System - Examples Sample tasks for each standard in the domain 6.NS.

Grade 6 Mathematics