DRAFT Curriculum Map

Grade 6 Mathematics

Vision for Assessment and InstructionAs 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 6, 2023________________________________________________________________________________________________

__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 2014-15The 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 QuestionsThought 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 LearningDiagnostic, 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 OutcomesSequence 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 LearningInstructional strategies that lead to desired learning outcomes.

Differentiation, e.g. EL, SpEd, GATEAn illustrative list of high-leverage strategies to meet the needs of all studentsStrategies for differentiation for special populations – To be completed by district experts)

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

Page 1Grade 6 Mathematics

GRADE 6 MATHEMATICS: Year-at-a-Glance

Month Units

Content Standards(Bold =Major

Standards)

District Bench-marks

Sept/Oct

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 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.

enVisionMATH, Topics 1-3

6.EE.16.EE.2a6.EE.2b6.EE.2c6.EE.36.EE.46.EE.56.EE.66.EE.76.EE.86.EE.96.NS.4

BM #1 Standards

to be assessed: 6.EE.1-9(BM #1 to be given between

Nov. 9 and Dec. 4)

BM #2 Standards

to be addressed6.NS.1-8

(BM#2 to be given

between Feb 22 and

Mar. 18)

Nov/Dec/Jan

Unit 2: Computation with Multi-digit Numbers, Decimals, and Fractions

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.

enVisionMATH, Topics 4-6

6.NS.16.NS.26.NS.36.NS.4

Jan/Feb

Unit 3: The System of Rational Numbers

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.

enVisionMATH, Topics 7 and 8

6.NS.56.NS.6a,b,c6.NS.7a,b6.NS.7c,d

6.NS.86.G.36.EE.9

Feb/March

Unit 4: Ratios and Proportional Relationships

Students will develop understandings of ratios and unit rates. They will connect ratio, rate, and percentage to whole-number multiplication and division and

will use concepts of ratios and 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.

6.RP.16.RP.2

6.RP.3a6.RP.3b6.RP.3c6.RP.3d6.EE.9

2

GRADE 6 MATHEMATICS: Year-at-a-Glance

Month Units

Content Standards(Bold =Major

Standards)

District Bench-marks

enVisionMATH, Topics 9-11

March/April

Unit 5: Geometry

Students will solve real-world and mathematical problems involving area, surface area, and volume.

enVisionMATH, Topics 12, 13

6.G.16.G.26.G.4

BM #3 (Required 1st grade, Optional 2nd – 6th)Standards

to be assessed

6.RP.1-36.G.1-4

BM#3 to be given

between May 9 – June 3)

May/June

Unit 6: Statistics and Probability

Students will develop understanding of statistical thinking. They will work with statistical variability and will represent and analyze data distributions. Students

will focus on the characterization of data distributions by measures of center and spread. They will learn to describe and summarize data sets. Students will

develop the ability to use statistical reasoning.

enVisionMATH, Topic 14

6.SP.16.SP.26.SP.36.SP.4

6.SP.5a,b,c,d

3

OVERVIEWUnit 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:

4

OVERVIEWUnit 1: Expressions and Equations

(Approx. 8 weeks)Documents to Better Understand the Standards in this UnitCA 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 programsCA Mathematics Framework s “Grade 6”

pages 30 – 37, Expressions and Equations

Kansas Association of Teachers of Mathematics (KATM) 6th FlipbookProvides illustrated examples, instructional strategies, additional resources/tools and misconceptions by standard.

pages 48 - 67, Expressions and Equations

Progressions Document for the CCSMNarrative 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.

pages 29 – 42, Expressions and EquationsContent StandardsCommon Core State Standards–MathematicsExpressions and Equations 6.EEApply 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.

5

OVERVIEWUnit 1: Expressions and Equations

(Approx. 8 weeks)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)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.NSCompute 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 PracticeSMP 1 Make sense of problems and persevere in solving them. Students seek the meaning of a problem and look for efficient ways to represent and solve it using algebraic expressions and equations. Particularly in their work with 6.EE.9 when students make sense of real-world problems and the relationships between verbal descriptions, graphs, tables, and equations.

SMP 2 Reason abstractly and quantitatively. Students represent a wide variety of real-world contexts by using rational numbers and variables in mathematical expressions, equations, and inequalities. Students contextualize to understand the meaning of the number or variable as related to the problem 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 of others. Students construct arguments with verbal or written explanations accompanied by expressions, equations, inequalities, models, graphs, and tables.

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.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 mathematical communication skills by using clear and precise

SMP 7 Look for and make use of structure. Students apply properties to generate equivalent expressions and to solve

6

OVERVIEWUnit 1: Expressions and Equations

(Approx. 8 weeks)language in their discussions with others and in their own reasoning. In this unit specifically students focus on use of vocabulary to describe parts of algebraic expressions – see 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”.

equations. Students develop an understanding of the structure of expressions and use this structure to simplify and evaluate expressions.

ELD Standards to Support UnitPart I: Interacting in Meaningful WaysA. 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 WorksB. 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-awarenessSelf-managementSocial awarenessRelationship skillsResponsible decision making

7

OVERVIEWUnit 1: Expressions and Equations

(Approx. 8 weeks)

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

“Universal Design for Learning ” from CAST, the Center for Applied Special Technology

Use of Formative Assessment Process including actionable feedback.

Use of math journals for differentiation and formative assessment (use link below) https://www.teachingchannel.org/videos/math-journals

Flexible grouping:1. Content2. Interest3. Project/product4. Level (Heterogeneous/ Homogeneous)

Tiered: Independent Management Plan (Must

Do/May Do) Grouping

o Contento Rigor w/in the concepto Project-based learningo Homeworko Grouping

Anchor Activities:1. Content-related2. Tasks for early finishers

1. Game2. Investigation3. Partner Activity4. Stations

Depth and Complexity Prompts/Icons:1. Depth

o Language of the Disciplineo Patternso Unanswered Questionso Ruleso Trendso Big Ideaso Complexity

8

SEQUENCE OF INSTRUCTIONUnit 1: Expressions and Equations

(Approx. 8 weeks)Sequence of Learning Outcomes

As a result of the learning experience, students will be able to…in order to…

ResourcesIntended 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.

6.EE.1

What is the meaning of exponents? How are exponential expressions evaluated?

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 materials in Topic 1 include only expressions with a whole-number as the base.

enVisionMATH: Lesson 1-1

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 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.

Compare and contrast the meaning of 4(3) as the sum of 4 terms of 3 (3 + 3 + 3 + 3) and 43 as the product of 3 factors of 4 (4 x 4 x 4) – (emphasize the distinction - repeated addition for multiplication vs. exponents for repeated multiplication).

2. Write expressions using numbers and variables in order to model operations.

6.EE.2a 6.EE.6

How can operations be represented using

enVisionMATH: Lesson 1-6

Math Worksheets Land: write expressions from phrases - matching

worksheet with 7 pairs to match write expressions from phrases

worksheet with 10 phrases

9

SEQUENCE OF INSTRUCTIONUnit 1: Expressions and Equations

(Approx. 8 weeks)numbers and variables?

How can real-world situations be modeled using algebraic expressions?

How do different forms of notation help to make algebraic expressions clear and concise?

Using Variables to Represent Numbers - matching

5 real-world scenarios Using Variables to Represent Numbers

10 real-world scenariosIllustrative 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, 3x instead of 3 x x which is potentially confusing.

Students also learn that x ÷ 2 can be written as x2 . 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 and inequalities) to model real-world situations.

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 structure of expressions.

66.EE.2b 6.EE.6

How can we identify and describe the parts of an algebraic expression?

What is the usefulness of being able to describe parts of algebraic expressions?

enVisionMATH: Lesson 1-7

Mathematical vocabulary used by students to describe expressions should include the following: term, constant, coefficient, variable, sum, difference, factor, product, and quotient.

Decompose each part of an expression into its base parts and recompose where appropriate.10

SEQUENCE OF INSTRUCTIONUnit 1: Expressions and Equations

(Approx. 8 weeks) For example, describe the expression 2(8+7) as a product of two factors (2 and 15); view (8+7) as both a single entity and a sum of two terms.

Students should consider expressions similar to the following example from the CA Framework.

4. Perform arithmetic operations, including those involving whole number exponents, in the conventional order. (Order of Operations)

6.EE.2c

How do we determine the order of operations to be used when simplifying or evaluating expressions?

Why do we have a conventional Order of Operations?

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 Topic 1 include only expressions with a whole-number as the base.

enVisionMATH: Lesson 1- 3

Georgia Department of Education: Rules for Exponents

This task incorporates the use of technology ( basic and scientific calculators) to facilitate students’ exploration of the rules regarding operations with exponents.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 Progressions document.)

5. Perform arithmetic operations in order to evaluate expressions at specific values of their variables, including expressions that arise from real-

enVisionMATH: Lessons: 1-5 and 1-8

11

SEQUENCE OF INSTRUCTIONUnit 1: Expressions and Equations

(Approx. 8 weeks)world problems.

6.EE.2c6.EE.6

What does it mean to evaluate an algebraic expression?

How do the properties of operations apply to variables?

Georgia Department of Education: Writing Expressions & Writing and Evaluating

Expressions

Howard County Public School SystemMy 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)

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 different values; this will support the later development of the concept of a function.

6. Apply the properties of operations in order to generate equivalent expressions.

6.EE.3

How can equivalent algebraic expressions be generated?

enVisionMATH, Lessons: 1-10 and 1-11

Georgia Department of Education: Conjectures About Properties The Magic of Algebra parts 1, 2, & 3

Series of three lessons explores patterns and “tricks” algebraically.

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

Common Core Sheets: Applying Properties of Operations

12

SEQUENCE OF INSTRUCTIONUnit 1: Expressions and Equations

(Approx. 8 weeks)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) Visual Model of Distributive Property

Students should generate equivalent expressions using the associative, commutative, and distributive properties.

Provide opportunities for students to develop an understanding of why an expression like 3n + 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 non-equivalence.

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

Apply distributive property by using a visual model video from LearnZillion.7. Use substitution as well as the properties of operations in order to identify when two expressions are equivalent.

6.EE.4 What is equivalence? How can we prove that two expressions are

equivalent?

enVisionMATH, Lessons: 1-12

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: 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 13

SEQUENCE OF INSTRUCTIONUnit 1: Expressions and Equations

(Approx. 8 weeks) 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.

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.

8. Use tables and algebraic expressions in order to solve problems.

6.EE.2a6.EE.2c6.EE.6

How can algebraic expressions be used to describe patterns?

How can tables and algebraic expressions be used together to solve problems?

enVisionMATH, Lessons: 1-9 and 1-13

Georgia Department of Education: Visual Patterns

“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

enVisionMATH, Lesson: 2-1

14

SEQUENCE OF INSTRUCTIONUnit 1: Expressions and Equations

(Approx. 8 weeks)specified set, if any, make the equation true?” Use substitution to determine whether a given number is a solution.

6.EE.5 What does it mean to solve an equation? How can we determine whether or not a

given value is a solution?

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 x + 21 = 32 students know that 21 + 9 = 30 therefore the solution must be 2 more than 9 or 11, so x = 11.

Use of open sentences (i.e. equations with a box to be filled or letter to be replaced e.g. 3.4 + 5 = 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.

Refer to Teaching Student-Centered Mathematics 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.)

10. 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 non-negative rational numbers.

6.EE.7 6.EE.6

How can algebraic expressions and equations be used to solve 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?

enVisionMATH, Lessons: 2-3, 2-4, 2-5, 2-6, and 2-9

Math Worksheets Land: Solve real-world problems

Worksheet with 10 scenarios. Solving equations with fractions and

decimals – worksheet 1 Solving equations with fractions and

decimals - worksheet 2Illustrative Mathematics:

Morning Walk Firefighter Allocation

LearnZillion videos-Solving equations using tape diagrams/bar modeling:

Addition : Subtraction: Multiplication: DivisionEach 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.

15

SEQUENCE OF INSTRUCTIONUnit 1: Expressions and Equations

(Approx. 8 weeks) 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. When solving equations of the form px = q students should use both division by p and

multiplication by the reciprocal ( 1p ).

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.

Problems should begin with whole-number values for p and q and should progress to include positive fraction and decimal values. Solutions should not require the use of operations with negative numbers.

Use of substitution to check a solution reinforces understanding of 6.EE.5. 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.) 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” but that it represents a value e.g. “t is the number of minutes”.

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

6.EE.8 6.EE.6

How can we represent real-world constraints and conditions algebraically?

NOTE: 6.EE.8 specifies that students should work with inequalities of the forms x > c and x < c. It is not inappropriate to introduce and include inequalities of the forms x ≥ c and x ≤ c .These forms can be used to support students’ development of conceptual understanding but should not assessed for grading purposes.

enVisionMATH, Lesson: 2-7

NOTE: The 6th grade standard 6.EE.8 is to work with the forms x < c and x > c. Lesson 2-7 also includes ≤, ≥, and ≠.

Common Core Sheets: Writing Inequalities

Writing from mathematical statements.Opus:

6.EE.8: Write an inequality of the form x > c or x < c to represent...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).

16

SEQUENCE OF INSTRUCTIONUnit 1: Expressions and Equations

(Approx. 8 weeks) 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

a < x < c. (Note: ≤ and or ≥ may also be included in these situations.) 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. 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 substitution to determine whether a given number in a specified set is a solution.

6.EE.86.EE.5

How can we represent and interpret solutions of inequalities?

NOTE: 6.EE.8 specifies that students should work with inequalities of the forms x > c and x < c. It is not inappropriate to introduce and include inequalities of the forms x ≥ c and x ≤ c .These forms can be used to support students’ development of conceptual understanding but should not assessed for grading purposes.

enVisionMATH, Lesson: 2-8

Math Worksheets Land: Solving Equations and Inequalities -

Matching

Solving Equations and Inequalities

Common Core Sheets: Expressing Inequalities on a Number line Matching Inequalities to Number lines Writing Inequalities from a Number line

Illustrative Mathematics: Fishing Adventures

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.

Students should recognize that inequalities of the form x > c and x < c 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 a < x < c. (Note: ≤ and or ≥ may also be included in these situations.)

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

17

SEQUENCE OF INSTRUCTIONUnit 1: Expressions and Equations

(Approx. 8 weeks)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.

13. Identify independent and dependent variables.

6.EE.9 What does it mean for one quantity to

depend on another?

enVisionMATH, Lesson: 3-1

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 enVisionMATH, Lessons: 3-2, 3-3, and 3-418

SEQUENCE OF INSTRUCTIONUnit 1: Expressions and Equations

(Approx. 8 weeks)real-world problem that change in relation to one another; write an equation to represent the relationship; analyze the relationship using graphs and tables and relate these to the equation.

6.EE.9

How can we represent relationships between quantities?

How do different representations help us analyze relationships between quantities?

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.

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

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.

Consider the following example from pages 36 and 37 of Grade 6 CA Framework . 19

SEQUENCE OF INSTRUCTIONUnit 1: Expressions and Equations

(Approx. 8 weeks)

20

SEQUENCE OF INSTRUCTIONUnit 1: Expressions and Equations

(Approx. 8 weeks)Additional Resources for Unit 1

Teaching Student-Centered Mathematics Developmentally Appropriate Instruction for Grades 6 – 8 Chapter 12 “Exploring Algebraic Thinking, Expressions, and Equations”, pages 222 – 259.

Inside Mathematics boxes.pdf - Performance Task This task provides an opportunity for students to reason about

equivalence and equations 6.EE.4 and 6.EE.5.

Achieve the Core extending-previous-understandings-of-properties-mini-assessment This mini-assessment is

designed to illustrate the cluster 6.EE.1 through 6.EE.4.

Georgia Department of Education 6th-Math-Unit-3.pdf Addresses 6.EE.1 through 6.EE.4. 6th-Math-Unit-4.pdf Addresses 6.EE.5 through 6.EE.9 and 6.RP.3.

EngageNY Grade 6 Mathematics Module 4 | EngageNY

Erie 1 BOCES- Deep Curriculum Alignment Project for Mathematics Expressions and Equations Samples Sample tasks for each standard in the domain 6.EE.

OVERVIEWUnit 2: Computation with Multi-digit Numbers, Decimals, and Fractions

21

(Approx. 7-8 weeks)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.

.

Over-Arching Essential Questions for Unit 2: How do we decompose numbers when they have fractions/decimals in them? How do we use factors and multiples to help us in computations with fractions and decimals? How do we use estimation to assist in computations using fractions/decimals? How are whole number and fraction division similar?

In this unit students will: Decompose whole numbers into their factors, both prime and composite. Find the Greatest Common Factor and Lowest Common Multiple of two or more numbers. 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. Find common denominators for easier calculation of fractions/decimals. Find the denominator of decimal when converting to fractions.

Culminating Task:

Documents to Better Understand the Standards in this UnitCA Mathematics Framework

“Instructional Strategies” chapter provides research-based strategies for teaching math, K-12

22

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

CA Mathematics Framework s “Grade 6” pages 17 – 26, The Number Systems

Kansas Association of Teachers of Mathematics (KATM) 6th FlipbookProvides illustrated examples, instructional strategies, additional resources/tools and misconceptions by standard.

pages 25-38, Number Sense

Progressions Document for the CCSMNarrative 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

North Carolina Unpacked Standards 6 th Grade Provides illustrated examples, instructional strategies, additional resources/tools and misconceptions by standard.

pages 14 – 22, The Number SystemContent StandardsCommon Core State Standards–MathematicsThe Number System 6.NS

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.1. 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 model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (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 land with length 3/4 mi and area 1/2 square mi?

Compute fluently with multi-digit numbers and find common factors and multiples.2. Fluently divide multi-digit numbers using the standard algorithm.3. 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. For example, express 36 + 8 as 4 (9 + 2).

Standards for Mathematical PracticeSMP 1 Make sense of problems and persevere in solving them. Students seek the meaning of a problem and look for efficient ways to represent and solve it using fractions and decimals. Particularly 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

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?”

23

fractions/decimals.SMP 3 Construct viable arguments and critique the reasoning of others. Students construct arguments with verbal or written explanations accompanied by numbers including fractions/decimals with and without whole numbers.

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.

SMP 6 Attend to precision. 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 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 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.

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 fractions and generate the “multiply by the reciprocal” when dividing by a fraction.

ELD Standards to Support UnitPart I: Interacting in Meaningful WaysB. 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 WorksB. 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-awarenessSelf-managementSocial awarenessRelationship skillsResponsible decision making

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

“Universal Design for Learning ” from CAST, the Center for Applied Special Technology

Use of Formative Assessment Process including actionable feedback.

Use of math journals for differentiation and formative assessment (use link below) https://www.teachingchannel.org/videos/math-journals

Flexible grouping:5. Content6. Interest7. Project/product8. Level (Heterogeneous/ Homogeneous)

Tiered: Independent Management Plan (Must

Do/May Do) Grouping

o Contento Rigor w/in the concepto Project-based learningo Homeworko Grouping

Anchor Activities:3. Content-related4. Tasks for early finishers

5. Game6. Investigation7. Partner Activity8. Stations

Depth and Complexity Prompts/Icons:2. Depth

o Language of the Disciplineo Patternso Unanswered Questionso Ruleso Trendso Big Ideaso Complexity

SEQUENCE OF INSTRUCTIONUnit 2: Computation with Multi-digit Numbers, Decimals, and Fractions

(Approx. 7-8 weeks)Sequence of Learning Outcomes

As a result of the learning experience, students will be able to…in order to…

ResourcesIntended to be used in/during class for teaching and learning

Strategies for Teaching and Learning

1. 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

enVisionMATH: ”Adding, Subtracting and Multiplying Decimals”

Lesson 4-1, 4-2, 4-3

Math Goodies Adding and Subtracting Decimals

25

(e.g. 6.32 =

6+32100

=6+30100

+2100

=

6+310

+2100

)

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.

6.NS.3

What are the multiple ways to decompose 3.125 as a fraction?

How do you know what the denominator is when writing a decimal number as a fraction?

Math Work Sheets Land Adding Decimals

EngageNY Use topics 6-15 for 6.NS.3

SCUSD Resource Development 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.

6.NS.3

How do you know what the denominator is when writing a decimal number as a fraction?

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?

enVisionMATH: ”Adding, Subtracting and Multiplying Decimals”

Lesson 4-1, 4-2, 4-3Illustrative Mathematics:

“Jayden’s Snacks”

When working with decimals or fractions, be sure to include problems with:o Same terminating place value decimal without re-grouping (e.g. 6.32 + 3.15) or borrowing,o Different terminating place value without regrouping (6.3 + 3.561)o Same and different terminating place value with regrouping (6.79 + 3.54 and 6.3 – 4.83)

Make connections to strategies for addition and subtraction with multi-digit whole numbers. When performing all operations with decimals and fractions, incorporate estimation into the

experience throughout the problem. Decompose numbers into expanded form

SCUSD Resource Development: Using Place Value to Add and Subtract Decimals Changing decimals and fraction addition and subtraction into multi-digit whole number addition

26

and subtraction Adding Up

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 the exact answer.

6.NS.3

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)

enVisionMATH: ”Adding, Subtracting and Multiplying Decimals”

Lesson 4-4

Illustrative Mathematics: Reasoning about Multiplication and Division

and Place Value, Part I

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.SCUSD Resource Development

Multiplying Decimals 4. Write decimals as fractions and multiply, using the denominator of the product to determine place value.

6.NS.3

How do you know what the denominator is when writing a decimal number as a fraction?

For multiplication, why is finding a common denominator not helpful?

enVisionMATH: ”Adding, Subtracting and Multiplying Decimals”

Lesson 4-5

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

5. Divide multi-digit whole numbers division recognizing place value throughout the process.

6.NS.2

Which method for dividing multi-digit whole numbers do you prefer – “scaffolded” or “stacking” –and why?

enVisionMATH: “Dividing Whole Numbers and Decimals”

Lesson 5-1, 5-2, 5-3Illustrative Mathematics

Baking Cookies Price per Pound and Pound per Dollar Reasoning about Multiplication and

Division and Place Value Part I The Florist Shop

SCUSD Resource Development Scaffold Division

Show Me.com Long Division Scaffolded

YouTube Partial Quotients (Stacked for Division)

Students must use all three methods to divide including:o Stacked long division – stacked division is a method of using repeated subtraction for division

27

which connects place value to the standards algorithm.o Scaffold long divisiono division algorithm

28

Recognize the connection between division and repeated subtraction. Write quotients as mixed numbers where appropriate. Reinforce place value whenever possible.6. Divide a decimal number by a whole number where the quotient is a decimal (e.g. 56.58÷3=18.86 ¿. Divide decimal by a decimal number where the quotient is a decimal (e.g. 3.3÷1.2=2.75

6.NS.3 How do we temporarily remove the

decimal point when doing division? How do we re-establish the decimal point

(place value)? Why are we allowed to add zeroes to the

end of a number to the right of the decimal?

Why can’t we add zeroes to the left of the decimal?

enVisionMATH: “Dividing Whole Numbers and Decimals”

Lessons: 5-4Illustrative Mathematics

Grandma’s Gifts EngageNY

Dividing Decimals (with adding zeroes) SCUSD - student examples of:

Dividing decimals

All decimals should be terminating. Make sure to include stacked division, scaffolded division, as well as the traditional algorithm. Pay close attention to place value. Verify that extending the place value by adding zeroes does not change the value of the dividend.7. 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, completing the division, and then dividing by the same power of 10 in order to maintain identity. Example:

2.4÷6=( 2.46 × 101 )× 110=246 × 110=4× 110= 410

enVisionMATH: “Dividing Whole Numbers and Decimals”

Lessons: 5-5Illustrative Mathematics

Reasoning Around Multiplication and Division and Place Value Part II

29

6.NS.3

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?

Multiply by powers of 10 to make the divisor a whole number. 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.SCUSD Resource Development

Dividing a Decimal by a Whole Number 8. Divide a decimal number by a decimal number where the quotient is a decimal (e.g. 16.728÷3.4=4.92. Divide multi-digit decimals with different terminating place values by writing as two fractions, finding a common denominator, then dividing straight across in order to perform long division with whole numbers.

6.NS.3

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

How are dividing with whole numbers and dividing with decimals similar and different?

How can dividing with decimals be made easier by converting to fractions?

enVisionMATH: “Dividing Whole Numbers and Decimals”

Lessons: 5-5Illustrative Mathematics

Buying Gas

SCUSD Resource Development Examples of Dividing Multi-digit Decimals With Whole Number Quotients

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 common factor of two numbers (1-100).

6.NS.4

How does finding all the prime factors of two numbers help to find all common factors including the GCF?

enVisionMATH: “Dividing Fractions” Lesson 6-1

Illustrative Mathematics: Adding Multiples

SCUSD Resource Development Find Common Factors

Common Core Sheets Greatest Common Factor

30

Students should use the greatest common factor to solve real world problems. Students use all prime factors to be able to list all factor pairs. Students should be reminded of using the distributive property to demonstrate factoring 2 (4×3 )=8×6=48

SCUSD Resource Development Strategy for Using Primes to find the GCF Strategy for Finding Factor Pairs Through Prime Factorization

*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?

Solution: 2 boxes of 16 and 3 boxes of 16 for a total of 5 boxes of 16 cupcakes.

10. Find the least common multiple of two numbers (1-12).

6.NS.4 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?

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?

How does finding the GCF of two numbers lead you to find the LCM of two numbers?

enVisionMATH, “Dividing Fractions” Lessons: 6-2

Illustrative Mathematics: Bake Sale

Mathematics Assessment Resource Service (MARS)

Factors and Multiples Common Core Sheets

Least Common Multiple

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.

31

11. Divide fractions with common denominators with models.

6.NS.1

Why is finding a common denominator helpful for dividing fractions by fractions?

How do you know what the denominator is when writing a decimal number as a fraction?

enVisionMATH, “Dividing Fractions” 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 Part I 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:

Students may divide straight across when denominators are already equivalent:SCUSD Resource Development

Strategies for Dividing Fractions

12. Divide fractions without common denominators with models to find a common denominator and to show that division means how many of one quantity goes into another.

6.NS.1

Why is finding a common denominator helpful for dividing fractions by fractions?

Why and when do we use common denominators with addition, subtraction, and division?

enVisionMATH, “Dividing Fractions” Lessons: 6-8

Illustrative Mathematics Baking Cookies Price per Pound and Pound per Dollar Reasoning about Multiplication and

Division and Place Value Part I 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) :

32

SCUSD Resource Development Strategies for Dividing Fractions

13. Divide fractions by fractions - straight across.

6.NS.1

Why is finding a common denominator helpful for dividing fractions by fractions?

Why and when do we use common denominators with addition, subtraction, and division?

enVisionMATH, “Dividing Fractions” Lessons: 6-5

Students must find a common denominator(s) to divide straight acrossSCUSD Resource Development

Strategies for Dividing Fractions 14. Analyze different cases of dividing fractions by fractions to generalize that one can multiply by the reciprocal.

6.NS.1

When can you multiply by the reciprocal when dividing fractions?

enVisionMATH, “Dividing Fractions” Lessons: 6-6

“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 Framework p.21)

SCUSD Resource Development Strategies for Dividing Fractions

Additional Resources for Unit 2

Teaching Student-Centered Mathematics Developmentally Appropriate Instruction for Grades 6 – 8 Chapters 8 & 9, “Fraction Concepts and Computation” & Decimal Concepts and Computation“ ,

pages 104 – 167.

Achieve the Core Cup of Rice This mini-assessment is designed to illustrate the cluster 6.NS.1 through 6.NS.4.

33

Georgia Department of Education 6th-Math-Unit-1.pdf Addresses 6.NS.1 through 6.NS.4.

EngageNY Grade 6 Mathematics Module 2 | EngageNY

Erie 1 BOCES- Deep Curriculum Alignment Project for Mathematics Number System Examples - Sample tasks from all standards in 6.NS

OVERVIEWUnit 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

34

OVERVIEWUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)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.

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:

35

OVERVIEWUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)

Documents to Better Understand the Standards in this UnitCA 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 Framework s “Grade 6” pages 27-31, The Number System

Kansas Association of Teachers of Mathematics (KATM) 6th FlipbookProvides illustrated examples, instructional strategies, additional resources/tools and misconceptions by standard.

pages 39 - 47, Expressions and Equations

Progressions Document for the CCSMNarrative 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

North Carolina Unpacked Standards 6 th Grade Provides illustrated examples, instructional strategies, additional resources/tools and misconceptions by standard.

pages 23 – 28, The Number SystemContent 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

36

OVERVIEWUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)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 PracticeSMP 1 Make sense of problems and persevere in solving them. Students seek the meaning of a problem and look for efficient ways to represent and solve problems using both vertical and horizontal number lines. They will understand the meaning of zero and opposites on a number line(s). Particularly in their work with 6.EE.9 when students make sense of real-world problems and the relationships between verbal descriptions, graphs, tables, and equations.

SMP 2 Reason abstractly and quantitatively. Students represent a wide variety of real-world contexts by using rational numbers and variables in mathematical expressions, equations, and inequalities. Students contextualize to understand the meaning of the number or variable as related to the problem and decontextualize to operate with symbolic representations. This is particularly true of students’ work with the standards 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 of others. Students construct arguments with verbal or written explanations accompanied by expressions, equations, inequalities, models, graphs, and tables.

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.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.

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OVERVIEWUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)SMP 6 Attend to precision. 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 unit specifically students focus on use of vocabulary (positive, negative, zero, absolute value, and magnitude) to describe numbers on a number line.

SMP 7 Look for and make use of structure. Students apply properties to generate number lines and graphs to solve equations. Students develop an understanding of vertical and horizontal number lines and subsequently, the coordinate plane. Students will use this for graphing, finding distance, and locating objects.

ELD Standards to Support UnitPart I: Interacting in Meaningful WaysC. 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 WorksB. 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-awarenessSelf-managementSocial awarenessRelationship skillsResponsible decision making

38

OVERVIEWUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)

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

“Universal Design for Learning ” from CAST, the Center for Applied Special Technology

Use of Formative Assessment Process including actionable feedback.

Use of math journals for differentiation and formative assessment (use link below) https://www.teachingchannel.org/videos/math-journals

Flexible grouping:9. Content10. Interest11. Project/product12. Level (Heterogeneous/ Homogeneous)

Tiered: Independent Management Plan (Must

Do/May Do) Grouping

o Contento Rigor w/in the concepto Project-based learningo Homeworko Grouping

Anchor Activities:5. Content-related6. Tasks for early finishers

9. Game10. Investigation11. Partner Activity12. Stations

Depth and Complexity Prompts/Icons:3. Depth

o Language of the Disciplineo Patternso Unanswered Questionso Ruleso Trendso Big Ideaso Complexity

39

SEQUENCE OF INSTRUCTIONUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)Sequence of Learning Outcomes

As a result of the learning experience, students will be able to…in order to…

ResourcesIntended to be used in/during class for teaching and learning

Strategies for Teaching and Learning

1) Make sense of positive and negative integers in context of a real-world situation, explaining the meaning of zero in each situation. Students should understand that positive and negative numbers as numbers on opposite sides of 0 on the number line, e.g. –(-3) as the opposite of (-3) otherwise known as 3. Graph opposites on a number line.

6.NS.56.NS.6a

What is the relationship between positive and negative numbers?

Is zero positive or negative? Explain. Where do we see negative numbers in the

real world?

enVisionMATH: ”Integers and Other Rational Numbers”

Lesson 7-1 Lesson 7-6

EngageNY Temperature - Exit Ticket pg. 41

LearnZillion Understanding Negative Numbers on a

Number Line Opposites

SCUSD Resource Development: Contest Winner

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

40

SEQUENCE OF INSTRUCTIONUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)

2. Compare and order integers on number lines and by using inequality symbols.

enVisionMATH: “Integers and Other Rational Numbers”

Lesson 7-241

SEQUENCE OF INSTRUCTIONUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)6.NS.7a 6.NS.7b

Why are bigger negative numbers really smaller?

Illustrative Mathematics: Mile High Warmer in Miami Integers on the Number Line I 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 zero (absolute value is the distance of a number from zero – and distance is always a positive

enVisionMATH: “Integers and Other Rational Numbers”

Lesson 7-3

42

SEQUENCE OF INSTRUCTIONUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)number). Understand absolute value as magnitude for a positive or negative quantity in a real-world situation, e.g. owing 30 dollars represents the magnitude of|−30|.

6.NS.7

What is absolute value? When is absolute value used in the real

world? What does it mean for numbers to be

opposites?

Lesson 7-1EngageNY

Absolute Value - Begins on p. 98From Illustrative Mathematics:

Comparing Temperatures Jumping Flea Above and Below Sea Level

CA Framework 6th Grade p. 31

4. Graph all different types of rational numbers on a number line, paying attention to the order of the numbers. Plot rational numbers on number lines, both horizontal and vertical, in context of a real world situation such as temperature or elevation.

6.NS.6c

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

a magnitude of a number?

enVisionMATH: “Integers and Other Rational Numbers”

Lesson 7-4Illustrative Mathematics:

Plotting Points in the Coordinate Plane Extending the Number Line

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.

43

SEQUENCE OF INSTRUCTIONUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)

5. Write, interpret, and explain inequality statements with rational numbers, using a number line to justify reasoning.

6.NS.7

Why are bigger negative numbers really smaller?

enVisionMATH: “Integers and Other Rational Numbers”

Lessons: 7-2LearnZillion

Comparing Rationals using inequalities Comparing Part II

From Common Core Toolbox: 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 problems using both positive and negative integers

enVisionMATH, “Integers and Other Rational Numbers”

44

SEQUENCE OF INSTRUCTIONUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)and rational numbers.

6.NS.5

When and why is it appropriate to work backwards in word problems?

Lessons: 7-6 LearnZillion

Understand Relationship Between Positive and Negative Numbers

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 real world contexts. E.g. x>|−10| may represent a debt greater than 10 dollars.

6.NS.7 How can we represent inequalities that have

absolute value in them?

LearnZillion Understand positive and negative numbers

in real life situations

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 quadrants of the coordinate plane.

6.NS.66.NS.8

How do we represent the four quadrants of the coordinate plane?

What is the origin? How do we use graphing integers in real-

world contexts?

enVisionMATH, “Coordinate Geometry” Lessons: 8-1

From SCUSD Resource Development: Amusement Park

From Internet for Classrooms: Coordinate Planes

I recommend using:Billy Bug (quadrant I only)Billy Bug 2 (all four Qs)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 four quadrants of the coordinate plane.

6.NS.6

enVisionMATH, “Coordinate Geometry” Lesson 8-2

Inside Mathematics:

45

SEQUENCE OF INSTRUCTIONUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)6.NS.8

How does graphing rational numbers differ from graphing integers?

Percent Cards LearnZillion:

Planning a Town Howard County Public School System

Pre-Post Assessment 6.NS.8 From Internet for Classrooms:

Coordinate Planes I recommend using:Graphing AppletGraphing 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 absolute value.

6.NS.8 Why is distance always positive? Why do we use absolute value to find

distance?

enVisionMATH, “Coordinate Geometry” Lessons: 8-3

IXL.com Distance Between Two Points (no

conceptual understanding - only practice problems)

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 coordinate plane.

6.NS.8 6.G.3

What other polygons can you make from triangles?

How can you combine (recomposing, as opposed to decomposing) polygons to figure out the areas of more complex shapes?

enVisionMATH, “Coordinate Geometry” Lessons: 8-4

Illustrative Mathematics: Distance Between Points Polygons in the Coordinate Plane

GeoGebra Geogebra

1. Start using - Geometry2. Use the toggle (triangle-circle) bar on the

right to bring up the coordinate plane and grid

3. Start creatingInternet for Classrooms:

Coordinate Planes

46

SEQUENCE OF INSTRUCTIONUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)Spy Guys Interactive

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 dependent and independent variables.

6.EE.9 How can we represent and interpret

independent and dependent points on the coordinate plane?

What kinds of representations (models) can we make for the relationship between independent and dependent variables?

enVisionMATH, “Coordinate Geometry” Lessons: 8-5

From Illustrative Mathematics: Chocolate Bar Sales

From Inside Mathematics: Gym

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 relationships between two variables.

6.EE.9 What does it mean for one quantity to

enVisionMATH, “Coordinate Geometry” Lessons: 8-6

NOTE: Remember that the objectives of the enVisionMATH lessons in Topic 3 focused on the use of tables and equations. Those lessons did not

47

SEQUENCE OF INSTRUCTIONUnit 3: The System of Rational Numbers

(Approx. 3-4 weeks)depend on another?

Why can we have two relationships between two variables? Can we have more than two? (lead toward 7th grade standards)

address the full content of the standard 6.EE.9.These lessons is Topic 8 do fully address 6.EE.9

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.

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