6th Grade Math Guide

Instructional Guide 2024-2025

Math

Grade

Instructional Guide 2024-2025

Introduction

What’s New and Updated in

6th grade math

What’s New

This section contains a listing of pages in the map that are new this year.

Page Number

Description

Added two different scope and sequence options for 6th grade. The units stay the same, but there are two scope and sequences. School PLC’s have the choice of option.

What’s Updated

This section contains a listing of pages in the page that have received substantial content updates for this year.

Description

Updated dates for all units

Grade 6

Math Overview

ORGANIZATION OF STANDARDS

The Utah Core Standards are organized into strands, which represent significant areas of learning within content areas. Depending on the core area, these strands may be designated by time periods, thematic principles, modes of practice, or other organizing principles. Within each strand are standards. A standard is an articulation of the demonstrated proficiency to be obtained. A standard represents an essential element of learning that is expected. While some standards within a strand may be more comprehensive than others, all standards are essential for mastery.

UNDERSTANDING MATHEMATICS

These standards define what students should understand and be able to do in their study of mathematics. Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness. The standards set grade-specific standards but do not dictate curriculum or teaching methods, nor do they define the intervention methods or materials necessary to support students who are well below or well above grade-level expectations. It is also beyond the scope of the Standards to define the full range of supports appropriate for English language learners and for students with special needs. At the same time, all students must have the opportunity to learn and meet the same high standards if they are to access the knowledge and skills necessary in their post-school lives. The standards should be read as allowing for the widest possible range of students to participate fully from the outset, along with appropriate accommodations to ensure maximum participation of students with special education needs. No set of grade-specific standards can fully reflect the great variety in abilities, needs, learning rates, and achievement levels of students in any given classroom. However, the standards do provide clear signposts along the way to the goal of college and career readiness for all students. What students can learn at any particular grade level depends upon what they have learned before. Ideally then, each standard in this document might have been phrased in the form, "Students who already know… should next come to learn ..." Grade placements for specific topics have been made on the basis of state and international comparisons and the collective experience and collective professional judgment of educators, researchers and mathematicians. Learning opportunities will

continue to vary across schools and school systems, and educators should make every effort to meet the needs of individual students based on their current understanding.

USBE Course Overview Grade 6 The purpose of this document is to provide a brief overview of the most essential content in the grade level along with a progression of how the content was addressed in the prior grade level and will prepare students for content in the future grade level. This is not a comprehensive list of content in the grade level, but rather highlights the major work of the grade level.

Major Work of Grade Band: Grades 6 - 8 ● Apply and use operations with rational numbers ● Understand ratio concepts and apply proportional reasoning ● Simplify expressions and solve equations ● Represent and analyze relationships

Major Work and Vertical Alignment

Major work: Operations with Rational Numbers Grade 6: Apply and extend understanding of operations with rational numbers : Apply previous understanding of all four operations with rational numbers (6.NS.1-3), with the extension of dividing fractions by fractions. Students are introduced to integers via opposite signs, value, and direction; number line models; and absolute value (6.NS.5-7). ● Prior grades : Students understand patterns in place value including decimals and powers of ten (5.NBT.1-3). Add, subtract, multiply and divide decimals to hundredths (5.NBT.7). Multiply a fraction or whole number by a fraction including real-world problems (5.NF.4,6). Divide unit fractions by whole numbers and whole numbers by unit fractions using reasoning about the relationship between multiplication and division (5.NF.7). Fluently multiply multi-digit whole numbers using the standard algorithm (5.NBT.5) and divide whole numbers with up to four-digit dividends and two-digit divisors (5.NBT.6). ● Future Grades : Students will apply previous understanding of operations with rational numbers to include integers in grade 7 (7.NS.1-3) and irrational numbers in grade 8 (8.NS.1-3). In Secondary Math II, students will expand the number system to include imaginary numbers (II.N.CN.1, 2, 7-9).

Major work: Ratio and Rate Reasoning Grade 6: Understand ratio concepts and apply proportional reasoning: Understand ratio concepts (6.RP.1) and understand the concept of unit rate (6.RP.2). Use multiple representations to solve

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ratio/rate problems (tables of equivalent ratios, equations, and plot values on a coordinate plane in all four quadrants) (6.RP.3). ● Prior Grades: In grades 4 and 5, students have created equivalent fractions (4.NF, 5.NF) and use equivalent fractions as a strategy to add and subtract fractions with unlike denominators including mixed numbers (5.NF.1–2). Fractions are interpreted as division of the numerator by the denominator (5.NF.3). Interpret multiplication as scaling (5.NF.5). ● Future Grades: In grade 7, students will recognize and represent proportional relationships between quantities using multiple representations (7.RP.1-2), and use proportional relationships to solve multi-step and percent problems (7.RP.3). In grade 8, students will extend their understanding of proportional relationships to linear equations, recognizing slope as the proportional relationship between quantities (8.EE.5) and that linear functions have a vertical shift of b units (8.EE.5-6, 8.F). In high school, students will identify functions based on rates of change (High School Functions standards). Major work: Simplify Expressions and Solve One Step Equations with one variable one step simple equations Grade 6: Simplify expressions and solve equations: Apply and extend previous understandings of arithmetic to using variables and generating equivalent algebraic expressions (6.EE.1-4). Reason about and, for the first time in their math education, formally solve simple one-variable equations and inequalities, for example: (x+q < r) (6.EE.5-8). ● Prior Grades: Students solve for unknown values starting in the early grades (K.OA.4, 1.OA.1, 2.OA.1, etc.), are introduced to equality and inequality symbols (1.NBT.3) and analyze patterns and relationships (5.OA.3). (1.NBT.3) students introduced to inequality symbols <, >, = ● Future Grades: In grade 7, students will apply properties of operations to factor, expand, and convert between forms and assess reasonableness of an answer (7.EE.1-3). Students will use variables to represent quantities to construct and solve simple equations and inequalities (for example: px+q < r ) (7.EE.4). In grade 8, students will solve complex linear equations and inequalities (8.EE.7). Students solve equations throughout high school and justify why solutions work (High School Algebra standards). Major work: Represent and Analyze Relationships Grade 6: Represent and analyze relationships: Solve simple problems using numerical and algebraic expressions (6.EE.5-8); represent and analyze quantitative relationships between dependent and independent variables and graph the relationship on a coordinate plane (6.NS.8, 6.EE.9).

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● Prior Grades: In grade 5, students will generate two numerical patterns using two given rules (5.OA.3). Students also understand concepts of geometric measurement and relate volume to multiplication and to addition (5.MD.3-5). ● Future Grades: In grade 7, students will solve problems using numerical and algebraic expressions (7.EE-4), draw references between two populations (7. SP.3-4), and investigate probability models (7. SP.5-8). Students will represent two variable relationships, compare quantities, and analyze relationships throughout their mathematics career. In grade 8 and high school, students will continue to study and compare how multiple quantities interact and relate in all strands.

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Instructional Guide 2024-2025

Scope and Sequence

Grade 6

YEAR AT A GLANCE Option 1

Illustrative Unit

Unit 1 Area and

Unit 2 Introducing Ratios (17 sections)

Unit 3 Unit Rates and Percentages (17 sections)

Unit 4 Dividing Fractions (17 sections)

Unit 5 Arithmetic in Base Ten (15 sections)

Unit 6 Expressions and Equations (19 sections)

Unit 7 Rational Numbers (19 sections)

Unit 8 Data Sets and Distributions (18 sections)

Unit 9 Putting it All Together

Surface Area (19 sections)

Suggested Pacing

Aug 19 - Sept 26 (28 days)

Sept 30 - Oct 30 (20 days)

Oct 31 - Nov 26 (19 days)

Dec 2 - Jan 10 (20 days)

Jan 13 - Feb 7 (18 days)

Feb 10 - Mar 12 (21 days)

Mar 13 - Apr 18 (21 days)

Apr 21 - May 16 (20 days)

May 17 - May 30

6.G.1 6.G.2 6.G.4 6.EE.1 6.EE.2

6.RP.1 6.RP.2 6.RP.3

6.RP.2 6.RP.3

6.NS.1 6.G.3

6.NS.2 6.NS.3

6.EE.1 6.EE.2 6.EE.3 6.EE.4 6.EE.5 6.EE.6 6.EE.7 6.EE.9

6.EE.8 6.NS.4 6.NS.5 6.NS.6 6.NS.7 6.NS.8

6.SP.1 6.SP.2 6.SP.3 6.SP.4 6.SP.5

6.G 6.RP 6.NS

Practice Standards

Practice Standards

Practice Standards

Practice Standards

Practice Standards

Standards

Practice Standards

Practice Standards

Practice Standards

Practice Standards

These standards will be assessed along with all other standards on the RISE

DWSBA & Testing Window

DWSBA #1

DWSBA #2

DWSBA #3

Aug 19 - Sept 25

Dec 4 - Jan 16

MAP Window

Apr 1 - May 9

SALTA Extensions

“Are You Ready for More?”

Accessing the District-Wide Standards-Based Assessment (DWSBA)

The DWSBA’s will be done through Canvas on Derivita Instructions to access the DWSBA can be found here.

Grade 6

YEAR AT A GLANCE Option 2

Illustrative Unit

Unit 2 Introducing Ratios (17 sections)

Unit 3 Unit Rates and

Unit 1 Area and

Unit 4 Dividing Fractions (17 sections)

Unit 5 Arithmetic in Base Ten (15 sections)

Unit 6 Expressions and Equations (19 sections)

Unit 7 Rational Numbers (19 sections)

Unit 8 Data Sets and Distributions (18 sections)

Unit 9 Putting it All Together

Surface Area (19 sections)

Percentages (17 sections)

Suggested Pacing

Aug 19 - Sept 24 (26 days)

Sept 25 - Oct 25 (19 days)

Oct 28 - Nov 26 (22 days)

Dec 2 - Jan 10 (20 days)

Jan 13 - Feb 7 (18 days)

Feb 10 - Mar 12 (21 days)

Mar 13 - Apr 18 (21 days)

Apr 21 - May 16 (20 days)

May 17 - May 30

6.RP.1 6.RP.2 6.RP.3

6.RP.2 6.RP.3

6.G.1 6.G.2 6.G.4 6.EE.1 6.EE.2

6.NS.1 6.G.3

6.NS.2 6.NS.3

6.EE.1 6.EE.2 6.EE.3 6.EE.4 6.EE.5 6.EE.6 6.EE.7 6.EE.9

6.EE.8 6.NS.4 6.NS.5 6.NS.6 6.NS.7 6.NS.8

6.SP.1 6.SP.2 6.SP.3 6.SP.4 6.SP.5

6.G 6.RP 6.NS

Practice Standards

Practice Standards

Practice Standards

Practice Standards

Practice Standards

Standards

Practice Standards

Practice Standards

Practice Standards

Practice Standards

These standards will be assessed along with all other standards on the RISE

DWSBA & Testing Window

DWSBA #1

DWSBA #2

DWSBA #3

Aug 19 - Sept 25

Dec 4 - Jan 16

MAP Window

Apr 1 - May 9

SALTA Extensions

“Are You Ready for More?”

Accessing the District-Wide Standards-Based Assessment (DWSBA)

The DWSBA’s will be done through Canvas on Derivita Instructions to access the DWSBA can be found here.

AREA AND SURFACE AREA

Unit 1

CALCULATOR

PACING

KEY LANGUAGE USES

Yes

Option 1: August 19 - September 26 (28 days) Option 2: October 18 - November 26 (22 days)

EXPLAIN

STANDARDS

Standard 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing and decomposing into rectangles, triangles and/or other shapes; apply these techniques in the context of solving real-world and mathematical problems. Standard 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. Standard 6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

Standard 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

END OF UNIT COMPETENCY WITH LANGUAGE SUPPORTS

I can explain how to find the surface area of a figure. Language Supports: ● Vocabulary (Surface area, centimeters squared, inches squared, etc.) I can explain how to find the area of a figure by decomposing and rearranging. Language Supports: ● Vocabulary (Area, decompose, rearrange, move, etc.)

DIFFERENTIATION IN ACTION

Skill Building

In Lesson 5, practice verbal use of the mathematical language “base” and “height.” As students are comparing parallelograms and examples/non -examples of correct heights, encourage them to re- voice their partners’ reasoning. Reinforce

the meaning of “perpendicular” by using visuals (e.g., manipulatives, drawings, gestures) of right angles. From Lesson 12 Are You Ready for More? How many sticky notes are needed to cover the outside of 2 cabinets pushed together (including the bottom)? What about 3 cabinets? 20 cabinets?

Extension

RESOURCES

1.4 - 1.6 are shorter and can be combined 1.9-1.10 are shorter and can be combined Keep 1.19 as optional, possibly in emergency sub plans Unit 1 Vocabulary Information about Practice Problems

ILLUSTRATIVE MATHEMATICS AND CORE ALIGNMENT

Standard

Section(s)

6.G.1

1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 1.10, 1.11, 1.19

6.G.2

1.15, 1.16, 1.17

6.G.4

1.12, 1.13, 1.14, 1.15, 1.16, 1.18, 1.19

6.EE.1

1.17, 1.18

Pre-Assessment

Previous Standards with Example Problems

Number on Pre Assessment

Section

Section

Unfinished Learning Standard

1

1.17

1.2

3.MD.5b

2

1.3

1.3

3.MD.7d

3

1.7

1.4, 1.11

4.G.2

4

1.4

1.7

5.NF.4b

5

1.7, 1.11

1.11

4.G.1

6

1.17, 1.18

1.14

5.MD.5

7

1.11

1.15, 1.16

3.MD.5, 4.MD.1

1.16

5.MD.3b, 5.MD.4

1.16, 1.17

5.MD.5a

LEARNING INTENTIONS

● Find the area of right triangles by composing or decomposing rectangles. ● Find area of polygons by composing and decomposing into basic shapes (rectangles, triangles, and other shapes). ● Solve real-world area problems by decomposing and composing polygons. ● Extend background knowledge of volume with whole units (5.MD.3-5) and tiling area with fractional units (5.NF.4) to find volume of right rectangular prisms with fractional edge lengths. ● Recognize the relationship between the volume formula and filling a right rectangular prism with cubes. ● Describe how finding the volume is the same process whether edge lengths are whole units or fractional units. ● Solve real-world volume problems. ● Visualize how nets relate to three-dimensional figures. ● Understand how area of two-dimensional figures relates to surface area of three-dimensional figures.

KEY VOCABULARY

● Area ● Region

● Base ● Height ● Opposite Vertex ● Quadrilateral ● Vertex (Vertices) ● Edge

● Side ● Polygon ● Surface Area ● Face ● Net ● Polyhedron (polyhedra)

● Prism ● Pyramid ● Square of a number ● Cube of a number ● Exponent

● Rearrange ● Compose ● Decompose ● Parallelogram

INTRODUCING RATIOS

Unit 2

CALCULATOR

PACING

KEY LANGUAGE USES

Yes

Option 1: September 30 - October 30 (20 days) Option 2: August 19 - September 24 (26 days)

EXPLAIN

STANDARDS

Standard 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

Standard 6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0 and use rate language in the context of a ratio relationship.

Standard 6.RP.3 Use ratio and rate reasoning to solve real-world (with a context) and mathematical (void of context) problems, using strategies such as reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations involving unit rate problems. a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. b. Solve unit rate problems including those involving unit pricing and constant speed. c. Find a percent of a quantity as a rate per 100. Solve problems involving finding the whole, given a part and the percent. d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

END OF UNIT COMPETENCY WITH LANGUAGE SUPPORTS

I can explain the relationship between two numbers using ratio language, including equivalent ratios. Language Supports: ● Vocabulary (ratio, equivalent)

I can represent equivalent traits on a number line, bar diagram, or table. Language Supports: ● Vocabulary (number line, bar diagram, table)

DIFFERENTIATION IN ACTION

Skill Building

In Activity 7.3, Briefly review the meaning of the following terms: " parallel," "equal increments," and "line up." Use visuals to show what these terms are in the context of the problem. From Lesson 14 “Are you ready for more?” The ratio of cats to dogs in a room is 2:3. Five more cats enter the room, and then the ratio of cats to dogs is 9:11. How many cats and dogs were in the room to begin with?

Extension

RESOURCES

2.1-2.2 can be combined Keep 2.17 optional Unit 2 Vocabulary Information about Practice Problems (Go to the tab marked Unit 2)

ILLUSTRATIVE MATHEMATICS AND CORE ALIGNMENT

Standard

Section(s)

6.RP.1

2.1, 2.2, 2.3, 2.4, 2.5

6.RP.2

2.10

6.RP.3

2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 2.15, 2.16, 2.17

Pre-Assessment

Previous Standards with Example Problems

Number on Pre Assessment

Section

Section

Unfinished Learning Standard

1

2.8

2.1

3.MD.6

2

2.5

2.2

5.NF.3

3

2.10, 2.14

2.3

4.OA.1, 4.NF.4c

4

2.7

2.4

4.NBT.5

5

2.2

2.6, 2.7

4.NF.4b

6

2.3

2.8

4.NBT.1

7

2.9

2.9

5.NBT.1

2.11

5.OA.3

2.15

3.OA.5, 5.NF.7

LEARNING INTENTIONS

● Understand the concept of a ratio as a way of expressing relationships between quantities. ● Distinguish when a ratio is describing part-to-part or part-to-whole comparison. ● Communicate ratio relationships flexibly moving between ratio notation (2:3, 2 to 3, 2/3) and ratio language (two for every three). ● Understand that a rate is a special ratio that compares two quantities with different units of measure. ● Understand that unit rates are the ratio of two measurements in which the second term is one (e.g., x miles per one hour). ● Understand that when using / b to represent a rate, “b” cannot be 0 (because division by 0 is undefined). ● Understand rate language (per, each, or the @ symbol) and correctly use ratio notation and models to represent relationships between quantities. ● Use various representations such as tables of equivalent ratios, tape diagrams and/or double number line diagrams to support the development of ratio and rate reasoning and to solve problems. ● Use a table to compare ratios and find missing values using ratios. ● Understand that establishing connections between tables and plotted points on the coordinate plane allows for extended reasoning and synthesis of the concept of ratios and rates. ● Solve problems with and without context that include unit rate, percent, and measurement conversions using ratio reasoning. ● Understand percent as a rate per 100. ● Use rate reasoning to find the percent of a number. ● Use rate reasoning to solve problems involving finding the whole, given a part and the percent. *Teacher Note: This standard is not about setting up proportional relationships algebraically, but focuses on ratio reasoning

KEY VOCABULARY

● Ratio ● Equivalent Ratios

● Unit price ● Meters per second

● Tape Diagram ● Same rate ● Table

● Double

number line diagram

● Per

UNIT RATES & PERCENTAGES

Unit 3

CALCULATOR

PACING

KEY LANGUAGE USES

Yes

Option 1: October 30 - November 26 Option 2: September 25 - October 25 (19 days)

EXPLAIN

STANDARDS

Standard 6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0 and use rate language in the context of a ratio relationship.

Standard 6.RP.3 Use ratio and rate reasoning to solve real-world (with a context) and mathematical (void of context) problems, using strategies such as reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations involving unit rate problems. a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. b. Solve unit rate problems including those involving unit pricing and constant speed. c. Find a percent of a quantity as a rate per 100. Solve problems involving finding the whole, given a part and the percent. d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

END OF UNIT COMPETENCY WITH LANGUAGE SUPPORTS

I can find a unit rate. Language Supports: ● Vocabulary (rate, unit rate, speed)

I can use my knowledge of ratios and unit rates to find a percent of a number and explain why it works. Language Supports: ● Vocabulary (percent, percentage)

DIFFERENTIATION IN ACTION

Skill Building

From activity 4.2 Ask students to display their approaches to determine whether or not Elena’s mom was speeding. As students share their work, encourage them to explain the meaning of each quantity they use. For example, if they convert 80 miles per hour into kilometers per hour, where 80 is multiplied by 5/8, ask what 5/8 means in this context and why they decide to multiply it by 80. If students

used a table or a double number line, ask how these representations connect with other strategies. This will help students make sense of the various approaches to reason about equivalent ratios which can be used for reasoning about converting one unit of measure to another. Lesson 7 Are You Ready for more?Jada eats 2 scoops of ice cream in 5 minutes. Noah eats 3 scoops of ice cream in 5 minutes. How long does it take them to eat 1 scoop of ice cream working together (if they continue eating ice cream at the same rate they do individually)?

Extension

RESOURCES

3.1 is a mini-lesson and can be done as an intro to the unit 3.2-3.4 can be done in 2 days 3.14-3.16 should be extended over 5 - 6 days Don’t do 3.17, do a review instead Unit 3 Vocabulary Information about Practice Problems (Go to the tab marked Unit 3)

ILLUSTRATIVE MATHEMATICS AND CORE ALIGNMENT

Standard

Section(s)

6.RP.2

3.4, 3.5, 3.6, 3.7

6.RP.3

3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.14, 3.15, 3.16

6.G, 6.RP

3.17

Pre-Assessment

Previous Standards with Example Problems

Number on Pre Assessment

Section

Section

Unfinished Learning Standard

1

3.2

3.1

4.MD.1, 5.MD.1

2

3.3

3.2

4.MD.1

3

3.4

3.3

2.MD.2

4

3.9

3.4

5.NF.4a

5

3.10

3.5

5.NF.3

3.10

2.MD.8, 5.NBT.3

3.14

5.NBT.7

3.15

5.NBT.6

LEARNING INTENTIONS

● Understand that a rate is a special ratio that compares two quantities with different units of measure. ● Understand that unit rates are the ratio of two measurements in which the second term is one (e.g., x miles per one hour). ● Understand that when using / b to represent a rate, “b” cannot be 0 (because division by 0 is undefined). ● Understand rate language (per, each, or the @ symbol) and correctly use ratio notation and models to represent relationships between quantities. ● Use various representations such as tables of equivalent ratios, tape diagrams and/or double number line diagrams to support the development of ratio and rate reasoning and to solve problems. ● Use a table to compare ratios and find missing values using ratios. ● Understand that establishing connections between tables and plotted points on the coordinate plane allows for extended reasoning and synthesis of the concept of ratios and rates. ● Solve problems with and without context that include unit rate, percent, and measurement conversions using ratio reasoning. ● Understand percent as a rate per 100. ● Use rate reasoning to find the percent of a number. ● Use rate reasoning to solve problems involving finding the whole, given a part and the percent. *Teacher Note: This standard is not about setting up proportional relationships algebraically, but focuses on ratio reasoning

KEY VOCABULARY

● Unit Rate ● Pace ● Speed

● Percent ● Percentage

DIVIDING FRACTIONS

Unit 4

CALCULATOR

PACING

KEY LANGUAGE USES

No

December 2 - January 10 (20 days)

EXPLAIN

STANDARDS

Standard 6.NS.1 Compute quotients of fractions by fractions.

a. Solve real-world problems involving division of fractions by fractions, and explain the meaning of quotients in fraction division problems. b. Apply strategies such as using visual fraction models, applying the relationship between multiplication and division, and using equations to represent such problems as: How much chocolate will each person get if 3 people share ½ 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? c. 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.) when multiplying or dividing quantities. Standard 6.G.2 Find the volume of a right rectangular prism with appropriate unit fraction edge lengths by packing it with cubes of the appropriate unit fraction edge lengths ( for example, 3 1/2 x 2 x 6), and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. (Note: Model the packing using drawings and diagrams)

END OF UNIT COMPETENCY WITH LANGUAGE SUPPORTS

I can divide fractions and explain why the algorithm works. Language Supports: ● Vocabulary (divide, reciprocal)

DIFFERENTIATION IN ACTION

Skill Building

From activity 6.2: As groups of 3-4 discuss the first question, circulate and

record language students use to explain how Andre’s tape diagram can be used to solve the equation. Listen for phrases such as “equal parts,” “same size,” and “group of ⅔ .” If groups are stuck, consider asking “How are the number of groups represented in the tape diagram?”, “Where are the values in the equation represented in the diagram?”, or “What do the blue and white parts represent?” Post the collected language in the front of the room so that students can refer to it throughout the rest of the activity and lesson. This will help students develop mathematical language to explain how a tape diagram can be used to solve a division problem. Lesson 11 “Are You Ready for More?” You have a pint of grape juice and a pint of milk. Transfer 1 tablespoon from the grape juice into the milk and mix it up. Then transfer 1 tablespoon of the mixture back to the grape juice. Which mixture is more contaminated?

Extension

RESOURCES

4.1-4.3 can take 2 days 4.4-4.9 can take 2 days

4.10-4.11 can take 4 days 4.12-4.15 can take 4 days 4.16-4.17 are optional Unit 4 Vocabulary Information about Practice Problems (Go to the tab marked Unit 4)

ILLUSTRATIVE MATHEMATICS AND CORE ALIGNMENT

Standard

Section(s)

6.NS.1

4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10, 4.11, 4.12, 4.13, 4.14, 4.16*

6.G.2

4.13, 4.14, 4.15, 4.17*

Pre-Assessment

Previous Standards with Example Problems

Number on Pre Assessment

Section

Section

Unfinished Learning Standard

1

4.1

4.1

5.NBT.6

2

4.2

4.2

3.OA.2

3

4.7

4.3

5.NF.6

4

4.6

4.4, 4.7, 4.10, 4.11

5.NF.4

5

4.6

4.5, 4.10

5.NF.7

6

4.10

4.6

5.NF.3

7

4.15

4.10

5.NF.1

4.12

3.OA.5

4.13

5.NF.4b

4.15

5.MD.5

LEARNING INTENTIONS

● Model division of fractions with manipulatives, visual diagrams, and word problems. ● Interpret what the quotient represents in problems. ● Discover how to find the lengths of sides of polygons using the coordinates of the vertices having the same first coordinate (or second coordinate) and generalize a technique to apply in solving problems.

ARITHMETIC IN BASE 10

Unit 5

CALCULATOR

PACING

KEY LANGUAGE USES

No

January 13 - February 7 (18 days)

ARGUE

STANDARDS

Standard 6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.

Standard 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. a. Fluently divide multi-digit decimals using the standard algorithm, limited to a whole number dividend with a decimal divisor or a decimal dividend with a whole number divisor. b. Solve division problems in which both the dividend and the divisor are multi-digit decimals; develop the standard algorithm by using models, the meaning of division, and place value understanding.

END OF UNIT COMPETENCY WITH LANGUAGE SUPPORTS

I can explain how to multiply and divide decimals. Language Supports: ● Vocabulary (product, dividend)

DIFFERENTIATION IN ACTION

Skill Building

In Activity 5.2: MLR 2 Collect and Display Use this routine while students are working through the first two questions. As partners work, circulate and listen to student talk about the connections they see between the problems. Record student language and written representations on a visual display. Listen for language like “the same,” “reciprocal,” and “inverse operation.” Continue to add to the display as students work through other problems during the next three lessons. This will help students read and use mathematical language during their paired and whole group discussions. Lesson 12 “Are You Ready for More?” A distant, magical land uses jewels for their bartering system. The jewels are valued and ranked in order of their rarity. Each jewel is worth 3 times the jewel immediately below it in the ranking. The ranking is red, orange, yellow, green, blue, indigo, and violet. So a red jewel is worth 3

Extension

orange jewels, a green jewel is worth 3 blue jewels, and so on. A group of 4 craftsmen are paid 1 of each jewel. If they split the jewels evenly amongst themselves, which jewels does each craftsman get?

RESOURCES

In 5.2 definitely emphasize the represent 5.15 serves as a review for all students, and you can pull small groups for other intentional review Unit 5 Vocabulary

Other resources for unit 5 lessons Information about Practice Problems (Go to the tab marked Unit 5)

ILLUSTRATIVE MATHEMATICS AND CORE ALIGNMENT

Standard

Section(s)

6.NS.2

5.9, 5.10, 5.11

6.NS.3

5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.12, 5.13, 5.14, 5.15*

Pre-Assessment

Previous Standards with Example Problems

Number on Pre Assessment

Section

Section

Unfinished Learning Standard

1

5.2

5.1, 5.2, 5.6

5.NBT.7

2

5.2

5.2

5.NBT.1

3

5.10, 5.11

5.5

5.NBT.2

4

5.3

5.7

4.NBT.5

5

5.6

5.9

5.NBT.6

6

5.10

5.10, 5.11

4.NBT.6

7

5.9

5.11

4.NF.6

5.14

5.OA.2

LEARNING INTENTIONS

● Understand the role of place value when dividing multi-digit numbers. ● Fluently (flexibly, accurately, efficiently, and appropriately) divide multi-digit numbers.

Unit 6 EXPRESSIONS & EQUATIONS

CALCULATOR PACING

KEY LANGUAGE USES

February 10 - March 12 (21 days)

No

EXPLAIN

STANDARDS

Standard 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

Standard 6.EE.2 Write, read, and evaluate expressions in which letters represent numbers. a. Write expressions that record operations with numbers and with letters representing numbers. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coeffcient); view one or more parts of an expression as a single entity and a sum of two terms. c. Evaluate expressions at specifc values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, applying the Order of Operations when there are no parentheses to specify a particular order.

Standard 6.EE.3 Apply the properties of operations to generate equivalent expressions.

Standard 6.EE.4 Identify when two expressions are equivalent.

Standard 6.EE.5 Understand solving an equation or inequality as a process of answering the question: which values from a specifed set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specifed set makes an equation or inequality true. Standard 6.EE.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, and number in a specifed set.

Standard 6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + a = b and ax= b for cases in which a, b and x are all non-negative rational numbers.

Standard 6.EE.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 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.

END OF UNIT COMPETENCY WITH LANGUAGE SUPPORTS

I can describe how parts of an equation represent parts of a story. Language Supports: ● Vocabulary (equation) I can explain strategies for determining whether expressions are equivalent. Language Supports: ● Vocabulary (expressions, equivalent)

DIFFERENTIATION IN ACTION

Skill Building

From Activity 5.3: Writing, Speaking: MLR 1 Stronger and Clearer Each Time. Use this routine with successive pair shares to give students a structured opportunity to revise and refne their writing. For this activity, students should use the story for the equation they chose to solve. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help teams strengthen their ideas and clarify their language (e.g., “How are the parts of the equation represented in your story?”, “Can you say more about how your solution fts in your story?”). Provide students with time to complete a fnal draft based on the feedback they receive about language and clarity. Lesson 16 “Are You Ready for More?” A fruit stand sells apples, peaches, and tomatoes. Today, they sold 4 apples for every 5 peaches. They sold 2 peaches for every 3 tomatoes. They sold 132 pieces of fruit in total. How many of each fruit did they sell?

Extension

RESOURCES

Make sure you emphasize 6.11 6.18 can serve as a review and most kids can do it. Unit 6 Vocabulary

Other resources for unit 6 lessons Information about Practice Problems (Go to the tab marked Unit 6)

ILLUSTRATIVE MATHEMATICS AND CORE ALIGNMENT

Standard

Section(s)

6.EE.1

6.12, 6.13, 6.14, 6.15

6.EE.2

6.6, 6.10, 6.11, 6.14, 6.15

6.EE.3

6.9, 6.10, 6.11

6.EE.4

6.8, 6.9, 6.10, 6.11

6.EE.5

6.2, 6.3, 6.4, 6.5, 6.8, 6.15

6.EE.6

6.1, 6.3, 6.4, 6.5, 6.6, 6.7

6.EE.7

6.3, 6.4, 6.5, 6.7

6.EE.9

6.16, 6.17, 6.18

Pre-Assessment

Previous Standards with Example Problems

Number on Pre-Assessment

Section

Section

Unfnished Learning Standard

1

6.12

6.1

1.OA.4, 2.MD.5

2

6.16

6.2

3.OA.6

3

6.13

6.9

3.MD.7c, 3.OA.5

4

6.8

6.16

5.G.2

5

6.1

6

6.1

LEARNING INTENTIONS

● Understand the meaning of exponents and exponential notation. ● Write numerical expressions involving whole-number exponents. ● Evaluate numerical expressions involving whole-number exponents. ● Translate verbal expressions into numerical expressions and numerical expressions into verbal expressions. ● Identify parts of an expression (sum, term, product, factor, quotient, coeffcient). ● View one or more parts of an expression to fexibly recognize the structure of the expression (see examples from b). ● Evaluate expressions at specifc values of their variables. ● Understand that the properties used with numbers also apply to expressions with variables. ● Apply the properties of operations with expressions involving variables to generate equivalent expressions. ● Identify when two expressions are equivalent. ● Reason that two expressions are equivalent by combining like terms. ● Understand how to manipulate an expression to identify a different, yet equivalent form. ● Understand that a solution is any value or values that make an equation or inequality true. ● Use substitution to determine whether the given value makes the equation or inequality true. ● Recognize that solutions to inequalities represent a range of possible values rather than a single solution. ● Represent real-world scenarios with variable expressions, identifying what the variable represents. ● Understand that a variable can represent a number or any number in a specifed set of numbers. ● Solve equations that represent real-world and mathematical problems. ● Write and solve equations of the form x + a = b. ● Write and solve equations of the form ax = b. ● Utilize graphs and tables to recognize that a change in the independent variable creates a change in the dependent variable. ● Write an equation to express the relationship of the quantities in terms of the dependent and independent variables.

KEY VOCABULARY

● Solution to an equation ● Variable ● Coeffcient

● Equivalent Expressions ● Exponent

● Dependent Variable ● Independent Variable

Unit 7

RATIONAL NUMBERS

CALCULATOR PACING

KEY LANGUAGE USES

March 13- April 18 (21 days)

No

EXPLAIN

STANDARDS Standard 6.EE.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 of the form x > c or x < c have infnitely many solutions; represent solutions of such inequalities on number line diagrams. Standard 6.NS.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 commune factor. For example, express 36 + 8 as 4(9 + 2). Standard 6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (for example, 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. Standard 6.NS.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, for example: -(-3) = 3, and that 0 is its own opposite. b. Understand that the signs of numbers in ordered pairs indicate their location in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by refections across one or both axes. c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; fnd and position pairs of integers and other rational numbers on a coordinate plane. 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 o C>–7 o C to express the fact that –3 o C is warmer than –7 o 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 context. Standard 6.NS.7 Understand ordering and absolute value of rational numbers.

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. Standard 6.NS.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 fnd distances between points with the same x-coordinate or the same y-coordinate. END OF UNIT COMPETENCY WITH LANGUAGE SUPPORTS

I can reason about a situation involving negative numbers Language Supports: ● Vocabulary (negative, positive) I can explain what the solution to an inequality means. Language Supports: ● Vocabulary (solution, inequality) DIFFERENTIATION IN ACTION

Skill Building

From Activity 8.2: MLR 7 Compare and Connect. To foster students’ meta-awareness of language as they compare and contrast different mathematical situations, display these two sentences: “A fshing boat can hold fewer than 9 people” and “A magician will perform her magic tricks only if there are at least 9 people in the audience.” Ask students what is the same and what is different about these two situations. Invite students to work with a partner to fnd values can be true for each statement. This discussion makes explicit the language of inequalities (e.g., fewer, at least), and offers an opportunity to connect verbal descriptions to solution sets of numbers. From Lesson 13 “Are You Ready for More?”: The point (0, 4) is 5 taxicab units away from (-4, 3) and 5 taxicab units away from (2, 1). 1. Find as many other points as you can that are 4 taxicab units away from both (-4, 3) and (2, 1). 2. Are there any points that are 3 taxicab units away from both points?

Extension

RESOURCES

Each lesson takes one day Unit 7 Vocabulary Information about Practice Problems (Go to the tab marked Unit 7)

ILLUSTRATIVE MATHEMATICS AND CORE ALIGNMENT

Standard

Section(s)

6.EE.8

7.8, 7.9, 7.10

6.NS.4

7.16, 7.17, 7.18

6.NS.5

7.1, 7.5

6.NS.6

7.1, 7.2, 7.4, 7.7, 7.11, 7.12, 7.13, 7.14, 7.15

6.NS.7

7.3, 7.4, 7.6, 7.7, 7.8, 7.9, 7.13

6.NS.8

7.11, 7.13, 7.14, 7.15

Pre-Assessment

Previous Standards with Example Problems

Number on Pre-Assessment

Section

Section

Unfnished Learning Standard

1

7.4

7.2

3.NF.2

2

7.2

7.3

4.NBT.2

3

7.2

7.4

5.NBT.3b

4

7.4

7.6

4.NF.2

5

7.7

7.7

4.NF.6

6

7.11

7.11, 7.12, 7.15

5.G.1

7.13

7

7.13

5.G.2

LEARNING INTENTIONS

● Understand inequalities as a constraint or condition. ● Write inequalities based on real-world and mathematical problems. ● Recognize that inequalities of the form x > c or x < c have infnitely many solutions. ● Represent solutions of inequalities on number line diagrams. ● Discover that every negative integer is less than zero. ● Use integers to represent situations in real-world context. ● Understand the meaning of zero is determined by the context. ● Understand that a factor is a whole number that divides without a remainder into another number. Find GCF for a given pair of numbers. ● Understand that a multiple is a whole number that is a product of the whole number and any other factor. Find LCM for a given pair of numbers. ● Use knowledge of common factors and common multiples to create equivalent expressions by factoring and distributing quantities.

● Understand the meaning of the term opposite (see 6.NS.6.a) and plot opposites on a number line. ● Recognize that the opposite of the opposite of a number is the number itself. ● Understand and position rational numbers on horizontal and vertical number lines. ● Plot pairs of integers and other rational numbers on the coordinate plane. ● Plot points in the coordinate plane, recognizing that the signs of numbers in ordered pairs indicate their location in the quadrants. ● Recognize when ordered pairs are refections across an axis. For example, (x, y) refected over the x-axis becomes the point (x, -y). ● Order rational numbers on a number line. ● Compare rational numbers using inequality symbols and justify orally and/or in writing the inequality symbol used. ● Understand, compare, and interpret rational numbers found in real-world scenarios. ● Discover absolute value of a rational number as its distance from 0 on the number line. ● Model absolute value with number lines. ● Understand that quantities could have a negative value based on the scenario (e.g. debt, sea level, temperature). ● Discover how to fnd the length of a line segment using the coordinates that have the same frst coordinate (or second coordinate). ● Understand that the distance from a point on a coordinate plane to an axis is an absolute value.

KEY VOCABULARY

● Negative Number ● Positive Number ● Opposite ● Rational Number ● Sign

● Absolute Value ● Solution to an Inequality ● Quadrant ● x - coordinate ● y - coordinate

● Coordinates ● Coordinate Plane ● Less than ● Greater than ● Common Factor

● Greatest Common Factor ● Least Common Multiple ● Common Multiple