Teacher Training Revised ELA and Math Standards Math 3–5 · multiplication table, how to add,...

$: Teacher Training Revised ELA and Math Standards Math 3–5 · multiplication table, how to add, subtract, multiply, and divide basic numbers, how to work with simple fractions and$
Teacher TrainingRevised ELA and Math Standards

Math 3–5Tennessee Department of Education | 2017 Summer Teacher Training

Welcome, Teachers!

We are excited to welcome you to this summer’s teacher training on the Revised ELA standards. We appreciate your dedication to the students in your classroom and your growth as an educator. As you interact with the ELA standards over the next two days, we hope you are able to find ways to connect this new content to your own classroom. Teachers perform outstanding work every school year, and our hope is that the knowledge you gain this week will enhance the high-quality instruction you provide Tennessee's children every day.

We are honored that the content of this training was developed by and with Tennessee educators for Tennessee educators. We believe it is important for professional development to be informed by current educators, who work every day to cultivate every student’s potential.

We’d like to thank the following educators for their contribution to the creation and review of this content:

Dr. Holly Anthony, Tennessee Technological UniversityMichael Bradburn, Alcoa City SchoolsDr. Jo Ann Cady, University of TennesseeSherry Cockerham, Johnson City SchoolsDr. Allison Clark, Arlington Community SchoolsKimberly Herring, Cumberland County SchoolsDr. Joseph Jones, Cheatham County SchoolsDr. Emily Medlock, Lipscomb University

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Key Ideas for Teacher Training

Strong Standards High Expectations

Instructional Shifts

Aligned Materials and Assessments

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Standards Review Process The graphic below illustrates Tennessee’s standards review process. Here you can see the various stakeholders involved throughout the process.

• The process begins with a website for public feedback.• Tennessee educators who are experts in their content area and grade band serve

on the advisory panels. These educators review all the public feedback and thecurrent standards, then use their content expertise and knowledge of Tennesseestudents to draft a revised set of standards.

• The revised standards are posted for a second feedback collection fromTennessee’s stakeholders.

• The Standards Recommendation Committee (SRC) consists of 10 membersappointed by legislators. This group looks at all the feedback from the website,the current standards, and revised drafts. Recommendations are then made foradditional revisions if needed.

• The SRC recommends the final draft to the State Board of Education for approval.

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Educator Advisory Team MembersEvery part of the state was represented with multiple voices.

47Timeline of Standards Adoptions and Aligned Assessments Implementation

Subject

Math

ELA

Science

Social Studies

Standards Proposed to State Board

for Final Read

April 2016

April 2016

October 2016

July 2017

LEAs: New Standards in Classrooms

2017–18

2017–18

2018–19

2019–20

Aligned Assessment

2017–18

2017–18

2018–19

2019–20

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Standards Revision Key Points• The instructional shifts remain the same and are still the focus of

the standards.

• The revised standards represent a stronger foundation that willsupport the progression of rigorous standards throughout the gradelevels.

• The revised standards improve connections:

• within a single grade level, and• between multiple grade levels.

“Districts and schools in Tennessee will exemplify excellence and equity such that

all students are equipped with the knowledge and skills to successfully

embark upon their chosen path in life.”

What is your role in ensuring that all students are college and career ready?

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2016%–%18%Curriculum%Revision%Plan%KCS%Mathematics%Department%

Design'Process' Implementation'Plan'

Time'Line' Person(s)'Responsible'

Action'Step(s)'

Introduce%Plan%at%Coaches%Network%

December%9,%2016%

• KCS%MathematicsSupervisor

• KCS%MathematicsSpecialist

Introduce%and%revise%plan%(if%necessary)%with%KCS%Numeracy%Coaches%Create%tentative%curriculum%committees.%

Curriculum%Revision%Work%

Days%

January%20,%2017%

February%13,%2017%

March%6,%2017%



• KCS%NumeracyCoaches

• Select%KCSMathematicsTeachers

January%20th%%(KQ1st,%6th,%Algebra%I)%February%13th%(2nd%–%3rd,%7th%–%7th%H,%Geo.)%March%6th%(4th%–%5th,%8th,%Algebra%II)%

DistrictQWide%Teacher%Training%

%June%and%July%2017%



TN%DOE%and%KCS%Standards%Institutes%

August%2017%




• Select%KCSMathematicsTeachers

Workshops(s)%at%the%2017%August%District%Learning%Day:%%Training%from%Curriculum%Associates%for%Ready%Math%Implementation%KQ5,%%Curriculum%Planning%sessions,%%Choice%sessions.%

2017Q18%DLD%




• Select%KCSMathematicsTeachers.

Assessing%Student%Understanding%and%Instructional%Planning%with%Aligned%Instructional%Materials%

Goals• Reinforce the continued expectations of the Tennessee Math Academic

Standards.

• Revisit the three instructional shifts and their continued and connected role inthe revised standards.

• Review the overarching changes of the revised Tennessee Math AcademicStandards.

High Expectations

We have a continued goal to prepare students to be college and career ready.

Strong Standards

Standardsarethebricksthatshouldbemasterfullylaidthroughqualityinstructiontoensurethatallstudentsreachtheexpectationofthestandards.


Theinstructionalshiftsareanessentialcomponentofthestandardsandprovideguidanceforhowthestandardsshouldbetaughtandimplemented.

Aligned Materials and Assessments

Educatorsplayakeyroleinensuringthatourstandards,classroominstructionalmaterials,andassessmentsarealigned.

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Setting the StageDirections:

1. Read and annotate the General Introduction to the TN Math Standards (page 1–2)focusing on the “Mathematically Prepared” and “Conceptual Understanding,Procedural Fluency, and Application” sections.

2. After reading and annotating the two parts, write the sentence or phrase you feltwas the most important in the box below and your rationale for choosing it.

Most Important Idea:

Rationale:

Key Ideas from Discussion:

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What Has NOT Changed

• Students prepared for college and career

• K–12 learning progressions

• Traditional and integrated pathways (for high school)

• Standards for Mathematical Practice

• Instructional shifts

What HAS Changed• Category Change

• Revised Structured

• Coding & Nomenclature

• Literacy Skills for Mathematical Proficiency

Notes:

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Instructional Expectations

Focus1. In your grade-level groups, discuss ways you could respond if someone asks you

the following question: “Why focus? There’s so much math that students could belearning. Why limit them?”

2. Review the table below and answer the questions, “Which two of the followingrepresent areas of major focus for the indicated grade?”

K Compare numbers Use tally marksUnderstand the meaning of additionand subtraction

1 Add and subtract within 20

Measure lengths indirectly and by iterating length units

Create and extend patterns and sequences

2

Represent and solve problems involving addition and subtraction

Understand place valueIdentify line of symmetry in two dimensions

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Instructional Expectations

Focus1. In your grade-level groups, discuss ways you could respond if someone asks you

the following question: “Why focus? There’s so much math that students could belearning. Why limit them?”

2. Review the table below and answer the questions, “Which two of the followingrepresent areas of major focus for the indicated grade?”

3 Multiply and divide within 100

Identify the measures of central tendency and distribution

Develop understanding of fractions as numbers

4Examine transformations on the coordinate plane

Generalize place value understanding for multi-digit whole numbers

Extend understanding of fraction equivalence and ordering

5Understand and calculate probability of single events

Understand the place value system

Apply and extend previous understandings of multiplication and division to multiply and divide fractions

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Coherence

In the space below, copy all of the standards for your assigned domain and note how coherence is evident in the vertical progression of these standards.

Grade Standard Summary of the Standard (If the standard has sub-parts,summarize each sub-part.)

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Rigor 1. Make a true statement: Rigor = _______________ + _______________ + _______________

2. In your groups, discuss ways to respond to one of the following comments: “Thesestandards are expecting that we just teach rote memorization. Seems like a stepbackwards to me.” Or “I’m not going to spend time on fluency—it should just be anatural outcome of conceptual understanding.”

3. The shift towards rigor is required by the standards. Find and copy in the spacebelow standards which specifically set expectations for each component of rigor.

Standard Evidence

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4.OA.A.1

What HAS Changed

Coding and Nomenclature

Notes:

4

OA

A

1

5.NBT.A.1

5

NTB

A

2

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Revisions to the Math Standards

Specific to K–5

• Refined for clarity

• Increased fluency expectations

• Revised examples

Overarching Revisions

• Supporting and additional work of the grade is combined as supporting work ofthe grade


Increased Fluency Expectations

Former Standard Current Standard

Kindergarten K.OA.5 Fluently add and subtractwithin 5.

K.OA.A.5 Fluently add and subtract within 10 using mental strategies.

FirstGrade

1.OA.6. Add and subtract within 20, demonstrating fluency for addition

and subtraction within 10.

1.OA.C.6 Fluently add and subtractwithin 20 using mental strategies. By

the end of Grade 1, know frommemory all sums up to 10.

SecondGrade

2.OA.2 Fluently add and subtractwithin 20 using mental strategies.

By end of Grade 2, know frommemory all sums of two one-digit

numbers.

2.OA.B.2 Fluently add and subtractwithin 30 using mental strategies. By

the end of Grade 2, know frommemory all sums of two one-digit numbers and related subtraction

facts.

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Specific to K–5




Overarching Revisions

• Added/shifted a small number of standards to strengthen coherence across gradelevels

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Specific to K–5




Overarching Revisions• Revised language to provide clarity and continuity

• Highlighted chart for–grade level mastery expectation for addition, subtraction,multiplication and division

Former Standard 2.NBT.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

Current Standard 2.NBT.A.3 Read and write numbers to 1000 using standard form, word form, and expanded form.

Former Standard 4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.

Current Standard 4.NBT.A.3 Round multi-digit whole numbers to any place (up to and including the hundred-thousand place) using understanding of place value.

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Focusing on Fluency in K–5

One-minute Free Write: What is fluency?

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Focusing on Fluency in K–5

All students should be able to recall and use their math education when the need arises. That is, a student should know certain math facts and concepts such as the multiplication table, how to add, subtract, multiply, and divide basic numbers, how to work with simple fractions and percentages, etc. There is a level of procedural fluency that a student’s K–12 math education should provide him or her along with conceptual understanding so that this can be recalled and used throughout his or her life.

—Tennessee Academic Standards for Mathematics

• The ability to apply procedures __________________________ .

• Recognizing when one strategy or procedure is _________________________ to applythan another.

• Having opportunities to justify both informal strategies and commonly usedprocedures through distributed practice.

• Procedural fluency includes computational fluency with the four arithmeticoperations. In the early grades, students are expected to develop fluency withwhole numbers in addition, subtraction, multiplication, and division.

What is Fluency?

Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently.

The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships.

Definition of Fluency

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Fluency Progression Chart

Focus K 1 2 3 4 5

Add

Within 10 using

mental strategies

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Standards Comparison Activity

Compare the former standards to the current standards

Directions:

1. Highlight any changes you notice between the former standards and the currentstandards in the column on the right.

2. Categorize the changes:

• If a standard is in the 2016–17 document, but not in the 2017–18 document, placea check in the “Dropped from Course” column.

• If a standard was not in the 2016–17 document, but is now in the 2017–18document, place a check in the “Added to Course” column.

• If a standard was revised in any way (recoded, changes to the standard itself,moving examples from the standard to “Scope and Clarifications,” etc.), place acheck in the “Revised Or Refined” column.

• If a standard was not revised in any way, place a check in the “No Change”column.

Notes:

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Grade 3 Standards Comparison Activity

Coding Former TN Standards Revised TN Standards 3.OA.A.1 Interpret products of whole numbers, e.g.,

interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.

3.OA.A.1 Interpret the factors and products inwhole number multiplication equations (e.g., 4 x 7 is 4 groups of 7 objects with a total of 28 objects or 4 strings measuring 7 inches each with a total of 28 inches.)

3.OA.A.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

3.OA.A.2 Interpret the dividend, divisor, andquotient in whole number division equations (e.g., 28 ÷ 7 can be interpreted as 28 objects divided into 7 equal groups with 4 objects in each group or 28 objects divided so there are 7 objects in each of the 4 equal groups).

3.OA.A.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem

3.OA.A.3 Multiply and divide within 100 to solvecontextual problems, with unknowns in all positions, in situations involving equal groups, arrays, and measurement quantities using strategies based on place value, the properties of operations, and the relationship between multiplication and division (e.g., contexts including computations such as 3 x ? = 24, 6 x 16 = ?, ? ÷ 8 = 3, or 96 ÷ 6 = ?) (See Table 2 - Multiplication and Division Situations).

3.OA.A.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ? ÷ 3, 6 × 6 = ?.

3.OA.A.4 Determine the unknown whole numberin a multiplication or division equation relating three whole numbers within 100. For example, determine the unknown number that makes the equation true in each of the equations: 8 x ? = 48, 5 = ? ÷ 3, 6 x 6 =?

3.OA.B.5 Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3× 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) (Students need not use formal terms for these properties.)

3.OA.B.5 Apply properties of operations asstrategies to multiply and divide. (Students need not use formal terms for these properties.) Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known (Commutative property of multiplication). 3 x 5 x 2 can be solved by (3 x 5) x 2 or 3 x (5 x 2) (Associative property of multiplication). One way to find 8 x 7 is by using 8 x (5 + 2) = (8 x 5) + (8 x 2). By knowing that 8 x 5 = 40 and 8 x 2 = 16, then 8 x 7 = 40 + 16 = 56 (Distributive property of multiplication over addition).

3.OA.B.6 Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

3.OA.B.6 Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by

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finding the number that makes 32 when multiplied by 8.

3.OA.C.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

3.OA.C.7 Fluently multiply and divide within 100,using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of 3rd grade, know from memory all products of two one-digit numbers and related division facts.

3.OA.D.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (This standard is limited to problems posed with whole numbers and having whole number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).)

3.OA.D.8 Solve two-step contextual problemsusing the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding (See Table 1 - Addition and Subtraction Situations and Table 2 - Multiplication and Division Situations).

3.OA.D.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

3.OA.D.9 Identify arithmetic patterns (includingpatterns in the addition and multiplication tables) and explain them using properties of operations. For example, analyze patterns in the multiplication table and observe that 4 times a number is always even (because 4 x 6 = (2 x 2) x 6 = 2 x (2 x 6), which uses the associative property of multiplication) (See Table 3 - Properties of Operations).

3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100.

3.NBT.A.1 Round whole numbers to the nearest10 or 100 using understanding of place value.

3.NBT.A.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

3.NBT.A.2 Fluently add and subtract within 1000using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

3.NBT.A.3 Multiply one-digit whole numbers bymultiples of 10 in the range 10–90 (e.g., 9 x 80, 5 x 60) using strategies based on place value andproperties of operations.

3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

3.NF.A.1 Understand a fraction, 1/!, as thequantity formed by 1 part when a whole is partitioned into b equal parts (unit fraction); understand a fraction "/! as the quantity formed by a parts of size 1/!. For example, 3/4 represents a quantity formed by 3 parts of size 1/4.

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3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. a. Represent a fraction 1/b on a number line

diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. b. Represent a fraction a/b on a number line

diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

3.NF.A.2 Understand a fraction as a number onthe number line. Represent fractions on a number line. a. Represent a fraction 1/! on a number line

diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/! and that the endpoint locates the number 1/! on the number line. For example, on a number line from 0 to 1, students can partition it into 4 equal parts and recognize that each part represents a length of 1/4 and the first part has an endpoint at ¼ on the number line. b. Represent a fraction "/! on a number line diagram by marking off a lengths 1/! from 0. Recognize that the resulting interval has size "/! and that its endpoint locates the number "/! on the number line. For example, 5/3 is the distance from 0 when there are 5 iterations of 1/3.

3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal)

if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalentfractions, (e.g., 1/2 =2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. c. Express whole numbers as fractions, andrecognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. d. Compare two fractions with the same numerator orthe same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

3.NF.A.3 Explain equivalence of fractions andcompare fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal) ifthey are the same size or the same point on a number line. b. Recognize and generate simple equivalent

fractions (e.g., 1/2 = 2/4, 4/6 = 2/3) and explain why the fractions are equivalent using a visual fraction model. c. Express whole numbers as fractions and recognize

fractions that are equivalent to whole numbers. For example, express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point on a number line diagram. d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Use the symbols >, =, or < to show the relationship and justify the conclusions.

3.MD.A.1 Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

3.MD.A.1 Tell and write time to the nearestminute and measure time intervals in minutes. Solve contextual problems involving addition and subtraction of time intervals in minutes. For example, students may use a number line to determine the difference between the start time and the end time of lunch.

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3.MD.A.2 Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).6 Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.

3.MD.A.2 Measure the mass of objects and liquidvolume using standard units of grams (g), kilograms (kg), milliliters (ml), and liters (l). Estimate the mass of objects and liquid volume using benchmarks. For example, a large paper clip is about one gram, so a box of about 100 large clips is about 100 grams. Therefore, ten boxes would be about 1 kilogram.

3.MD.B.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

3.MD.B.3 Draw a scaled pictograph and a scaledbar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled graphs.

3.MD.B.4 Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.

3.MD.B.4 Generate measurement data bymeasuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units: whole numbers, halves, or quarters.

3.MD.C.5 Recognize area as an attribute of plane figures and understand concepts of area measurement.

a. A square with side length 1 unit, called “a unitsquare,” is said to have “one square unit” of area, and can be used to measure area. b. A plane figure which can be covered withoutgaps or overlaps by n unit squares is said to have an area of n square units.

3.MD.C.5 Recognize that plane figures have anarea and understand concepts of area measurement.

a. Understand that a square with side length 1unit, called "a unit square," is said to have "one square unit" of area and can be used to measure area. b. Understand that a plane figure which can becovered without gaps or overlaps by n unit squares is said to have an area of n square units.

3.MD.C.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

3.MD.C.6Measure areas by counting unit squares (square centimeters, square meters, square inches, square feet, and improvised units).

3.MD.C.7 Relate area to the operations of multiplication and addition. a. Find the area of a rectangle with whole-numberside lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

3.MD.C.7 Relate area of rectangles to theoperations of multiplication and addition. a. Find the area of a rectangle with whole-number side lengths by tiling it and show that the area is the same as would be found by multiplying the side lengths.

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b. Multiply side lengths to find areas of rectangleswith whole number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. c. Use tiling to show in a concrete case that thearea of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. d. Recognize area as additive. Find areas ofrectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

b. Multiply side lengths to find areas ofrectangles with whole number side lengths in the context of solving real-world and mathematical problems and represent whole-number products as rectangular areas in mathematical reasoning. c. Use tiling to show in a concrete case that thearea of a rectangle with whole-number side lengths a and b + c is the sum of a x b and a x c. Use area models to represent the distributive property in mathematical reasoning. For example, in a rectangle with dimensions 4 by 6, students can decompose the rectangle into 4 x 3 and 4 x 3 to find the total area of 4 x 6. (See Table 3 - Properties of Operations) d. Recognize area as additive. Find areas of

rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real-world problems.

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

3.MD.D.8 Solve real-world and mathematicalproblems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

3.G.A.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

3.G.A.1 Understand that shapes in differentcategories may share attributes and that the shared attributes can define a larger category. Recognize rhombuses, rectangles, and squares as examples of quadrilaterals and draw examples of quadrilaterals that do not belong to any of these subcategories.

3.G.A.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

3.G.A.2 Partition shapes into parts with equalareas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area and describe the area of each part as 1/4 of the area of the shape. 3.G.A.3 Determine if a figure is a polygon.

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Notation Former TN Standards Revised TN Standards 4.OA.1 Interpret a multiplication equation as a

comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

4.OA.A.1 Interpret a multiplication equation as acomparison (e.g., interpret 35 = 5 x 7 as a

statement that 35 is 5 times as many as 7 and 7 times as many as 5). Represent verbal statements of multiplicative comparisons as

multiplication equations.

4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

4.OA.A.2 Multiply or divide to solve contextualproblems involving multiplicative comparison, and distinguish multiplicative comparison from additive comparison. For example, school A has 300 students and school B has 600 students: to say that school B has two times as many students is an example of multiplicative comparison; to say that school B has 300 more students is an example of additive comparison.

4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

4.OA.A.3 Solve multi-step contextual problemsposed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

4.OA.B.4 Find all factor pairs for a wholenumber in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

4.OA.C.5 Generate a number or shape patternthat follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

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4.NBT.A.1 1. Recognize that in a multi-digit whole number,a digit in one place represents ten times what it represents in the place to its right. For example, recognize that700 ÷ 70 = 10 by applying concepts of place value and division.

4.NBT.A.1 Recognize that in a multi-digit wholenumber (less than or equal to 1,000,000), a digit in one place represents 10 times as much as it represents in the place to its right. For example, recognize that 7 in 700 is 10 times bigger than the 7 in 70 because 700 ÷ 70 = 10 and 70 x 10 = 700.

4.NBT.A.2 2. Read and write multi-digit whole numbersusing base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

4.NBT.A.2 Read and write multi-digit wholenumbers (less than or equal to 1,000,000) using standard form, word form, and expanded form (e.g. the expanded form of 4256 is written as 4 x 1000 + 2 x 100 + 5 x 10 + 6 x 1). Compare two multi- digit numbers based on meanings of the digits in each place and use the symbols >, =, and < to show the relationship.

4.NBT.A.3 3. Use place value understanding to roundmulti-digit whole numbers to any place.

4.NBT.A.3 Round multi-digit whole numbers toany place (up to and including the hundred-thousand place) using understanding of place value.

4.NBT.B.4 4. Fluently add and subtract multi-digit wholenumbers using the standard algorithm.

4.NBT.B.4 Fluently add and subtract within1,000,000 using appropriate strategies and algorithms.

4.NBT.B.5 5. Multiply a whole number of up to four digitsby a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

4.NBT.B.5 Multiply a whole number of up tofour digits by a one-digit whole number and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

4.NBT.B.6 6. Find whole-number quotients andremainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

4.NBT.B.6 Find whole-number quotients andremainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

4.NF.A.1 Explain why a fraction !" is

equivalent to a fraction #×%&×% or #÷%&÷% by using

visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. For example,

29


() =

(×+)×+ =

,-

4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

4.NF.A.2 Compare two fractions with differentnumerators and different denominators by creating common denominators or common numerators or by comparing to a benchmark

fraction such as .+. Recognize that

comparisons are valid only when the two fractions refer to the same whole. Use the symbols >, =, or < to show the relationship and justify the conclusions.

4.NF.33. Understand a fraction a/b with a > 1 as a

sum of fractions 1/b.a. Understand addition and subtraction

of fractions as joining and separatingparts referring to the same whole.

b. Decompose a fraction into a sum offractions with the samedenominator in more than oneway, recording each decompositionby an equation. Justifydecompositions, e.g., by using avisual fraction model. Examples:3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 +2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8+ 1/8.

c. Add and subtract mixed numberswith like denominators, e.g., byreplacing each mixed number withan equivalent fraction, and/or byusing properties of operations andthe relationship between additionand subtraction.

d. Solve word problems involvingaddition and subtraction offractions referring to the samewhole and having likedenominators, e.g., by using visualfraction models and equations torepresent the problem.

4.NF.B.3 Understand a fraction #& with a>1 as a

sum of fractions .&. For example, )/ =./ +

./ +

./ +

./.

a. Understand addition and subtraction offractions as joining and separating parts referring to the same whole.

b. Decompose a fraction into a sum offractions with the same denominator in more than one way (e.g. (- =

.- +

.- +

.-;

(- =

.- +

+-;

2 .- = 1 + 1 + .-) recording each decomposition by

an equation. Justify decompositions by using a visual fraction model.

c. Add and subtract mixed numbers withlike denominators by replacing eachmixed number with an equivalentfraction and/or by using properties ofoperations and the relationshipbetween addition and subtraction.

d. Solve contextual problems involvingaddition and subtraction of fractions referring to the same whole and having like denominators

4.NF.4 4. Apply and extend previous understandingsof multiplication to multiply a fraction by awhole number.a. Understand a fraction a/b as a

multiple of 1/b. For example, use avisual fraction model to represent5/4 as the product 5 × (1/4),

4.NF.B.4 Apply and extend previousunderstandings of multiplication as repeated addition to multiply a whole number by a fraction.

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recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/bas a multiple of 1/b, and usethis understanding tomultiply a fraction by a wholenumber. For example, use avisual fraction model toexpress 3 × (2/5) as 6 × (1/5),recognizing this product as6/5. (In general, n × (a/b) = (n ×a)/b.)

c. Solve word problemsinvolving multiplication of afraction by a whole number,e.g., by using visual fractionmodels and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

a. Understand a fraction #& as a multiple of .&. For example, use a

visual fraction model to represent /) as the

product 5× .), recording the conclusion by the

equation /) = 5× .).

b. Understand a multiple of #& as a multiple

of .& and use this understanding to multiply a whole number by a fraction. For example, use a visual fraction model to express 3× +

/ as 6× ./,

recognizing this product as ,/ . (In general, × #& =

%×#& = 7×8 × .

& .

c. Solve contextual problems involvingmultiplication of a whole number by afraction (e.g., by using visual fractionmodels and equations to represent theproblem). For example, if each person at a

party will eat 3 of a pound of roast beef,and there will be 4 people at the party,how many pounds of roast beef will beneeded? Between what two wholenumbers does your answer lie?

4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

(Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.)

4.NF.C.5 Express a fraction with denominator10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For

example, express (.9 as (9.99 and add (.9 +

).99 =

().99

4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

4.NF.C.6 Read and write decimal notation for fractions with denominators 10 or 100. Locate these decimals on a number line.

4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals

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results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

refer to the same whole. Use the symbols >, =, or < to show the relationship and justify the conclusions.

4.MD.A.1 1. Know relative sizes of measurement unitswithin one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...

4.MD.A.1 Measure and estimate to determinerelative sizes of measurement units within a single system of measurement involving length, liquid volume, and mass/weight of objects using customary and metric units.

4.MD.A.2 2. Use the four operations to solve wordproblems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

4.MD.A.2 Solve one- or two-step real-worldproblems involving whole number measurements with all four operations within a single system of measurement including problems involving simple fractions.

4.MD.A.3 3. Apply the area and perimeter formulas forrectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

4.MD.A.3 Know and apply the area and perimeterformulas for rectangles in real- world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

4.MD.B.4 4. Make a line plot to display a data set ofmeasurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

4.MD.B.4 Make a line plot to display a data set ofmeasurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

4.MD.C.5(including parts a and b)

4. Recognize angles as geometric shapesthat are formed wherever two raysshare a common endpoint, andunderstand concepts of anglemeasurement:a. An angle is measured with reference

to a circle with its center at thecommon endpoint of the rays, byconsidering the fraction of thecircular arc between the points

4.MD.C.5 Recognize angles as geometricshapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement.

a. Understand that an angle is measuredwith reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle.

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where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.

b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

b. Understand that an angle that turnsthrough 1/360 of a circle is called a"one-degree angle," and can be used tomeasure angles. An angle that turnsthrough n one-degree angles is said tohave an angle measure of n degreesand represents a fractional portion ofthe circle.

4.MD.C.6 6. Measure angles in whole-number degreesusing a protractor. Sketch angles of specified measure.

4.MD.C.6 Measure angles in whole-numberdegrees using a protractor. Sketch angles of specified measure.

4.MD.C.7 7. Recognize angle measure as additive. Whenan angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

4.MD.C.7 Recognize angle measure as additive.When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems (e.g., by using an equation with a symbol for the unknown angle measure).

4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two- dimensional figures.

4.G.A.1 Draw points, lines, line segments,rays, angles (right, acute, obtuse, straight, reflex), and perpendicular and parallel lines. Identify these in two- dimensional figures.

4.G.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

4.G.A.2 Classify two-dimensional figuresbased on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size. Recognize right triangles as a category and identify right triangles.

4.G.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

4.G.A.3 Recognize and draw lines of symmetry fortwo-dimensional figures.

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Coding Former TN Standards Revised TN Standards 5.OA.A.1 Use parentheses, brackets, or braces in

numerical expressions, and evaluate expressions with these symbols.

5.OA.A.1 Use parentheses, brackets, or bracesin numerical expressions, and evaluate expressions with these symbols.

5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 x (8 + 7). Recognize that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

5.OA.A.2 Write simple expressions that recordcalculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 x (8 + 7). Recognize that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

5.OA.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

5.OA.B.3 Generate two numerical patternsusing two given rules. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences. a. Identify relationships betweencorresponding terms in two numerical patterns. For example, observe that the terms in one sequence are twice the corresponding terms in the other sequence. b. Form ordered pairs consisting ofcorresponding terms from two numerical patterns and graph the ordered pairs on a coordinate plane.

5.NBT.A.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

5.NBT.A.1 Recognize that in a multi-digitnumber, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

5.NBT.A.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

5.NBT.A.2 Explain patterns in the number ofzeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denotepowers of 10.

5.NBT.A.3 Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandthsusing base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 x (1/10) + 9 x (1/100) + 2 x (1/1000). b. Compare two decimals to thousandths basedon meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

5.NBT.A.3 Read and write decimals tothousandths using standard form, word form, and expanded form (e.g., the expanded form of 347.392 is written as 3 x 100 + 4 x 10 + 7 x 1 + 3 x (1/10) + 9 x (1/100) + 2 x (1/1000)). Compare two decimals to thousandths based on meanings of the digits in each place and use the symbols >, =, and < to show the relationship.

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5.NBT.A.4 Use place value understanding to round decimals to any place.

5.NBT.A.4 Round decimals to the nearesthundredth, tenth, or whole number using understanding of place value.

5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

5.NBT.B.5 Fluently multiply multi-digit wholenumbers (up to three-digit by four-digit factors) using appropriate strategies and algorithms.

5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

5.NBT.B.6 Find whole-number quotients andremainders of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

5.NBT.B.7 Add, subtract, multiply, and dividedecimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between operations; assess the reasonableness of answers using estimation strategies. (Limit division problems so that either the dividend or the divisor is a whole number.)

5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

5.NF.A.1 Add and subtract fractions with unlikedenominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models orequations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

5.NF.A.2 Solve contextual problems involvingaddition and subtraction of fractions referring to the same whole, including cases of unlike denominators. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

5.NF.B.3 Interpret a fraction as division of the numerator by the denominator a/b=a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction

5.NF.B.3 Interpret a fraction as division of thenumerator by the denominator (a/b=a ÷ b). Solve contextual problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers by using visual

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models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

fraction models or equations to represent the problem. For example, if 8 people want to share 49 sheets of construction paper equally, how many sheets will each person receive? Between what two whole numbers does your answer lie?

5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

a. Interpret the product (a/b) x q (3/5 x 6) as aparts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q ÷ b . For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same with (2/3) x (4/5) = 8/15. (In general, (a/b) x (c/d) = ac/bd.)

5.NF.B.4 Apply and extend previousunderstandings of multiplication to multiply a fraction by a whole number or a fraction by a fraction. a. Interpret the product a/b x q as a x (q ÷ b)(partition the quantity q into b equal parts and then multiply by a). Interpret the product a/b x q as (a x q) ÷ b (multiply a times the quantity q and then partition the product into b equal parts). For example, use a visual fraction model or write a story context to show that 3/4 x 16 can be interpreted as 3 x (16 ÷ 4) or (3 x 16) ÷ 4. Do the same with 2/3 x 4/5 = 8/15. (In general, a/b x c/d = ac/bd.)

5.NF.B.5 Interpret multiplication as scaling (resizing), by:

a. Comparing the size of a product to the sizeof one factor on the basis of the size of the other factor, without performing the indicated multiplication. b. Explaining why multiplying a given numberby a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (nxa)/(nxb) to the effect of multiplying a/b by 1.

5.NF.B.5 Interpret multiplication as scaling(resizing). a. Compare the size of a product to the size ofone factor on the basis of the size of the other factor, without performing the indicated multiplication. For example, the product of 1/2 and 1/4 will be smaller than each of the factors. b. Explain why multiplying a given number by afraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relate the principle of fraction equivalence to the

effect of multiplying by 1. 5.NF.B.6 Solve real world problems involving

multiplication of fractions and mixed numbers, e.g., by using visual fraction models orequations to represent the problem.

5.NF.B.6 Solve real world problems involvingmultiplication of fractions and mixed numbers by using visual fraction models or equations to represent the problem.

5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

5.NF.B.7 Apply and extend previousunderstandings of division to divide unit fractions by whole numbers and whole

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(Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.) a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3. b. Interpret division of a whole number by aunit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4. c. Solve real world problems involving divisionof unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.) a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. Use visual models and the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3. b. Interpret division of a whole number by aunit fraction, and compute such quotients. Use visual models and the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4. c. Solve real world problems involving divisionof unit fractions by non-zero whole numbers and division of whole numbers by unit fractions by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

5.MD.A.1 Convert customary and metricmeasurement units within a single system by expressing measurements of a larger unit in terms of a smaller unit. Use these conversions to solve multi- step real world problems involving distances, intervals of time, liquid volumes, masses of objects, and money (including problems involving simple fractions or decimals). For example, 3.6 liters and 4.1 liters can be combined as 7.7 liters or 7700 milliliters.

5.MD.B.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

5.MD.B.2 Make a line plot to display a data setof measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

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5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a "unitcube," is said to have "one cubic unit" of volume, and can be used to measure volume. b. A solid figure which can be packed withoutgaps or overlaps using n unit cubes is said to have a volume of n cubic units.

5.MD.C.3 Recognize volume as an attribute ofsolid figures and understand concepts of volume measurement. a. Understand that a cube with side length 1unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume. b. Understand that a solid figure which can bepacked without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

5.MD.C.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

5.MD.C.4 Measure volumes by counting unit cubes, usingcubic cm, cubic in, cubic ft, and improvised units.

5.MD.C.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prismwith whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property ofmultiplication. b. Apply the formulas V = l x w x h and V = b x hfor rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems. c. Recognize volume as additive. Find volumesof solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

5.MD.C.5 Relate volume to the operations ofmultiplication and addition and solve real world and mathematical problems involving volume of right rectangular prisms. a. Find the volume of a right rectangular prismwith whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent whole-number products of three factors as volumes (e.g., to represent the associative property of multiplication). b. Apply the formulas V = l x w x h andV = B x h (where B represents the area of the base) for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems. c. Recognize volume as additive. Find volumesof solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

5.G.A.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number

5.G.A.1 Graph and label points using the firstquadrant of the coordinate plane. Understand that the first number indicates the horizontal distance traveled along the x-axis from the origin, and the second number indicates the vertical distance traveled along the y-axis with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

3738


indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

5.G.A.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

5.G.A.2 Represent real world and mathematicalproblems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

5.G.B.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

5.G.B.3 Classify two-dimensional figures in ahierarchy based on properties. Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

5.G.B.4 Classify two-dimensional figures in a hierarchy based on properties.

3839

Standards Comparison ChartStandard

CodingDropped

from CourseAdded to

CourseRevised or

Refined No change

40


Standards Comparison Activity

1. If you had to summarize the revisions to these selected standards in twentywords or less, what would you say?

Notes:

Small Group Consensus:

Whole Group Consensus:

41

Closer Look

Take a few minutes to read the overview page for your grade level and think about how this relates to the overarching revisions we have just seen.

Notes:

Now summarize your course in 140 characters. Write your tweet to inform others regarding what is included in your grade.

My Tweet:

42

Reflect on ways literacy skills are already present in your mathematics classroom.

Literacy in your Math Classroom

Literacy Skills for Math ProficiencyCommunication in mathematics requires literacy skills in reading, vocabulary, speaking, listening, and writing.

43

Reading

Vocabulary

Speaking & Listening

Writing

Literacy Skills for Math ProficiencyCategorize the strategies you listed and discussed with your table partners in the chart below.

44

Literacy Skills for Mathematical Proficiency

1. Read and annotate your assigned section from pages 13–14 of the TN MathStandards. Work with your group to present this information to your colleagues.

2. Use the chart below to take notes and highlight the main ideas of each section.

Reading

Vocabulary

Speaking & Listening

Writing

45

46

READING STANDARDS: Craft and Structure – Standard #4 R.CS.4

Cornerstone: Interpret words and phrases as they are used in a text, including technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone.

GRADE SPAN LITERATURE INFORMATIONAL TEXT

11-12

11-12.RL.CS.4 Determine the meaning of words and phrases as they are used in a text, including figurative and connotative meanings; analyze the impact of specific word choices on meaning and tone, including words with multiple meanings and language that is stylistically poignant and engaging.

11-12.RI.CS.4 Determine the meaning of words and phrases as they are used in a text, including figurative, connotative, and technical meanings; analyze how an author uses and refines the meaning of a key term or terms over the course of a text.

9 - 10

9-10.RL.CS.4 Determine the meaning of words and phrases as they are used in a text, including figurative and connotative meanings; analyze the cumulative impact of specific word choices on meaning and tone, such as how language evokes a sense of time and place, and how it communicates an informal or formal tone.

9-10.RI.CS.4 Determine the meaning of words and phrases as they are used in a text, including figurative, connotative, and technical meanings; analyze the cumulative impact of specific word choices on meaning and tone.

8

8.RL.CS.4 Determine the meaning of wordsand phrases as they are used in a text, including figurative and connotative meanings; analyze the impact of specific word choices on meaning and tone, including allusions to other texts, repetition of words and phrases, and analogies.

8.RI.CS.4 Determine the meaning of wordsand phrases as they are used in a text, including figurative, connotative, and technical meanings; analyze the impact of a specific word choice on meaning and tone, including analogies and allusions to other texts.

7

7.RL.CS.4 Determine the meaning of wordsand phrases as they are used in a text, including figurative and connotative meanings; analyze the impact of specific word choices on meaning and tone, including allusions to other texts and repetition of words and phrases.

7.RI.CS.4 Determine the meaning of wordsand phrases as they are used in a text, including figurative and connotative meanings; analyze the impact of specific word choices on meaning and tone, including allusions to other texts and repetition of words and phrases.

6

6.RL.CS.4 Determine the meaning of wordsand phrases as they are used in a text, including figurative and connotative meanings; analyze the impact of specific word choices on meaning and tone, including allusions to other texts.

6.RI.CS.4 Determine the meaning of wordsand phrases as they are used in a text, including figurative, connotative, and technical meanings.

55.RL.CS.4 Determine the meaning of wordsand phrases as they are used in a text, including figurative language with emphasis on similes and metaphors; analyze the impact of sound devices on meaning and tone.

5.RI.CS.4 Determine the meaning of wordsand phrases as they are used in a text relevant to a grade 5 topic or subject area, including figurative, connotative, and technical meanings.

47

44.RL.CS.4 Determine the meaning of wordsand phrases as they are used in a text, including those that refer to significant characters and situations found in literature and history.

4.RI.CS.4 Determine the meaning of wordsand phrases as they are used in a text relevant to a grade 4 topic or subject area, including figurative, connotative, and technical meanings.

33.RL.CS.4 Determine the meaning of wordsand phrases as they are used in a text, distinguishing literal from nonliteral language (e.g., feeling blue versus the color blue).

3.RI.CS.4 Determine the meaning of wordsand phrases in a text relevant to a grade 3 topic or subject area.

2 2.RL.CS.4 Describe how words and phrasessupply meaning in a story, poem, or song.


11.RL.CS.4 Identify words and phrases instories and poems that suggest feelings or appeal to the senses.


KK.RL.CS.4 With prompting and support, ask and answer questions about unknown words in text.

K.RI.CS.4 With prompting and support, determine the meaning of words and phrases in a text relevant to a Kindergarten topic or subject area.

48

SPEAKING AND LISTENING STANDARDS: Comprehension and Collaboration – Standard #1

SL.CC.1

Cornerstone: Prepare for and participate effectively in a range of conversations and collaborations with varied partners, building on others’ ideas and expressing their own clearly and persuasively.

GRADE SPAN STANDARDS LINKING STANDARDS

11-12 11-12.SL.CC.1 Initiate and participate effectively with varied partners in a range of collaborative discussions on appropriate 11th - 12th grade topics, texts, and issues, building on others’ ideas and expressing their own clearly and persuasively.

RL.1-7, 9, 10 RI.1-10

W.6

9-10 9-10.SL.CC.1 Initiate and participate effectively with varied partners in a range of collaborative discussions on appropriate 9th- 10th grade topics, texts, and issues, building on others’ ideas and expressing their own clearly and persuasively.

RL.1-7, 9, 10 RI.1-10,

W.6

8 8.SL.CC.1 Prepare for collaborative discussions on 8th gradelevel topics and texts; engage effectively with varied partners, building on others’ ideas and expressing their own ideas clearly.

RL.1-7, 9, 10 RI.1-10 W.5-6


RL.1-7, 9, 10 RI.1-10 W.5-6


RL.1-7, 9, 10 RI.1-10 W.5-6


FL.F.5 RL.1-7, 9, 10

RI.1-10 W.5-6


FL.F.5 RL.1-7, 9, 10

RI.1-10 W.5-6

3 3.SL.CC.1 Prepare for collaborative discussions on 3rd gradelevel topics and texts; engage effectively with varied partners, building on others’ ideas and expressing their own ideas clearly.

FL.F.5 RL.1-7, 9, 10 RI.1-10

W.4-6

2 2.SL.CC.1 Participate with varied peers and adults incollaborative conversations in small or large groups about appropriate 2nd grade topics and texts.

FL.F.5 RL.1-7, 9, 10

RI.1-10 W.5-8

1 1.SL.CC.1 Participate with varied peers and adults incollaborative conversations in small or large groups about appropriate 1st grade topics and texts.

FL.F.5 RL.1-7, 9, 10

RI.1-10 W.1-3, 5-8

K K.SL.CC.1 Participate with varied peers and adults in collaborative conversations in small or large groups about appropriate Kindergarten topics.

FL.F.5 RL.1- 7, 9,10

RI.1-10 W.1-3, 5-8

49

SPEAKING AND LISTENING STANDARDS: Comprehension and Collaboration – Standard #2

SL.CC.2

Cornerstone: Integrate and evaluate information presented in diverse media formats, such as visual, quantitative, and oral formats.

GRADE SPAN STANDARDS LINKING STANDARDS

11-12

11-12.SL.CC.2 Integrate multiple sources of information presented in diverse media formats in order to make informed decisions and solve problems; evaluate the credibility and accuracy of each source and note any discrepancies among the data.

L.VAU.5-6 Reading Cornerstone Standards 1 and 10.

RL/RI.7 W.8

9-10 9-10.SL.CC.2 Integrate and evaluate multiple sources of information presented in diverse media formats; evaluate the credibility and accuracy of each source.


RL/RI.7 W.8

8 8.SL.CC.2 Analyze the purpose of information presented indiverse media formats; evaluate the motives, such as social, commercial, and political, behind its presentation.


RL/RI.7 W.8

7 7.SL.CC.2 Analyze the main ideas and supporting detailspresented in diverse media formats; explain how this clarifies a topic, text, or issue under study.


RL/RI.7 W.8

6 6.SL.CC.2 Interpret information presented in diverse mediaformats; explain how source information contributes to a topic, text, or issue under study.


RL/RI.7 W.8

5 5.SL.CC.2 Summarize a text presented in diverse media such asvisual, quantitative, and oral formats.

FL.VAC.7 Reading Cornerstone Standards 1 and 10

RL/RI.7 W.8

4 4.SL.CC.2 Paraphrase portions of a text presented in diversemedia such as visual, quantitative, and oral formats.


RL/RI.7 W.8

3 3.SL.CC.2 Determine the main ideas and supporting details of atext presented in diverse media such as visual, quantitative, and oral formats.


RL/RI.7 W.8

2 2.SL.CC.2 Recount or describe key ideas or details from a textread aloud or information presented orally or through other media.


RL/RI.7 W.8

A Guide to Rigor in Mathem

atics

In order to provide a quality mathem

atical education for students, instruction must be rigorous, focused, and coherent. This

document provides explanations and a standards-based alignm

ent to assist teachers in providing the first of those: a rigorous education. W

hile this document w

ill help teachers identify the explicit component(s) of rigor called for by each of the Louisiana

Student Standards for Mathem

atics (LSSM), it is up to the teacher to ensure his/her instruction aligns to the expectations of the

standards, allowing for the proper developm

ent of rigor in the classroom.

This rigor document is considered a “living” docum

ent as we believe that teachers and other educators w

ill find w

ays to improve the docum

ent as they use it. Please send feedback to [email protected] so that w

e m

ay use your input when updating this guide.

Updated January 18, 2017. Click here to see updates.

1

A Guide to Rigor

Table of Contents Introduction

Definitions of the Components of Rigor .....…

……

… ...................................................................................... …

……

……

……

2 A Special N

ote on Procedural Skill and Fluency ……

……

……

……

……

……

……

……

… ................................................................. 2

Recognizing the Com

ponents of Rigor ……

……

……

……

……

……

……

……

……

….…

….. ................................................................. 3

Focus in the standards …

……

……

……

……

……

……

……

……

……

……

……

............................................ ……

……

……

…3

Rigor by Grade/Content Level

Kindergarten ……

……

……

……

……

……

……

……

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……

…..…

…..…

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……

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… ................................................................. 4

1

st Grade .................................................................................. ……

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..……

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……

.6

2nd Grade …

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……

……

….…

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…. ................................................................ 8

3

rd Grade ................................................................... ……

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……

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……

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……

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……

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……

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……

..……

.10

4th Grade ................................................................. …

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……

……

……

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……

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……

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……

……

……

……

……

……

……

...14

5th Grade ................................................................. …

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……

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……

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……

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...18

6th Grade ................................................................... …

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…...22

7

th Grade ................................................................ ……

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……

.26

8th Grade …

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..……

…. ............................................................. 30

Algebra I ........................................................................................................................................................................... 34

Geom

etry ……

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............................................................. 38

Algebra II ................................................................. ……

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..….42

2

A Guide to Rigor Definitions of the Com

ponents of Rigor Rigorous teaching in m

athematics does not sim

ply mean increasing the difficulty or com

plexity of practice problems.

Incorporating rigor into classroom instruction and student learning m

eans exploring at a greater depth, the standards and ideas w

ith which students are grappling. There are three com

ponents of rigor that will be expanded upon in this docum

ent, and each is equally im

portant to student mastery: Conceptual Understanding, Procedural Skill and Fluency, and Application.

� Conceptual Understanding refers to understanding m

athematical concepts, operations, and relations. It is m

ore than know

ing isolated facts and methods. Students should be able to m

ake sense of why a m

athematical idea is im

portant and the kinds of contexts in w

hich it is useful. It also allows students to connect prior know

ledge to new ideas and concepts.

� Procedural Skill and Fluency is the ability to apply procedures accurately, efficiently, and flexibly. It requires speed and accuracy in calculation w

hile giving students opportunities to practice basic skills. Students’ ability to solve more com

plex application tasks is dependent on procedural skill and fluency.

� Application provides valuable content for learning and the opportunity to solve problem

s in a relevant and a m

eaningful way. It is through real-w

orld application that students learn to select an efficient method to find a solution,

determine w

hether the solution makes sense by reasoning, and develop critical thinking skills.

A Special Note on Procedural Skill and Fluency W

hile speed is definitely a component of fluency, it is not necessarily speed in producing an answ

er; rather, fluency can be observed by w

atching the speed with w

hich a student engages with a particular problem

. Furthermore, fluency does not

require the most efficient strategy. The standards specify grade-level appropriate strategies or types of strategies w

ith which

students should demonstrate fluency (e.g., 1.O

A.C.6 allows for students to use counting on, m

aking ten, creating equivalent but easier or know

n sums, etc.). It should also be noted that teachers should expect som

e procedures to take longer than others (e.g., fluency w

ith the standard algorithm for division, 6.N

S.B.2, as compared to fluently adding and subtracting w

ithin 10, 1.O

A.C.6).

Standards identified as targeting procedural skill and fluency do not all have an expectation of automaticity and/or rote recall.

Only tw

o standards, 2.OA.B.2 and 3.O

A.C.7, have explicit expectations of students knowing facts from

mem

ory. Other

standards targeting procedural skill and fluency do not require students to reach automaticity. For exam

ple, in 4.G.A.2,

students do not need to reach automaticity in classifying tw

o-dimensional figures.

3

A Guide to Rigor

Recognizing the Components of Rigor

In the LSSM each standard is aligned to one or m

ore components of rigor, m

eaning that each standard aims to prom

ote student grow

th in conceptual understanding, procedural skill and fluency, and/or application. Key words and phrases in the

standards indicate which com

ponent(s) of rigor the standard is targeting: conceptual understanding standards often use terms

like understand, recognize, or interpret; procedural skill and fluency standards tend to use words like fluently, find, or solve;

and application standards typically use phrases like word problem

s or real-world problem

s. Key words and phrases are

underlined in each standard to help clarify the identified component(s) of rigor for each standard.

Focus in the Standards N

ot all content in a given grade is emphasized equally in the standards. Som

e clusters require greater emphasis than others

based on the depth of the ideas, the time that they take to m

aster, and/or their importance to future m

athematics or the

demands of college and career readiness. M

ore time in these areas is also necessary for students to m

eet the Louisiana Standards for M

athematical Practice. To say that som

e things have greater emphasis is not to say that anything in the

standards can safely be neglected in instruction. Neglecting m

aterial will leave gaps in student skill and understanding and

may leave students unprepared for the challenges of a later grade. Students should spend the large m

ajority of their time on

the major w

ork of the grade (�). Supporting w

ork (�) and, w

here appropriate, additional work (�

) can engage students in the m

ajor work of the grade.

4

A Guide to Rigor

Kindergarten LSS

M –

Kin

de

rg

arte

n

Exp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill

an

d F

lue

ncy

Ap

plic

atio

n

K.C

C.A

.1

Co

un

t t

o 1

00

by o

ne

s a

nd

by t

en

s.

✔

K.C

C.A

.2

Co

un

t f

orw

ard

be

gin

nin

g f

ro

m a

giv

en

nu

mb

er w

ith

in t

he

kn

ow

n s

eq

ue

nce

(in

ste

ad

of h

avin

g t

o b

eg

in a

t 1

).

✔

K.C

C.A

.3

Writ

e n

um

be

rs f

ro

m 0

to

20

. Re

pre

se

nt a

nu

mb

er o

f o

bje

cts w

ith

a w

rit

te

n n

um

era

l 0-2

0 (

wit

h 0

re

pre

se

ntin

g a

co

un

t o

f n

o o

bje

cts).

✔

K.C

C.B

.4

Un

de

rsta

nd

th

e r

ela

tio

nsh

ip b

etw

ee

n n

um

be

rs a

nd

qu

an

tit

ies; c

on

ne

ct c

ou

ntin

g t

o c

ard

ina

lity.

✔

K.C

C.B

.4a

W

he

n c

ou

ntin

g o

bje

cts in

sta

nd

ard

ord

er, s

ay t

he

nu

mb

er n

am

es a

s t

he

y r

ela

te

to

ea

ch

ob

ject in

th

e g

ro

up

,

de

mo

nstra

tin

g o

ne

-to

-o

ne

co

rre

sp

on

de

nce

. ✔

K.C

C.B

.4b

U

nd

ersta

nd

th

at t

he

last n

um

be

r n

am

e s

aid

te

lls t

he

nu

mb

er o

f o

bje

cts c

ou

nte

d. T

he

nu

mb

er o

f o

bje

cts is

th

e s

am

e r

eg

ard

less o

f t

he

ir a

rra

ng

em

en

t o

r t

he

ord

er in

wh

ich

th

ey w

ere

co

un

te

d.

✔

K.C

C.B

.4c

Un

de

rsta

nd

th

at e

ach

su

cce

ssiv

e n

um

be

r n

am

e r

efe

rs t

o a

qu

an

tit

y t

ha

t is

on

e la

rg

er.

✔

K.C

C.B

.5

Co

un

t t

o a

nsw

er “

Ho

w m

an

y?

” q

ue

stio

ns.

✔

K.C

C.B

.5a

C

ou

nt o

bje

cts u

p t

o 2

0, a

rra

ng

ed

in a

line

, a r

ecta

ng

ula

r a

rra

y, o

r a

cir

cle

.

✔

K.C

C.B

.5b

C

ou

nt o

bje

cts u

p t

o 1

0 in

a s

ca

tte

re

d c

on

fig

ura

tio

n.

✔

K.C

C.B

.5c

Wh

en

giv

en

a n

um

be

r f

ro

m 1

-2

0, c

ou

nt o

ut t

ha

t m

an

y o

bje

cts.

✔

K.C

C.C

.6

Ide

ntif

y w

he

th

er t

he

nu

mb

er o

f o

bje

cts in

on

e g

ro

up

is g

re

ate

r t

ha

n, le

ss t

ha

n, o

r e

qu

al t

o t

he

nu

mb

er o

f

ob

jects in

an

oth

er g

ro

up

, e.g

., by u

sin

g m

atch

ing

an

d c

ou

ntin

g s

tra

te

gie

s.

✔

K.C

C.C

.7

Co

mp

are

tw

o n

um

be

rs b

etw

ee

n 1

an

d 1

0 p

re

se

nte

d a

s w

rit

te

n n

um

era

ls.

✔

K.O

A.A

.1

Re

pre

se

nt a

dd

itio

n a

nd

su

btra

ctio

n w

ith

ob

jects, f

ing

ers, m

en

ta

l ima

ge

s, d

ra

win

gs, s

ou

nd

s (

e.g

., cla

ps),

actin

g o

ut s

itu

atio

ns, v

erb

al e

xp

lan

atio

ns, e

xp

re

ssio

ns, o

r e

qu

atio

ns.

✔

K.O

A.A

.2

So

lve

ad

dit

ion

an

d s

ub

tra

ctio

n w

ord

pro

ble

ms, a

nd

ad

d a

nd

su

btra

ct w

ith

in 1

0, e

.g., b

y u

sin

g o

bje

cts o

r

dra

win

gs t

o r

ep

re

se

nt t

he

pro

ble

m.

✔

✔

K.O

A.A

.3

De

co

mp

ose

nu

mb

ers le

ss t

ha

n o

r e

qu

al t

o 1

0 in

to

pa

irs in

mo

re

th

an

on

e w

ay, e

.g., b

y u

sin

g o

bje

cts o

r

dra

win

gs, a

nd

re

co

rd

ea

ch

de

co

mp

osit

ion

by a

dra

win

g o

r e

qu

atio

n (

e.g

., 5 =

2 +

3 a

nd

5 =

4 +

1).

✔

K.O

A.A

.4

Fo

r a

ny n

um

be

r f

ro

m 1

to

9, f

ind

th

e n

um

be

r t

ha

t m

ake

s 1

0 w

he

n a

dd

ed

to

th

e g

ive

n n

um

be

r, e

.g., b

y

usin

g o

bje

cts o

r d

ra

win

gs, a

nd

re

co

rd

th

e a

nsw

er w

ith

a d

ra

win

g o

r e

qu

atio

n.

✔

K.O

A.A

.5

Flu

en

tly

ad

d a

nd

su

btra

ct w

ith

in 5

.

✔

K.N

BT

.A.1

G

ain

un

de

rsta

nd

ing

of p

lace

va

lue

. ✔

K.N

BT

.A.1

a

Un

de

rsta

nd

th

at t

he

nu

mb

ers 1

1–

19

are

co

mp

ose

d o

f t

en

on

es a

nd

on

e, t

wo

, th

re

e, f

ou

r, f

ive

, six

, se

ve

n,

eig

ht, o

r n

ine

on

es.

✔

5

A Guide to Rigor

LSS

M –

Kin

de

rg

arte

n

Exp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill

an

d F

lue

ncy

Ap

plic

atio

n

K.N

BT

.A.1

a

Un

de

rsta

nd

th

at t

he

nu

mb

ers 1

1–

19

are

co

mp

ose

d o

f t

en

on

es a

nd

on

e, t

wo

, th

re

e, f

ou

r, f

ive

, six

, se

ve

n,

eig

ht, o

r n

ine

on

es.

✔

K.N

BT

.A.1

b

Co

mp

ose

an

d d

eco

mp

ose

nu

mb

ers 1

1 t

o 1

9 u

sin

g p

lace

va

lue

(e

.g., b

y u

sin

g o

bje

cts o

r d

ra

win

gs).

✔

K.N

BT

.A.1

c

Re

co

rd

ea

ch

co

mp

osit

ion

or d

eco

mp

osit

ion

usin

g a

dra

win

g o

r e

qu

atio

n (

e.g

., 18

is o

ne

te

n a

nd

eig

ht o

ne

s,

18

= 1

te

n +

8 o

ne

s, 1

8 =

10

+ 8

).

✔

K.M

D.A

.1

De

scrib

e m

ea

su

ra

ble

attrib

ute

s o

f o

bje

cts, s

uch

as le

ng

th

or w

eig

ht. D

escrib

e s

eve

ra

l me

asu

ra

ble

attrib

ute

s

of a

sin

gle

ob

ject.

✔

K.M

D.A

.2

Dir

ectly

co

mp

are

tw

o o

bje

cts w

ith

a m

ea

su

ra

ble

attrib

ute

in c

om

mo

n, t

o s

ee

wh

ich

ob

ject h

as “

mo

re

of”/“le

ss o

f” t

he

attrib

ute

, an

d d

escrib

e t

he

dif

fe

re

nce

. For example, directly com

pare the heights of two

children and describe one child as taller/shorter. ✔

K.M

D.B

.3

Cla

ssif

y o

bje

cts in

to

giv

en

ca

te

go

rie

s b

ase

d o

n t

he

ir a

ttrib

ute

s; c

ou

nt t

he

nu

mb

ers o

f o

bje

cts in

ea

ch

ca

te

go

ry a

nd

so

rt t

he

ca

te

go

rie

s b

y c

ou

nt.

✔

✔

K.M

D.C

.4

Re

co

gn

ize

pe

nn

ies, n

icke

ls, d

ime

s, a

nd

qu

arte

rs b

y n

am

e a

nd

va

lue

(e

.g., T

his

is a

nic

ke

l an

d it

is w

orth

5

ce

nts.)

✔

K.G

.A.1

D

escrib

e o

bje

cts in

th

e e

nvir

on

me

nt u

sin

g n

am

es o

f s

ha

pe

s, a

nd

de

scrib

e t

he

re

lativ

e p

osit

ion

s o

f t

he

se

ob

jects u

sin

g t

erm

s s

uch

as a

bo

ve

, be

low

, be

sid

e, in

fro

nt o

f, b

eh

ind

, an

d n

ext t

o.

✔

✔

K.G

.A.2

C

orre

ctly

na

me

sh

ap

es r

eg

ard

less o

f t

he

ir o

rie

nta

tio

ns o

r o

ve

ra

ll siz

e.

✔

K.G

.A.3

Id

en

tif

y s

ha

pe

s a

s t

wo

-d

ime

nsio

na

l (ly

ing

in a

pla

ne

, “fla

t”) o

r t

hre

e-d

ime

nsio

na

l (“so

lid”).

✔

K.G

.B.4

An

aly

ze

an

d c

om

pa

re

tw

o- a

nd

th

re

e-d

ime

nsio

na

l sh

ap

es, in

dif

fe

re

nt s

ize

s a

nd

orie

nta

tio

ns, u

sin

g in

fo

rm

al

lan

gu

ag

e t

o d

escrib

e t

he

ir s

imila

rit

ies, d

iffe

re

nce

s, p

arts (

e.g

., nu

mb

er o

f s

ide

s a

nd

ve

rtic

es/“co

rn

ers”) a

nd

oth

er a

ttrib

ute

s (

e.g

., ha

vin

g s

ide

s o

f e

qu

al le

ng

th

).

✔

K.G

.B.5

M

od

el s

ha

pe

s in

th

e w

orld

by b

uild

ing

sh

ap

es f

ro

m c

om

po

ne

nts (

e.g

., stic

ks a

nd

cla

y b

alls

) a

nd

dra

win

g

sh

ap

es.

✔

K.G

.B.6

C

om

po

se

sim

ple

sh

ap

es t

o f

orm

larg

er s

ha

pe

s. For exam

ple, "Can you join these tw

o triangles with full sides

touching to make a rectangle?

✔

6

A Guide to Rigor

1st Grade

LSSM – 1

st Grade Explicit Com

ponent(s) of Rigor

Code Standard

Conceptual U

nderstanding Procedural Skill

and Fluency Application

1.OA.A.1

Use addition and subtraction w

ithin 20 to solve word problem

s involving situations of adding to, taking from

, putting together, taking apart, and com

paring, with unknow

ns in all positions, e.g., by using objects, drawings, and equations w

ith a sym

bol for the unknown num

ber to represent the problem.

✔

1.OA.A.2

Solve word problem

s that call for addition of three whole num

bers whose sum

is less than or equal to 20, e.g., by using objects, draw

ings, and equations with a sym

bol for the unknown num

ber to represent the problem

.

✔

1.OA.B.3

Apply properties of operations to add and subtract. Examples: If 8 + 3 = 11 is know

n, then 3 + 8 = 11 is also know

n. (Comm

utative property of addition.) To add 2 + 6 + 4, the second two num

bers can be added to m

ake a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) ✔

1.OA.B.4

Understand subtraction as an unknow

n-addend problem. For exam

ple, subtract 10 – 8 by finding the num

ber that makes 10 w

hen added to 8. ✔

1.OA.C.5

Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). ✔

1.OA.C.6

Add and subtract within 20, dem

onstrating fluency for addition and subtraction within 10. U

se strategies such as counting on; m

aking ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a num

ber leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship betw

een addition and subtraction (e.g., know

ing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or know

n sums (e.g.,

adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

✔

✔

1.OA.D.7

Understand the m

eaning of the equal sign, and determine if equations involving addition and subtraction

are true or false. For example, w

hich of the following equations are true and w

hich are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

✔

✔

1.OA.D.8

Determine the unknow

n whole num

ber in an addition or subtraction equation relating three whole

numbers. For exam

ple, determine the unknow

n number that m

akes the equation true in each of the equations 8 + ? = 11, 5 = ? – 3, 6 + 6 = ?.

✔

1.NBT.A.1

Count to 120, starting at any number less than 120. In this range, read and w

rite numerals and represent a

number of objects w

ith a written num

eral. ✔

✔

1.NBT.B.2

Understand that the tw

o digits of a two-digit num

ber represent amounts of tens and ones. U

nderstand the follow

ing as special cases: ✔

1.NBT.B.2a

10 can be thought of as a bundle of ten ones — called a “ten.”

✔

1.N

BT

.B.2

b T

he

nu

mb

ers f

ro

m 1

1 t

o 1

9 a

re

co

mp

ose

d o

f a

te

n a

nd

on

e, t

wo

, th

re

e, f

ou

r, f

ive

, six

, se

ve

n, e

igh

t, o

r n

ine

on

es.

✔

1.N

BT

.B.2

c

Th

e n

um

be

rs 1

0, 2

0, 3

0, 4

0, 5

0, 6

0, 7

0, 8

0, 9

0 r

efe

r t

o o

ne

, tw

o, t

hre

e, f

ou

r, f

ive

, six

, se

ve

n, e

igh

t, o

r n

ine

te

ns (

an

d 0

on

es).

✔

7

A Guide to Rigor

LSS

M –

1st G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

1.N

BT

.B.3

C

om

pa

re

tw

o t

wo

-d

igit

nu

mb

ers b

ase

d o

n m

ea

nin

gs o

f t

he

te

ns a

nd

on

es d

igit

s, r

eco

rd

ing

th

e r

esu

lts o

f

co

mp

aris

on

s w

ith

th

e s

ym

bo

ls >

, =, a

nd

<.

✔

1.N

BT

.C.4

A

dd

wit

hin

10

0, in

clu

din

g a

dd

ing

a t

wo

-d

igit

nu

mb

er a

nd

a o

ne

-d

igit

nu

mb

er, a

nd

ad

din

g a

tw

o-d

igit

nu

mb

er a

nd

a m

ult

iple

of 1

0.

✔

1.N

BT

.C.4

a

Use

co

ncre

te

mo

de

ls o

r d

ra

win

gs a

nd

stra

te

gie

s b

ase

d o

n p

lace

va

lue

, pro

pe

rtie

s o

f o

pe

ra

tio

ns, a

nd

/o

r t

he

re

latio

nsh

ip b

etw

ee

n a

dd

itio

n a

nd

su

btra

ctio

n; r

ela

te

th

e s

tra

te

gy t

o a

nu

mb

er s

en

te

nce

; ju

stif

y t

he

re

aso

nin

g u

se

d w

ith

a w

rit

te

n e

xp

lan

atio

n.

✔

1.N

BT

.C.4

b

Un

de

rsta

nd

th

at in

ad

din

g t

wo

-d

igit

nu

mb

ers, o

ne

ad

ds t

en

s a

nd

te

ns, o

ne

s a

nd

on

es; a

nd

so

me

tim

es it

is

ne

ce

ssa

ry t

o c

om

po

se

a t

en

. ✔

1.N

BT

.C.5

G

ive

n a

tw

o-d

igit

nu

mb

er, m

en

ta

lly f

ind

10

mo

re

or 1

0 le

ss t

ha

n t

he

nu

mb

er, w

ith

ou

t h

avin

g t

o c

ou

nt;

exp

lain

th

e r

ea

so

nin

g u

se

d.

✔

✔

1.N

BT

.C.6

Su

btra

ct m

ult

iple

s o

f 1

0 in

th

e r

an

ge

10

-9

0 f

ro

m m

ult

iple

s o

f 1

0 in

th

e r

an

ge

10

-9

0 (

po

sit

ive

or z

ero

dif

fe

re

nce

s), u

sin

g c

on

cre

te

mo

de

ls o

r d

ra

win

gs a

nd

stra

te

gie

s b

ase

d o

n p

lace

va

lue

, pro

pe

rtie

s o

f

op

era

tio

ns, a

nd

/o

r t

he

re

latio

nsh

ip b

etw

ee

n a

dd

itio

n a

nd

su

btra

ctio

n; r

ela

te

th

e s

tra

te

gy t

o a

writ

te

n

me

th

od

an

d e

xp

lain

th

e r

ea

so

nin

g u

se

d.

✔

1.M

D.A

.1

Ord

er t

hre

e o

bje

cts b

y le

ng

th

; c

om

pa

re

th

e le

ng

th

s o

f t

wo

ob

jects in

dir

ectly

by u

sin

g a

th

ird

ob

ject.

✔

1.M

D.A

.2

Exp

re

ss t

he

len

gth

of a

n o

bje

ct a

s a

wh

ole

nu

mb

er o

f le

ng

th

un

its, b

y la

yin

g m

ult

iple

co

pie

s o

f a

sh

orte

r

ob

ject (

th

e le

ng

th

un

it) e

nd

to

en

d; u

nd

ersta

nd

th

at t

he

len

gth

me

asu

re

me

nt o

f a

n o

bje

ct is

th

e n

um

be

r o

f

sa

me

-siz

e le

ng

th

un

its t

ha

t s

pa

n it

wit

h n

o g

ap

s o

r o

ve

rla

ps. Lim

it to contexts where the object being

measured is spanned by a w

hole number of length units w

ith no gaps or overlaps.

✔

✔

1.M

D.B

.3

Te

ll an

d w

rit

e t

ime

in h

ou

rs a

nd

ha

lf-h

ou

rs u

sin

g a

na

log

an

d d

igit

al c

locks.

✔

✔

1.M

D.C

.4

Org

an

ize

, re

pre

se

nt, a

nd

inte

rp

re

t d

ata

wit

h u

p t

o t

hre

e c

ate

go

rie

s; a

sk a

nd

an

sw

er q

ue

stio

ns a

bo

ut t

he

to

ta

l nu

mb

er o

f d

ata

po

ints, h

ow

ma

ny in

ea

ch

ca

te

go

ry, a

nd

ho

w m

an

y m

ore

or le

ss a

re

in o

ne

ca

te

go

ry

th

an

in a

no

th

er.

✔

✔

1.M

D.D

.5

De

te

rm

ine

th

e v

alu

e o

f a

co

llectio

n o

f c

oin

s u

p t

o 5

0 c

en

ts. (

Pe

nn

ies, n

icke

ls, d

ime

s, a

nd

qu

arte

rs in

iso

latio

n; n

ot t

o in

clu

de

a c

om

bin

atio

n o

f d

iffe

re

nt c

oin

s.)

✔

1.G

.A.1

D

istin

gu

ish

be

tw

ee

n d

efin

ing

attrib

ute

s (

e.g

., tria

ng

les a

re

clo

se

d a

nd

th

re

e-sid

ed

) v

ersu

s n

on

-d

efin

ing

attrib

ute

s (

e.g

., co

lor, o

rie

nta

tio

n, o

ve

ra

ll siz

e); b

uild

an

d d

ra

w s

ha

pe

s t

o p

osse

ss d

efin

ing

attrib

ute

s.

✔

1.G

.A.2

Co

mp

ose

tw

o-d

ime

nsio

na

l sh

ap

es (

re

cta

ng

les, s

qu

are

s, t

ra

pe

zo

ids, t

ria

ng

les, h

alf

-cir

cle

s, a

nd

qu

arte

r-

cir

cle

s) a

nd

th

re

e-d

ime

nsio

na

l sh

ap

es (

cu

be

s, r

igh

t r

ecta

ng

ula

r p

ris

ms, r

igh

t c

ircu

lar c

on

es, a

nd

rig

ht

cir

cu

lar c

ylin

de

rs) t

o c

re

ate

a c

om

po

sit

e s

ha

pe

, an

d c

om

po

se

ne

w s

ha

pe

s f

ro

m t

he

co

mp

osit

e s

ha

pe

.

✔

1.G

.A.3

Pa

rtit

ion

cir

cle

s a

nd

re

cta

ng

les in

to

tw

o a

nd

fo

ur e

qu

al s

ha

re

s, d

escrib

e t

he

sh

are

s u

sin

g t

he

wo

rd

s h

alv

es,

fo

urth

s, a

nd

qu

arte

rs, a

nd

use

th

e p

hra

se

s h

alf

of, f

ou

rth

of, a

nd

qu

arte

r o

f. D

escrib

e t

he

wh

ole

as t

wo

of,

or f

ou

r o

f t

he

sh

are

s. U

nd

ersta

nd

fo

r t

he

se

exa

mp

les t

ha

t d

eco

mp

osin

g in

to

mo

re

eq

ua

l sh

are

s c

re

ate

s

sm

alle

r s

ha

re

s.

✔

✔

8

A Guide to Rigor

2nd Grade

LSS

M –

2n

d G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill

an

d F

lue

ncy

Ap

plic

atio

n

2.O

A.A

.1

Use

ad

dit

ion

an

d s

ub

tra

ctio

n w

ith

in 1

00

to

so

lve

on

e- a

nd

tw

o-ste

p w

ord

pro

ble

ms in

vo

lvin

g s

itu

atio

ns o

f

ad

din

g t

o, t

akin

g f

ro

m, p

uttin

g t

og

eth

er, t

akin

g a

pa

rt, a

nd

co

mp

arin

g, w

ith

un

kn

ow

ns in

all p

osit

ion

s, e

.g.,

by u

sin

g d

ra

win

gs a

nd

eq

ua

tio

ns w

ith

a s

ym

bo

l fo

r t

he

un

kn

ow

n n

um

be

r t

o r

ep

re

se

nt t

he

pro

ble

m.

✔

2.O

A.B

.2

Flu

en

tly

ad

d a

nd

su

btra

ct w

ith

in 2

0 u

sin

g m

en

ta

l stra

te

gie

s. B

y e

nd

of G

ra

de

2, k

no

w f

ro

m m

em

ory

all

su

ms o

f t

wo

on

e-d

igit

nu

mb

ers.

✔

2.O

A.C

.3

De

te

rm

ine

wh

eth

er a

gro

up

of o

bje

cts (

up

to

20

) h

as a

n o

dd

or e

ve

n n

um

be

r o

f m

em

be

rs, e

.g., b

y p

air

ing

ob

jects o

r c

ou

ntin

g t

he

m b

y 2

s; w

rit

e a

n e

qu

atio

n t

o e

xp

re

ss a

n e

ve

n n

um

be

r a

s a

su

m o

f t

wo

eq

ua

l

ad

de

nd

s.

✔

2.O

A.C

.4

Use

ad

dit

ion

to

fin

d t

he

to

ta

l nu

mb

er o

f o

bje

cts a

rra

ng

ed

in r

ecta

ng

ula

r a

rra

ys w

ith

up

to

5 r

ow

s a

nd

up

to

5 c

olu

mn

s; w

rit

e a

n e

qu

atio

n t

o e

xp

re

ss t

he

to

ta

l as a

su

m o

f e

qu

al a

dd

en

ds.

✔

2.N

BT

.A.1

U

nd

ersta

nd

th

at t

he

th

re

e d

igit

s o

f a

th

re

e-d

igit

nu

mb

er r

ep

re

se

nt a

mo

un

ts o

f h

un

dre

ds, t

en

s, a

nd

on

es;

e.g

., 70

6 e

qu

als

7 h

un

dre

ds, 0

te

ns, a

nd

6 o

ne

s. U

nd

ersta

nd

th

e f

ollo

win

g a

s s

pe

cia

l ca

se

s:

✔

2.N

BT

.A.1

a

10

0 c

an

be

th

ou

gh

t o

f a

s a

bu

nd

le o

f t

en

te

ns —

ca

lled

a “

hu

nd

re

d.”

✔

2.N

BT

.A.1

b

Th

e n

um

be

rs 1

00

, 20

0, 3

00

, 40

0, 5

00

, 60

0, 7

00

, 80

0, 9

00

re

fe

r t

o o

ne

, tw

o, t

hre

e, f

ou

r, f

ive

, six

, se

ve

n,

eig

ht, o

r n

ine

hu

nd

re

ds (

an

d 0

te

ns a

nd

0 o

ne

s).

✔

2.N

BT

.A.2

C

ou

nt w

ith

in 1

00

0; s

kip

-co

un

t b

y 5

s, 1

0s, a

nd

10

0s.

✔

2.N

BT

.A.3

R

ea

d a

nd

writ

e n

um

be

rs t

o 1

00

0 u

sin

g b

ase

-te

n n

um

era

ls, n

um

be

r n

am

es, a

nd

exp

an

de

d f

orm

. ✔

2.N

BT

.A.4

C

om

pa

re

tw

o t

hre

e-d

igit

nu

mb

ers b

ase

d o

n m

ea

nin

gs o

f t

he

hu

nd

re

ds, t

en

s, a

nd

on

es d

igit

s, u

sin

g >

, =,

an

d <

sym

bo

ls t

o r

eco

rd

th

e r

esu

lts o

f c

om

pa

ris

on

s.

✔

2.N

BT

.B.5

F

lue

ntly

ad

d a

nd

su

btra

ct w

ith

in 1

00

usin

g s

tra

te

gie

s b

ase

d o

n p

lace

va

lue

, pro

pe

rtie

s o

f o

pe

ra

tio

ns, a

nd

/o

r

th

e r

ela

tio

nsh

ip b

etw

ee

n a

dd

itio

n a

nd

su

btra

ctio

n.

✔

2.N

BT

.B.6

A

dd

up

to

fo

ur t

wo

-d

igit

nu

mb

ers u

sin

g s

tra

te

gie

s b

ase

d o

n p

lace

va

lue

an

d p

ro

pe

rtie

s o

f o

pe

ra

tio

ns.

✔

2.N

BT

.B.7

Ad

d a

nd

su

btra

ct w

ith

in 1

00

0, u

sin

g c

on

cre

te

mo

de

ls o

r d

ra

win

gs a

nd

stra

te

gie

s b

ase

d o

n p

lace

va

lue

,

pro

pe

rtie

s o

f o

pe

ra

tio

ns, a

nd

/o

r t

he

re

latio

nsh

ip b

etw

ee

n a

dd

itio

n a

nd

su

btra

ctio

n; ju

stif

y t

he

re

aso

nin

g

use

d w

ith

a w

rit

te

n e

xp

lan

atio

n. U

nd

ersta

nd

th

at in

ad

din

g o

r s

ub

tra

ctin

g t

hre

e- d

igit

nu

mb

ers, o

ne

ad

ds o

r

su

btra

cts h

un

dre

ds a

nd

hu

nd

re

ds, t

en

s a

nd

te

ns, o

ne

s a

nd

on

es; a

nd

so

me

tim

es it

is n

ece

ssa

ry t

o c

om

po

se

or d

eco

mp

ose

te

ns o

r h

un

dre

ds.

✔

2.N

BT

.B.8

M

en

ta

lly a

dd

10

or 1

00

to

a g

ive

n n

um

be

r 1

00

–9

00

, an

d m

en

ta

lly s

ub

tra

ct 1

0 o

r 1

00

fro

m a

giv

en

nu

mb

er

10

0–

90

0.

✔

9

A Guide to Rigor

LSS

M –

2n

d G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill

an

d F

lue

ncy

Ap

plic

atio

n

2.N

BT

.B.9

E

xp

lain

wh

y a

dd

itio

n a

nd

su

btra

ctio

n s

tra

te

gie

s w

ork, u

sin

g p

lace

va

lue

an

d t

he

pro

pe

rtie

s o

f o

pe

ra

tio

ns.

✔

2.M

D.A

.1

Me

asu

re

th

e le

ng

th

of a

n o

bje

ct b

y s

ele

ctin

g a

nd

usin

g a

pp

ro

pria

te

to

ols

su

ch

as r

ule

rs, y

ard

stic

ks, m

ete

r

stic

ks, a

nd

me

asu

rin

g t

ap

es.

✔

2.M

D.A

.2

Me

asu

re

th

e le

ng

th

of a

n o

bje

ct t

wic

e, u

sin

g le

ng

th

un

its o

f d

iffe

re

nt le

ng

th

s f

or t

he

tw

o m

ea

su

re

me

nts;

de

scrib

e h

ow

th

e t

wo

me

asu

re

me

nts r

ela

te

to

th

e s

ize

of t

he

un

it c

ho

se

n.

✔

✔

2.M

D.A

.3

Estim

ate

len

gth

s u

sin

g u

nit

s o

f in

ch

es, f

ee

t, c

en

tim

ete

rs, a

nd

me

te

rs.

✔

2.M

D.A

.4

Me

asu

re

to

de

te

rm

ine

ho

w m

uch

lon

ge

r o

ne

ob

ject is

th

an

an

oth

er, e

xp

re

ssin

g t

he

len

gth

dif

fe

re

nce

in

te

rm

s o

f a

sta

nd

ard

len

gth

un

it.

✔

2.M

D.B

.5

Use

ad

dit

ion

an

d s

ub

tra

ctio

n w

ith

in 1

00

to

so

lve

wo

rd

pro

ble

ms in

vo

lvin

g le

ng

th

s t

ha

t a

re

giv

en

in t

he

sa

me

un

its, e

.g., b

y u

sin

g d

ra

win

gs (

su

ch

as d

ra

win

gs o

f r

ule

rs) a

nd

eq

ua

tio

ns w

ith

a s

ym

bo

l fo

r t

he

un

kn

ow

n n

um

be

r t

o r

ep

re

se

nt t

he

pro

ble

m.

✔

2.M

D.B

.6

Re

pre

se

nt w

ho

le n

um

be

rs a

s le

ng

th

s f

ro

m 0

on

a n

um

be

r lin

e d

iag

ra

m w

ith

eq

ua

lly s

pa

ce

d p

oin

ts

co

rre

sp

on

din

g t

o t

he

nu

mb

ers 0

, 1, 2

, ..., an

d r

ep

re

se

nt w

ho

le-n

um

be

r s

um

s a

nd

dif

fe

re

nce

s w

ith

in 1

00

on

a n

um

be

r lin

e d

iag

ra

m.

✔

2.M

D.C

.7

Te

ll an

d w

rit

e t

ime

fro

m a

na

log

an

d d

igit

al c

locks t

o t

he

ne

are

st f

ive

min

ute

s, u

sin

g a

.m. a

nd

p.m

. ✔

✔

2.M

D.C

.8

So

lve

wo

rd

pro

ble

ms in

vo

lvin

g d

olla

r b

ills, q

ua

rte

rs, d

ime

s, n

icke

ls, a

nd

pe

nn

ies, u

sin

g $

an

d ¢

sym

bo

ls

ap

pro

pria

te

ly. Exam

ple: If you have 2 dimes and 3 pennies, how

many cents do you have?

✔

2.M

D.D

.9

Ge

ne

ra

te

me

asu

re

me

nt d

ata

by m

ea

su

rin

g le

ng

th

s o

f s

eve

ra

l ob

jects t

o t

he

ne

are

st w

ho

le u

nit

, or b

y

ma

kin

g r

ep

ea

te

d m

ea

su

re

me

nts o

f t

he

sa

me

ob

ject. S

ho

w t

he

me

asu

re

me

nts b

y m

akin

g a

line

plo

t, w

he

re

th

e h

oriz

on

ta

l sca

le is

ma

rke

d o

ff in

wh

ole

-n

um

be

r u

nit

s.

✔

2.M

D.D

.10

Dra

w a

pic

tu

re

gra

ph

an

d a

ba

r g

ra

ph

(w

ith

sin

gle

-u

nit

sca

le) t

o r

ep

re

se

nt a

da

ta

se

t w

ith

up

to

fo

ur

ca

te

go

rie

s. S

olv

e s

imp

le p

ut- t

og

eth

er, t

ake

-a

pa

rt, a

nd

co

mp

are

pro

ble

ms u

sin

g in

fo

rm

atio

n p

re

se

nte

d in

a b

ar g

ra

ph

.

✔

2.G

.A.1

R

eco

gn

ize

an

d d

ra

w s

ha

pe

s h

avin

g s

pe

cif

ied

attrib

ute

s, s

uch

as a

giv

en

nu

mb

er o

f a

ng

les o

r a

giv

en

nu

mb

er o

f e

qu

al f

ace

s. Id

en

tif

y t

ria

ng

les, q

ua

drila

te

ra

ls, p

en

ta

go

ns, h

exa

go

ns, a

nd

cu

be

s.

✔

2.G

.A.2

P

artit

ion

a r

ecta

ng

le in

to

ro

ws a

nd

co

lum

ns o

f s

am

e-siz

e s

qu

are

s a

nd

co

un

t t

o f

ind

th

e t

ota

l nu

mb

er o

f

th

em

. ✔

✔

2.G

.A.3

Pa

rtit

ion

cir

cle

s a

nd

re

cta

ng

les in

to

tw

o, t

hre

e, o

r f

ou

r e

qu

al s

ha

re

s, d

escrib

e t

he

sh

are

s u

sin

g t

he

wo

rd

s

ha

lve

s, t

hir

ds, h

alf

of, a

th

ird

of, e

tc., a

nd

de

scrib

e t

he

wh

ole

as t

wo

ha

lve

s, t

hre

e t

hir

ds, f

ou

r f

ou

rth

s.

Re

co

gn

ize

th

at e

qu

al s

ha

re

s o

f id

en

tic

al w

ho

les n

ee

d n

ot h

ave

th

e s

am

e s

ha

pe

.

✔

✔

10

A Guide to Rigor

3rd Grade

LSS

M –

3rd G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

3.O

A.A

.1

Inte

rp

re

t p

ro

du

cts o

f w

ho

le n

um

be

rs, e

.g., in

te

rp

re

t 5

× 7

as t

he

to

ta

l nu

mb

er o

f o

bje

cts in

5 g

ro

up

s o

f 7

ob

jects e

ach

. For example, describe a context in w

hich a total number of objects can be expressed as 5 ×

7.

✔

3.O

A.A

.2

Inte

rp

re

t w

ho

le-n

um

be

r q

uo

tie

nts o

f w

ho

le n

um

be

rs, e

.g., in

te

rp

re

t 5

6 ÷

8 a

s t

he

nu

mb

er o

f o

bje

cts in

ea

ch

sh

are

wh

en

56

ob

jects a

re

pa

rtit

ion

ed

eq

ua

lly in

to

8 s

ha

re

s, o

r a

s a

nu

mb

er o

f s

ha

re

s w

he

n 5

6

ob

jects a

re

pa

rtit

ion

ed

into

eq

ua

l sh

are

s o

f 8

ob

jects e

ach

. For exam

ple, describe a context in which a

number of shares o

r a number of groups can be expressed as 56 ÷ 8

✔

3.O

A.A

.3

Use

mu

ltip

lica

tio

n a

nd

div

isio

n w

ith

in 1

00

to

so

lve

wo

rd

pro

ble

ms in

sit

ua

tio

ns in

vo

lvin

g e

qu

al g

ro

up

s,

arra

ys, a

nd

me

asu

re

me

nt q

ua

ntit

ies, e

.g., b

y u

sin

g d

ra

win

gs a

nd

eq

ua

tio

ns w

ith

a s

ym

bo

l fo

r t

he

un

kn

ow

n n

um

be

r t

o r

ep

re

se

nt t

he

pro

ble

m.

✔

3.O

A.A

.4

De

te

rm

ine

th

e u

nkn

ow

n w

ho

le n

um

be

r in

a m

ult

iplic

atio

n o

r d

ivis

ion

eq

ua

tio

n r

ela

tin

g t

hre

e w

ho

le

nu

mb

ers. For exam

ple, determine the unknow

n number that m

akes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?

✔

3.O

A.B

.5

Ap

ply

pro

pe

rtie

s o

f o

pe

ra

tio

ns a

s s

tra

te

gie

s t

o m

ult

iply

an

d d

ivid

e.2 Exam

ples: If 6 × 4 = 24 is known, then

4 × 6 = 24 is also known. (C

omm

utative property of m

ultiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10

, then 3 × 10 = 30. (Asso

ciative property of m

ultiplication.) Knowing that

8 × 5 = 40 and 8 × 2 = 16, one can

find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (D

istributive property.)

✔

3.O

A.B

.6

Un

de

rsta

nd

div

isio

n a

s a

n u

nkn

ow

n-fa

cto

r p

ro

ble

m. For exam

ple, find 32 ÷ 8 by finding the number that

makes 32 w

hen multiplied by 8

. ✔

3.O

A.C

.7

Flu

en

tly

mu

ltip

ly a

nd

div

ide

wit

hin

10

0, u

sin

g s

tra

te

gie

s s

uch

as t

he

re

latio

nsh

ip b

etw

ee

n m

ult

iplic

atio

n

an

d d

ivis

ion

(e

.g., k

no

win

g t

ha

t 8

× 5

= 4

0, o

ne

kn

ow

s 4

0 ÷

5 =

8) o

r p

ro

pe

rtie

s o

f o

pe

ra

tio

ns. B

y t

he

en

d

of G

ra

de

3, k

no

w f

ro

m m

em

ory a

ll pro

du

cts o

f t

wo

on

e-d

igit

nu

mb

ers.

✔

3.O

A.D

.8

So

lve

tw

o-ste

p w

ord

pro

ble

ms u

sin

g t

he

fo

ur o

pe

ra

tio

ns. R

ep

re

se

nt t

he

se

pro

ble

ms u

sin

g e

qu

atio

ns w

ith

a le

tte

r s

ta

nd

ing

fo

r t

he

un

kn

ow

n q

ua

ntit

y. A

sse

ss t

he

re

aso

na

ble

ne

ss o

f a

nsw

ers u

sin

g m

en

ta

l

co

mp

uta

tio

n a

nd

estim

atio

n s

tra

te

gie

s in

clu

din

g r

ou

nd

ing

.

✔

✔

3.O

A.D

.9

Ide

ntif

y a

rit

hm

etic

pa

tte

rn

s (

inclu

din

g p

atte

rn

s in

th

e a

dd

itio

n t

ab

le o

r m

ult

iplic

atio

n t

ab

le), a

nd

exp

lain

th

em

usin

g p

ro

pe

rtie

s o

f o

pe

ra

tio

ns. For exam

ple, observe that 4 times a num

ber is always even, and

explain w

hy 4 times a num

ber can be decomposed into tw

o equal addends. ✔

3.N

BT

.A.1

U

se

pla

ce

va

lue

un

de

rsta

nd

ing

to

ro

un

d w

ho

le n

um

be

rs t

o t

he

ne

are

st 1

0 o

r 1

00

. ✔

11

A Guide to Rigor

LSS

M –

3rd G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

3.N

BT

.A.2

F

lue

ntly

ad

d a

nd

su

btra

ct w

ith

in 1

00

0 u

sin

g s

tra

te

gie

s a

nd

alg

orit

hm

s b

ase

d o

n p

lace

va

lue

, pro

pe

rtie

s o

f

op

era

tio

ns, a

nd

/o

r t

he

re

latio

nsh

ip b

etw

ee

n a

dd

itio

n a

nd

su

btra

ctio

n.

✔

3.N

BT

.A.3

M

ult

iply

on

e-d

igit

wh

ole

nu

mb

ers b

y m

ult

iple

s o

f 1

0 in

th

e r

an

ge

10

–9

0 (

e.g

., 9 ×

80

, 5 ×

60

) u

sin

g

stra

te

gie

s b

ase

d o

n p

lace

va

lue

an

d p

ro

pe

rtie

s o

f o

pe

ra

tio

ns.

✔

3.N

F.A

.1

Un

de

rsta

nd

a f

ra

ctio

n 1

/b,

wit

h d

en

om

ina

to

rs 2

, 3, 4

, 6, a

nd

8, a

s t

he

qu

an

tit

y f

orm

ed

by 1

pa

rt w

he

n a

wh

ole

is p

artit

ion

ed

into

b e

qu

al p

arts; u

nd

ersta

nd

a f

ra

ctio

n a

/b

as t

he

qu

an

tit

y f

orm

ed

by a

pa

rts o

f

siz

e 1

/b

.

✔

3.N

F.A

.2

Un

de

rsta

nd

a f

ra

ctio

n w

ith

de

no

min

ato

rs 2

, 3, 4

, 6, a

nd

8 a

s a

nu

mb

er o

n t

he

nu

mb

er lin

e; r

ep

re

se

nt

fra

ctio

ns o

n a

nu

mb

er lin

e d

iag

ra

m.

✔

3.N

F.A

.2a

Re

pre

se

nt a

fra

ctio

n 1

/b

on

a n

um

be

r lin

e d

iag

ra

m b

y d

efin

ing

th

e in

te

rva

l fro

m 0

to

1 a

s t

he

wh

ole

an

d

pa

rtit

ion

ing

it in

to

b e

qu

al p

arts. R

eco

gn

ize

th

at e

ach

pa

rt h

as s

ize

1/b

an

d t

ha

t t

he

en

dp

oin

t o

f t

he

pa

rt

ba

se

d a

t 0

loca

te

s t

he

nu

mb

er 1

/b

on

th

e n

um

be

r lin

e.

✔

3.N

F.A

.2b

R

ep

re

se

nt a

fra

ctio

n a

/b

on

a n

um

be

r lin

e d

iag

ra

m b

y m

arkin

g o

ff a

len

gth

s 1

/b

fro

m 0

. Re

co

gn

ize

th

at

th

e r

esu

ltin

g in

te

rva

l ha

s s

ize

a/b

an

d t

ha

t it

s e

nd

po

int lo

ca

te

s t

he

nu

mb

er a

/b

on

th

e n

um

be

r lin

e.

✔

3.N

F.A

.3

Exp

lain

eq

uiv

ale

nce

of f

ra

ctio

ns w

ith

de

no

min

ato

rs 2

, 3, 4

, 6, a

nd

8 in

sp

ecia

l ca

se

s, a

nd

co

mp

are

fra

ctio

ns b

y r

ea

so

nin

g a

bo

ut t

he

ir s

ize

. ✔

3.N

F.A

.3a

U

nd

ersta

nd

tw

o f

ra

ctio

ns a

s e

qu

iva

len

t (

eq

ua

l) if

th

ey a

re

th

e s

am

e s

ize

, or t

he

sa

me

po

int o

n a

nu

mb

er

line

. ✔

3.N

F.A

.3b

R

eco

gn

ize

an

d g

en

era

te

sim

ple

eq

uiv

ale

nt f

ra

ctio

ns, e

.g., 1

/2

= 2

/4

, 4/6

= 2

/3

). E

xp

lain

wh

y t

he

fra

ctio

ns

are

eq

uiv

ale

nt, e

.g., b

y u

sin

g a

vis

ua

l fra

ctio

n m

od

el.

✔

3.N

F.A

.3c

Exp

re

ss w

ho

le n

um

be

rs a

s f

ra

ctio

ns, a

nd

re

co

gn

ize

fra

ctio

ns t

ha

t a

re

eq

uiv

ale

nt t

o w

ho

le n

um

be

rs.

Examples: Express 3 in the form

3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a

number line diagram

. ✔

3.N

F.A

.3d

Co

mp

are

tw

o f

ra

ctio

ns w

ith

th

e s

am

e n

um

era

to

r o

r t

he

sa

me

de

no

min

ato

r b

y r

ea

so

nin

g a

bo

ut t

he

ir s

ize

.

Re

co

gn

ize

th

at c

om

pa

ris

on

s a

re

va

lid o

nly

wh

en

th

e t

wo

fra

ctio

ns r

efe

r t

o t

he

sa

me

wh

ole

. Re

co

rd

th

e

re

su

lts o

f c

om

pa

ris

on

s w

ith

th

e s

ym

bo

ls >

, =, o

r <

, an

d ju

stif

y t

he

co

nclu

sio

ns, e

.g., b

y u

sin

g a

vis

ua

l

fra

ctio

n m

od

el.

✔

3.M

D.A

.1

Un

de

rsta

nd

tim

e t

o t

he

ne

are

st m

inu

te

. ✔

3.M

D.A

.1a

T

ell a

nd

writ

e t

ime

to

th

e n

ea

re

st m

inu

te

an

d m

ea

su

re

tim

e in

te

rva

ls in

min

ute

s, w

ith

in 6

0 m

inu

te

s, o

n

an

an

alo

g a

nd

dig

ita

l clo

ck.

✔

✔

3.M

D.A

.1b

C

alc

ula

te

ela

pse

d t

ime

gre

ate

r t

ha

n 6

0 m

inu

te

s t

o t

he

ne

are

st q

ua

rte

r a

nd

ha

lf h

ou

r o

n a

nu

mb

er lin

e

dia

gra

m.

✔

3.M

D.A

.1c

So

lve

wo

rd

pro

ble

ms in

vo

lvin

g a

dd

itio

n a

nd

su

btra

ctio

n o

f t

ime

inte

rva

ls in

min

ute

s, e

.g., b

y r

ep

re

se

ntin

g

th

e p

ro

ble

m o

n a

nu

mb

er lin

e d

iag

ra

m.

✔

12

A Guide to Rigor

LSS

M –

3rd G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

3.M

D.A

.2

Me

asu

re

an

d e

stim

ate

liqu

id v

olu

me

s a

nd

ma

sse

s o

f o

bje

cts u

sin

g s

ta

nd

ard

un

its o

f g

ra

ms (

g), k

ilog

ra

ms

(kg

), a

nd

lite

rs (

l). A

dd

, su

btra

ct, m

ult

iply

, or d

ivid

e t

o s

olv

e o

ne

-ste

p w

ord

pro

ble

ms in

vo

lvin

g m

asse

s o

r

vo

lum

es t

ha

t a

re

giv

en

in t

he

sa

me

un

its, e

.g., b

y u

sin

g d

ra

win

gs (

su

ch

as a

be

ake

r w

ith

a m

ea

su

re

me

nt

sca

le) t

o r

ep

re

se

nt t

he

pro

ble

m.

✔

✔

✔

3.M

D.B

.3

Dra

w a

sca

led

pic

tu

re

gra

ph

an

d a

sca

led

ba

r g

ra

ph

to

re

pre

se

nt a

da

ta

se

t w

ith

se

ve

ra

l ca

te

go

rie

s. S

olv

e

on

e- a

nd

tw

o-ste

p “

ho

w m

an

y m

ore

” a

nd

“h

ow

ma

ny le

ss” p

ro

ble

ms u

sin

g in

fo

rm

atio

n p

re

se

nte

d in

sca

led

ba

r g

ra

ph

s. For exam

ple, draw

a bar graph in which each sq

uare in the bar graph might represent

5 pets.

✔

3.M

D.B

.4

Ge

ne

ra

te

me

asu

re

me

nt d

ata

by m

ea

su

rin

g le

ng

th

s u

sin

g r

ule

rs m

arke

d w

ith

ha

lve

s a

nd

fo

urth

s o

f a

n

inch

. Sh

ow

th

e d

ata

by m

akin

g a

line

plo

t, w

he

re

th

e h

oriz

on

ta

l sca

le is

ma

rke

d o

ff in

ap

pro

pria

te

un

its—

wh

ole

nu

mb

ers, h

alv

es, o

r q

ua

rte

rs.

✔

3.M

D.C

.5

Re

co

gn

ize

are

a a

s a

n a

ttrib

ute

of p

lan

e f

igu

re

s a

nd

un

de

rsta

nd

co

nce

pts o

f a

re

a m

ea

su

re

me

nt.

✔

3.M

D.C

.5a

A

sq

ua

re

wit

h s

ide

len

gth

1 u

nit

, ca

lled

“a

un

it s

qu

are

,” is

sa

id t

o h

ave

“o

ne

sq

ua

re

un

it” o

f a

re

a, a

nd

ca

n

be

use

d t

o m

ea

su

re

are

a.

✔

3.M

D.C

.5b

A

pla

ne

fig

ure

wh

ich

ca

n b

e c

ove

re

d w

ith

ou

t g

ap

s o

r o

ve

rla

ps b

y n

un

it s

qu

are

s is

sa

id t

o h

ave

an

are

a o

f

n s

qu

are

un

its.

✔

3.M

D.C

.6

Me

asu

re

are

as b

y c

ou

ntin

g u

nit

sq

ua

re

s (

sq

ua

re

cm

, sq

ua

re

m, s

qu

are

in, s

qu

are

ft, a

nd

imp

ro

vis

ed

un

its).

✔

3.M

D.C

.7

Re

late

are

a t

o t

he

op

era

tio

ns o

f m

ult

iplic

atio

n a

nd

ad

dit

ion

. ✔

3.M

D.C

.7a

F

ind

th

e a

re

a o

f a

re

cta

ng

le w

ith

wh

ole

-n

um

be

r s

ide

len

gth

s b

y t

iling

it, a

nd

sh

ow

th

at t

he

are

a is

th

e

sa

me

as w

ou

ld b

e f

ou

nd

by m

ult

iply

ing

th

e s

ide

len

gth

s.

✔

✔

3.M

D.C

.7b

Mu

ltip

ly s

ide

len

gth

s t

o f

ind

are

as o

f r

ecta

ng

les w

ith

wh

ole

- n

um

be

r s

ide

len

gth

s in

th

e c

on

te

xt o

f s

olv

ing

re

al-

wo

rld

an

d m

ath

em

atic

al p

ro

ble

ms, a

nd

re

pre

se

nt w

ho

le-n

um

be

r p

ro

du

cts a

s r

ecta

ng

ula

r a

re

as in

ma

th

em

atic

al r

ea

so

nin

g.

✔

✔

✔

3.M

D.C

.7c

Use

tilin

g t

o s

ho

w in

a c

on

cre

te

ca

se

th

at t

he

are

a o

f a

re

cta

ng

le w

ith

wh

ole

-n

um

be

r s

ide

len

gth

s a

an

d b

+ c is

th

e s

um

of a + b

an

d a + c

. Use

are

a m

od

els

to

re

pre

se

nt t

he

dis

trib

utiv

e p

ro

pe

rty in

ma

th

em

atic

al

re

aso

nin

g.

✔

3.M

D.D

.8

So

lve

re

al-

wo

rld

an

d m

ath

em

atic

al p

ro

ble

ms in

vo

lvin

g p

erim

ete

rs

of p

oly

go

ns, in

clu

din

g f

ind

ing

th

e p

erim

ete

r g

ive

n t

he

sid

e le

ng

th

s, f

ind

ing

an

un

kn

ow

n s

ide

len

gth

, an

d

exh

ibit

ing

re

cta

ng

les w

ith

th

e s

am

e p

erim

ete

r a

nd

dif

fe

re

nt a

re

as o

r w

ith

th

e s

am

e a

re

a a

nd

dif

fe

re

nt

pe

rim

ete

rs.

✔

✔

3.M

D.E

.9

So

lve

wo

rd

pro

ble

ms in

vo

lvin

g p

en

nie

s, n

icke

ls, d

ime

s, q

ua

rte

rs, a

nd

bills

gre

ate

r t

ha

n o

ne

do

llar, u

sin

g

th

e d

olla

r a

nd

ce

nt s

ym

bo

ls a

pp

ro

pria

te

ly.

✔

13

A Guide to Rigor

LSS

M –

3rd G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

3.G

.A.1

Un

de

rsta

nd

th

at s

ha

pe

s in

dif

fe

re

nt c

ate

go

rie

s (

e.g

., rh

om

bu

se

s, r

ecta

ng

les, a

nd

oth

ers) m

ay s

ha

re

attrib

ute

s (

e.g

., ha

vin

g f

ou

r s

ide

s), a

nd

th

at t

he

sh

are

d a

ttrib

ute

s c

an

de

fin

e a

larg

er c

ate

go

ry (

e.g

.,

qu

ad

rila

te

ra

ls). R

eco

gn

ize

rh

om

bu

se

s, r

ecta

ng

les, a

nd

sq

ua

re

s a

s e

xa

mp

les o

f q

ua

drila

te

ra

ls, a

nd

dra

w

exa

mp

les o

f q

ua

drila

te

ra

ls t

ha

t d

o n

ot b

elo

ng

to

an

y o

f t

he

se

su

bca

te

go

rie

s.

✔

3.G

.A.2

Pa

rtit

ion

sh

ap

es in

to

pa

rts w

ith

eq

ua

l are

as. E

xp

re

ss t

he

are

a o

f e

ach

pa

rt a

s a

un

it f

ra

ctio

n o

f t

he

wh

ole

.

For example, partition a shape into 4 parts w

ith equal area, and describe the area of each part as 1/4 of

the area of the shape. ✔

✔

14

A Guide to Rigor

4th Grade

LSS

M –

4th G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

4.O

A.A

.1

Inte

rp

re

t a

mu

ltip

lica

tio

n e

qu

atio

n a

s a

co

mp

aris

on

an

d r

ep

re

se

nt v

erb

al s

ta

te

me

nts o

f m

ult

iplic

ativ

e

co

mp

aris

on

s a

s m

ult

iplic

atio

n e

qu

atio

ns, e

.g., in

te

rp

re

t 3

5 =

5 x

7 a

s a

sta

te

me

nt t

ha

t 3

5 is

5 t

ime

s a

s

ma

ny a

s 7

, an

d 7

tim

es a

s m

an

y a

s 5

.

✔

4.O

A.A

.2

Mu

ltip

ly o

r d

ivid

e t

o s

olv

e w

ord

pro

ble

ms in

vo

lvin

g m

ult

iplic

ativ

e c

om

pa

ris

on

, e.g

., by u

sin

g d

ra

win

gs

an

d e

qu

atio

ns w

ith

a s

ym

bo

l fo

r t

he

un

kn

ow

n n

um

be

r t

o r

ep

re

se

nt t

he

pro

ble

m, d

istin

gu

ish

ing

mu

ltip

lica

tiv

e c

om

pa

ris

on

fro

m a

dd

itiv

e c

om

pa

ris

on

(E

xa

mp

le: 6

tim

es a

s m

an

y v

s. 6

mo

re

th

an

).

✔

4.O

A.A

.3

So

lve

mu

lti-

ste

p w

ord

pro

ble

ms p

ose

d w

ith

wh

ole

nu

mb

ers a

nd

ha

vin

g w

ho

le-n

um

be

r a

nsw

ers u

sin

g t

he

fo

ur o

pe

ra

tio

ns, in

clu

din

g p

ro

ble

ms in

wh

ich

re

ma

ind

ers m

ust b

e in

te

rp

re

te

d. R

ep

re

se

nt t

he

se

pro

ble

ms

usin

g e

qu

atio

ns w

ith

a le

tte

r s

ta

nd

ing

fo

r t

he

un

kn

ow

n q

ua

ntit

y. A

sse

ss t

he

re

aso

na

ble

ne

ss o

f a

nsw

ers

usin

g m

en

ta

l co

mp

uta

tio

n a

nd

estim

atio

n s

tra

te

gie

s in

clu

din

g r

ou

nd

ing

. Example: Tw

enty-five people are going to the m

ovies. Four people fit in each car. How

many cars a

re needed to get all 25 people to the theater at the sam

e time?

✔

✔

4.O

A.B

.4

Usin

g w

ho

le n

um

be

rs in

th

e r

an

ge

1–

10

0,

4.O

A.B

.4a

F

ind

all f

acto

r p

air

s f

or a

giv

en

wh

ole

nu

mb

er.

✔

4.O

A.B

.4b

R

eco

gn

ize

th

at a

giv

en

wh

ole

nu

mb

er is

a m

ult

iple

of e

ach

of it

s f

acto

rs.

✔

4.O

A.B

.4c

De

te

rm

ine

wh

eth

er a

giv

en

wh

ole

nu

mb

er is

a m

ult

iple

of a

giv

en

on

e-d

igit

nu

mb

er.

✔

4.O

A.B

.4d

D

ete

rm

ine

wh

eth

er a

giv

en

wh

ole

nu

mb

er is

prim

e o

r c

om

po

sit

e.

✔

4.O

A.C

.5

Ge

ne

ra

te

a n

um

be

r o

r s

ha

pe

pa

tte

rn

th

at f

ollo

ws a

giv

en

ru

le. Id

en

tif

y a

pp

are

nt f

ea

tu

re

s o

f t

he

pa

tte

rn

th

at w

ere

no

t e

xp

licit

in t

he

ru

le it

se

lf. For exam

ple, given the rule "Add 3" and the starting num

ber 1, generate term

s in the resulting sequence and observe that the terms appear to alternate betw

een odd and even num

bers. Explain informally w

hy the numbers w

ill continue to alternate in this way

.

✔

4.N

BT

.A.1

Re

co

gn

ize

th

at in

a m

ult

i-d

igit

wh

ole

nu

mb

er le

ss t

ha

n o

r e

qu

al t

o 1

,00

0,0

00

, a d

igit

in o

ne

pla

ce

re

pre

se

nts t

en

tim

es w

ha

t it

re

pre

se

nts in

th

e p

lace

to

its r

igh

t. Exam

ples: (1) recognize that 700 ÷ 70 = 10; (2) in the num

ber 7,246, the 2 represents 200, but in the num

ber 7,426 the 2 represents 20, recognizing that 200 is ten tim

es as large as 20, by applying concepts of place value and division.

✔

4.N

BT

.A.2

Re

ad

an

d w

rit

e m

ult

i-d

igit

wh

ole

nu

mb

ers le

ss t

ha

n o

r e

qu

al t

o 1

,00

0,0

00

usin

g b

ase

-te

n n

um

era

ls,

nu

mb

er n

am

es, a

nd

exp

an

de

d f

orm

. Co

mp

are

tw

o m

ult

i-d

igit

nu

mb

ers b

ase

d o

n m

ea

nin

gs o

f t

he

dig

its

in e

ach

pla

ce

, usin

g >

, =, a

nd

< s

ym

bo

ls t

o r

eco

rd

th

e r

esu

lts o

f c

om

pa

ris

on

s.

✔

4.N

BT

.A.3

U

se

pla

ce

va

lue

un

de

rsta

nd

ing

to

ro

un

d m

ult

i-d

igit

wh

ole

nu

mb

ers, le

ss t

ha

n o

r e

qu

al t

o 1

,00

0,0

00

, to

an

y p

lace

.

✔

4.N

BT

.B.4

F

lue

ntly

ad

d a

nd

su

btra

ct m

ult

i-d

igit

wh

ole

nu

mb

ers, w

ith

su

ms le

ss t

ha

n o

r e

qu

al t

o 1

,00

0,0

00, u

sin

g

th

e s

ta

nd

ard

alg

orit

hm

.

✔

15

A Guide to Rigor

LSS

M –

4th G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

4.N

BT

.B.5

Mu

ltip

ly a

wh

ole

nu

mb

er o

f u

p t

o f

ou

r d

igit

s b

y a

on

e-d

igit

wh

ole

nu

mb

er, a

nd

mu

ltip

ly t

wo

tw

o-d

igit

nu

mb

ers, u

sin

g s

tra

te

gie

s b

ase

d o

n p

lace

va

lue

an

d t

he

pro

pe

rtie

s o

f o

pe

ra

tio

ns. Illu

stra

te

an

d e

xp

lain

th

e c

alc

ula

tio

n b

y u

sin

g e

qu

atio

ns, r

ecta

ng

ula

r a

rra

ys, a

nd

/o

r a

re

a m

od

els

.

✔

4.N

BT

.B.6

Fin

d w

ho

le-n

um

be

r q

uo

tie

nts a

nd

re

ma

ind

ers w

ith

up

to

fo

ur-d

igit

div

ide

nd

s a

nd

on

e-d

igit

div

iso

rs, u

sin

g

stra

te

gie

s b

ase

d o

n p

lace

va

lue

, th

e p

ro

pe

rtie

s o

f o

pe

ra

tio

ns, a

nd

/o

r t

he

re

latio

nsh

ip b

etw

ee

n

mu

ltip

lica

tio

n a

nd

div

isio

n. Illu

stra

te

an

d e

xp

lain

th

e c

alc

ula

tio

n b

y u

sin

g e

qu

atio

ns, r

ecta

ng

ula

r a

rra

ys,

an

d/o

r a

re

a m

od

els

.

✔

4.N

F.A

.1

Exp

lain

wh

y a

fra

ctio

n a

/b is

eq

uiv

ale

nt t

o a

fra

ctio

n (n

× a

)/(n

× b

) b

y u

sin

g v

isu

al f

ra

ctio

n m

od

els

, wit

h

atte

ntio

n t

o h

ow

th

e n

um

be

r a

nd

siz

e o

f t

he

pa

rts d

iffe

r e

ve

n t

ho

ug

h t

he

tw

o f

ra

ctio

ns t

he

mse

lve

s a

re

th

e s

am

e s

ize

. Use

th

is p

rin

cip

le t

o r

eco

gn

ize

an

d g

en

era

te

eq

uiv

ale

nt f

ra

ctio

ns. (

De

no

min

ato

rs a

re

limit

ed

to

2, 3

, 4, 5

, 6, 8

, 10

, 12

, an

d 1

00

.)

✔

✔

4.N

F.A

.2

Co

mp

are

tw

o f

ra

ctio

ns w

ith

dif

fe

re

nt n

um

era

to

rs a

nd

dif

fe

re

nt d

en

om

ina

to

rs, e

.g., b

y c

re

atin

g c

om

mo

n

de

no

min

ato

rs o

r n

um

era

to

rs, o

r b

y c

om

pa

rin

g t

o a

be

nch

ma

rk f

ra

ctio

n s

uch

as 1

/2

. Re

co

gn

ize

th

at

co

mp

aris

on

s a

re

va

lid o

nly

wh

en

th

e t

wo

fra

ctio

ns r

efe

r t

o t

he

sa

me

wh

ole

. Re

co

rd

th

e r

esu

lts o

f

co

mp

aris

on

s w

ith

sym

bo

ls >

, =, o

r <

, an

d ju

stif

y t

he

co

nclu

sio

ns, e

.g., b

y u

sin

g a

vis

ua

l fra

ctio

n m

od

el.

(D

en

om

ina

to

rs a

re

limit

ed

to

2, 3

, 4, 5

, 6, 8

, 10

, 12

, an

d 1

00

.)

✔

4.N

F.B

.3

Un

de

rsta

nd

a f

ra

ctio

n a

/b w

ith

a >

1 a

s a

su

m o

f f

ra

ctio

ns 1

/b. (

De

no

min

ato

rs a

re

limit

ed

to

2, 3

, 4, 5

, 6,

8, 1

0, 1

2, a

nd

10

0.)

✔

4.N

F.B

.3a

U

nd

ersta

nd

ad

dit

ion

an

d s

ub

tra

ctio

n o

f f

ra

ctio

ns a

s jo

inin

g a

nd

se

pa

ra

tin

g p

arts r

efe

rrin

g t

o t

he

sa

me

wh

ole

. Example: 3/4 = 1/4 + 1/4 + 1/4

. ✔

4.N

F.B

.3b

De

co

mp

ose

a f

ra

ctio

n in

to

a s

um

of f

ra

ctio

ns w

ith

th

e s

am

e d

en

om

ina

to

r in

mo

re

th

an

on

e w

ay,

re

co

rd

ing

ea

ch

de

co

mp

osit

ion

by a

n e

qu

atio

n. J

ustif

y d

eco

mp

osit

ion

s, e

.g., b

y u

sin

g a

vis

ua

l fra

ctio

n

mo

de

l. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8

.

✔

4.N

F.B

.3c

Ad

d a

nd

su

btra

ct m

ixe

d n

um

be

rs w

ith

like

de

no

min

ato

rs, e

.g., b

y r

ep

lacin

g e

ach

mix

ed

nu

mb

er w

ith

an

eq

uiv

ale

nt f

ra

ctio

n, a

nd

/o

r b

y u

sin

g p

ro

pe

rtie

s o

f o

pe

ra

tio

ns a

nd

th

e r

ela

tio

nsh

ip b

etw

ee

n a

dd

itio

n a

nd

su

btra

ctio

n.

✔

4.N

F.B

.3d

S

olv

e w

ord

pro

ble

ms in

vo

lvin

g a

dd

itio

n a

nd

su

btra

ctio

n o

f f

ra

ctio

ns r

efe

rrin

g t

o t

he

sa

me

wh

ole

an

d

ha

vin

g lik

e d

en

om

ina

to

rs, e

.g., b

y u

sin

g v

isu

al f

ra

ctio

n m

od

els

an

d e

qu

atio

ns t

o r

ep

re

se

nt t

he

pro

ble

m.

✔

4.N

F.B

.4

Mu

ltip

ly a

fra

ctio

n b

y a

wh

ole

nu

mb

er. (

De

no

min

ato

rs a

re

limit

ed

to

2, 3

, 4, 5

, 6, 8

, 10

, 12

, an

d 1

00

.)

✔

4.N

F.B

.4a

U

nd

ersta

nd

a f

ra

ctio

n a

/b a

s a

mu

ltip

le o

f 1

/b. For exam

ple, use a visual fraction model to represent 5/4

as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4

).

✔

4.N

F.B

.4b

Un

de

rsta

nd

a m

ult

iple

of a

/b

as a

mu

ltip

le o

f 1

/b

, an

d u

se

th

is u

nd

ersta

nd

ing

to

mu

ltip

ly a

fra

ctio

n b

y a

wh

ole

nu

mb

er. For exam

ple, use a visual fraction model to express 3 × (2/5) as 6 × (1/5

), recognizing this product as 6/5. (In general, n × (a

/b) = (n × a)/b.)

✔

4.N

F.B

.4c

So

lve

wo

rd

pro

ble

ms in

vo

lvin

g m

ult

iplic

atio

n o

f a

fra

ctio

n b

y a

wh

ole

nu

mb

er, e

.g., b

y u

sin

g v

isu

al

fra

ctio

n m

od

els

an

d e

qu

atio

ns t

o r

ep

re

se

nt t

he

pro

ble

m. For exam

ple, if each person at a party will eat

3/8 of a pound of roast beef, and there will be 5 people at the party, how

many pounds of roast beef w

ill

✔

16

A Guide to Rigor

LSS

M –

4th G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

be needed? Betw

een what tw

o whole num

bers does your answer lie?

4.N

F.C

.5

Exp

re

ss a

fra

ctio

n w

ith

de

no

min

ato

r 1

0 a

s a

n e

qu

iva

len

t f

ra

ctio

n w

ith

de

no

min

ato

r 1

00

, an

d u

se

th

is

te

ch

niq

ue

to

ad

d t

wo

fra

ctio

ns w

ith

re

sp

ectiv

e d

en

om

ina

to

rs 1

0 a

nd

10

0. For exam

ple, express 3/10 as 30/100, and add 3/10 + 4/100 = 3

4/100.

✔

4.N

F.C

.6

Use

de

cim

al n

ota

tio

n f

or f

ra

ctio

ns w

ith

de

no

min

ato

rs 1

0 o

r 1

00

. For example, rew

rite 0.62 as 62/100;

describe a length as 0.62 meters; locate 0.62 on a num

ber line diagram; represent 62/100 of a dollar as

$0.62. ✔

4.N

F.C

.7

Co

mp

are

tw

o d

ecim

als

to

hu

nd

re

dth

s b

y r

ea

so

nin

g a

bo

ut t

he

ir s

ize

. Re

co

gn

ize

th

at c

om

pa

ris

on

s a

re

va

lid o

nly

wh

en

th

e t

wo

de

cim

als

re

fe

r t

o t

he

sa

me

wh

ole

. Re

co

rd

th

e r

esu

lts o

f c

om

pa

ris

on

s w

ith

th

e

sym

bo

ls >

, =, o

r <

, an

d ju

stif

y t

he

co

nclu

sio

ns, e

.g., b

y u

sin

g a

vis

ua

l mo

de

l.

✔

4.M

D.A

.1

Kn

ow

re

lativ

e s

ize

s o

f m

ea

su

re

me

nt u

nit

s w

ith

in o

ne

syste

m o

f u

nit

s in

clu

din

g: f

t, in

; k

m, m

, cm

; k

g, g

; lb

,

oz.;

l, ml;

hr, m

in, s

ec. W

ith

in a

sin

gle

syste

m o

f m

ea

su

re

me

nt, e

xp

re

ss m

ea

su

re

me

nts in

a la

rg

er u

nit

in

te

rm

s o

f a

sm

alle

r u

nit

. (C

on

ve

rsio

ns a

re

limit

ed

to

on

e-ste

p c

on

ve

rsio

ns.)

Re

co

rd

me

asu

re

me

nt

eq

uiv

ale

nts in

a t

wo

-co

lum

n t

ab

le. For exam

ple, know that 1 ft is 12 tim

es as long as 1 in. Express the length of a 4 ft snake as 48 in. G

enerate a conversion table for feet and inches listing the number pairs (1,

12), (2, 24), (3, 36), ...

✔

✔

4.M

D.A

.2

Use

th

e f

ou

r o

pe

ra

tio

ns t

o s

olv

e w

ord

pro

ble

ms in

vo

lvin

g d

ista

nce

s, in

te

rva

ls o

f t

ime

, liqu

id v

olu

me

s,

ma

sse

s o

f o

bje

cts, a

nd

mo

ne

y, in

clu

din

g p

ro

ble

ms in

vo

lvin

g w

ho

le n

um

be

rs a

nd

/o

r s

imp

le f

ra

ctio

ns

(a

dd

itio

n a

nd

su

btra

ctio

n o

f f

ra

ctio

ns w

ith

like

de

no

min

ato

rs a

nd

mu

ltip

lyin

g a

fra

ctio

n t

ime

s a

fra

ctio

n

or a

wh

ole

nu

mb

er),

an

d p

ro

ble

ms t

ha

t r

eq

uir

e e

xp

re

ssin

g m

ea

su

re

me

nts g

ive

n in

a la

rg

er u

nit

in t

erm

s

of a

sm

alle

r u

nit

. Re

pre

se

nt m

ea

su

re

me

nt q

ua

ntit

ies u

sin

g d

iag

ra

ms s

uch

as n

um

be

r lin

e d

iag

ra

ms t

ha

t

fe

atu

re

a m

ea

su

re

me

nt s

ca

le.

✔

✔

4.M

D.A

.3

Ap

ply

th

e a

re

a a

nd

pe

rim

ete

r f

orm

ula

s f

or r

ecta

ng

les in

re

al-

wo

rld

an

d m

ath

em

atic

al p

ro

ble

ms. For

example, find the w

idth of a rectangular room

given the area of the flooring and the length, by viewing

the area formula as a m

ultiplication equation with an unknow

n factor.

✔

✔

4.M

D.B

.4

Ma

ke

a lin

e p

lot t

o d

isp

lay a

da

ta

se

t o

f m

ea

su

re

me

nts in

fra

ctio

ns o

f a

un

it (

1/2

, 1/4

, 1/8

). S

olv

e

pro

ble

ms in

vo

lvin

g a

dd

itio

n a

nd

su

btra

ctio

n o

f f

ra

ctio

ns b

y u

sin

g in

fo

rm

atio

n p

re

se

nte

d in

line

plo

ts. For

example, from

a line plot find and interpret the difference in length between the longest and shortest

specimens in an insect collection

.

✔

4.M

D.C

.5

Re

co

gn

ize

an

gle

s a

s g

eo

me

tric

sh

ap

es t

ha

t a

re

fo

rm

ed

wh

ere

ve

r t

wo

ra

ys s

ha

re

a c

om

mo

n e

nd

po

int,

an

d u

nd

ersta

nd

co

nce

pts o

f a

ng

le m

ea

su

re

me

nt.

✔

4.M

D.C

.5a

A

n a

ng

le is

me

asu

re

d w

ith

re

fe

re

nce

to

a c

ircle

wit

h it

s c

en

te

r a

t t

he

co

mm

on

en

dp

oin

t o

f t

he

ra

ys, b

y

co

nsid

erin

g t

he

fra

ctio

n o

f t

he

cir

cu

lar a

rc b

etw

ee

n t

he

po

ints w

he

re

th

e t

wo

ra

ys in

te

rse

ct t

he

cir

cle

✔

17

A Guide to Rigor

LSS

M –

4th G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

4.M

D.C

.5b

A

n a

ng

le t

ha

t t

urn

s t

hro

ug

h 1

/3

60

of a

cir

cle

is c

alle

d a

"o

ne

-d

eg

re

e a

ng

le,"

an

d c

an

be

use

d t

o m

ea

su

re

an

gle

s.

✔

4.M

D.C

.5c

An

an

gle

th

at t

urn

s t

hro

ug

h n

on

e-d

eg

re

e a

ng

les is

sa

id t

o h

ave

an

an

gle

me

asu

re

of n

de

gre

es.

✔

4.M

D.C

.6

Me

asu

re

an

gle

s in

wh

ole

-n

um

be

r d

eg

re

es u

sin

g a

pro

tra

cto

r. S

ke

tch

an

gle

s o

f s

pe

cif

ied

me

asu

re

. ✔

✔

4.M

D.C

.7

Re

co

gn

ize

an

gle

me

asu

re

as a

dd

itiv

e. W

he

n a

n a

ng

le is

de

co

mp

ose

d in

to

no

n-o

ve

rla

pp

ing

pa

rts, t

he

an

gle

me

asu

re

of t

he

wh

ole

is t

he

su

m o

f t

he

an

gle

me

asu

re

s o

f t

he

pa

rts. S

olv

e a

dd

itio

n a

nd

su

btra

ctio

n

pro

ble

ms t

o f

ind

un

kn

ow

n a

ng

les o

n a

dia

gra

m in

re

al-

wo

rld

an

d m

ath

em

atic

al p

ro

ble

ms, e

.g., b

y u

sin

g

an

eq

ua

tio

n w

ith

a le

tte

r f

or t

he

un

kn

ow

n a

ng

le m

ea

su

re

.

✔

✔

✔

4.M

D.D

.8

Re

co

gn

ize

are

a a

s a

dd

itiv

e. F

ind

are

as o

f r

ectilin

ea

r f

igu

re

s b

y d

eco

mp

osin

g t

he

m in

to

no

n-o

ve

rla

pp

ing

re

cta

ng

les a

nd

ad

din

g t

he

are

as o

f t

he

no

n-o

ve

rla

pp

ing

pa

rts, a

pp

lyin

g t

his

te

ch

niq

ue

to

so

lve

re

al-

wo

rld

pro

ble

ms.

✔

✔

✔

4.G

.A.1

D

ra

w p

oin

ts, lin

es, lin

e s

eg

me

nts, r

ays, a

ng

les (

rig

ht, a

cu

te

, ob

tu

se

), a

nd

pe

rp

en

dic

ula

r a

nd

pa

ra

llel lin

es.

Ide

ntif

y t

he

se

in t

wo

-d

ime

nsio

na

l fig

ure

s.

✔

4.G

.A.2

Cla

ssif

y t

wo

-d

ime

nsio

na

l fig

ure

s b

ase

d o

n t

he

pre

se

nce

or a

bse

nce

of p

ara

llel o

r p

erp

en

dic

ula

r lin

es, o

r

th

e p

re

se

nce

or a

bse

nce

of a

ng

les o

f a

sp

ecif

ied

siz

e. R

eco

gn

ize

rig

ht t

ria

ng

les a

s a

ca

te

go

ry, a

nd

ide

ntif

y

rig

ht t

ria

ng

les

✔

4.G

.A.3

Re

co

gn

ize

a lin

e o

f s

ym

me

try f

or a

tw

o-d

ime

nsio

na

l fig

ure

as a

line

acro

ss t

he

fig

ure

su

ch

th

at t

he

fig

ure

ca

n b

e f

old

ed

alo

ng

th

e lin

e in

to

ma

tch

ing

pa

rts. Id

en

tif

y lin

e-sym

me

tric

fig

ure

s a

nd

dra

w lin

es o

f

sym

me

try.

✔

18

A Guide to Rigor

5th Grade

LSS

M –

5th G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

5.O

A.A

.1

Use

pa

re

nth

ese

s o

r b

ra

cke

ts in

nu

me

ric

al e

xp

re

ssio

ns, a

nd

eva

lua

te

exp

re

ssio

ns w

ith

th

ese

sym

bo

ls.

✔

✔

5.O

A.A

.2

Writ

e s

imp

le e

xp

re

ssio

ns t

ha

t r

eco

rd

ca

lcu

latio

ns w

ith

wh

ole

nu

mb

ers, f

ra

ctio

ns a

nd

de

cim

als

, an

d

inte

rp

re

t n

um

eric

al e

xp

re

ssio

ns w

ith

ou

t e

va

lua

tin

g t

he

m. For exam

ple, express the calculation "add 8 and 7, then m

ultiply by 2" as 2 × (8 + 7). Recognize that 3 × (18,932 + 9.21) is three tim

es as large as 18,932 + 9.21, w

ithout having to calculate the indicated sum or product

.

✔

5.O

A.B

.3

Ge

ne

ra

te

tw

o n

um

eric

al p

atte

rn

s u

sin

g t

wo

giv

en

ru

les. Id

en

tif

y a

pp

are

nt r

ela

tio

nsh

ips b

etw

ee

n

co

rre

sp

on

din

g t

erm

s. F

orm

ord

ere

d p

air

s c

on

sis

tin

g o

f c

orre

sp

on

din

g t

erm

s f

ro

m t

he

tw

o p

atte

rn

s, a

nd

gra

ph

th

e o

rd

ere

d p

air

s o

n a

co

ord

ina

te

pla

ne

. For example, given the rule "A

dd 3" and the starting num

ber 0, and given the rule "Ad

d 6" and the starting number 0, generate term

s in the resulting sequences, and observe that the term

s in one sequence are twice th

e corresponding terms in the other

sequence. Explain informally w

hy this is so.

✔

5.N

BT

.A.1

R

eco

gn

ize

th

at in

a m

ult

i-d

igit

nu

mb

er, a

dig

it in

on

e p

lace

re

pre

se

nts 1

0 t

ime

s a

s m

uch

as it

re

pre

se

nts

in t

he

pla

ce

to

its r

igh

t a

nd

1/1

0 o

f w

ha

t it

re

pre

se

nts in

th

e p

lace

to

its le

ft.

✔

5.N

BT

.A.2

Exp

lain

an

d a

pp

ly p

atte

rn

s in

th

e n

um

be

r o

f z

ero

s o

f t

he

pro

du

ct w

he

n m

ult

iply

ing

a n

um

be

r b

y p

ow

ers

of 1

0. E

xp

lain

an

d a

pp

ly p

atte

rn

s in

th

e v

alu

es o

f t

he

dig

its in

th

e p

ro

du

ct o

r t

he

qu

otie

nt, w

he

n a

de

cim

al is

mu

ltip

lied

or d

ivid

ed

by a

po

we

r o

f 1

0. U

se

wh

ole

-n

um

be

r e

xp

on

en

ts t

o d

en

ote

po

we

rs o

f

10

. For example, 10

0 = 1, 10

1 = 10 ... and 2.1 x 10

2 = 210.

✔

5.N

BT

.A.3

R

ea

d, w

rit

e, a

nd

co

mp

are

de

cim

als

to

th

ou

sa

nd

th

s.

✔

5.N

BT

.A.3

a

Re

ad

an

d w

rit

e d

ecim

als

to

th

ou

sa

nd

th

s u

sin

g b

ase

-te

n n

um

era

ls, n

um

be

r n

am

es, a

nd

exp

an

de

d f

orm

,

e.g

., 34

7.3

92

= 3

× 1

00

+ 4

× 1

0 +

7 ×

1 +

3 ×

(1

/1

0) +

9 ×

(1

/1

00

) +

2 ×

(1

/1

00

0).

✔

5.N

BT

.A.3

b

Co

mp

are

tw

o d

ecim

als

to

th

ou

sa

nd

th

s b

ase

d o

n m

ea

nin

gs o

f t

he

dig

its in

ea

ch

pla

ce

, usin

g >

, =, a

nd

<

sym

bo

ls t

o r

eco

rd

th

e r

esu

lts o

f c

om

pa

ris

on

s.

✔

5.N

BT

.A.4

U

se

pla

ce

va

lue

un

de

rsta

nd

ing

to

ro

un

d d

ecim

als

to

an

y p

lace

. ✔

5.N

BT

.B.5

F

lue

ntly

mu

ltip

ly m

ult

i-d

igit

wh

ole

nu

mb

ers u

sin

g t

he

sta

nd

ard

alg

orit

hm

.

✔

5.N

BT

.B.6

Fin

d w

ho

le-n

um

be

r q

uo

tie

nts o

f w

ho

le n

um

be

rs w

ith

up

to

fo

ur-d

igit

div

ide

nd

s a

nd

tw

o-d

igit

div

iso

rs,

usin

g s

tra

te

gie

s b

ase

d o

n p

lace

va

lue

, th

e p

ro

pe

rtie

s o

f o

pe

ra

tio

ns, s

ub

tra

ctin

g m

ult

iple

s o

f t

he

div

iso

r,

an

d/o

r t

he

re

latio

nsh

ip b

etw

ee

n m

ult

iplic

atio

n a

nd

div

isio

n. Illu

stra

te

an

d/o

r e

xp

lain

th

e c

alc

ula

tio

n b

y

usin

g e

qu

atio

ns, r

ecta

ng

ula

r a

rra

ys, a

re

a m

od

els

, or o

th

er s

tra

te

gie

s b

ase

d o

n p

lace

va

lue

.

✔

19

A Guide to Rigor

LSS

M –

5th G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

5.N

BT

.B.7

Ad

d, s

ub

tra

ct, m

ult

iply

, an

d d

ivid

e d

ecim

als

to

hu

nd

re

dth

s, u

sin

g c

on

cre

te

mo

de

ls o

r d

ra

win

gs a

nd

stra

te

gie

s b

ase

d o

n p

lace

va

lue

, pro

pe

rtie

s o

f o

pe

ra

tio

ns, a

nd

/o

r t

he

re

latio

nsh

ip b

etw

ee

n a

dd

itio

n a

nd

su

btra

ctio

n;

justif

y t

he

re

aso

nin

g u

se

d w

ith

a w

rit

te

n e

xp

lan

atio

n.

✔

5.N

F.A

.1

Ad

d a

nd

su

btra

ct f

ra

ctio

ns w

ith

un

like

de

no

min

ato

rs (

inclu

din

g m

ixe

d n

um

be

rs) b

y r

ep

lacin

g g

ive

n

fra

ctio

ns w

ith

eq

uiv

ale

nt f

ra

ctio

ns in

su

ch

a w

ay a

s t

o p

ro

du

ce

an

eq

uiv

ale

nt s

um

or d

iffe

re

nce

of

fra

ctio

ns w

ith

like

de

no

min

ato

rs. For exam

ple, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d =

(ad + bc)/bd.)

✔

✔

5.N

F.A

.2

So

lve

wo

rd

pro

ble

ms in

vo

lvin

g a

dd

itio

n a

nd

su

btra

ctio

n o

f f

ra

ctio

ns.

✔

5.N

F.A

.2a

So

lve

wo

rd

pro

ble

ms in

vo

lvin

g a

dd

itio

n a

nd

su

btra

ctio

n o

f f

ra

ctio

ns r

efe

rrin

g t

o t

he

sa

me

wh

ole

,

inclu

din

g c

ase

s o

f u

nlik

e d

en

om

ina

to

rs, e

.g., b

y u

sin

g v

isu

al f

ra

ctio

n m

od

els

or e

qu

atio

ns t

o r

ep

re

se

nt

th

e p

ro

ble

m.

✔

5.N

F.A

.2b

Use

be

nch

ma

rk f

ra

ctio

ns a

nd

nu

mb

er s

en

se

of f

ra

ctio

ns t

o e

stim

ate

me

nta

lly a

nd

justif

y t

he

re

aso

na

ble

ne

ss o

f a

nsw

ers. For exam

ple, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

✔

5.N

F.B

.3

Inte

rp

re

t a

fra

ctio

n a

s d

ivis

ion

of t

he

nu

me

ra

to

r b

y t

he

de

no

min

ato

r (a

/b =

a ÷

b). S

olv

e w

ord

pro

ble

ms

invo

lvin

g d

ivis

ion

of w

ho

le n

um

be

rs le

ad

ing

to

an

sw

ers in

th

e f

orm

of f

ra

ctio

ns o

r m

ixe

d n

um

be

rs, e

.g.,

by u

sin

g v

isu

al f

ra

ctio

n m

od

els

or e

qu

atio

ns t

o r

ep

re

se

nt t

he

pro

ble

m. For exam

ple, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 m

ultiplied by 4 equals 3, and that when 3 w

holes are shared equally am

ong 4 people each person has a share of size 3/4. If 9 people w

ant to share a 50-pound sack of rice equally by w

eight, how m

any pounds of rice should each person get? Betw

een what tw

o whole

numbers does your answ

er lie?

✔

✔

5.N

F.B

.4

Ap

ply

an

d e

xte

nd

pre

vio

us u

nd

ersta

nd

ing

s o

f m

ult

iplic

atio

n t

o m

ult

iply

a f

ra

ctio

n o

r w

ho

le n

um

be

r b

y a

fra

ctio

n.

✔

5.N

F.B

.4a

Inte

rp

re

t t

he

pro

du

ct (m

/n) x

q a

s m

pa

rts o

f a

pa

rtit

ion

of q

into

n e

qu

al p

arts; e

qu

iva

len

tly

, as t

he

re

su

lt o

f a

se

qu

en

ce

of o

pe

ra

tio

ns, m

x q

÷ n

. For example, use a visual fraction m

odel to show

understanding, and create a story context for (m/n

) x q. ✔

5.N

F.B

.4b

C

on

stru

ct a

mo

de

l to

de

ve

lop

un

de

rsta

nd

ing

of t

he

co

nce

pt o

f m

ult

iply

ing

tw

o f

ra

ctio

ns a

nd

cre

ate

a

sto

ry c

on

te

xt f

or t

he

eq

ua

tio

n. [

In g

en

era

l, (m/n

) x

(c/d

) =

(mc)/(nd

).]

✔

5.N

F.B

.4c

Fin

d t

he

are

a o

f a

re

cta

ng

le w

ith

fra

ctio

na

l sid

e le

ng

th

s b

y t

iling

it w

ith

un

it s

qu

are

s o

f t

he

ap

pro

pria

te

un

it f

ra

ctio

n s

ide

len

gth

s, a

nd

sh

ow

th

at t

he

are

a is

th

e s

am

e a

s w

ou

ld b

e f

ou

nd

by m

ult

iply

ing

th

e s

ide

len

gth

s.

✔

✔

5.N

F.B

.4d

M

ult

iply

fra

ctio

na

l sid

e le

ng

th

s t

o f

ind

are

as o

f r

ecta

ng

les, a

nd

re

pre

se

nt f

ra

ctio

n p

ro

du

cts a

s

re

cta

ng

ula

r a

re

as.

✔

✔

20

A Guide to Rigor

LSS

M –

5th G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

5.N

F.B

.5

Inte

rp

re

t m

ult

iplic

atio

n a

s s

ca

ling

(re

siz

ing

)

✔

5.N

F.B

.5a

C

om

pa

rin

g t

he

siz

e o

f a

pro

du

ct t

o t

he

siz

e o

f o

ne

fa

cto

r o

n t

he

ba

sis

of t

he

siz

e o

f t

he

oth

er f

acto

r,

wit

ho

ut p

erfo

rm

ing

th

e in

dic

ate

d m

ult

iplic

atio

n.

✔

5.N

F.B

.5b

E

xp

lain

ing

wh

y m

ult

iply

ing

a g

ive

n n

um

be

r b

y a

fra

ctio

n g

re

ate

r t

ha

n 1

re

su

lts in

a p

ro

du

ct g

re

ate

r t

ha

n

th

e g

ive

n n

um

be

r (

re

co

gn

izin

g m

ult

iplic

atio

n b

y w

ho

le n

um

be

rs g

re

ate

r t

ha

n 1

as a

fa

milia

r c

ase

).

✔

5.N

F.B

.5c

Exp

lain

ing

wh

y m

ult

iply

ing

a g

ive

n n

um

be

r b

y a

fra

ctio

n le

ss t

ha

n 1

re

su

lts in

a p

ro

du

ct s

ma

ller t

ha

n t

he

giv

en

nu

mb

er.

✔

5.N

F.B

.5d

R

ela

tin

g t

he

prin

cip

le o

f f

ra

ctio

n e

qu

iva

len

ce

a/b

= (n

x a

)/(n

x b

) t

o t

he

effe

ct o

f m

ult

iply

ing

a/b

by 1

. ✔

5.N

F.B

.6

So

lve

re

al-

wo

rld

pro

ble

ms in

vo

lvin

g m

ult

iplic

atio

n o

f f

ra

ctio

ns a

nd

mix

ed

nu

mb

ers, e

.g., b

y u

sin

g v

isu

al

fra

ctio

n m

od

els

or e

qu

atio

ns t

o r

ep

re

se

nt t

he

pro

ble

m.

✔

5.N

F.B

.7

Ap

ply

an

d e

xte

nd

pre

vio

us u

nd

ersta

nd

ing

s o

f d

ivis

ion

to

div

ide

un

it f

ra

ctio

ns b

y w

ho

le n

um

be

rs a

nd

wh

ole

nu

mb

ers b

y u

nit

fra

ctio

ns.

✔

5.N

F.B

.7a

Inte

rp

re

t d

ivis

ion

of a

un

it f

ra

ctio

n b

y a

no

n-ze

ro

wh

ole

nu

mb

er, a

nd

co

mp

ute

su

ch

qu

otie

nts. For

example, create a story context for (1/3) ÷ 4, and use a visual fraction m

odel to show the quotient. U

se the relationship betw

een multiplication and division to explain that (1/3) ÷ 4 = 1/12 b

ecause (1/12) × 4 =

1/3.

✔

✔

5.N

F.B

.7b

Inte

rp

re

t d

ivis

ion

of a

wh

ole

nu

mb

er b

y a

un

it f

ra

ctio

n, a

nd

co

mp

ute

su

ch

qu

otie

nts. For exam

ple, create a story context for 4 ÷ (1/5

), and use a visual fraction m

odel to show the quotient. U

se the relationship betw

een multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1

/5) = 4.

✔

✔

5.N

F.B

.7c

So

lve

re

al-

wo

rld

pro

ble

ms in

vo

lvin

g d

ivis

ion

of u

nit

fra

ctio

ns b

y n

on

-ze

ro

wh

ole

nu

mb

ers a

nd

div

isio

n o

f

wh

ole

nu

mb

ers b

y u

nit

fra

ctio

ns, e

.g., b

y u

sin

g v

isu

al f

ra

ctio

n m

od

els

an

d e

qu

atio

ns t

o r

ep

re

se

nt t

he

pro

ble

m. For exam

ple, how m

uch chocolate will each person get if 3 people share 1/2 lb of chocolate

equally? How

many 1/3-cup servings are in 2 cups of raisins?

✔

5.M

D.A

.1

Co

nve

rt a

mo

ng

dif

fe

re

nt-siz

ed

sta

nd

ard

me

asu

re

me

nt u

nit

s w

ith

in a

giv

en

me

asu

re

me

nt a

nd

use

th

ese

co

nve

rsio

ns in

so

lvin

g m

ult

i-ste

p, r

ea

l-w

orld

pro

ble

ms (

e.g

., co

nve

rt 5

cm

to

0.0

5 m

; 9

ft t

o 1

08

in).

✔

✔

5.M

D.B

.2

Ma

ke

a lin

e p

lot t

o d

isp

lay a

da

ta

se

t o

f m

ea

su

re

me

nts in

fra

ctio

ns o

f a

un

it (

1/2

, 1/4

, 1/8

). U

se

op

era

tio

ns o

n f

ra

ctio

ns f

or t

his

gra

de

to

so

lve

pro

ble

ms in

vo

lvin

g in

fo

rm

atio

n p

re

se

nte

d in

line

plo

ts. For exam

ple, given different measurem

ents of liquid in identical beakers, find the amount of liquid

each beaker would contain if the total am

ount in all the beakers were redistributed equally

.

✔

✔

5.M

D.C

.3

Re

co

gn

ize

vo

lum

e a

s a

n a

ttrib

ute

of s

olid

fig

ure

s a

nd

un

de

rsta

nd

co

nce

pts o

f v

olu

me

me

asu

re

me

nt.

✔

5.M

D.C

.3a

A

cu

be

wit

h s

ide

len

gth

1 u

nit

, ca

lled

a "

un

it c

ub

e,"

is s

aid

to

ha

ve

"o

ne

cu

bic

un

it" o

f v

olu

me

, an

d c

an

be

use

d t

o m

ea

su

re

vo

lum

e.

✔

5.M

D.C

.3b

A

so

lid f

igu

re

wh

ich

ca

n b

e p

acke

d w

ith

ou

t g

ap

s o

r o

ve

rla

ps u

sin

g n

un

it c

ub

es is

sa

id t

o h

ave

a v

olu

me

of n

cu

bic

un

its.

✔

21

A Guide to Rigor

LSS

M –

5th G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

5.M

D.C

.4

Me

asu

re

vo

lum

es b

y c

ou

ntin

g u

nit

cu

be

s, u

sin

g c

ub

ic c

m, c

ub

ic in

, cu

bic

ft, a

nd

imp

ro

vis

ed

un

its.

✔

5.M

D.C

.5

Re

late

vo

lum

e t

o t

he

op

era

tio

ns o

f m

ult

iplic

atio

n a

nd

ad

dit

ion

an

d s

olv

e r

ea

l-w

orld

an

d m

ath

em

atic

al

pro

ble

ms in

vo

lvin

g v

olu

me

. ✔

✔

✔

5.M

D.C

.5a

Fin

d t

he

vo

lum

e o

f a

rig

ht r

ecta

ng

ula

r p

ris

m w

ith

wh

ole

-n

um

be

r s

ide

len

gth

s b

y p

ackin

g it

wit

h u

nit

cu

be

s, a

nd

sh

ow

th

at t

he

vo

lum

e is

th

e s

am

e a

s w

ou

ld b

e f

ou

nd

by m

ult

iply

ing

th

e e

dg

e le

ng

th

s,

eq

uiv

ale

ntly

by m

ult

iply

ing

th

e h

eig

ht b

y t

he

are

a o

f t

he

ba

se

. Re

pre

se

nt t

hre

efo

ld w

ho

le-n

um

be

r

pro

du

cts a

s v

olu

me

s, e

.g., t

o r

ep

re

se

nt t

he

asso

cia

tiv

e p

ro

pe

rty o

f m

ult

iplic

atio

n.

✔

✔

5.M

D.C

.5b

Ap

ply

th

e f

orm

ula

s V

= l ×

w ×

h a

nd

V =

b ×

h f

or r

ecta

ng

ula

r p

ris

ms t

o f

ind

vo

lum

es o

f r

igh

t r

ecta

ng

ula

r

pris

ms w

ith

wh

ole

-n

um

be

r e

dg

e le

ng

th

s in

th

e c

on

te

xt o

f s

olv

ing

re

al-

wo

rld

an

d m

ath

em

atic

al

pro

ble

ms.

✔

✔

5.M

D.C

.5c

Re

co

gn

ize

vo

lum

e a

s a

dd

itiv

e. F

ind

vo

lum

es o

f s

olid

fig

ure

s c

om

po

se

d o

f t

wo

no

n-o

ve

rla

pp

ing

rig

ht

re

cta

ng

ula

r p

ris

ms b

y a

dd

ing

th

e v

olu

me

s o

f t

he

no

n-o

ve

rla

pp

ing

pa

rts, a

pp

lyin

g t

his

te

ch

niq

ue

to

so

lve

re

al-

wo

rld

pro

ble

ms.

✔

✔

✔

5.G

.A.1

Use

a p

air

of p

erp

en

dic

ula

r n

um

be

r lin

es, c

alle

d a

xe

s, t

o d

efin

e a

co

ord

ina

te

syste

m, w

ith

th

e

inte

rse

ctio

n o

f t

he

line

s (

th

e o

rig

in) a

rra

ng

ed

to

co

incid

e w

ith

th

e 0

on

ea

ch

line

an

d a

giv

en

po

int in

th

e

pla

ne

loca

te

d b

y u

sin

g a

n o

rd

ere

d p

air

of n

um

be

rs, c

alle

d it

s c

oo

rd

ina

te

s. U

nd

ersta

nd

th

at t

he

fir

st

nu

mb

er in

th

e o

rd

ere

d p

air

ind

ica

te

s h

ow

fa

r t

o t

ra

ve

l fro

m t

he

orig

in in

th

e d

ire

ctio

n o

f o

ne

axis

, an

d

th

e s

eco

nd

nu

mb

er in

th

e o

rd

ere

d p

air

ind

ica

te

s h

ow

fa

r t

o t

ra

ve

l in t

he

dir

ectio

n o

f t

he

se

co

nd

axis

,

wit

h t

he

co

nve

ntio

n t

ha

t t

he

na

me

s o

f t

he

tw

o a

xe

s a

nd

th

e c

oo

rd

ina

te

s c

orre

sp

on

d (

e.g

., x-a

xis

an

d x

-

co

ord

ina

te

, y-a

xis

an

d y

-co

ord

ina

te

).

✔

5.G

.A.2

R

ep

re

se

nt r

ea

l-w

orld

an

d m

ath

em

atic

al p

ro

ble

ms b

y g

ra

ph

ing

po

ints in

th

e f

irst q

ua

dra

nt o

f t

he

co

ord

ina

te

pla

ne

, an

d in

te

rp

re

t c

oo

rd

ina

te

va

lue

s o

f p

oin

ts in

th

e c

on

te

xt o

f t

he

sit

ua

tio

n.

✔

✔

✔

5.G

.B.3

Un

de

rsta

nd

th

at a

ttrib

ute

s b

elo

ng

ing

to

a c

ate

go

ry o

f t

wo

-d

ime

nsio

na

l fig

ure

s a

lso

be

lon

g t

o a

ll

su

bca

te

go

rie

s o

f t

ha

t c

ate

go

ry. For exam

ple, all rectangles have four right angles and squares are

rectangles, so all squares have four right angles.

✔

5.G

.B.4

C

lassif

y q

ua

drila

te

ra

ls in

a h

iera

rch

y b

ase

d o

n p

ro

pe

rtie

s. (

Stu

de

nts w

ill de

fin

e a

tra

pe

zo

id a

s a

qu

ad

rila

te

ra

l wit

h a

t le

ast o

ne

pa

ir o

f p

ara

llel s

ide

s.)

✔

22

A Guide to Rigor

6th Grade

LSS

M –

6th G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

6.R

P.A

.1

Un

de

rsta

nd

th

e c

on

ce

pt o

f a

ra

tio

an

d u

se

ra

tio

lan

gu

ag

e t

o d

escrib

e a

ra

tio

re

latio

nsh

ip b

etw

ee

n t

wo

qu

an

tit

ies. For exam

ple, "The ratio of wings to beaks in the bird house at the zoo w

as 2:1, because for every 2 w

ings there was 1 beak." "For every vote candidate A

received, candidate C received nea

rly three votes."

✔

6.R

P.A

.2

Un

de

rsta

nd

th

e c

on

ce

pt o

f a

un

it r

ate

a/b a

sso

cia

te

d w

ith

a r

atio a

:b w

ith

b ≠ 0, and u

se

ra

te

lan

gu

ag

e

in t

he

co

nte

xt o

f a

ra

tio

re

latio

nsh

ip. For exam

ple, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "W

e paid $75 for 15 hamburgers, w

hich is a

rate of $5 per hamburg

er."

✔

6.R

P.A

.3

Use

ra

tio

an

d r

ate

re

aso

nin

g t

o s

olv

e r

ea

l-w

orld

an

d m

ath

em

atic

al p

ro

ble

ms, e

.g., b

y r

ea

so

nin

g a

bo

ut

ta

ble

s o

f e

qu

iva

len

t r

atio

s, t

ap

e d

iag

ra

ms, d

ou

ble

nu

mb

er lin

e d

iag

ra

ms, o

r e

qu

atio

ns.

✔

✔

✔

6.R

P.A

.3a

M

ake

ta

ble

s o

f e

qu

iva

len

t r

atio

s r

ela

tin

g q

ua

ntit

ies w

ith

wh

ole

-n

um

be

r m

ea

su

re

me

nts, f

ind

mis

sin

g

va

lue

s in

th

e t

ab

les, a

nd

plo

t t

he

pa

irs o

f v

alu

es o

n t

he

co

ord

ina

te

pla

ne

. Use

ta

ble

s t

o c

om

pa

re

ra

tio

s.

✔

✔

6.R

P.A

.3b

So

lve

un

it r

ate

pro

ble

ms in

clu

din

g t

ho

se

invo

lvin

g u

nit

pric

ing

an

d c

on

sta

nt s

pe

ed

. For example, if it

took 7 hours to mow

4 lawns, then at that rate, how

many law

ns could be mow

ed in 35 hours? At w

hat unit rate w

ere lawns being m

owed?

✔

✔

6.R

P.A

.3c

Fin

d a

pe

rce

nt o

f a

qu

an

tit

y a

s a

ra

te

pe

r 1

00

(e

.g., 3

0%

of a

qu

an

tit

y m

ea

ns 3

0/1

00

tim

es t

he

qu

an

tit

y);

so

lve

pro

ble

ms in

vo

lvin

g f

ind

ing

th

e w

ho

le, g

ive

n a

pa

rt a

nd

th

e p

erce

nt.

✔

✔

6.R

P.A

.3d

U

se

ra

tio

re

aso

nin

g t

o c

on

ve

rt m

ea

su

re

me

nt u

nit

s; m

an

ipu

late

an

d t

ra

nsfo

rm

un

its a

pp

ro

pria

te

ly w

he

n

mu

ltip

lyin

g o

r d

ivid

ing

qu

an

tit

ies.

✔

✔

6.N

S.A

.1

Inte

rp

re

t a

nd

co

mp

ute

qu

otie

nts o

f f

ra

ctio

ns, a

nd

so

lve

wo

rd

pro

ble

ms in

vo

lvin

g d

ivis

ion

of f

ra

ctio

ns b

y

fra

ctio

ns, e

.g., b

y u

sin

g v

isu

al f

ra

ctio

n m

od

els

an

d e

qu

atio

ns t

o r

ep

re

se

nt t

he

pro

ble

m. For exam

ple, create a story context for (2/3) ÷ (3/4) and use a visual fraction m

odel to show the quotient; use the

relationship between m

ultiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 o

f 8/9 is 2/3. (In gen

eral, (a/b) ÷ (c/d

) = ad/bc.) H

ow m

uch 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 cu

p of yogurt? How

wide is a

rectangular strip of land with length 3/4 m

i and area 1/2 square mi?

.

✔

✔

✔

6.N

S.B

.2

Flu

en

tly

div

ide

mu

lti-

dig

it n

um

be

rs u

sin

g t

he

sta

nd

ard

alg

orit

hm

.

✔

6.N

S.B

.3

Flu

en

tly

ad

d, s

ub

tra

ct, m

ult

iply

, an

d d

ivid

e m

ult

i-d

igit

de

cim

als

usin

g t

he

sta

nd

ard

alg

orit

hm

fo

r e

ach

op

era

tio

n.

✔

6.N

S.B

.4

Fin

d t

he

gre

ate

st c

om

mo

n f

acto

r o

f t

wo

wh

ole

nu

mb

ers le

ss t

ha

n o

r e

qu

al t

o 1

00

an

d t

he

lea

st

co

mm

on

mu

ltip

le o

f t

wo

wh

ole

nu

mb

ers le

ss t

ha

n o

r e

qu

al t

o 1

2. U

se

th

e d

istrib

utiv

e p

ro

pe

rty t

o

exp

re

ss a

su

m o

f t

wo

wh

ole

nu

mb

ers 1

-1

00

wit

h a

co

mm

on

fa

cto

r a

s a

mu

ltip

le o

f a

su

m o

f t

wo

wh

ole

nu

mb

ers w

ith

no

co

mm

on

fa

cto

r. For exam

ple, express 36 + 8 as 4 (9 + 2).

✔

✔

23

A Guide to Rigor

LSS

M –

6th G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

6.N

S.C

.5

Un

de

rsta

nd

th

at p

osit

ive

an

d n

eg

ativ

e n

um

be

rs a

re

use

d t

og

eth

er t

o d

escrib

e q

ua

ntit

ies h

avin

g

op

po

sit

e d

ire

ctio

ns o

r v

alu

es (

e.g

., te

mp

era

tu

re

ab

ove

/b

elo

w z

ero

, ele

va

tio

n a

bo

ve

/b

elo

w s

ea

leve

l,

cre

dit

s/d

eb

its, p

osit

ive

/n

eg

ativ

e e

lectric

ch

arg

e); u

se

po

sit

ive

an

d n

eg

ativ

e n

um

be

rs t

o r

ep

re

se

nt

qu

an

tit

ies in

re

al-

wo

rld

co

nte

xts, e

xp

lain

ing

th

e m

ea

nin

g o

f 0

in e

ach

sit

ua

tio

n.

✔

6.N

S.C

.6

Un

de

rsta

nd

a r

atio

na

l nu

mb

er a

s a

po

int o

n t

he

nu

mb

er lin

e. E

xte

nd

nu

mb

er lin

e d

iag

ra

ms a

nd

co

ord

ina

te

axe

s f

am

iliar f

ro

m p

re

vio

us g

ra

de

s t

o r

ep

re

se

nt p

oin

ts o

n t

he

line

an

d in

th

e p

lan

e w

ith

ne

ga

tiv

e n

um

be

r c

oo

rd

ina

te

s.

✔

6.N

S.C

.6a

Re

co

gn

ize

op

po

sit

e s

ign

s o

f n

um

be

rs a

s in

dic

atin

g lo

ca

tio

ns o

n o

pp

osit

e s

ide

s o

f 0

on

th

e n

um

be

r lin

e;

re

co

gn

ize

th

at t

he

op

po

sit

e o

f t

he

op

po

sit

e o

f a

nu

mb

er is

th

e n

um

be

r it

se

lf, e

.g., -

(-3

) =

3, a

nd

th

at 0

is

its o

wn

op

po

sit

e.

✔

6.N

S.C

.6b

Un

de

rsta

nd

sig

ns o

f n

um

be

rs in

ord

ere

d p

air

s a

s in

dic

atin

g lo

ca

tio

ns in

qu

ad

ra

nts o

f t

he

co

ord

ina

te

pla

ne

; r

eco

gn

ize

th

at w

he

n t

wo

ord

ere

d p

air

s d

iffe

r o

nly

by s

ign

s, t

he

loca

tio

ns o

f t

he

po

ints a

re

re

late

d

by r

efle

ctio

ns a

cro

ss o

ne

or b

oth

axe

s.

✔

6.N

S.C

.6c

Fin

d a

nd

po

sit

ion

inte

ge

rs a

nd

oth

er r

atio

na

l nu

mb

ers o

n a

ho

riz

on

ta

l or v

ertic

al n

um

be

r lin

e d

iag

ra

m;

fin

d a

nd

po

sit

ion

pa

irs o

f in

te

ge

rs a

nd

oth

er r

atio

na

l nu

mb

ers o

n a

co

ord

ina

te

pla

ne

. ✔

6.N

S.C

.7

Un

de

rsta

nd

ord

erin

g a

nd

ab

so

lute

va

lue

of r

atio

na

l nu

mb

ers.

✔

6.N

S.C

.7a

Inte

rp

re

t s

ta

te

me

nts o

f in

eq

ua

lity a

s s

ta

te

me

nts a

bo

ut t

he

re

lativ

e p

osit

ion

of t

wo

nu

mb

ers o

n a

nu

mb

er lin

e d

iag

ra

m. For exam

ple, interpret -3 > -7 as a statement that -3 is located to the right of -7 on

a number line oriented from

left to right.

✔

6.N

S.C

.7b

W

rit

e, in

te

rp

re

t, a

nd

exp

lain

sta

te

me

nts o

f o

rd

er f

or r

atio

na

l nu

mb

ers in

re

al-

wo

rld

co

nte

xts. For

example, w

rite -3 oC > -7 oC

to express the fact that -3 oC

is warm

er than -7 oC.

✔

6.N

S.C

.7c

Un

de

rsta

nd

th

e a

bso

lute

va

lue

of a

ra

tio

na

l nu

mb

er a

s it

s d

ista

nce

fro

m 0

on

th

e n

um

be

r lin

e; in

te

rp

re

t

ab

so

lute

va

lue

as m

ag

nit

ud

e f

or a

po

sit

ive

or n

eg

ativ

e q

ua

ntit

y in

a r

ea

l-w

orld

sit

ua

tio

n. For exam

ple, for an account balance o

f -30 dollars, write |-30| = 30 to describe the size of the debt in dollars

.

✔

6.N

S.C

.7d

D

istin

gu

ish

co

mp

aris

on

s o

f a

bso

lute

va

lue

fro

m s

ta

te

me

nts a

bo

ut o

rd

er. For exam

ple, recognize that an account balance less than -30 dollars represents a debt greater tha

n 30 dollars.

✔

6.N

S.C

.8

So

lve

re

al-

wo

rld

an

d m

ath

em

atic

al p

ro

ble

ms b

y g

ra

ph

ing

po

ints in

all f

ou

r q

ua

dra

nts o

f t

he

co

ord

ina

te

pla

ne

. Inclu

de

use

of c

oo

rd

ina

te

s a

nd

ab

so

lute

va

lue

to

fin

d d

ista

nce

s b

etw

ee

n p

oin

ts w

ith

th

e s

am

e

fir

st c

oo

rd

ina

te

or t

he

sa

me

se

co

nd

co

ord

ina

te

.

✔

✔

6.E

E.A

.1

Writ

e a

nd

eva

lua

te

nu

me

ric

al e

xp

re

ssio

ns in

vo

lvin

g w

ho

le-n

um

be

r e

xp

on

en

ts.

✔

✔

6.E

E.A

.2

Writ

e, r

ea

d, a

nd

eva

lua

te

exp

re

ssio

ns in

wh

ich

lette

rs s

ta

nd

fo

r n

um

be

rs.

✔

✔

6.E

E.A

.2a

W

rit

e e

xp

re

ssio

ns t

ha

t r

eco

rd

op

era

tio

ns w

ith

nu

mb

ers a

nd

wit

h le

tte

rs s

ta

nd

ing

fo

r n

um

be

rs. For

example, exp

ress the calculation "Subtract y from 5" as 5 - y

. ✔

24

A Guide to Rigor

LSS

M –

6th G

ra

de

E

xp

licit

Co

mp

on

en

t(s) o

f R

igo

r

Co

de

S

ta

nd

ard

C

on

ce

ptu

al

Un

de

rsta

nd

ing

Pro

ce

du

ra

l Skill a

nd

Flu

en

cy

Ap

plic

atio

n

6.E

E.A

.2b

Ide

ntif

y p

arts o

f a

n e

xp

re

ssio

n u

sin

g m

ath

em

atic

al t

erm

s (

su

m, t

erm

, pro

du

ct, f

acto

r, q

uo

tie

nt,

co

effic

ien

t); v

iew

on

e o

r m

ore

pa

rts o

f a

n e

xp

re

ssio

n a

s a

sin

gle

en

tit

y. For exam

ple, describe the expression 2 (8 + 7

) as a product of two factors; view

(8 + 7) as both a single entity and a sum of tw

o term

s.

✔

6.E

E.A

.2c

Eva

lua

te

exp

re

ssio

ns a

t s

pe

cif

ic v

alu

es o

f t

he

ir v

aria

ble

s. In

clu

de

exp

re

ssio

ns t

ha

t a

ris

e f

ro

m f

orm

ula

s

use

d in

re

al-

wo

rld

pro

ble

ms. P

erfo

rm

arit

hm

etic

op

era

tio

ns, in

clu

din

g t

ho

se

invo

lvin

g w

ho

le-n

um

be

r

exp

on

en

ts, in

th

e c

on

ve

ntio

na

l ord

er w

he

n t

he

re

are

no

pa

re

nth

ese

s t

o s

pe

cif

y a

pa

rtic

ula

r o

rd

er

(O

rd

er o

f O

pe

ra

tio

ns). For exam

ple, use the formulas V

= s3 and A

= 6 s2 to find the volum

e and surface area of a cub

e with sides of length s = 1/2

.

✔

6.E

E.A

.3

Ap

ply

th

e p

ro

pe

rtie

s o

f o

pe

ra

tio

ns t

o g

en

era

te

eq

uiv

ale

nt e

xp

re

ssio

ns. For exam

ple, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); app

ly properties of operations to y + y + y to produce the equivalent expression 3y

.

✔

6.E

E.A

.4

Ide

ntif

y w

he

n t

wo

exp

re

ssio

ns a

re

eq

uiv

ale

nt (

i.e., w

he

n t

he

tw

o e

xp

re

ssio

ns n

am

e t

he

sa

me

nu

mb

er

re

ga

rd

less o

f w

hic

h v

alu

e is

su

bstit

ute

d in

to

th

em

). For exam

ple, the expressions y + y + y and 3y are equivalent because they nam

e the same num

ber regardless of w

hich number y stands for.

✔

6.E

E.B

.5

Un

de

rsta

nd

so

lvin

g a

n e

qu

atio

n o

r in

eq

ua

lity a

s a

pro

ce

ss o

f a

nsw

erin

g a

qu

estio

n: w

hic

h v

alu

es f

ro

m a

sp

ecif

ied

se

t, if

an

y, m

ake

th

e e

qu

atio

n o

r in

eq

ua

lity t

ru

e?

Use

su

bstit

utio

n t

o d

ete

rm

ine

wh

eth

er a

giv

en

nu

mb

er in

a s

pe

cif

ied

se

t m

ake

s a

n e

qu

atio

n o

r in

eq

ua

lity t

ru

e.

✔

✔

6.E

E.B

.6

Use

va

ria

ble

s t

o r

ep

re

se

nt n

um

be

rs a

nd

writ

e e

xp

re

ssio

ns w

he

n s

olv

ing

a r

ea

l-w

orld

or m

ath

em

atic

al

pro

ble

m; u

nd

ersta

nd

th

at a

va

ria

ble

ca

n r

ep

re

se

nt a

n u

nkn

ow

n n

um

be

r, o

r, d

ep

en

din

g o

n t

he

pu

rp

ose

at h

an

d, a

ny n

um

be

r in

a s

pe

cif

ied

se

t.

✔

✔

✔

6.E

E.B

.7

So

lve

re

al-

wo

rld

an

d m

ath

em

atic

al p

ro

ble

ms b

y w

rit

ing

an

d s

olv

ing

eq

ua

tio

ns a

nd

ine

qu

alit

ies o

f t

he

fo

rm

x +

p =

q a

nd

px =

q f

or c

ase

s in

wh

ich

p, q

an

d x

are

all n

on

ne

ga

tiv

e r

atio

na

l nu

mb

ers. In

eq

ua

litie

s

will in

clu

de

<, >, ≤, and ≥.

✔

✔

6.E

E.B

.8

Writ

e a

n in

eq

ua

lity o

f t

he

fo

rm

x >

c o

r x

< c

to

re

pre

se

nt a

co

nstra

int o

r c

on

dit

ion

in a

re

al-

wo

rld

or

ma

th

em

atic

al p

ro

ble

m. R

eco

gn

ize

th

at in

eq

ua

litie

s o

f t

he

fo

rm

x >

c o

r x <

c h

ave

infin

ite

ly m

an

y

so

lutio

ns; r

ep

re

se

nt s

olu

tio

ns o

f s

uch

ine

qu

alit

ies o

n n

um

be

r lin

e d

iag

ra

ms.

✔

✔

✔

6.E

E.C

.9

Use

va

ria

ble

s t

o r

ep

re

se

nt t

wo

qu

an

tit

ies in

a r

ea

l-w

orld

pro

ble

m t

ha

t c

ha

ng

e in

re

latio

nsh

ip t

o o

ne

an

oth

er; w

rit

e a

n e

qu

atio

n t

o e

xp

re

ss o

ne

qu

an

tit

y, t

ho

ug

ht o

f a

s t

he

de

pe

nd

en

t v

aria

ble

, in t

erm

s o

f

th

e o

th

er q

ua

ntit

y, t

ho

ug

ht o

f a

s t

he

ind

ep

en

de

nt v

aria

ble

. An

aly

ze

th

e r

ela

tio

nsh

ip b

etw

ee

n t

he

de

pe

nd

en

t a

nd

ind

ep

en

de

nt v

aria

ble

s u

sin

g g

ra

ph

s a

nd

ta

ble

s, a

nd

re

late

th

ese

to

th

e e

qu

atio

n. For

example, in a problem

involving motion at constant speed, list and graph ordered pairs o

f distances and tim

es, and write the equation d = 65t to represent the relationship betw

een distance and time.

✔

✔

6.G

.A.1

Fin

d t

he

are

a o

f r

igh

t t

ria

ng

les, o

th

er t

ria

ng

les, s

pe

cia

l qu

ad

rila

te

ra

ls, a

nd

po

lyg

on

s b

y c

om

po

sin

g in

to

re

cta

ng

les o

r d

eco

mp

osin

g in

to

tria

ng

les a

nd

oth

er s

ha

pe

s; a

pp

ly t

he

se

te

ch

niq

ue

s in

th

e c

on

te

xt o

f

so

lvin

g r

ea

l-w

orld

an

d m

ath

em

atic

al p

ro

ble

ms.

✔

✔

✔

Teacher Training Revised ELA and Math Standards Math 3–5 · multiplication table, how to add,...

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Transcript of Teacher Training Revised ELA and Math Standards Math 3–5 · multiplication table, how to add,...