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%MathematicsSupervisor
• KCS%MathematicsSpecialist
• 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%
• KCS%MathematicsSupervisor
• KCS%MathematicsSpecialist
TN%DOE%and%KCS%Standards%Institutes%
August%2017%
• KCS%MathematicsSupervisor
• KCS%MathematicsSpecialist
• KCS%NumeracyCoaches
• 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%
• KCS%MathematicsSupervisor
• KCS%MathematicsSpecialist
• KCS%NumeracyCoaches
• 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.
Instructional Shifts
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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Instructional Shifts
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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Instructional Shifts
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
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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Revisions to the Math Standards
Specific to K–5
• Refined for clarity
• Increased fluency expectations
• Revised examples
Overarching Revisions
• Added/shifted a small number of standards to strengthen coherence across gradelevels
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Revisions to the Math Standards
Specific to K–5
• Refined for clarity
• Increased fluency expectations
• Revised examples
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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Revisions to the Math Standards
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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Grade 3 Standards Comparison Activity
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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Grade 3 Standards Comparison Activity
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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Grade 3 Standards Comparison Activity
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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Grade 3 Standards Comparison Activity
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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Grade 4 Standards Comparison Activity
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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Grade 4 Standards Comparison Activity
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
Grade 4 Standards Comparison Activity
() =
(×+)×+ =
,-
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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Grade 4 Standards Comparison Activity
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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Grade 4 Standards Comparison Activity
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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Grade 4 Standards Comparison Activity
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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Grade 5 Standards Comparison Activity
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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Grade 5 Standards Comparison Activity
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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Grade 5 Standards Comparison Activity
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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Grade 5 Standards Comparison Activity
(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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Grade 5 Standards Comparison Activity
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
Grade 5 Standards Comparison Activity
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
Revisions to the Math Standards
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.
2.RI.CS.4 Determine the meaning of wordsand phrases in a text relevant to a grade 2 topic or subject area.
11.RL.CS.4 Identify words and phrases instories and poems that suggest feelings or appeal to the senses.
1.RI.CS.4 Determine the meaning of wordsand phrases in a text relevant to a grade 1 topic or subject area.
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
7 7.SL.CC.1 Prepare for collaborative discussions on 7th 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
6 6.SL.CC.1 Prepare for collaborative discussions on 6th 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
5 5.SL.CC.1 Prepare for collaborative discussions on 5th 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.5-6
4 4.SL.CC.1 Prepare for collaborative discussions on 4th 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.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.
L.VAU.5-6 Reading Cornerstone Standards 1 and 10.
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.
L.VAU.5-6 Reading Cornerstone Standards 1 and 10.
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.
L.VAU.5-6 Reading Cornerstone Standards 1 and 10.
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.
L.VAU.5-6 Reading Cornerstone Standards 1 and 10.
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.
FL.VAC.7 Reading Cornerstone Standards 1 and 10
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.
FL.VAC.7 Reading Cornerstone Standards 1 and 10
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.
FL.VAC.7 Reading Cornerstone Standards 1 and 10
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 .....…
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… ...................................................................................... …
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2 A Special N
ote on Procedural Skill and Fluency ……
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Recognizing the Com
ponents of Rigor ……
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Focus in the standards …
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…3
Rigor by Grade/Content Level
Kindergarten ……
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1
st Grade .................................................................................. ……
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.6
2nd Grade …
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3
rd Grade ................................................................... ……
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.10
4th Grade ................................................................. …
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...14
5th Grade ................................................................. …
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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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Algebra I ........................................................................................................................................................................... 34
Geom
etry ……
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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
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