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Pojoaque Valley Intermediate SchoolMathematics CCSS Pacing Guide

4th Grade

*Skills adapted from Kentucky Department of Education

Math Deconstructed Standards** Evidence of attainment/assessment,

Vocabulary, Knowledge, Skills and Essential Elements adapted from

Wisconsin Department of Education and Standards Insights Computer-Based Program

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Pojoaque Valley SchoolsMath Common Core Pacing Guide Introduction

The Pojoaque Valley Schools pacing guide documents are intended to guide teachers’ use of Common Core State Standards (CCSS) over the course of an instructional school year. The guides identify the focus standards by quarter. Teachers should understand that the focus standards emphasize deep instruction for that timeframe. However, because a certain quarter does not address specific standards, it should be understood that previously taught standards should be reinforced while working on the focus standards for any designated quarter. Some standards will recur across all quarters due to their importance and need to be addressed on an ongoing basis.

The Math pacing guides are grounded in four key components: the key fluency expectations for each grade level, the critical areas designated in the CCSS Math Standards, the Common Core Standards for Mathematics and the integration of the Standards for Mathematical Practice. In planning instruction it is important that math teachers incorporate the 8 mathematical practices for mathematics to ensure that the Common Core standards are mastered by all students.

The Math CCSS pacing guides contain the following elements: Grade Level: Identify the grade level of the intended standard Standard with code: Defines the knowledge and skills for students. The code contains the grade level,

domain and standard number. Domain: Larger groups of related standards. Standards from different domains may sometimes be

closely related. Cluster: Summarize groups of related standards. Skills and Knowledge: Identified as subsets of the standard and appear in one or more quarters. Define

the skills and knowledge embedded in the standard to meet the full intent of the standard itself.

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Version 2 of the Pojoaque Valley School District Pacing guides for Reading Language Arts and Mathematics are based on the done by staff and teachers of the school district using the Kentucky model, and a synthesis of the excellent work done by Wisconsin Cooperative Educational Service Agency 7 (CESA 7) School Improvement Services, Green Bay, WI. (2010), Standards Insight project.

Standards Insight was developed to give educators a tool for in depth investigation of the Common Core State Standards (CCSS). The CCSS are “unpacked” or dissected, identifying specific knowledge, skills, vocabulary, understandings, and evidence of student attainment for each standard. Standards Insight may be used by educators to gain a thorough grasp of the CCSS or as a powerful collaborative tool supporting educator teams through the essential conversations necessary for developing shared responsibility for student attainment of all CCSS. . . . serves as a high-powered vehicle to help educators examine the standards in a variety of ways.

The Version 2 Pojoaque Valley School District Pacing guides present the standard with levels of detail and then the necessary skills by quarter based on the Kentucky model. On the second page for each standard, the synthesis of the Standards Insight project is presented in a way that further defines and refines the standard such that teachers may use the information to refine their teaching practices.

Based on this synthesis of work and the purpose for the unpacking, the following fields were selected as most helpful to aid in understanding of the Common Core Standards that will lead to shifts in instruction:

1. Evidence of Student Attainment: “What could students do to show attainment of the standard?”

2. Vocabulary: “What are key terms in the standard that are essential for interpretation and understanding in order for students to learn the content?”

3. Knowledge: “What does the student need to know in order to aid in attainment of this standard?”

4. Skills and Understanding: “What procedural skill(s) does the student need to demonstrate for attainment of this standard?”, and “What will students understand to attain the standard?”

The following fields are included in Version 2:

Evidence of Student Attainment: This field describes what the standard may look like in student work. Specific expectations are listed in performance terms showing what students will say or do to demonstrate attainment of the standard.

Standards Vocabulary: This field lists words and phrases specific to each standard. Shared interpretation and in depth understanding of standards vocabulary are essential for consistent instruction across and within grade levels and content areas.

Knowledge: The knowledge field lists what students will need to know in order to master each standard (facts, vocabulary, and definitions).

Skills and Understanding: The skills field identifies the procedural knowledge students apply in order to master each standard (actions, applications, strategies), as well as the overarching understanding that connects the standard, knowledge, and skills. Understandings included in Standards Insight synthesize ideas and have lasting value.

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Instructional Achievement Level Descriptors: This field lists, by level what a teacher can expect to see in a student who achieves at a particular level. Additionally teachers can use this filed to differentiate instruction to provide further growth for student’s in moving from one level to another. This field can be used to provide specific teaching approaches to the standard in question.

A Note About High School Standards: The high school standards are listed in conceptual categories. Conceptual categories portray a coherent view of high school instruction that crosses traditional course boundaries. We have done everything possible, with teacher input, to link individual standards to the appropriate pacing guides,

References to Tables: References to tables within the standards in the Standards Insight tool refer to Tables 1-5 found in the glossary of the Mathematics Common Core State Standards document found at www.corestandards.org.

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Quarterly View of Standards 4 th Grade Mathematics Pacing Guide

Quarter 1 2 3 44.OA.1 Interpret a multiplication equation as a comparison, 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.

X X X X

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.

X

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.

X X

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.

X

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.

X X

4.NBT.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 that 700 ÷ 70 = 10 by applying concepts of place value and division.

X X

4.NBT.2 Read and write multi-digit whole numbers using 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.

X

4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place X X4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm. X X X X4.NBT.5 Multiply a whole number of up to four 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 explainthe calculation by using equations, rectangular arrays, and/or area models.

X X

4.NBT.6 Find whole-number quotients and remainders 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.

X X

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.

X

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.

X

4.NF.3a 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 separating parts referring to the same whole.

X

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Quarter 1 2 3 44.NF.3b Understand a fraction a/b with a > 1 as a sum of fractions 1/b.b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual 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.

X

4.NF.3c Understand a fraction a/b with a > 1 as a sum of fractions 1/b.c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

X

4.NF.3d Understand a fraction a/b with a > 1 as a sum of fractions 1/b.d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

X

4.NF.4a Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.a. Understand a fraction a/b as a multiple of 1/b. For example, 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).

X

4.NF.4b Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, 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.)

X

4.NF.4c Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models 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?

X

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 100.4 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

X

Standard with code: 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.

X

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

X

4.MD.1 Know relative sizes of measurement units within 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), ...

X

4.MD.2 Use the four operations to solve word problems 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.

X

4.MD.3 Apply the area and perimeter formulas 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.

X

4.MD.4 Make a line plot to display a data set of measurements 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.

X

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Quarter 1 2 3 44.MD.5ab Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:a. An angle is measured with 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. 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.

X

4.MD.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. X4.MD.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.

X

4.MD.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.

X

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.

X

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.

X

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CCSS Math Pacing GuideGrade 4

Grade Level: 4th Standard with code: 4.OA.1 Interpret a multiplication equation as a comparison, 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.

Domain: Operation and Algebraic Thinking Cluster: Use the four operations with whole numbers to solve problems.

Quarter 1: Know multiplication

strategies. Interpret a multiplication equation as a

comparison (e.g. 18 = 3 times as many as 6.

Represent verbal statements of multiplicative comparisons as multiplication equations.

Quarter 2: Know multiplication strategies. Interpret a multiplication equation as a

comparison (e.g. 18 = 3 times as many as 6. Represent verbal statements of multiplicative

comparisons as multiplication equations.

Quarter 3: Know multiplication strategies. Interpret a multiplication equation

as a comparison (e.g. 18 = 3 times as many as 6.


Quarter 4: Know multiplication strategies. Interpret a multiplication equation

as a comparison (e.g. 18 = 3 times as many as 6.


Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeatedreasoning.

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Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Common Core

Essential Elements Instructional Achievement Level Descriptors

Students:Given a multiplication equation,

Create and explain a corresponding verbal multiplicative comparison statement (Table 2).

Given a verbal (written or oral) representation of a multiplicative comparison,

Write and solve the related multiplication equation (e.g., given "Johnny has 7 cards and Shawna has 5 times as many cards as Johnny," the student will write 5 x 7 and accurately find the number of cards Shawna has to be 35).

Multiplicative comparison

See Table 2 for problem types.

Students know:

Characteristics of multiplicative comparisons (Table 2).

Students understand that/are able to:

Use mathematical language to communicate the relationship between verbal representations of multiplicative comparisons and the related multiplication equations,

Write multiplication equations that correspond to given multiplicative comparison statements,

Write verbal multiplicative comparison descriptions given a multiplication equation.

Multiplicative comparisons relate the size of two quantities and a scale factor,

Factors in multiplication problems have different roles from each other in the context of comparison

EE4.OA.1-2. Demonstrate the connection between repeated addition and multiplication.

Level IV Students will:EE4.OA.1-2. Apply repeated addition to solve a multiplication problem represented with numbers.Ex. Presented with a multiplication problem such as 3 x 6, use egg cartons and concrete objects to create arrays (e.g., sort three objects into six egg slots or six objects into three slots).Ex. When presented with two choices of arrays on the smart board and a multiplication problem, identify (i.e., eye gaze) the correct array.Ex. Use skip counting on a number line to solve multiplication problems (e.g., move two digits five times for the problem 2 x 5).

Level III Students will:EE4.OA.1-2. Demonstrate the connection between repeated addition and multiplication.Ex. Skip count by two, five, and 10 to solve multiplication problems.Ex. Using three groups of two objects, communicate that 2 + 2 + 2 is equal to 3 x 2.Ex. Using plastic eggs and an egg carton to hold the eggs in place, place an object in each egg to illustrate 6 + 6 = 12 or 6 x 2 = 12.Ex. Represent the chairs in a class with three rows of four chairs in each (e.g., identify 4 + 4 + 4).

Level II Students will:EE4.OA.1-2. Demonstrate repeated addition to sums of 10.Ex. Skip count by two and five to 10.Ex. Add 1 + 1 + 1.Ex. Add 2 + 2 + 2.Ex. Add 3 + 3 + 3.Ex. Add 2 + 2 + 2 + 2 + 2 to equal 10.Ex. Presented with a picture of two chairs in a row and given four pictures of individual chairs, arrange the additional four chairs into equal rows and count all of the chairs.

Level I Students will:EE4.OA.1-2. Make a set of 10 and count to 10.Ex. Using fingers count to 10.Ex. Using a 10 frame, place a cube in each square.Ex. Use a switch to count to 10.Ex. Count like objects to make a set of 10.

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problems.PVIS CCSS Math Pacing GuideGrade 4

Grade Level: 4th Standard with code: 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.Domain: Operation and Algebraic Thinking Cluster: Use the four operations with whole numbers to solve problems.

Quarter 1: Quarter 2: Multiply or divide to solve word problems. Describe multiplicative comparison. Describe additive comparison. Determine appropriate operation and solve

word problems involving multiplicative comparison.

Determine and use a variety of representations to model a problem involving multiplicative comparison.

Distinguish between multiplicative comparison and additive comparison (repeated addition).

Quarter 3: Quarter 4:









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Students:Given multiplication and division problems involving multiplicative comparisons,

Find, explain and justify solutions using connections between pictorial representations and related equations involving a single unknown.

Given a mixture of multiplicative comparison and additive comparison problems,

Apply their understanding of operations and a variety of representations to explain and justify the choice of operation in solving the problem.

Multiplicative comparison

Additive comparison

See (Tables 1 and 2) for problem types.

Students know:

Characteristics of multiplicative comparison problems and additive comparison problems,

Addition, subtraction, multiplication, and division strategies.


Compare and contrast mathematical contexts in order to determine the types of mathematical comparisons present,

Represent multiplicative comparison contexts physically, pictorially, or symbolically,

Strategically choose and apply a variety of representations to solve multiplicative comparison problems,

Use symbols to represent unknown quantities in multiplicative comparison equations,

Accurately compute products and quotients,

Use mathematical language to communicate the connections among contexts involving all

EE4.OA.1-2. Demonstrate the connection between repeated addition and multiplication.

Level IV Students will:EE4.OA.1-2. Apply repeated addition to solve a multiplication problem represented with numbers.Ex. Presented with a multiplication problem such as 3 x 6, use egg cartons and concrete objects to create arrays (e.g., sort three objects into six egg slots or six objects into three slots).Ex. When presented with two choices of arrays on the smart board and a multiplication problem, identify (i.e., eye gaze) the correct array.Ex. Use skip counting on a number line to solve multiplication problems (e.g., move two digits five times for the problem 2 x 5).

Level III Students will:EE4.OA.1-2. Demonstrate the connection between repeated addition and multiplication.Ex. Skip count by two, five, and 10 to solve multiplication problems.Ex. Using three groups of two objects, communicate that 2 + 2 + 2 is equal to 3 x 2.Ex. Using plastic eggs and an egg carton to hold the eggs in place, place an object in each egg to illustrate 6 + 6 = 12 or 6 x 2 = 12.Ex. Represent the chairs in a class with three rows of four chairs in each (e.g., identify 4 + 4 + 4).

Level II Students will:EE4.OA.1-2. Demonstrate repeated addition to sums of 10.Ex. Skip count by two and five to 10.Ex. Add 1 + 1 + 1.Ex. Add 2 + 2 + 2.Ex. Add 3 + 3 + 3.Ex. Add 2 + 2 + 2 + 2 + 2 to equal 10.Ex. Presented with a picture of two chairs in a row and given four pictures of individual chairs, arrange the additional four chairs into equal rows and count all of the chairs.

Level I Students will:EE4.OA.1-2. Make a set of 10 and count to 10.Ex. Using fingers count to 10.Ex. Using a 10 frame, place a cube in each square.Ex. Use a switch to count to 10.Ex. Count like objects to make a set of 10.

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four operations and related physical, pictorial, or symbolic representations and justify solutions/solution paths

The operation of multiplication represents contexts of putting together equal sized groups or multiplicative comparisons,

The operation of division represents contexts of partitioning into equal-sized shares or contexts of partitioning equally among a given number of groups or contexts involving multiplicative comparisons,

The operation of subtraction represents taking apart, taking from, and additive comparison contexts,

Mathematical problems (four basic operations) can be solved using a variety of strategies, models,

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representations,

Variables represent unknown quantities when modeling mathematical situations algebraically..

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PVIS CCSS Math Pacing GuideGrade 4

Grade Level: 4th Standard with code: 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.Domain: Operation and Algebraic Thinking Cluster: Use the four operations with whole numbers to solve problems.

Quarter 1: Quarter 2: Divide whole numbers including division

with remainders. Represent multi-step word problems

using equations with a letter standing for the unknown quantity.

Interpret multistep word problems (including problems in which remainders must be interpreted) and determine the appropriate operation(s) to solve

Assess the reasonableness of an answer in solving a multistep word problem using mental math and estimation strategies (including rounding).

Quarter 3: Divide whole numbers including

division with remainders. Represent multi-step word

problems using equations with a letter standing for the unknown quantity.

Interpret multistep word problems (including problems in which remainders must be interpreted) and determine the appropriate operation(s) to solve

Assess the reasonableness of an answer in solving a multistep word problem using mental math and estimation strategies (including rounding).

Quarter 4:









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Students:Given a variety of multistep word problems involving all four operations on whole numbers (including problems in which remainders must be interpreted),

Explain and justify solutions using connections between the problems and related equations involving a single (letter) unknown,

Apply their understanding of operations and estimation strategies including rounding to evaluate the reasonableness of their solutions.

"Problems in which remainders must be interpreted"

Students know:

Characteristics (see Table 1 and 2) of addition, subtraction, multiplication and division contexts,

Addition, subtraction, multiplication, and division strategies,

Strategies for mentally computing and estimating sums, differences, products, and quotients.


Represent quantities and operations (addition, subtraction, multiplication, and division of whole numbers) physically, pictorially, or symbolically,

Strategically choose and apply a variety of representations to solve addition, subtraction, multiplication, and division multi-step word problems,

Use symbols to represent unknown quantities in equations that represent multi-step word problems,

Use logical reasoning and connections between physical/pictorial representations to justify solutions and solution paths and to interpret remainders,

Estimate answers in addition, subtraction, multiplication and

EE4.OA.3. Solve one-step word problems using addition or subtraction.

Level IV Students will:EE4.OA.3. Solve two-step problems using addition or subtraction when a number in the problem is unknown (result, start, change, difference).Ex. Use a number line to solve two-step problems.Ex. Use a hundreds chart to solve a two-step problem.Ex. Solve a two-step word problem involving addition (e.g., “If Amy has 10 sheets of paper and you have 10 more sheets than Amy, how many sheets do you have? [addition – compare total unknown]).Ex. Solve a two-step word problem involving subtractions (e.g., “Sandi has 10 cats and 20 dogs – does she have more cats or dogs? How many more?” [subtraction - compare difference unknown]).

Level III Students will:EE4.OA.3. Solve one-step problems using addition or subtraction.Ex. Use manipulatives to add or subtract two groups.Ex. Use manipulatives on a number line to solve addition or subtraction problems.Ex. Solve one-step word problem involving addition (e.g., “If Sam gave away 10 apples and has five apples left how many did he start with?” [addition – start unknown]).Ex. Solve one-step word problem involving subtractions (e.g., “If June had 50 dollars and spent ten, how much does she have left?” [subtraction – classic take away]).Ex. Solve one-step word problem involving addition (e.g., “If Jessie had 20 cakes and bought five more, how many does he have now?” [addition join-part/part – whole]).Ex. Solve one-step word problem involving subtractions (e.g., “If Sandy wanted to collect 35 cards and she already has 15, how many more does she need?” [subtraction deficit missing amount]).

Level II Students will:EE4.OA.3. Solve one-step addition or subtraction problems when there is an unknown (result, start, change, difference) up to 10.Ex. Given a group of five items, determine how many more are need to make 10.Ex. Given a group of eight items, determine how many to take away to make five.

Level I Students will:EE4.OA.3. Add up to five.Ex. Given a group of two, add objects to a total of five.Ex. Given a group of three, add objects to a total of five.

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division problems,

Evaluate the reasonableness of answers by comparing actual answers to estimates.

The operation of addition represents both putting together and adding to contexts,

The operation of subtraction represents taking apart, taking from, and additive comparison contexts,

The operation of multiplication represents contexts of putting together equal sized groups,

The operation of division represents contexts of partitioning into equal-sized shares or contexts of partitioning equally among a given number of groups,

The interpretation of the remainder in a division problem is dependent upon the original context and question,

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Variables represent unknown quantities when modeling mathematical situations algebraically,

Solutions can be evaluated by using reasoning to compare the actual solution with estimated solutions.

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Grade Level: 4th Standard with code: 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.Domain: Operation and Algebraic Thinking Cluster: Gain familiarity with factors and multiples.

Quarter 1:

Quarter 2: Define prime and composite numbers. Know strategies to determine whether a

whole number is prime or composite. Identify all factor pairs for any given

number 1-100. Recognize that a whole number is a

multiple of each of its factors. Determine if a given whole number (1 -

100) is a multiple of a given one-digit number.


Make sense of problems and persevere insolving them.







Look for and express regularity in repeated reasoning.

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Students:Given any whole number from 1-100,

Use knowledge of multiplication and division, models for multiplication, and logical reasoning to decompose the given number into all possible factor pairs,

Determine if it is a multiple of a given single-digit number,

Use knowledge and vocabulary of factors, factor pairs, and multiples to justify the classification of numbers as prime and composite.

Factor

Factor pair

Multiple

Prime

Composite

Students know:

Strategies for finding factor pairs,

Vocabulary: factor, multiple, factor pair, prime, composite.


Use models and logical reasoning to determine all possible factor pairs for a whole number between 1 - 100,

Accurately compute products and quotients,

Use an understanding of prime and composite to classify numbers.

A whole number is a multiple of each of its factors,

Numbers can be classified as prime, composite, or neither, based on their properties and characteristics.

EE4.OA.4. Show one way to arrive at product.

Level IV Students will:EE4.OA.4. Show multiple ways to arrive at the same product.Ex. Given a product, use manipulatives to create groups that represent the product.Ex. Given a number (product) of the day, match their factor cards to another student’s factor card to equal the product.Ex. Given an equation on a dry erase board (e.g., 2 x 4 = 8), make equal groups to show possible factors for eight (e.g., one group of eight, two groups of four, four groups of two).

Level III Students will:EE4.OA.4. Show one way to arrive at a product.Ex. Using a group of manipulatives, separate into equal groups.Ex. Provided with counters, pieces of string, or yarn and a work map, make equal sets to arrive at the product.Ex. Given eight objects that represent the product, make equal sets to represent the factors (e.g., 2 + 2 + 2 + 2) and count to arrive at the product (e.g., 8).

Level II Students will:EE4.OA.4. Make equal sets and count to determine the product.Ex. Using two spinners, spin first spinner to determine the number of groups and the second spinner to determine how many in each group. Supply the numbers from the spinners as factors in the multiplication equation (e.g., __ x __ = ___).

Level I Students will:EE4.OA.4. Replicate one way to arrive at a product.Ex. Copy a teacher-created model using manipulatives.Ex. Given a set, replicate the equal set. ** why don't we use ___ instead of underline?

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Grade Level: 4th Standard with code: 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.Domain: Operation and Algebraic Thinking Cluster: Generate and analyze patterns.

Quarter 1: Identify a number or shape pattern Generate a number or shape pattern that

follows a given rule. Analyze a pattern to determine features not

apparent in the rule (always odd or even, alternates between odd and even, etc.).

Quarter 2: Identify a number or shape pattern Generate a number or shape pattern that

follows a given rule. Analyze a pattern to determine features not

apparent in the rule (always odd or even, alternates between odd and even, etc.).










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Students:Given a number or shape pattern in the form of a rule,

Generate successive members of the pattern and identify apparent features of the pattern that were not explicit in the rule itself, (e.g., 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.

Students know:

Strategies for generating and recording number or shape patterns from rules,

Strategies for identifying and communicating shape and number patterns.


Generate and record number and shape patterns from rules,

Use logical reasoning and informal language to explain relationships between successive terms in a pattern.

Patterns in the number system can be used with logical reasoning to make conjectures and solve problems,

Identifying patterns in the number system leads to a deeper understanding of numbers, their characteristics, and their properties.

EE4.OA.5. Use repeating patterns to make predictions.

Level IV Students will:EE4.OA.5. Create a pattern based on a given rule and their prediction of what comes next.Ex. Given a BCBC rule, create a pattern based on the rule.Ex. Given a die with plus two, or plus three, rolls the die and creates a number pattern based on the outcome.

Level III Students will:EE4.OA.5. Use repeating patterns to make predictions.Ex. Using a number line, predict what the next number will be when you apply the rule “add 2.”Ex. Using a shape pattern (e.g., squares, circles, triangles) predict what will come next in the series of three shapes.Ex. Given a simple ABCABC pattern, indicate, “What comes next?”

Level II Students will:EE4.OA.5. Replicate a pattern.Ex. Using wooden beads, copy a pattern.Ex. Rhythmic or tactile patterns.

Level I Students will:EE4.OA.5. Differentiate between a pattern and a non-pattern.Ex. A pile of blocks vs. an ABAB pattern of blocks.Ex. Play listening game to determine rhythmic patterns versus non-patterns.

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Grade Level: 4th Standard with code: 4.NBT.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 that 700 ÷ 70 = 10 by applying concepts of place value and division.Domain: Number and Operations in Base Ten Cluster: Generalize place value understanding for multi-digit whole numbers.

Quarter 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.

Quarter 2: 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.










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Students:

Explain, when asked, how the value of a digit differs in two successive place values, that the one to the right is 1/10 of the one to its left or that the one on the left is 10 times the one on the right.

Students know:

Place values

Place value models


Use logical reasoning to explain the relationship between two successive place values.

Values of digits in any multi-digit number are based on patterns within a base-10 place value system,

Patterns created by the use of 10 digits in a place value system make a place value to the right 1/10 of the previous place value and a place value to the left 10 times the previous place value.

EE4.NBT.1. Compare numbers to each other based on place value groups by composing and decomposing to 50.

Level IV Students will:EE4.NBT.1. Compare numbers to each other based on place value groups by composing and decomposing greater than 50.Ex. Given a number over 50, use place value blocks to indicate the value of each digit.Ex. Using popsicle sticks with beans glued to it in groups of 10 and loose beans, illustrate a multi-digit number.Ex. Show a number on the number line and answer the number of tens and ones in the given number.Ex. Decompose numbers to 50 in multiple ways (e.g., 36 is three 10s and six ones, or two 10s and 16 ones, or 36 ones).

Level III Students will:EE4.NBT.1. Compare numbers to each other based on place value groups by composing and decomposing to 50.Ex. Given a two digit number up to 50, use place value blocks to indicate the tens value and the ones value.Ex. Use money (dimes and pennies) to represent place value.Ex. Decompose numbers to 50 (e.g., 15 is one 10 and five ones, 22 is two 10s and two ones, 36 is three 10s and six ones, 41 is four 10s and a one, 57 is five 10s and seven ones).Ex. Decompose numbers in one way (e.g. 36 is three sets of 10 and six ones).

Level II Students will:EE4.NBT.1. Compose and decompose whole numbers to 20.Ex. Given 15 pennies, create a group of one 10 and a group of five ones.Ex. Use a number balance to determine what two numbers are needed to equal the number on the other side.

Level I Students will:EE4.NBT.1. Identify whole numbers to 10.Ex. Given sets, pair with numbers.Ex. Given numbers, match to sets.

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Grade Level: 4th Standard with code: 4.NBT.2 Read and write multi-digit whole numbers using 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.Domain: Number and Operations in Base Ten Cluster: Generalize place value understanding for multi-digit whole numbers.

Quarter 1: Read and write multi-digit whole numbers

using 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.

Quarter 2: Quarter 3: Quarter 4:









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Students:Given multi-digit whole numbers orally or in written form,

Represent quantities in a variety of ways including words, base-ten numerals, and expanded form,

Explain relationships among representations.

Given two numbers less than 1000,

Use place value terminology and concepts to explain and justify the placement of <, =, > to compare the numbers and create true equalities and inequalities.

Expanded form

<, =, and > symbols

Students know:

Place values,

Meanings and appropriate use of the mathematical symbols: <, =, >.


Represent quantities in a number of forms including words, base-ten numerals, and expanded form,

Compare whole numbers in equalities and inequalities.

The same quantity can be represented with words, mathematical models, and expanded form based on the place value of the digits,

The value of a digit in a multi-digit number depends on the place value spot it holds.

EE4.NBT.2. Compare whole numbers (<, >, =).

Level IV Students will:EE4.NBT.2. Compare whole numbers using symbols (<, >, =).Ex. Utilize a number line to compare two numbers greater than 50 and place a card with the correct symbol on the line to show the relationship (<, >).Ex. During P.E., compare scores of a game to determine the winner. Use the symbol to show the relationship between the scores.Ex. State or match meaning of >, <, and = as greater than, less than, or equal to.

Level III Students will:EE4.NBT.2. Compare whole numbers (<, >, =).Ex. Given two groups of blocks, close or equal in value, determine which is greater, less, or equal.Ex. Using a floor number line, two students stand on two different numbers and determine which is greater or less than.

Level II Students will:EE4.NBT.2. Compare whole numbers (<, >, =) from 0-20.Ex. Given two groups of objects, seven blocks and 10 blocks, determine which is greater or which is less.Ex. Play a fish game: One fish and two ponds, each with a certain number of bugs, turn fish towards the pond with the most bugs.

Level I Students will:EE4.NBT.2. Compare whole numbers (<, >) from 0-10.Ex. Use a 10 frame with two tactile dots and a 10 frame with 10 tactile dots, determine which is more or less.

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Grade Level: 4th Standard with code: 4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.Domain: Number and Operations in Base Ten Cluster: Generalize place value understanding for multi-digit whole numbers.

Quarter 1: Round multi-digit whole numbers to any

place using place value.

Quarter 2: Round multi-digit whole numbers to any

place using place value.










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Students:Given any whole number,

Justify the rounding of the number to a designated place value using models and place value vocabulary (e.g., 3,456 rounded to the nearest ten is 3460 because it is between the two tens 3450 and 3460, but closer to 3460).

Students know:

Place value vocabulary: ones, tens, hundreds, thousands, ten-thousands, hundred-thousands, millions,

Place value models (e.g., number lines, place value blocks),

Place value strategies for comparing and ordering numbers.


Count by 10s, 100s, 1000s, 10,000s, etc.,

Determine what is halfway between two consecutive multiples of powers of 10 (360 and 370, 36,000 and 37,000),

Compare whole numbers,

Use place value vocabulary, models, and logical reasoning to justify solutions to rounding problems.

Rounding aids estimation of quantities by changing the original number to the closest multiple of a power of 10.

EE4.NBT.3. Round one- and two-digit whole numbers from 0—50 to the nearest 10.

Level IV Students will:EE4.NBT.3. Round one- and two-digit numbers, greater than 50, to the nearest 10.Ex. Roll the dice to count up the rounding tape and state the nearest 10.Ex. Using a hundreds chart and a given number between 50-100, round to the nearest tens place.

Level III Students will:EE4.NBT.3. Round single one- and two-digit whole numbers from 0-50 to the nearest 10.Ex. Poster boards, distributed around the room, labeled by tens up to 50, be given a number, and asked to go to the nearest 10.Ex. Using pennies earned, exchange for dimes.

Level II Students will:EE4.NBT.3. Round single one-digit numbers to the nearest 10.Ex. Using paper plates labeled zero and 10, given a card with a number zero to 10, place it on the correct plate.Ex. Use a number line to round to the nearest 10.

Level I Students will:EE4.NBT.3. Identify numbers that are more or less than five on a number line.Ex. Place their fingers on five on a number line and count to find a number greater than five.Ex. Shown five on a number line, identify a number that is less than five.

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Grade Level: 4th Standard with code: 4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.Domain: Number and Operations in Base Ten Cluster: Use place value understanding and properties to perform multi-digit arithmetic.

Quarter 1: Fluently add and subtract multi-digit whole

numbers less than or equal to 1,000,000using the standard algorithm.

Quarter 2: Fluently add and subtract multi-digit

whole numbers less than or equal to 1,000,000using the standard algorithm.

Quarter 3: Fluently add and subtract multi-

digit whole numbers less than or equal to 1,000,000using the standard algorithm.

Quarter 4: Fluently add and subtract multi-

digit whole numbers less than or equal to 1,000,000using the standard algorithm.











Students:Given a context which calls for the addition or subtraction of two whole numbers,

Choose the most appropriate strategy for computing the answer,

Produce accurate results efficiently using the standard algorithm when

Standard algorithms (addition and subtraction)

Students know:

Strategies for computing answers to addition and subtraction problems.


Strategically choose and apply appropriate methods for adding and subtracting,

Accurately find sums/differences using the standard addition and

EE4.NBT 4. Add and subtract double-digit whole numbers.

Level IV Students will:EE4.NBT.4. Add and subtract multi-digit whole numbers.Ex. Given base ten pieces, make exchanges to solve multi-digit addition and subtraction problems.Ex. Use a calculator and show how the problem is solved.

Level III Students will:EE4.NBT.4. Add and subtract double-digit whole numbers.Ex. Use a sorting box divided into two sections with manipulatives to add, subtract, and regroup to solve addition and subtraction problems.Ex. Use break-apart numbers (e.g., 20 + 30 = 50, 3 + 5 = 8, 40 + 8 = 48).Ex. Use a number line to demonstrate addition by tens.

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appropriate. subtraction algorithms.

Mathematical problems can be solved using a variety of strategies, models, and representations,

Efficient application of computation strategies is based on the numbers and operations in the problems.

The steps used in the standard algorithm for addition and subtraction can be justified by using properties of operations and understanding of place value.

Among all techniques and algorithms that may be chosen for accurately performing multi-digit computations, some procedures have been chosen with which all should be fluent for efficiency, communication, and

Level II Students will:EE4.NBT.4. Solve addition with numbers 20-50 and subtraction problems with numbers 0-20.Ex. Use counters to add and subtract.Ex. Use number lines to add or subtract.Ex. Produce addends to 10 fluently.Ex. The teacher orally states 14 – 1 = 13 and use magnetic symbols to display the problem.

Level I Students will:EE4.NBT.4. Solve single digit addition problems to add one to another number.Ex .Use counters to add one to another number.Ex .Use number lines to add one to another number.

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use in other mathematics situations.


Grade Level: 4th Standard with code: 4.NBT.5 Multiply a whole number of up to four 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 explainthe calculation by using equations, rectangular arrays, and/or area models.Domain: Number and Operations in Base Ten Cluster: Use place value understanding and properties to perform multi-digit arithmetic.

Quarter 1: Quarter 2: Multiply a whole number of up to four

digits by a one-digit whole number. Multiply two two-digit numbers. Use strategies based on place value and

the properties of operations to multiply whole numbers.

Illustrate and explain calculations by using written equations, rectangular arrays, and/or area models.

Quarter 3: Multiply a whole number of up to

four digits by a one-digit whole number.

Multiply two two-digit numbers. Use strategies based on place value

and the properties of operations to multiply whole numbers.

Illustrate and explain calculations by using written equations, rectangular arrays, and/or area models.

Quarter 4:









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Students:Given multiplication problems (four-digit whole number by one-digit whole number, or two-digit numbers by two-digit numbers),

Use strategies based on multiplication models (e.g., rectangular arrays, open arrays, area models), place value and properties of operations to find and justify solutions and solution paths.

Students know:

Place value models for multiplying numbers (e.g., area models, open arrays, place value blocks),

Strategies for finding products based on place value and properties of operations.


Use strategies based on an understanding of place value and properties of operations to find products,

Use a variety of place value models of multiplication problems to justify solutions and solution paths.

Multiplication problems can be solved using a variety of strategies, models, and representations.

Efficient application of multiplication computation strategies is based on the numbers and operations in the problems.

EE4.NBT 5. N/A (See EE. 4.OA.1.)

EE4.NBT 5. N/A (See EE. 4.OA.1.)

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Grade Level: 4th Standard with code: 4.NBT.6 Find whole-number quotients and remainders 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.Domain: Number and Operations in Base Ten Cluster: Use place value understanding and properties to perform multi-digit arithmetic.

Quarter 1: Quarter 2: Find whole number quotients and

remainders with up to four-digit dividends and one-digit divisors.

Use the strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.

Illustrate and explain the calculation by using written equations, rectangular arrays, and/or area models

Quarter 3: Find whole number quotients and

remainders with up to four-digit dividends and one-digit divisors.

Use the strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.

Illustrate and explain the calculation by using written equations, rectangular arrays, and/or area models

Quarter 4:











Students:Given division problems (up to four-digit dividends and one-digit divisors),

Students know:

Tools for modeling division problems,


Model division problems using

EE4.NBT 6. N/A EE4.NBT 6. N/A

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Find whole-number quotients and remainders using strategies that involve using representations based on place value, properties of operations, and/or the relationship between multiplication and division,

Justify solutions and solution paths through equations, rectangular arrays, and/or area models.

Strategies and methods for symbolically (numerically) recording strategies for solving division problems.

appropriate tools,

Record strategies for solving division problems,

Use logical reasoning to communicate the relationship between models and symbolic (numeric) representations of solutions to division problems,

Accurately compute quotients with remainders.

Division problems can be solved using a variety of strategies, models, and representations,

Efficient application of division computation strategies is based on the numbers and operations in the problems,

Relationships between models of division problems and symbolic recordings of those models can be used to justify solutions.

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PVIS CCSS Math Pacing GuideGrade 4Grade Level: 4th Standard with code: 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.Domain: Number and Operations - Factors Cluster: Extend understanding of fraction equivalence and ordering.

Quarter 1:X

Quarter 2: Quarter 3: Recognize and identify equivalent

fractions with unlike denominators. Explain why a/b is equal to

(nxa)/(nxb) by using fraction models with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. (Ex: Use fraction strips to show why ½=2/4=3/6=4/8)

Use visual fraction models to show why fractions are equivalent (ex: ¾ = 6/8)

Generate equivalent fractions using visual fraction models and explain why they can be called “equivalent.

Quarter 4:








Look for and express regularity in repeatedreasoning

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Students:Given a fraction a/b,

Use visual models to create equivalent fractions and explain the generalized pattern, a/b = (n x a) / (n x b),

Use the generalized pattern to generate equivalent fractions.

Students know:

Strategies for partitioning wholes,

Strategies for representing fractional parts of a whole,

Multiplication and division strategies.


Represent fractional quantities using visual models,

Write fractions related to visual models,

Generate equivalent fractions by modeling the original fraction and further partitioning shares,

Explain the equivalence of fractions and the generalization a/b = (n x a) / (n x b) using logical reasoning, patterns, and visual models,

Generate equivalent fractions using the generalization a/b = (n x a) / (n x b).

Two fractions are equivalent if they are the same size share

EE4.NF.1-2. Understand 2/4 = 1/2.

Level IV Students will:EE4.NF.1-2. Understand two fractions having unlike denominators are equivalent if they represent the same size portion of a whole.Ex. Given two squares of paper, one scored for 1/2s and one scored for 1/8s, fold the each paper as scored, then unfold the paper scored for 1/3s and compare to the one folded into 1/2 to find the same size portion (e.g., 4/8 = 1/2).*Ex. Use tangrams.

Level III Students will:EE4.NF.1-2. Understand 2/4 = 1/2.Ex. Given two rectangles, cut one rectangle into half and a second into fourths and compare the rectangles to determine how many fourths equal a half.Ex. Working with two rectangles of the same size, fold one rectangle in half and the other in fourths and compare to find how many fourths equal half.Ex. Using a picture of two circles, cut one in half and the other in fourths and compare them to find how many fourths equal half.

Level II Students will:EE4.NF.1-2. Understand 4/4 or 2/2 = 1.Ex. Complete two- and four-piece puzzles.Ex. File folder game with self-sticking non-adhesive pieces that make a whole.

Level I Students will:EE4.NF.1-2. Understand that two halves is equivalent to one whole.Ex. Wooden shapes are separated into halves and put back together into a whole.Ex. Plastic eggs are broken into halves and put back to whole.*Refer to the Common Core Essential Elements document for diagram.

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(represent the same amount) of the same whole or name the same point on a number line.

PVIS CCSS Math Pacing GuideGrade 4Version 3 2015-2016 36

Grade Level: 4th Standard with code: 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.Domain: Number and Operations - Factors Cluster: Extend understanding of fraction equivalence and ordering.

Quarter 1: Quarter 2: Quarter 3: Recognize fractions as being greater

than, less than, or equal to other fractions.

Record comparison results with symbols: <, >, =

Use benchmark fractions such as ½ for comparison purposes.

Make comparisons based on parts of the same whole.

Compare two fractions with different numerators, e.g. by comparing to a benchmark fraction such as ½.

Compare two fractions with different denominators, e.g. by creating common denominators, or by comparing to a benchmark fraction such as ½.

Justify the results of a comparison of two fractions, e.g. by using a visual fraction model.

Quarter 4:









Evidence of Student Vocabulary Knowledge Skills Common Core Instructional Achievement Level Descriptors

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Attainment/Assessment Essential ElementsStudents:Given two fractions (having denominators of 2,3, 4, 5, 6, 8, 10, 12, 100),

Use logical reasoning and a variety of models to represent and order the fractions (using <, =, >) and justify their answers,

Communicate the reason why it is not valid to make a comparison between fractions that refer to different wholes (e.g., why it may not be valid to say 1/2 > 1/4 if the 1/2 refers to a small pizza and the 1/4 refers to an extra-large pizza or "Susie said her 1/6 pizza was bigger than my 1/2 pizza. Is she correct?").

Benchmark fraction Students know:

Strategies for representing fractional quantities,

Strategies for comparing fractions (e.g., comparing numerators of like fractions, comparing denominators of fractions with like numerators, creating common denominators, and comparing to landmark fractions such as 1/2).


Strategically choose and apply representations to compare two fractions,

Record the comparison of two fractions using <, =, and > notation,

Use mathematical language and logical reasoning to justify solutions.

Two fractions are equivalent if they are the same size share (represent the same amount) of the same whole or name the same point on a number line,

Comparisons of fractions are valid only when the two fractions refer to the same whole.

EE4.NF.1-2. Understand 2/4 = 1/2.

Level IV Students will:EE4.NF.1-2. Understand two fractions having unlike denominators are equivalent if they represent the same size portion of a whole.Ex. Given two squares of paper, one scored for 1/2s and one scored for 1/8s, fold the each paper as scored, then unfold the paper scored for 1/3s and compare to the one folded into 1/2 to find the same size portion (e.g., 4/8 = 1/2).*Ex. Use tangrams.

Level III Students will:EE4.NF.1-2. Understand 2/4 = 1/2.Ex. Given two rectangles, cut one rectangle into half and a second into fourths and compare the rectangles to determine how many fourths equal a half.Ex. Working with two rectangles of the same size, fold one rectangle in half and the other in fourths and compare to find how many fourths equal half.Ex. Using a picture of two circles, cut one in half and the other in fourths and compare them to find how many fourths equal half.

Level II Students will:EE4.NF.1-2. Understand 4/4 or 2/2 = 1.Ex. Complete two- and four-piece puzzles.Ex. File folder game with self-sticking non-adhesive pieces that make a whole.

Level I Students will:EE4.NF.1-2. Understand that two halves is equivalent to one whole.Ex. Wooden shapes are separated into halves and put back together into a whole.Ex. Plastic eggs are broken into halves and put back to whole.*Refer to the Common Core Essential Elements document for diagram.

PVIS CCSS Math Pacing Guide

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Grade 4

Grade Level: 4th Standard with code: 4.NF.3a 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 separating parts referring to the same whole.Domain: Number and Operations - Factors Cluster: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Quarter 1: Quarter 2: Quarter 3: Accumulating unit fractions (1/b)

results in a fraction (a/b), where a is greater than 1.

Using fraction models, reason that addition of fractions is joining parts that are referring to the same whole.

Using fraction models, reason that subtraction of fractions is separating parts that are referring to the same whole.

Quarter 4:











Students:Given any fraction or mixed number,

Connect their understanding of unit fractions and their understanding of addition to decompose the given fraction or mixed number into the sum of smaller

Students know:

Characteristics of addition and subtraction contexts for whole numbers and like fractions,

Strategies for representing and solving addition and


Represent quantities (whole numbers and fractions) and operations (addition and subtraction) physically, pictorially, or symbolically,

EE4.NF.3. Differentiate between whole, half, and fourth.

Level IV Students will:EE4.NF.3. Differentiate fractional parts less than 1/4.Ex. With fraction bars labeled 1/2, compare the 1/2 to fraction bars less than.Ex. Using squares, fold it in 1/2, 1/4, 1/8, . . .

Level III Students will:EE4.NF.3. Differentiate between whole, half, and fourth.Ex. Use fraction strips and fraction tiles to identify whole and half, and which is more.Ex. Using squares of paper, fold it in 1/2 and 1/4 and identify the parts.

Level II Students will:

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fractions/mixed numbers, including unit fractions.

Given a variety of addition and subtraction word problems involving fractions with like denominators,

Explain and justify solutions using connections among unit fractions, pictorial representations, and related equations involving a single unknown.

subtraction problems involving fractions.

Strategically choose and apply a variety of representations to solve addition and subtraction word problems involving like fractions,

Use symbols to represent unknown quantities in addition and subtraction equations and solve such equations,

Accurately compute sums and differences of fractions,

Use logical reasoning and connections among representations to justify solutions and solution paths.

Addition and subtraction of fractions are applied to fractions referring to the same whole,

The operation of addition with whole numbers and/or fractions represents both putting together and adding to contexts,

The operation of subtraction with whole numbers and/or

EE4.NF.3. Differentiate between whole and half.Ex. Given a whole sandwich versus a half sandwich cut horizontally, vertically, and diagonally select the whole or half upon request.Ex. Show the halfway point on a number line.Ex. With pictures cut into halves and pictures not cut, sort the pictures into halves and wholes.

Level I Students will:EE4.NF.3. Recognize that fractions are part of a whole.Ex. Using a self-sticking non-adhesive shape, take apart and put together fractional parts of a whole.Ex. Utilize wooden shapes, separate into halves and put back together into whole.Ex. Shown pictures of the whole class and part of the class, select the picture that shows part of the class upon request.

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fractions represents taking apart, taking from, and additive comparison contexts,

The unit fraction (1/b) names the size of the unit with respect to the referenced whole and that the numerator counts the parts referenced and the denominator tells the number of parts into which the whole was partitioned,

The operations of addition and subtraction are performed on counts with like names/labels/denominators and that the sum or difference retains the same name/label/denominator

Mathematical problems (addition and subtraction of fractions) can be solved using a variety of strategies, models, representations, Variables represent unknown quantities when modeling mathematical situations algebraically.


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Grade Level: 4th Standard with code: 4.NF.3b Understand a fraction a/b with a > 1 as a sum of fractions 1/b.b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual 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.Domain: Number and Operations - Factors Cluster: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Quarter 1: Quarter 2: Quarter 3: Add and subtract fractions with

like denominators. Recognize multiple

representations of one whole using fractions with the same denominator.

Using visual fraction models, decompose a fraction into the sum of fractions with the same denominator in more than one way.

Record decompositions of fractions as an equation and explain the equation using visual fraction models.

Quarter 4:












Connect their understanding of unit fractions and their understanding of addition to decompose the given fraction or mixed number

Students know:


Strategies for representing and


Represent quantities (whole numbers and fractions) and operations (addition and subtraction) physically, pictorially, or




Version 3 2015-2016 42

into the sum of smaller fractions/mixed numbers, including unit fractions.



solving addition and subtraction problems involving fractions.

symbolically,







The operation of

Level II Students will:EE4.NF.3. Differentiate between whole and half.Ex. Given a whole sandwich versus a half sandwich cut horizontally, vertically, and diagonally select the whole or half upon request.Ex. Show the halfway point on a number line.Ex. With pictures cut into halves and pictures not cut, sort the pictures into halves and wholes.


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subtraction with whole numbers and/or fractions represents taking apart, taking from, and additive comparison contexts,




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Grade Level: 4th Standard with code: 4.NF.3c Understand a fraction a/b with a > 1 as a sum of fractions 1/b.c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.Domain: Number and Operations - Factors Cluster: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Quarter 1: Quarter 2: Quarter 3: Add and subtract mixed numbers

with like denominators by using properties of operations and the relationship between addition and subtraction.

Replace mixed numbers with equivalent fractions, using visual fraction models.

Replace improper fractions with a mixed number, using visual fraction models.

Quarter 4:












Connect their understanding of unit fractions and their understanding of addition to decompose the given fraction or mixed number

Students know:




Represent quantities (whole numbers and fractions) and operations (addition and subtraction) physically, pictorially, or




Version 3 2015-2016 45

into the sum of smaller fractions/mixed numbers, including unit fractions.



solving addition and subtraction problems involving fractions.

symbolically,







The operation of

Level II Students will:EE4.NF.3. Differentiate between whole and half.Ex. Given a whole sandwich versus a half sandwich cut horizontally, vertically, and diagonally select the whole or half upon request.Ex. Show the halfway point on a number line.Ex. With pictures cut into halves and pictures not cut, sort the pictures into halves and wholes.


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subtraction with whole numbers and/or fractions represents taking apart, taking from, and additive comparison contexts,




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Grade Level: 4th Standard with code: 4.NF.3d Understand a fraction a/b with a > 1 as a sum of fractions 1/b.d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.Domain: Number and Operations - Factors Cluster: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Quarter 1: Quarter 2: Quarter 3: Add and subtract fractions with

like denominators. Solve word problems involving

addition and subtraction of fractions referring to the same whole and having like denominators, by using visual fraction models and equations to represent the problem.

Quarter 4:












Connect their understanding of unit fractions and their understanding of addition to decompose the given fraction or mixed number into the sum of smaller fractions/mixed numbers,

Students know:


Strategies for representing and solving addition and subtraction problems


Represent quantities (whole numbers and fractions) and operations (addition and subtraction) physically, pictorially, or symbolically,

Strategically choose and




Level II Students will:EE4.NF.3. Differentiate between whole and half.

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including unit fractions.



involving fractions. apply a variety of representations to solve addition and subtraction word problems involving like fractions,






The operation of subtraction with whole numbers and/or fractions represents taking apart, taking from, and additive

Ex. Given a whole sandwich versus a half sandwich cut horizontally, vertically, and diagonally select the whole or half upon request.Ex. Show the halfway point on a number line.Ex. With pictures cut into halves and pictures not cut, sort the pictures into halves and wholes.


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comparison contexts,


The operations of addition and subtraction are performed on counts with like names/labels/denominators and that the sum or difference retains the same name/label/denominator,

Mathematical problems (addition and subtraction of fractions) can be solved using a variety of strategies, models, representations,

Variables represent unknown quantities when modeling mathematical situations algebraically.

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PVIS CCSS Math Pacing GuideGrade 4Grade Level: 4th Standard with code: 4.NF.4a Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.a. Understand a fraction a/b as a multiple of 1/b. For example, 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).Domain: Number and Operations - Factors Cluster: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Quarter 1: Quarter 2: Quarter 3: Represent a fraction a/b as a

multiple of 1/b (unit fractions). For example, represent 5/4 as an accumulation of five ¼’s.

Apply multiplication of whole umbers to multiplication of a fraction by a whole number using visual fraction models. (For example, just as students know that four 3’s can be represented by 4x3, students know that 5 ¼’s is 5 x ¼ which is 5/4.)

Quarter 4:











Students:Given any fraction,

Use their knowledge of multiples of whole numbers to connect a visual representation of a non-unit fraction to a product of a whole number and a unit

Multiple Students know:

Associative Property of Multiplication,

Characteristics of multiplication contexts for whole numbers and fractions,


Represent and rename fractional quantities as multiples of whole numbers and unit fractions,

Strategically choose and apply a variety of

EE4.NF.4. N/A (See EE. 4.OA.1-2.)

EE4.NF.4. N/A (See EE. 4.OA.1-2.)

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fraction.

Given a multiplication problem involving a whole number and a fraction,

Use a visual representation of the problem, understanding of unit fractions, and properties of the operation of multiplication to justify n x (a/b) = (n x a)/b,

Given a word problem involving the multiplication of a fraction by a whole number,

Explain and justify solutions and the reasonableness of solutions using connections among unit fractions, visual representations, and an understanding of multiplication.

Strategies for representing and solving multiplication problems involving whole numbers and fractions.

representations to solve multiplication word problems involving whole numbers and fractions,

Apply knowledge of the Associative Property of Multiplication with knowledge of unit fractions to accurately compute products of whole numbers and fractions,

Use logical reasoning and connections among representations to justify solutions, reasonableness of solutions, and solution paths.

A fraction a/b is a multiple of the unit fraction 1/b, (e.g.., a/b = a x 1/b),

Multiplication may be viewed as putting together equal-sized groups,

Mathematical problems (multiplication of whole numbers and fractions) can be solved using a variety

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of strategies, models, and representations,

A fractional quantity can be modeled using a variety of representations (e.g., part of a whole, part of a group, a distance on a numberline) each of which may reveal important features of given contexts.

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PVIS CCSS Math Pacing GuideGrade 4Grade Level: 4th Standard with code: 4.NF.4b Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, 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.)Domain: Number and Operations - Factors Cluster: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Quarter 1: Quarter 2: Quarter 3: Explain that a multiple of a/b is a

multiple of 1/b (unit fraction) using a visual fraction model.

Multiply a fraction by a whole number by using the idea that a/b is a multiple of 1/b. For example, use a visual fraction model to express 3 x (2/5) as 6 x (1/5) recognizing this product as (6/5).

Quarter 4:









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Use their knowledge of multiples of whole numbers to connect a visual representation of a non-unit fraction to a product of a whole number and a unit fraction.




Explain and justify solutions and the reasonableness of




Strategies for representing and solving multiplication problems involving whole numbers and fractions.



Strategically choose and apply a variety of representations to solve multiplication word problems involving whole numbers and fractions,



A fraction a/b is a multiple of the unit fraction 1/b, (e.g.., a/b =

EE4.NF.4. N/A (See EE. 4.OA.1-2.)

EE4.NF.4. N/A (See EE. 4.OA.1-2.)

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solutions using connections among unit fractions, visual representations, and an understanding of multiplication.

a x 1/b),


Mathematical problems (multiplication of whole numbers and fractions) can be solved using a variety of strategies, models, and representations,


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PVIS CCSS Math Pacing GuideGrade 4Grade Level: 4th Standard with code: 4.NF.4c Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models 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?Domain: Number and Operations - Factors Cluster: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Quarter 1: Quarter 2: Quarter 3: Multiply a fraction by a whole

number. Use fraction models and equations

to represent the problem. Solve word problems involving

multiplication of a fraction by a whole number.

Quarter 4:









Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills

Common Core Essential Elements

Instructional Achievement Level Descriptors


Use their knowledge of multiples of whole numbers to connect a visual representation of a non-unit fraction to a product of a whole number and a unit fraction.







Strategically choose and apply a variety of representations to solve multiplication word problems involving whole

EE4.NF.4. N/A (See EE. 4.OA.1-2.)

EE4.NF.4. N/A (See EE. 4.OA.1-2.)

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Explain and justify solutions and the reasonableness of solutions using connections among unit fractions, visual representations, and an understanding of multiplication.

solving multiplication problems involving whole numbers and fractions.

numbers and fractions,



A fraction a/b is a multiple of the unit fraction 1/b, (e.g.., a/b = a x 1/b),


Mathematical problems (multiplication of whole numbers and fractions) can be solved using a variety of strategies, models, and representations,


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Grade Level: 4th Standard with code: 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 100.4 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.Domain: Number and Operations - Factors Cluster: Understand decimal notation for fractions, and compare decimal fractions.

Quarter 1: Quarter 2: Quarter 3: Rename and recognize a fraction

with a denominator of 10 as a fraction with a denominator of 100.

Recognize that two fractions with unlike denominators can be equivalent.

Use knowledge of renaming tenths to hundredths to add two fractions with denominators 10 and 100.

Quarter 4:









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Students:Given a fraction with a denominator of 10,

Use visual models and the generalized pattern, a/b = (n x a) / (n x b) to find the equivalent fraction with a denominator of 100.

Given an addition problem with two fractions with respective denominators of 10 and 100,

Compute the sum by expressing the fraction with a denominator of 10 as an equivalent fraction with a denominator of 100 and then adding.

Students know:

Strategies for generating equivalent fractions,

Strategies for adding fractions with like denominators.


Represent fractional quantities using visual models,

Write fractions related to visual models,

Generate equivalent fractions using the generalization a/b = (n x a) / (n x b).

Accurately add fractions.

Addition may be viewed as joining or adding to,

Two fractions are equivalent if they are the same size share of the same whole or are the same point on a number line,

The operations of addition and subtraction are performed on counts with like names/denominators and that the sum or difference retains the same name/denominator.

EE4.NF.5-7. N/A (Decimals begin at grade 7).


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Grade Level: 4th Standard with code: 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.Domain: Number and Operations - Factors Cluster: Understand decimal notation for fractions, and compare decimal fractions.

Quarter 1: Quarter 2: Quarter 3: Explain the values of digits in the

decimal places. Read and write decimals through

hundredths. Rename fractions with 10 and 100in

the denominator as decimals. Recognize multiple representations

of fractions with denominators 10 or 100. Represent fractions with denominators 10 or 100 with multiple representations and decimal notation.

Explain how decimals and fractions relate.

Quarter 4:









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Students:Given a fraction with a denominator of 10 or 100,

Write the equivalent fraction using decimal notation.

Given a fraction in decimal notation (tenths or hundredths),

Write the equivalent fraction.

Given a fraction in decimal notation,

Create a number line diagram and justify the placement of the fraction on the number line.

Students know:

Decimal place value,

Decimal notation,

Fraction notation.


Represent fractional quantities including decimals using visual models,

Write fractions including decimals related to visual models.

Two fractions are equivalent if they are the same size share (represent the same amount) of the same whole or name the same point on a number line.



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Grade Level: 4th Standard with code: 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 results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.Domain: Number and Operations - Factors Cluster: Understand decimal notation for fractions, and compare decimal fractions.

Quarter 1: Quarter 2: Quarter 3: Recognize that comparisons are

valid only when the two decimals refer to the same whole.

Compare two decimals to hundredths by reasoning about their size.

Record the results of comparisons with the symbols >, =, or <.

Justify the conclusions using visual models and other methods.

Quarter 4:









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Evidence of Student Attainment/Assessment

Vocabulary Knowledge Skills Common Core Essential Elements


Students:Given two decimals,

Use logical reasoning and a variety of models to represent and order the decimals (using <, =, >) and justify their answers,

Communicate the reason why it is not valid to make a comparison between decimals that refer to different wholes (e.g., why it may not be valid to say 0.5 > 0.25 if 0.5 refers to a small pizza and the 0.25 refers to an extra-large pizza, or "Susie said her 0.25 pizza was bigger than my 0.5 pizza. Is she correct?").

Students know:

Strategies for representing decimal quantities,

Strategies for comparing decimals (e.g., comparing numerators of like decimals creating common denominators, comparing to landmark fractions such as 1/2).


Strategically choose and apply representations to compare two decimals,

Record the comparison of two decimals using <, =, and > notation,

Use mathematical language and logical reasoning to justify solutions.

Two fractions (decimals) are equivalent if they are the same size share (represent the same amount) of the same whole or name the same point on a number line,

Comparisons of fractions (decimals) are valid only when the two fractions (decimals) refer to the same whole.



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Grade Level: 4th Standard with code: 4.MD.1 Know relative sizes of measurement units within 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), ...Domain: Measurement and Data Cluster: Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

Quarter 1: Quarter 2: Quarter 3:

Quarter 4: Know relative size of

measurement units (km, m; kg, g; lb, oz; L, mL; hrs, min, sec)

Compare the different units within the same system of measurement (e.g. 1 ft = 12 in; 1 lb= 16 oz)

Convert larger units of measurement within the same system to smaller units and record conversions in a 2-column table.









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Students:Given a measurement in a relatively large unit (e.g., km, m, kg, lb., l, hr., min.),

Accurately convert the measurement to an equivalent measurement using smaller units (e.g., m, cm, g, oz., ml, min., sec.) within the same measurement system through the use of a two column table. (e.g., Express the length of a 4 ft. snake as 48 inches by generating a conversion table for feet and inches listing the number pairs (1,12), (2,24), etc.).

Students know:

Relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb., oz.; l, ml; hr., min., sec.,

Strategies for converting from relatively large units of measure to smaller units of measure within the same system including multiplication and two-column tables.


Multiply or use repeated addition to accurately generate number pairs for conversion tables,

Interpret tables to solve problems.

The relationships among units within a system of measurement (e.g., metric length, time, standard mass, etc.) are multiplicative comparisons.

EE4.MD.1. Identify the smaller measurement units that divide a larger unit within a measurement system.

Level IV Students will:EE4.MD.1. Solve problems by demonstrating whole units can be broken into smaller units.Ex. Use a one-cup measure to pour water into a pint jar to determine how many plants could be watered if each plant needs one cup of water.Ex. Pour soil from a 1/2-cup measuring cup into a pint to see how many starter pots could be filled with a pint of soil.Ex. Determine which is better for measuring a desktop, a ruler or a yardstick. Measure the tablet, mark the length on the ruler, and compare it to the yardstick.Ex. Pour tablespoons of water into a 1/2 cup a tablespoon at a time and determine how many one-tablespoon portions there are in a cup.

Level III Students will:EE4.MD.1. Identify the smaller measurement units that divide a larger unit within a measurement system.Ex. Identify how many inches are the smaller units on a ruler.Ex. Identify how many feet are the smaller units on a yardstick.Ex. Identify how many cups are the smaller units on a pint measuring cup.Ex. Given several measurement tools, match three rulers to one-yard stick.

Level II Students will:EE4.MD.1. Identify standard units of measurements.Ex. Use different measurement tools to measure sand in a tray.Ex. Use the inch worms on a foot ruler or yard stick to make the connection that while they both measure, one unit is smaller than the other.

Level I Students will:EE4.MD.1. Use measurement tools.Ex. Compare the length of a ruler to the length of a book.Ex. Use a balance scale to compare different sets of objects to determine which objects are <, >, or =.

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PVIS CCSS Math Pacing GuideGrade 4Grade Level: 4th Standard with code: 4.MD.2 Use the four operations to solve word problems 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.Domain: Measurement and Data Cluster: Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

Quarter 1: Quarter 2: Quarter 3: Quarter 4: Add, subtract, multiply, and divide

fractions and decimals. Express measurements given in a

larger unit in terms of a smaller unit.

Solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money. Solve word problems involving measurement that include simple fractions or decimals.

Solve word problems involving measurement that include simple fractions or decimals.

Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale









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Students:Given word problems involving distances (km, m, cm), intervals of time (hr., min., sec.), liquid volumes (l, ml), mass (kg, g, lb., oz.), and money, including problems involving simple fractions or decimals and problems involving measurements in different units (within the same measurement system, conversions from larger to smaller units only),

Justify choices of units, solve the problem, and justify solutions.

Students know:

Relative sizes of measurement units within one system of units including: km, m, cm; kg, g; lb., oz.; L, mL; hr., min., sec.,

Strategies for converting from relatively large units of measure to smaller units of measure within the same system including multiplication and two-column tables,

Strategies for solving word problems involving measurement including number line representations.


Strategically choose an appropriate common unit to use for computations, when working with problems that contain measurements in different units,

Strategically choose and apply representations and computation techniques for solving real life mathematical problems,

Accurately compute solutions,

Use logical reasoning to justify solution paths.

The relationships among units within a system of measurement (e.g., metric length, time, standard mass, etc.) are multiplicative comparisons,

EE4.MD.2.a. Tell time to the half hour using a digital or to the hour using an analog clock.EE4.MD.2.b. Select the appropriate measurement tool from two related options to solve problems.EE4.MD.2.c. Use standard measurement to compare lengths of objects.EE4.MD.2.d. Identify objects that have volume.EE4.MD.2.e. Identify coins (penny, nickel, dime, quarter) and their values.

Level IV Students will:EE4.MD.2.a. Tell time to the quarter hour using a digital or analog clock.Ex. Indicate time to the quarter hour on a digital clock.Ex. Place clock hands to show the quarter hour.EE4.MD.2.b. Use the appropriate measurement tools to solve problems.Ex. Select and use the appropriate measuring tool to measure different quantities for assigned tasks (e.g., cup for liquid and powder; scale for solids).Ex. On a field trip to the grocery store, use the scale to determine how much a bag of apples weighs.EE4.MD.2.c. Use standard measurements to compare length of objects and indicate how many each is by standard measures.Ex. Given a pencil and book, mark the length of each on a ruler to tell which is longer and approximately how many each is by inches.Ex. Given a tape measure, mark the length of a bookcase and the teacher’s desk on the tape measure to show which is longer and approximately how many each is by feet.EE4.MD.2.d. Determine volume of a cube by counting units of measure.Ex. Use cubes to fill a box (small number, how many) and count the number of cubes needed to fill the box.Ex. Use liquid to fill bowl (how much, one cup, etc.). EE4.MD.2.e. Identify relative value of different collections of coins.Ex. When asked what is worth five cents, chooses a nickel. When asked what is worth 25 cents, choose a quarter.Ex. Given two coins, identify the value of each and indicate which is more.Ex. Given 14 pennies and two dimes, indicate which set is worth more.

Level III Students will:EE4.MD.2.a. Tell time to the half hour using a digital clock or to the hour using an analog clock.Ex. Identify which clock shows a stated time on a digital clock (i.e., 2:30).Ex. Move hands on a clock to show a stated half hour.Ex. Say the hour on an analog clock.EE4.MD.2.b. Select the appropriate measurement tool from two related options to solve problems.Ex. During a science experiment, select the best tool to use to measure various ingredients (e.g., tablespoon or cup, ruler or yardstick).Ex. Given a book, select the appropriate measuring tool to use to measure its length (e.g., ruler or yardstick).EE4.MD.2.c. Use standard measurement to compare lengths of objects.Ex. Given a pencil and book, mark the length of each on a ruler to tell which is longer.

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The size of the unit of measurement and the number of units are inversely related,

Addition and subtraction of measurements require measurements in the same unit and that the unit is maintained in the answer.

Ex. Given a tape measure, mark the length of a bookcase and the teacher’s desk on the tape measure to show which is longer.EE4.MD.2.d. Identify objects that have volume.Ex. Given a group of pictures (cup, rock, fork), choose which one can be filled.Ex. Identify objects in the room that can be filled (cup, fish tank, etc.).Ex. Given a square and a cube, indicate cube.EE4.MD.2.e. Identify coins (penny, nickel, dime, quarter) and their values.Ex. Given two coins, choose correct coin by name and value.Ex. Shown a coin, names coin.Ex. Show relative values of penny, nickel, dime, quarter by arranging them in order from least to most.

Level II Students will:EE4.MD.2.a. Relate time to the hour to activities.Ex. Look at clock - 2:00 is time to go home.Ex. Identify activity on schedule by matching the hour on the schedule to the hour on the clock.Ex. Point to hour for next activity on personal schedule.EE4.MD.2.b. Select the appropriate measurement tool from two unrelated options to solve problems.Ex. Given options of unrelated measuring tools, choose the best tool for a particular task (e.g., “When making cookies, which would you use to measure flour, a cup or ruler?”).Ex. In a field trip to the grocery store, show which measuring tool should be used to weigh a bag of apples. Allow students to practice by choosing other fruits or vegetables to weigh.EE4.MD.2.c. Measure length of objects using standard tools, such as rulers, yardsticks, and meter sticks.Ex. Given an object and a measuring tool, use the tool to mark the length of the object.Ex. Given a ruler and sand in a bucket, mark the depth of the sand on a ruler.Ex. Given a yardstick, measure different lengths or widths of the room and record the length on the yardstick in number of yardsticks.EE4.MD.2.d. Demonstrate solid or full, empty and part full.Ex. Given a piece of paper and a cube, indicate, “Which one takes up more space?”Ex. Fill a cup half full from the water fountain.Ex. As the teacher is filling a cup, say stop when it is half full.EE4.MD.2.e. Match coins that are alike (penny, nickel, dime, quarter).Ex. Given a group of coins, match coins that are alike.Ex. Given a picture of a quarter, choose a quarter from a group of coins.

Level I Students will:EE4.MD.2.a. Differentiate a digital and analog clock from other measurement tools as

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a tool for telling time.Ex. Given a digital or analog clock and a ruler, identify the clock for telling time.Ex. Asked “How do we know when it is time to go to lunch?”, indicate a clock.EE4.MD.2.b. Identify measurement tools.Ex. Sort non-standard and standard measurement tools into two different groups.Ex. Using pictures of standard and non-standard tools, identify which can be used to measure different items.EE4.MD.2.c. Identify items as long or short.Ex. Given two different items, one much longer than the first, indicate long/short.Ex. After traveling to somewhere in the classroom and somewhere outside of room, indicate each distance as long or short.EE4.MD.2.d. Identify vocabulary related to volume (full, empty).Ex. Match picture of unopened bottle of soda to “full.”Ex. Identify an “empty” cup.Ex. Indicate which is full and/or which is empty when holding/feeling a full can of soda and an empty can of soda.EE4.MD.2.e. Select objects that are used for money.Ex. Given three pictures (two non-coins and one coin), identify which one is a coin.Ex. Given two choices, identify which one is a coin.

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Grade Level: 4th Standard with code: 4.MD.3 Apply the area and perimeter formulas 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.Domain: Measurement and Data Cluster: Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

Quarter 1: Quarter 2: Quarter 3: Quarter 4: Know that the formula for the

perimeter of a rectangle is 2L + 2W or L+L+W+W.

Know that the formula for the area of a rectangle is L x W.

Apply the formula for perimeter of a rectangle to solve real world and mathematical problems.

Apply the formula for area of a rectangle to solve real world and mathematical problems.

Solve area and perimeter problems in which there is an unknown factor (n).









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Students:Given real world and mathematical problems involving area and perimeter of rectangular regions,

Use a variety of representations (e.g., models, drawings, and equations) based on area and perimeter formulas to find and justify solutions and solution paths.

Area

Perimeter

Students know:

Strategies for representing contexts involving area and perimeter of rectangular regions,

Strategies including standard formulas (A = L x W, P = 2L + 2W, P = L + L + W + W or P = 2 (L +W)) for computing measurements related to the area and perimeter of rectangular regions.


Discriminate between contexts asking for perimeter and those asking for area measurements,

Strategically choose and apply appropriate methods for representing and calculating ,

Accurately compute measurements within area and perimeter of rectangular region problems.

Perimeter is measured in length units and is the distance around a 2-D figure,

The area of a plane figure is measured by the number of same-size squares that exactly cover the interior space of the figure and the formula for the area of a rectangle is a result of this understanding,

The length and width of a rectangular region are related to both the area and the perimeter

EE4.MD.3. N/A (Area begins at 6th grade and perimeter begins at 7th grade).

EE4.MD.3. N/A (Area begins at 6th grade and perimeter begins at 7th grade).

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of that region,

Addition and subtraction of measurements require measurements in the same unit and that the unit is maintained in the answer,

The multiplication and division of measurements result in the units also being multiplied or divided and that a new unit is created for the answer.

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Grade Level: 4th Standard with code: 4.MD.4 Make a line plot to display a data set of measurements 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.Domain: Measurement and Data Cluster: Represent and interpret data.

Quarter 1: Quarter 2: Quarter 3: Quarter 4: Add and subtract fractions. Analyze and interpret a line plot

to solve problems involving addition and subtraction of fractions.

Create a line plot to display a data set of measurements given in fractions of a unit.











Students:

Make and use line plots (with the scale matching the unit of measure) to represent data generated by measuring lengths (to the nearest eighth inch) of several objects or by making repeated measurements,

Use information from

Line plot Students know:

Techniques for constructing line plots,

Standard units and related tools for measuring length,

Strategies for adding and subtracting fractions.


Use standard units and related tools to measure length to the nearest eighth inch,

Organize and represent length measurement data on a line plot,

Choose and apply

EE4.MD.4.a. Insert data into a preconstructed bar graph template.EE4.MD.4.b. Interpret data from a variety of graphs to answer questions.

Level IV Students will:EE4.MD.4.a. Insert data into a graph to represent a data set with a scale equal to 10 (0 to 10 by ones).Ex. Using a bar graph, enter one unit for each student to show their favorite activity in the correct category (lunch, physical therapy, music, P.E.) to determine most popular and least popular.Ex. Go to the lost and found, categorize and count types of items and graph them to determine most and least.EE4.MD.4.b. Create their own questions that can be answered by the data on a picture and bar graph.Ex. Cut simple graphs from newspapers/magazines and glue them onto card stock, create questions/answers based on the graph.Ex. Create their own questions/answers based on the information from a graph showing class preferences between two different activities.

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data displays to generate questions and solve problems including problems that involve addition and subtraction of fractions.

appropriate strategies to solve problems generated by conjectures from examining data displays,

Apply strategies for solving problems involving adding and subtracting fractions.

Questions concerning mathematical contexts can be generated and answered by collecting, organizing, and analyzing data and data displays.

Level III Students will:EE4.MD.4.a. Insert data into a preconstructed bar graph template.Ex. Clean out desks, sort objects found into reusable, recyclable, reducible, or take home items and graph results with one bar for each – reuse, recycle, reduce, or take home.Ex. Given a preconstructed bar graph and data, enter the data on the bar graph by shading one unit of the bar for each piece of data.EE4.MD.4.b. Interpret data from a variety of graphs to answer questions.Ex. Answer questions based on information provided in a picture schedule.Ex. Tell how many sunny days there were in a month, based on a weather graph.

Level II Students will:EE4.MD.4.a. Identify an appropriate scale for the data set.Ex. Identify if it is appropriate to use degrees or ounces on a weather graph.Ex. Determine if it is appropriate to use inches or pounds on a height graph.EE4.MD.4.b. Make observational statements about data in a picture and bar graph.Ex. Tell you what they observe on a graph of students’ eye colors.Ex. Show students a graph of the Big 12 football teams and ask them what they think it is about.

Level I Students will:EE4.MD.4.a. Given a topic, identify appropriate data to collect.Ex. Using a weather graph, identify appropriate data given the choice between a picture of the sun and a picture of a shoe.Ex. Given the topic of snacks, determine whether jelly beans or books are appropriate for the graph.EE4.MD.4.b. Demonstrate awareness that symbols may be used to represent objects and events.Ex. Picture of ice cream represents a favorite flavor.Ex. Picture of snow represents a snowy day.

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Grade 4

Grade Level: 4th Standard with code: 4.MD.5ab Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:a. An angle is measured with 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. 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.Domain: Measurement and Data Cluster: Geometric measurement: understand concepts of angle and measure angles.

Quarter 1: Quarter 2: Quarter 3: Quarter 4: Define angle. Recognize a circle as a geometric

figure that has 360 degrees. Recognize and identify an angle as

a geometric shape formed from 2 rays with a common endpoint.

Recognize that an angle is a fraction of a 360 degree circle.

Explain the angle measurement in terms of degrees









Evidence of Student Vocabulary Knowledge Skills Common Core Instructional Achievement Level Descriptors

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Attainment/Assessment Essential ElementsStudents:

Justify the result of measuring angles (in isolation and as parts of polygons) as a number of one-degree angles contained between the rays that define the angle and including language to describe the number of degrees through which the angle has "turned."

Angle

Circular arc

Ray

Endpoint

Students know:

Measurable attributes of geometric shapes, specifically angle size,

Units of measurement, specifically one-degree angle (degrees).


Communicate the process of measuring angles and the relationship of the measurement to a one-degree angle as the unit of measure.

The rotation of an angle is measured by the number of one-degree angles that exactly cover the rotation of the angle

EE4.MD.5. Recognize angles in geometric shapes.

Level IV Students will:EE4.MD.5. Label different types of angles in geometric shapes.Ex. Construct geometric shapes using counting sticks. Then determine whether angles are right angles or not.Ex. Given a square, determine whether the angles are right angles or not and state a square has four angles.

Level III Students will:EE4.MD.5. Recognize angles in geometric shapes.Ex. Draw an arc to identify the angles after teacher draws a geometric shape on a whiteboard.Ex. Given pictures of different geometric shapes and angles that match the shapes, overlay shapes with matching angles.

Level II Students will:EE4.MD.5. Identify an angle.Ex. Wipe away the shape that does not contain an angle when teacher draws a shape with an angle and a circle.Ex. Identify as many angles as they can see or feel on the playground.Ex. Given an angle template, hold it to shapes in the classroom and tell if it matches.Ex. Given a set of four shapes (one with angles and three with no angles), indicate the shape with angles.

Level I Students will:EE4.MD.5. Identify shapes that contain angles.Ex. Given a square and a circle, identify the square.Ex. Find an object that is shaped like a square in the classroom.

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Grade Level: 4th Standard with code: 4.MD.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.Domain: Measurement and Data Cluster: Geometric measurement: understand concepts of angle and measure angles.

Quarter 1: Quarter 2: Quarter 3: Quarter 4: Recognize that angles are measured

in degrees (°). Read a protractor. Determine which scale on the

protractor to use, based on the direction the angle is open.

Determine the kind of angle based on the specified measure to decide reasonableness of the sketch

Measure angles in whole-number degrees using a protractor.

Sketch angles of specified measure.Make sense of problems and persevere insolving them.








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Students:Given a variety of angles,

Accurately measure the angles in whole number degrees using a protractor.

Given a variety of angle measurements,

Use a protractor and ruler to sketch the corresponding angles.

Angle

Protractor

Students know:


Units of measurement, specifically one-degree angle (degrees),

Procedures for using a protractor.


Use a protractor to measure angles in whole number degrees,

Use a protractor and ruler to sketch angles of a given measure.

The rotation of an angle is measured by the number of one-degree angles that exactly cover the rotation of the angle.

EE4.MD.6. Identify angles as larger and smaller.

Level IV Students will:EE4.MD.6. Construct angles of various sizes.Ex. Construct right and acute angles.Ex. Replicate angles from geometric shapes containing right and acute angles.

Level III Students will:EE4.MD.6. Identify angles as larger and smaller.Ex. Given an angle shaded to less than 45⁰ and one shaded to more than 120⁰, indicate “Which is larger?”Ex. Given two fraction puzzles pieces, one containing a significantly larger angle than the other, indicate “Which is smaller?”

Level II Students will:EE4.MD.6. Differentiate angles in shapes.Ex. Given an angle and a circle, indicates “Which is an angle?”Ex. Given a ball and a cube, indicate “Which has an angle?”

Level I Students will:EE4.MD.6. Replicate an angle.Ex. Use popsicle sticks to replicate a given angle.Ex. Bend a pipe cleaner to replicate a given angle.

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Grade Level: 4th Standard with code: 4.MD.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.Domain: Measurement and Data Cluster: Geometric measurement: understand concepts of angle and measure angles.

Quarter 1: Quarter 2: Quarter 3: Quarter 4: Recognize that an angle can be

divided into smaller angles Solve addition and subtraction

equations to find unknown angle measurements on a diagram.

Find an angle measure by adding the measurements of the smaller angles that make up the larger angle.

Find an angle measure by subtracting the measurements of the smaller angle from the larger angle.









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Students:Given real world and mathematical problems involving angle measurement,

Use a variety of representations (including diagrams and single unknown equations) to show angle measure as additive and to find and justify solutions and solution paths.

Angle Students know:


Units of measurement, specifically one-degree angle (degrees),

Strategies for representing and solving real world problems,

Strategies for finding sums, differences, products, and quotients of whole numbers.


Strategically choose and apply methods for finding sums, differences, products, and quotients of whole numbers,

Accurately compute sums, differences, products and quotients of whole numbers.

The rotation of an angle is measured by the number of one-degree angles that exactly cover the rotation of the angle,

Representations for solving problems are chosen based on the context and numbers in the problem.

EE4.MD.7. N/A (See EE4.MD.5.)

EE4.MD.7. N/A (See EE4.MD.5.)

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Grade Level: 4th Standard with code: 4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.Domain: Geometry Cluster: Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Quarter 1: Quarter 2: Quarter 3: Quarter 4: Draw points, lines, line segments,

rays, angles (right, acute, obtuse), and perpendicular and parallel lines.

Analyze two-dimensional figures to identify points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines.









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Students:Given a written or an oral prompt,

Strategically choose and use tools to draw points, lines, line segments, rays, angles (right, acute, obtuse), perpendicular lines, and parallel lines to specifications.

Given 2-D figures,

Trace or highlight specific components such as angles, line segments, rays, perpendicular lines, and parallel lines.

Lines

Line segments

Rays

Angles

Perpendicular

Parallel

Students know:

Defining characteristics of geometric figures: points, lines, line segments, angles (right, acute, and obtuse), parallel lines, and perpendicular lines.


Strategically choose and use tools to draw 2-D geometric figures,

Decompose 2-D figures in a variety of ways in order to name and identify component parts.

Shapes are categorized based on attributes they possess in common such as; angle size, side length, side relationships (parallel and perpendicular).

EE4.G.1. Distinguish between parallel and intersecting lines.

Level IV Students will:EE4.G.1. Create a representation of parallel and intersecting lines.Ex. Using popsicle sticks, create parallel and intersecting lines.Ex. Play “Simon Says” to illustrate parallel and intersecting lines with arm movements (or eye gaze a picture of students making the correct movements).

Level III Students will:EE4.G.1. Distinguish between parallel and intersecting lines.Ex. Using a road map rug, trace over the parallel lines and then trace over the intersecting lines.Ex. Using a map of the school on an interactive whiteboard, trace the classrooms that are in a parallel line and the hallways that intersect.Ex. Find parallel lines in shapes.

Level II Students will:EE4.G.1. Identify an intersecting line.Ex. Use sidewalk chalk to draw an intersecting line.Ex. Go on an environment hunt and identify intersecting lines.Ex. Trace intersecting lines (e.g., roads or hallways) on a map.

Level I Students will:EE4.G.1. Identify a line.Ex. Using yarn, stretch and glue a line on paper.Ex. Draw a line when directed.Ex. Walk on a line taped to the floor when directed.Ex. Given a line and a circle, indicate which is the line.

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Grade Level: 4th Standard with code: 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.Domain: Geometry Cluster: Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Quarter 1: Quarter 2: Quarter 3: Quarter 4: Identify parallel or perpendicular

lines in two dimensional figures. Recognize acute, obtuse, and right

angles. Identify right triangles. Classify two-dimensional figures

based on parallel or perpendicular lines and size of angles.

Classify triangles as right triangles or not right.









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Students:Given a variety of 2-D figures,

Justify the classification of the shapes based on the presence or absences of parallel lines or perpendicular lines, or the presence or absence of angles of a specified size.

Parallel

Perpendicular

Right triangle

Students know:

Defining characteristics of geometric figures: quadrilateral, trapezoid, rhombus, parallelogram, rectangle, square, right triangle, acute triangle, obtuse triangle, angles (right, acute, and obtuse), parallel lines, and perpendicular lines.


Justify classification of shapes based on the characteristics of their attributes.

Shapes are categorized based on attributes they possess in common such as: angle size, side length, side relationships (parallel and perpendicular).

EE4.G.2. Distinguish between different attributes of shapes (lines, curves, angles).

Level IV Students will:EE4.G.2. Classify shapes according to attributes.Ex. After reading “The Button Box,” determine which attributes can be used to sort geometric buttons (buttons can also be felt by visually impaired students or teacher can trace the shapes into the palm of a hand).Ex. Given several shapes, classify the shapes according to attributes such as shape and angles. (Teacher will trace geometric shape into student’s palm and, after given choices of shapes, activate a switch to indicate a category of attribute.)

Level III Students will:EE4.G.2. Distinguish between different attributes of shapes (lines, curves, angles).Ex. Sort different types of objects to show lines, curves, and angles.Ex. Find pictures that represent lines, angles, and curves.Ex. Draw a picture and identify the lines, angles, and curves used in the picture.

Level II Students will:EE4.G.2. Identify attributes of geometric shapes.Ex. Use attribute blocks to sort shapes.Ex. Assigned a shape, cut out magazine pictures to represent the assigned shape.

Level I Students will:EE4.G.2. Identify curves.Ex. Assemble a selection of curved items.Ex. Using a road map, use toy cars to find curves.Ex. Given a square and a circle, indicate which is curved/round.

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Grade Level: 4th Standard with code: 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.Domain: Geometry Cluster: Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Quarter 1: Quarter 2: Quarter 3: Quarter 4: Recognize lines of symmetry for a

two-dimensional figure. Recognize a line of symmetry as a

line across a figure that when folded along creates matching parts.

Draw lines of symmetry for two-dimensional figures.

Identify line-symmetric figure.Make sense of problems and persevere in solving them.








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Students:Given a variety of 2-D figures,

Justify the existence or non-existence of line symmetry within the figures by drawing the lines of symmetry.

Line symmetry Students know:

Defining characteristics of line symmetry.


Draw lines of symmetry and justify their placement.

A line of symmetry divides a shape into two parts such that when folded on the line the two parts match

EE4.G.3. Recognize a line of symmetry in a simple shape.

Level IV Students will:EE4.G.3. Locate the line of symmetry in a

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