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2 Related Standards
 MACC.4.NF.1.1: 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.Context Complexity: Level 3: Strategic Thinking & Complex Reasoning  Date Adopted or Revised: 12/10Remarks/Examples:
Examples of Opportunities for InDepth Focus
Extending fraction equivalence to the general case is necessary to extend arithmetic from whole numbers to fractions and decimals.  MACC.4.NF.1.2: 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.Context Complexity: Level 2: Basic Application of Skills & Concepts  Date Adopted or Revised: 12/10
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3 Core Content Connectors

Core Content Connector

MACC.4.NF.1.CCC.1a:
Determine equivalent fractions.
 Concrete:
 Equivalency (what is and what is not equivalent; this may begin with numbers/sets of objects: e.g., 3 = 3).
 Concept of fraction and decimal(part to whole).  Ability to identify two equivalent fractions, both represented either symbolically (i.e., in numbers) (e.g. — 1/2 = 2/4 ) or as a model (i.e., picture) (e.g. — pizza cut in halves and in fourths with 1/2 and 2/4 shaded in). If using symbolic representations, limit denominators to 2, 4, and 8 and numerators to1–7. Use a realworld context (e.g., pizza).
 Representation:
 Concrete:

MACC.4.NF.1.CCC.2a:
Use =, <, or > to compare two fractions (fractions with a denominator or 10 or less).
 Concrete:
 Understand the concept of a fraction (a fraction is less than a whole).
 Understand the concept of comparison (greater than, less than, equal).
 Identify concrete representation of a fractional part of a whole as greater than, less than, equal to another (divide a rectangle into fourths and compare ¼ to ¾). Can do by showing with parts of the whole (so relates to fractions; e.g., break into four equal parts…which is more the three parts or the one part?).
 Representation:
 Understand the symbols of <, >, and =.
 Label pictorial representations of fractions.
 Concrete:
 MACC.4.NF.1.CCC.2b: Compare up to two given fractions that have different denominators.
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Related Resources

Lesson Plan

Fun with Fractions: Making and Investigating Fraction Strips: This lesson has students discover relationships between different fractions. From equally cut pieces of paper (a whole), fraction strips are made into halves, thirds, fourths, sixths, and eighths. Students then compare them to one another, discovering their relationships, including equivalent fractions.
(from NCTM Illuminations) 
Fun with Fractions: Investigating Equivalent Fractions with Relationship Rods: This lesson is to follow Fun with Fractions  Investigating Fraction Relationships with Relationship Rods (see Related Resources.) Students will be asked to make the various differentcolored rods the "whole" in relation to the other rods, and then give the relative fraction name for the other nine rods, based on how many of each color it would take to "fill in" the whole. Students will develop problemsolving skills and reasoning as they explain the equivalencies of length.
(from NCTM Illuminations)

Equivalent Fractions: "This lesson helps students discover how to obtain equal fractions by using both fraction strips and playing a fraction matching game. Students will learn that to obtain equal fractions they may multiply the numerator and denominator by the same number." (from ALEX  Alabama Learning Exchange) 
Chocolate Fractions: Chocolate bars will be used to introduce equivalent fractions. Students will find patterns for equivalent fractions through the concreterepresentationalabstract process. 
The Brownie Breakdown: This lesson demonstrates the relationships between equivalent fractions and the size of the pieces that represent the fractions. The lesson moves from concrete activities to pictorial representations. The lesson begins by using a pan of brownies to represent equivalent fractions. The brownies will help to engage students as the lesson moves from the concrete to the pictorial representation of equivalent fractions. The brownies will assist students in understanding that the larger the denominator; the smaller the pieces become. 
Fun with Fractions: In this five lesson unit with overview from NCTM's Illuminations, student activities explore relationships among fractions through work with the length model. Students construct fraction strips and use fraction bars throughout the unit to make sense of basic fraction concepts, to compare fractions and order fractions and to work with equivalency in fractions. Specific learning objectives, a material list, an instructional plan, questions for the students, assessment options, extensions, and teacher reflections are given for each lesson.
Individual Lessons
 Lesson 1: Making and Investigating Fraction Strips
 This lesson has students discover relationships between different fractions. From equally cut pieces of paper (a whole), fraction strips are made into halves, thirds, fourths, sixths, and eighths. Students then compare them to one another, discovering their relationships, including equivalent fractions.
 Lesson 2: More Fun with Fraction Strips
 Students continue working with fraction strips, comparing and ordering them. Once organized from whole to eighths their attention is drawn to the relationship between size and the denominator.
 Lesson 3: Investigating Fraction Relationships with Relationship Rods
 This lesson centers around students using relationship rods (10 wooden/plastic rods ranging from 110cm, with each length having a different color). Students will use them to explore fraction relationships, which will lay the groundwork for more challenging concepts.
 Lesson 4: Investigating Equivalent Fractions with Relationship Rods
 Students will be asked to make the various differentcolored rods the "whole" in relation to the other rods, and then give the relative fraction name for the other nine rods, based on how many of each color it would take to "fill in" the whole. Students will develop problemsolving skills and reasoning as they explain the equivalencies of length.
 Lesson 5: Inch by Inch
 Students will build on the previous lessons by using a ruler to represent various fractions as lengths.

"What's the part? What's the whole?": In this lesson, students will correctly model and discover fractions and their whole relationships. 
Equivalent Fraction Dominoes: Students will identify equivalent fractions using an area model. They will reinforce their learning by playing equivalent fraction dominoes. 
Fraction Land: This is a Common Core 4th Grade Mathematics Lesson Plan based on the Standard MACC.4.NF.1.1: Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x 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.
This lesson is part of a series based on the above standard. All lessons in the series share the Fraction Land title and are available on CPALMS. By the end of the series, students will have created pieces for a game board and other items used to play. 
Fraction Land II: This is a Common Core 4th Grade Mathematics Lesson Plan based on the Standard MACC.4.NF.1.1: Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x 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
This lesson focuses on creating equivalent fractions using the numbers 2, 3, and 4. Students will practice multiplying the numerator and the denominator by 2, 3, or 4 to create equivalent fractions. 
Create a Quilt  Equivalent Fractions: In this lesson, students will work in cooperative pairs to design and construct quilts according to specified instructions. They will obtain the knowledge that fractions can be equivalent even though they may look different and are made up of different numbers. Students develop skills in reasoning as they defend and justify why two fractions are equivalent. 
Fun with Fractions: Eggsactly with a Dozen Eggs: Students will examine fractions as parts of a set of a dozen "eggs" in a carton. They will note what is different and similar about the other students' cartons, and develop problemsolving skills as they examine the relationship between the fractions of a set.
(from NCTM Illuminations) 
Fun with Fractions: More Fun with Fraction Strips: This lesson is intended to follow the lesson Fun with Fractions: Making and Investigating Fraction Strips. (See Related Resources tab) Students continue working with fraction strips, comparing and ordering them. Once organized from whole to eighths their attention is drawn to the relationship between size and the denominator. 
Fractional Clothesline: This lesson has the teacher stretch a lengthy string across the room, and then ask 5 students to equally distribute cards with the numbers 0, 1, 2, 3, and 4 written on them. Once this number line is done, students will use it to place cards with mixed numbers, proper fractions and improper fractions, which will help them realize the difference between these kinds of rational numbers. They will also discuss their strategies for placing the cards. Finally, students will play an estimation game practicing combining numbers from two cards. Exposure to looking at fractions in different ways will help their developing conceptual fraction sense. 
Fraction Lineup!: In this lesson, students will correctly model and compare fraction pairs and place on the inequality mat attached to this lesson. 
Gettin' Fancy with Fractions: In this lesson, students engage in problem solving, a fraction sort activity and play the game "Fraction War" to practice and demonstrate understanding of using benchmark fractions when comparing fractions with different numerators and denominators. 
Fraction War: This lesson is meant to be utilized as a means to enhance previous instruction of fractions that are greater than, or less than one. It is best utilized to build fluency, as this is meant to be a fast paced game to make learning interactive and engaging. 
Ordering Fractions: Students work in groups to arrange sets of fraction cards from least to greatest using benchmark fractions and pieces/parts comparisons. 
Out of Order?: This lesson is a way for students to use benchmark fractions to get a conceptual understanding of comparing and ordering fractions. 
Amazing Alice Cookies: Students will help Amazing Alice Cookies choose the perfect chocolate chip brand to use for their cookies. Students will be given data in the form of fractions and decimals. Fourth grade students will compare decimals and order and compare fractions. Students will write a letter describing their procedure to the client. 
Mrs. Thinkwell's Dilemma: Mrs. Thinkwell is a 4th grade teacher, but she is having a hard time keeping her students engaged during the science lessons. The science lectures are just not working. Of course, there are a few students who seem to be doing well, but there are so many who are underachieving. She could not figure out the problem. Her principal suggested giving the students a multiple intelligence (MI) assessment and possibly utilizing small groups for instruction. She decided to try the MI assessment and received the results; but she still was unsure of what that meant for her classroom. Mrs. Thinkwell wants to utilize small groups in her classroom, but did not know the best way to group the students based primarily on their multiple intelligences. Students will help Mrs. Thinkwell by creating groups of students based on a class data set of MI Assessment results.

Party Entertainment: In this MEA, students will decide which entertainer an owner of an entertainment company should hire. They will base their decisions on information provided on resumes. Students will calculate the cost of hiring the entertainer (multiplication of whole numbers) as well as compare the statistics of their talent competitions and attendance turnout (comparing fractions). Students will write letters to the owner of the entertainment company ranking the entertainers and providing explanation and justification of their strategy for doing so. 
Unit / Lesson Sequence

Learning and Teaching Ratio and Proportion: Research Implications: This is a chapter describing the theory and lesson planning for teaching ratio and rate problems. The authors describe how students use reasoning about multiplication and division to solve ratio and rate problems. 
Worksheet

Fraction Strips in Black and White: This worksheet models a whole as well as fractions with denominators 2 through 12. Teachers could use these, as appropriate to grade level's benchmark, to have students color various fractions, model equivalent fractions, or begin the instruction on operations with fractions. 
Virtual Manipulative

Pattern Blocks: This virtual manipulative offers pattern blocks of different shapes and in different colors which can be used for a range of activities. In addition, the resource offers descriptions of six example activities in the "Activities" section, and a sample lesson plan in the section "Parent/Teacher." 
Fraction Models: An interactive tool to represent a fraction circle, rectangle, or set model with numerators and denominators ranging from 1 to 100. The decimal and percent equivalents of the created fraction are also displayed. 
Fraction Game: This virtual manipulative allows individual students to work with fraction relationships. (There is also a link to a twoplayer version.) 
Fractions  Visualizing: Students use this applet to practice making visual models of fractions by dividing a shape into the given number of segments, then choosing the appropriate number to match the given fraction. Applet instructions and teaching ideas are included in the links at the top of the page. 
Fractions  Equivalent: Students use this virtual manipulative to visualize and name equivalent fractions. The applet presents a shape divided into equal parts, with some parts shaded. Students change the number of divisions of the shape to visualize equivalent fractions, name the fractions, and check their answers. Instructions for using the applet and teaching ideas for parents/teachers are available through the links at the top of the page. 
Conceptually Comparing Fractions and Decimals: This powerful interactive virtual manipulative tool supports teachers in building a conceptual understanding of fractions and decimal comparisons through manipulating fractions and representing them with a wide variety of visual models including strips, circles, grids, as well as symbolic representations (as numeric fractions and decimals). Up to 3 fractions or decimals can be represented, even fractions greater than one. The models can be moved next to each other or one above the other for visual comparison. Even fraction and decimal sums and differences can be compared. The video tutorial is highly recommended to understand all of the options. Included are a sample lesson plan and sample problems. Free Registration is required. A paid subscription is necessary to access other tools on the site.

Exploring Fractions: Match shapes and numbers to earn stars in this fractions game.  Match fractions using numbers and pictures
 make the same fractions using different numbers
 Match fractions in different picture patterns
 Compare fractions using numbers and patterns

Fraction Finder Number Line: In this activity, you will graphically determine the value of two given fractions represented as points on a number line. You will then graphically find a fraction whose value is between the two given fractions and determine its value. 
ProblemSolving Task

Running Laps: The purpose of this task is for students to compare two fractions that arise in a context. Because the fractions are equal, students need to be able to explain how they know that. Some students might stop at the secondtolast picture and note that it looks like they ran the same distance, but the explanation is not yet complete at that point. 
Money in the piggy bank: This task is designed to help students focus on the whole that a fraction refers to. It provides a context where there are two natural ways to view the coins: As equal parts of the set of coins in the piggy bank, and As money so each coin is assigned its monetary value. The important thing to realize here is that two different fractions can describe the same situation depending on what you choose to be the whole. 
Explaining Fraction Equivalence with Pictures: The purpose of this task is to provide students with an opportunity to explain fraction equivalence through visual models in a particular example. Students will need more opportunities to think about fraction equivalence with different examples and models, but this task represents a good first step. 
Using Benchmarks to Compare Fractions: This task is intended primarily for instruction. The goal is to provide examples for comparing two fractions, 1/5 and 2/7 in this case, by finding a benchmark fraction which lies in between the two. In Melissa's example, she chooses 1/4 as being larger than 1/5 and smaller than 2/7. 
Listing fractions in increasing size: The fractions for this task have been carefully chosen to encourage and reward different methods of comparison. The first solution judiciously uses each of the following strategies when appropriate: comparing to benchmark fractions, finding a common denominator, finding a common numerator. The second and third solution shown use only either common denominators or numerators. Teachers should encourage multiple approaches to solving the problem. This task is mostly intended for instructional purposes, although it has value as a formative assessment item as well. 
Educational Game

Ordering Fraction Monkeys: This is an online interactive game for students that reviews ordering fractions and equivalent fractions. 
Fractions Dolphin Racing Game: In this online interactive game, students are tasked with using a variety of strategies to quickly compare fractions. By choosing the largest fraction, the student's dolphin travels further faster. This game encourages students to interpret the meaning of fractions and rely on strategies that go beyond finding common denominators. 
Fraction Quiz: Test your fraction skills by answering questions on this site. This quiz asks you to simplify fractions, convert fractions to decimals and percentages, and answer algebra questions involving fractions. You can even choose difficulty level, question types, and time limit. 
Image / Photograph

Clipart ETC Fractions: Illustrations that can be used for teaching and demonstrating fractions. Fractional representations are modeled in wedges of circles ("pieces of pie") and parts of polygons. There are also clipart images of numerical fractions, both proper and improper, from halves to twelfths. Fraction charts and fraction strips found in this collection can be used as manipulatives and are ready to print for classroom use.
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13 Student Resources
 Pattern Blocks: This virtual manipulative offers pattern blocks of different shapes and in different colors which can be used for a range of activities. In addition, the resource offers descriptions of six example activities in the "Activities" section, and a sample lesson plan in the section "Parent/Teacher."
 Fraction Game: This virtual manipulative allows individual students to work with fraction relationships. (There is also a link to a twoplayer version.)
 Fractions  Visualizing: Students use this applet to practice making visual models of fractions by dividing a shape into the given number of segments, then choosing the appropriate number to match the given fraction. Applet instructions and teaching ideas are included in the links at the top of the page.
 Fractions  Equivalent: Students use this virtual manipulative to visualize and name equivalent fractions. The applet presents a shape divided into equal parts, with some parts shaded. Students change the number of divisions of the shape to visualize equivalent fractions, name the fractions, and check their answers. Instructions for using the applet and teaching ideas for parents/teachers are available through the links at the top of the page.
 Conceptually Comparing Fractions and Decimals: This powerful interactive virtual manipulative tool supports teachers in building a conceptual understanding of fractions and decimal comparisons through manipulating fractions and representing them with a wide variety of visual models including strips, circles, grids, as well as symbolic representations (as numeric fractions and decimals). Up to 3 fractions or decimals can be represented, even fractions greater than one. The models can be moved next to each other or one above the other for visual comparison. Even fraction and decimal sums and differences can be compared. The video tutorial is highly recommended to understand all of the options. Included are a sample lesson plan and sample problems. Free Registration is required. A paid subscription is necessary to access other tools on the site.
 Running Laps: The purpose of this task is for students to compare two fractions that arise in a context. Because the fractions are equal, students need to be able to explain how they know that. Some students might stop at the secondtolast picture and note that it looks like they ran the same distance, but the explanation is not yet complete at that point.
 Money in the piggy bank: This task is designed to help students focus on the whole that a fraction refers to. It provides a context where there are two natural ways to view the coins: As equal parts of the set of coins in the piggy bank, and As money so each coin is assigned its monetary value. The important thing to realize here is that two different fractions can describe the same situation depending on what you choose to be the whole.
 Explaining Fraction Equivalence with Pictures: The purpose of this task is to provide students with an opportunity to explain fraction equivalence through visual models in a particular example. Students will need more opportunities to think about fraction equivalence with different examples and models, but this task represents a good first step.
 Exploring Fractions: Match shapes and numbers to earn stars in this fractions game.
 Match fractions using numbers and pictures
 make the same fractions using different numbers
 Match fractions in different picture patterns
 Compare fractions using numbers and patterns
 Fraction Finder Number Line: In this activity, you will graphically determine the value of two given fractions represented as points on a number line. You will then graphically find a fraction whose value is between the two given fractions and determine its value.
 Fraction Quiz: Test your fraction skills by answering questions on this site. This quiz asks you to simplify fractions, convert fractions to decimals and percentages, and answer algebra questions involving fractions. You can even choose difficulty level, question types, and time limit.
 Using Benchmarks to Compare Fractions: This task is intended primarily for instruction. The goal is to provide examples for comparing two fractions, 1/5 and 2/7 in this case, by finding a benchmark fraction which lies in between the two. In Melissa's example, she chooses 1/4 as being larger than 1/5 and smaller than 2/7.
 Listing fractions in increasing size: The fractions for this task have been carefully chosen to encourage and reward different methods of comparison. The first solution judiciously uses each of the following strategies when appropriate: comparing to benchmark fractions, finding a common denominator, finding a common numerator. The second and third solution shown use only either common denominators or numerators. Teachers should encourage multiple approaches to solving the problem. This task is mostly intended for instructional purposes, although it has value as a formative assessment item as well.
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10 Parent Resources
 Pattern Blocks: This virtual manipulative offers pattern blocks of different shapes and in different colors which can be used for a range of activities. In addition, the resource offers descriptions of six example activities in the "Activities" section, and a sample lesson plan in the section "Parent/Teacher."
 Clipart ETC Fractions: Illustrations that can be used for teaching and demonstrating fractions. Fractional representations are modeled in wedges of circles ("pieces of pie") and parts of polygons. There are also clipart images of numerical fractions, both proper and improper, from halves to twelfths. Fraction charts and fraction strips found in this collection can be used as manipulatives and are ready to print for classroom use.
 Fractions  Visualizing: Students use this applet to practice making visual models of fractions by dividing a shape into the given number of segments, then choosing the appropriate number to match the given fraction. Applet instructions and teaching ideas are included in the links at the top of the page.
 Fractions  Equivalent: Students use this virtual manipulative to visualize and name equivalent fractions. The applet presents a shape divided into equal parts, with some parts shaded. Students change the number of divisions of the shape to visualize equivalent fractions, name the fractions, and check their answers. Instructions for using the applet and teaching ideas for parents/teachers are available through the links at the top of the page.
 Running Laps: The purpose of this task is for students to compare two fractions that arise in a context. Because the fractions are equal, students need to be able to explain how they know that. Some students might stop at the secondtolast picture and note that it looks like they ran the same distance, but the explanation is not yet complete at that point.
 Money in the piggy bank: This task is designed to help students focus on the whole that a fraction refers to. It provides a context where there are two natural ways to view the coins: As equal parts of the set of coins in the piggy bank, and As money so each coin is assigned its monetary value. The important thing to realize here is that two different fractions can describe the same situation depending on what you choose to be the whole.
 Explaining Fraction Equivalence with Pictures: The purpose of this task is to provide students with an opportunity to explain fraction equivalence through visual models in a particular example. Students will need more opportunities to think about fraction equivalence with different examples and models, but this task represents a good first step.
 Exploring Fractions: Match shapes and numbers to earn stars in this fractions game.
 Match fractions using numbers and pictures
 make the same fractions using different numbers
 Match fractions in different picture patterns
 Compare fractions using numbers and patterns
 Using Benchmarks to Compare Fractions: This task is intended primarily for instruction. The goal is to provide examples for comparing two fractions, 1/5 and 2/7 in this case, by finding a benchmark fraction which lies in between the two. In Melissa's example, she chooses 1/4 as being larger than 1/5 and smaller than 2/7.
 Listing fractions in increasing size: The fractions for this task have been carefully chosen to encourage and reward different methods of comparison. The first solution judiciously uses each of the following strategies when appropriate: comparing to benchmark fractions, finding a common denominator, finding a common numerator. The second and third solution shown use only either common denominators or numerators. Teachers should encourage multiple approaches to solving the problem. This task is mostly intended for instructional purposes, although it has value as a formative assessment item as well.
Don’t … Sort clusters from Major to Supporting, and then teach them in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.
Number: MACC.4.NF.1
Title: Extend understanding of fraction equivalence and ordering.
Type:
Cluster
Subject: CCSS: Mathematics
Grade: 4
Domain: Number and Operations  Fractions
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