# Cluster 1: Extend understanding of fraction equivalence and ordering. (Major Cluster)Archived

Clusters should not be sorted from Major to Supporting and then taught 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.

General Information
Number: MAFS.4.NF.1
Title: Extend understanding of fraction equivalence and ordering. (Major Cluster)
Type: Cluster
Subject: Mathematics - Archived
Domain-Subdomain: Number and Operations - Fractions

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.4.NF.1.AP.1a
Determine equivalent fractions using visual fraction models and a number line.
MAFS.4.NF.1.AP.2a
Use =, <, or > to compare two fractions (fractions with a denominator or 10 or less).
MAFS.4.NF.1.AP.2b
Compare 2 given fractions that have different denominators.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Educational Games

Flower Power: An Ordering of Rational Numbers Game:

This is a fun and interactive game that helps students practice ordering rational numbers, including decimals, fractions, and percents. You are planting and harvesting flowers for cash. Allow the bee to pollinate, and you can multiply your crops and cash rewards!

Type: Educational Game

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.

Type: Educational Game

## Formative Assessments

Equivalent Fractions on a Number Line:

Students scale number lines to locate given fractions, find equivalent fractions, and explain the relationship between equivalent fractions.

Type: Formative Assessment

Equivalence Using A Number Line:

Students use a number line to explain that one-half is equivalent to two-fourths.

Type: Formative Assessment

Eating Cake:

Students draw a visual fraction model to determine whether two fractions are equivalent.

Type: Formative Assessment

Corn Farms:

Students compare two fractions with unlike denominators in the context of a word problem and record the comparison using an inequality symbol.

Type: Formative Assessment

Comparing Fractions Using Benchmark Fractions:

Students compare two fractions using benchmark fractions on a number line and record the comparison using the less than or greater than symbol.

Type: Formative Assessment

Comparing Four-Fifths and Three-Fourths:

Students consider the correctness of a model for comparing four-fifths to three-fourths.

Type: Formative Assessment

Compare Fractions:

Students are given three sets of fractions to compare and are asked to record the comparisons using the less than, greater than, or equal to symbols.

Type: Formative Assessment

Are the Fractions Equivalent:

Students partition squares to model two fractions and then determine if the fractions are equivalent.

Type: Formative Assessment

## 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.

Type: Image/Photograph

## Lesson Plans

The Alternative Recipe:

The students follow a scaffold model starting with using concrete models of fractions with the fraction tiles to create equivalent fractions, and then by using prior knowledge of multiples to develop the algorithm for creating equivalent fractions.Â  The algorithm noted here is based on the identity property of multiplying a fraction times a fraction so you are multiplying by 2/2 or 3/3.Â Â

Type: Lesson Plan

Gardening In Schools:

This Model Eliciting Activity is written at a 4th grade level. In this open-ended problem, students must consider how to rank potting soil based on factors like fraction of ingredients, price, andÂ eco-friendliness. In teams, students determine their procedures and write letters back to the client.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal studentsâ€™ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

The Cookie Jar Wants a New Cookie!:

This lesson asks students to recommend which cookie the owners of The Cookie Jar should add to their menu. Before they make their decision, the students have to convert fractions so they have like denominators. Once they have converted the fractions they will be able to see exactly how many people voted for each cookie and they can factor in that information along with additional cookie facts to make their final recommendation.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal studentsâ€™ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

Fractions: Let's Compare:

The lesson is an application and extension of fraction comparison strategies, not an introduction.Â  While the beginning of the lesson has a review, the situational stories require students to read and analyze carefully.

Type: Lesson Plan

Comparing Fractions with Cupcakes:

In this MEA, students will compare fractions with different denominators to decide which cupcake a bakery should add to their menu.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal studentsâ€™ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

Create a Quilt - Equivalent Fractions:

In this lesson, students will apply their knowledge of identifying and creating equivalent fractions.Â  The use of tiles allows a visual check of their equivalent fraction creations using parts of a set.Â

Type: Lesson Plan

Wondrous Water Parks:

This activity requires students to apply their knowledge ofÂ unit conversions, speed calculation, and comparing fractions to solve the problem of which water park their class should choose to go on for their 5th grade class trip.

Type: Lesson Plan

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.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal studentsâ€™ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

Fraction Land II:

This Â is an application lesson for the studentsâ€™ knowledge of how to identify or create equivalent fractions. After a review, they are expected to multiply or divide to create equivalent fractions.

Type: Lesson Plan

Fraction Land:

This lesson uses paper strips to show equivalent fractions.Â  It discusses what happens to the denominator and numerator as students learn to multiply to make equivalent fractions.Â It is part of a series about identifying and creating equivalent fractions. All lessons in the series share the Fraction Land title and are available on CPALMS.Â

Type: Lesson Plan

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.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal studentsâ€™ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

Out of Order?:

This lesson is a way for students to use benchmark fractions to get a conceptual understanding of comparing and ordering fractions.

Type: Lesson Plan

"What's the part? What's the whole?":

This lesson provides a conceptual approach to multiplying a fraction times a whole number and a whole number times a fraction.Â  Students are to use an understanding of the meaning of the denominator and numerator to figure out a strategy for finding the solution.Â

Type: Lesson Plan

Equivalent Fraction Dominoes:

Students will identify equivalent fractionsÂ to 1/2 using manipulativesÂ and determine what happened to the numerator and denominator when an equivalent fraction was formed.Â  They will practice this idea by identifying equivalent fractions in a domino game.

Type: Lesson Plan

Fraction Line-up!:

Students will model and compare fraction pairsÂ by writing an inequality.Â Â

Type: Lesson Plan

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 benchmarkÂ quantities when comparing fractions with different numerators and denominators.Â  This lesson is not intended as initial instruction on using benchmark quantities to compare fractions.Â Â Instead, it may be useful for skill reinforcement, student engagement, and formative assessment of skill mastery.Â  Parts of this lesson could be revisited periodically as students build comfort and mastery comparing fractions.

Type: Lesson Plan

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.

Type: Lesson Plan

Chocolate Fractions:

Chocolate bars will be used to introduce equivalent fractions. Students will find patterns for equivalent fractions through the concrete-representational-abstract process.

Type: Lesson Plan

Equivalent Fraction Dominoes:

Students will identify equivalent fractionsÂ to 1/2 using manipulativesÂ and determine what happened to the numerator and denominator when an equivalent fraction was formed.Â  They will practice this idea by identifying equivalent fractions in a domino game.

Type: Lesson Plan

Ordering Fractions:

Students work in groups to arrange sets of fraction cards from least to greatest usingÂ multiple strategies.Â  Fractions include those greater than one.

Type: Lesson Plan

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 turn-out (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.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal studentsâ€™ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

## Original Student Tutorials

Comparing Fractions with Square Foot Gardens Part 2:

Use equivalent fractions to compare fractions in this garden-themed, interactive tutorials

This is Part 2 in a two-part series. Click to open Part 1,  “Mama’s Pizza, Butterflies, & Comparing Fractions.”

Type: Original Student Tutorial

Mama's Pizza, Butterflies, and Comparing Fractions Part 1:

Help a family settle an argument about who got the most pizza and which butterfly was longer by comparing fractions using benchmarks and area models, in this interactive tutorial.

Type: Original Student Tutorial

Equivalent Fractions Part 1: Sharing at the Shelter:

Learn how to create equivalent fractions and visually see how they are equivalent in this interactive tutorial.

This is part 1 of a 2-part series. Click HERE to open Part 2.

Type: Original Student Tutorial

Equivalent Fractions, Part 2: Patterns in the Multiplication Table:

Learn how to find equivalent fractions in a multiplication table in this interactive tutorial.

This is part 2 of a 2 part series. Click HERE to open Part 1.

Type: Original Student Tutorial

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.

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.

Comparing two different pizzas:

The focus of this task is on understanding that fractions, in an explicit context, are fractions of a specific whole. In this this problem there are three different wholes: the medium pizza, the large pizza, and the two pizzas taken together. This task is best suited for instruction. Students can practice explaining their reasoning to each other in pairs or as part of a whole group discussion.

Comparing Sums of Unit Fractions:

The purpose of this task is to help develop students' understanding of addition of fractions; it is intended as an instructional task. Notice that students are not asked to find the sum so this may be given to students who are limited to computing sums of fractions with the same denominator. Rather, they need to apply a firm understanding of unit fractions (fractions with one in the numerator) and reason about their relative size.

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.

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 second-to-last picture and note that it looks like they ran the same distance, but the explanation is not yet complete at that point.

## Tutorial

Equivalent Fractions: Visual Models:

This Khan Academy video illustrates that fraction a/b is equivalent to fraction (a x n)/(b x n).

Type: Tutorial

## Virtual Manipulatives

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

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

Fraction Game:

This virtual manipulative allows individual students to work with fraction relationships. (There is also a link to a two-player version.)

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Original Student Tutorials

Comparing Fractions with Square Foot Gardens Part 2:

Use equivalent fractions to compare fractions in this garden-themed, interactive tutorials

This is Part 2 in a two-part series. Click to open Part 1,  “Mama’s Pizza, Butterflies, & Comparing Fractions.”

Type: Original Student Tutorial

Mama's Pizza, Butterflies, and Comparing Fractions Part 1:

Help a family settle an argument about who got the most pizza and which butterfly was longer by comparing fractions using benchmarks and area models, in this interactive tutorial.

Type: Original Student Tutorial

Equivalent Fractions Part 1: Sharing at the Shelter:

Learn how to create equivalent fractions and visually see how they are equivalent in this interactive tutorial.

This is part 1 of a 2-part series. Click HERE to open Part 2.

Type: Original Student Tutorial

Equivalent Fractions, Part 2: Patterns in the Multiplication Table:

Learn how to find equivalent fractions in a multiplication table in this interactive tutorial.

This is part 2 of a 2 part series. Click HERE to open Part 1.

Type: Original Student Tutorial

## Educational Games

Flower Power: An Ordering of Rational Numbers Game:

This is a fun and interactive game that helps students practice ordering rational numbers, including decimals, fractions, and percents. You are planting and harvesting flowers for cash. Allow the bee to pollinate, and you can multiply your crops and cash rewards!

Type: Educational Game

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.

Type: Educational Game

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.

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.

Comparing two different pizzas:

The focus of this task is on understanding that fractions, in an explicit context, are fractions of a specific whole. In this this problem there are three different wholes: the medium pizza, the large pizza, and the two pizzas taken together. This task is best suited for instruction. Students can practice explaining their reasoning to each other in pairs or as part of a whole group discussion.

Comparing Sums of Unit Fractions:

The purpose of this task is to help develop students' understanding of addition of fractions; it is intended as an instructional task. Notice that students are not asked to find the sum so this may be given to students who are limited to computing sums of fractions with the same denominator. Rather, they need to apply a firm understanding of unit fractions (fractions with one in the numerator) and reason about their relative size.

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.

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 second-to-last picture and note that it looks like they ran the same distance, but the explanation is not yet complete at that point.

## Tutorial

Equivalent Fractions: Visual Models:

This Khan Academy video illustrates that fraction a/b is equivalent to fraction (a x n)/(b x n).

Type: Tutorial

## Virtual Manipulatives

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

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

Fraction Game:

This virtual manipulative allows individual students to work with fraction relationships. (There is also a link to a two-player version.)

Type: Virtual Manipulative

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

## 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.

Type: Image/Photograph

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.

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.

Comparing two different pizzas:

The focus of this task is on understanding that fractions, in an explicit context, are fractions of a specific whole. In this this problem there are three different wholes: the medium pizza, the large pizza, and the two pizzas taken together. This task is best suited for instruction. Students can practice explaining their reasoning to each other in pairs or as part of a whole group discussion.

Comparing Sums of Unit Fractions:

The purpose of this task is to help develop students' understanding of addition of fractions; it is intended as an instructional task. Notice that students are not asked to find the sum so this may be given to students who are limited to computing sums of fractions with the same denominator. Rather, they need to apply a firm understanding of unit fractions (fractions with one in the numerator) and reason about their relative size.

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.

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 second-to-last picture and note that it looks like they ran the same distance, but the explanation is not yet complete at that point.