# MA.3.FR.2.1

Plot, order and compare fractional numbers with the same numerator or the same denominator.

### Examples

The fraction is to the right of the fraction on a number line so is greater than .

### Clarifications

Clarification 1: Instruction includes making connections between using a ruler and plotting and ordering fractions on a number line.
Clarification 2: When comparing fractions, instruction includes an appropriately scaled number line and using reasoning about their size.

Clarification 3: Fractions include fractions greater than one, including mixed numbers, with denominators limited to 2, 3, 4, 5, 6, 8, 10 and 12.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Fractions
Status: State Board Approved

## Benchmark Instructional Guide

• Number Line

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

The purpose of this benchmark is for students to plot and order fractions with the same numerator (e.g., $\frac{\text{3}}{\text{4}}$, $\frac{\text{3}}{\text{2}}$, $\frac{\text{3}}{\text{8}}$) or fractions with the same denominator (e.g., $\frac{\text{3}}{\text{5}}$, $\frac{\text{10}}{\text{5}}$, $\frac{\text{7}}{\text{5}}$) to compare them by their location on a number line.
• During instruction, teachers should provide students opportunities to practice using the number line, which will assist students with understanding the difference in size when fractions have the same numerator (the size of the parts) and with comparing fractions with the same denominator (number of parts) (MTR.2.1).
• Through making connections to rulers, students see that appropriately scaled number lines allow for comparisons of fraction size. Students should also utilize open number lines as to practice creating their own appropriately scaled number lines (MTR.2.1).
• Instruction should model that fractional units on a number line represent intervals that are its unit fraction in size. For example, $\frac{\text{5}}{\text{3}}$ on a number line is represented by 5 units from 0 that are each one-third in length. Second, number lines help students see comparisons of fractions to the same whole and will continue as students compare fractions with different numerators and denominators in Grade 4. Finally, number lines reinforce Clarification 3 for MA.3.FR.1.3, that fractions are numbers (MTR.2.1, MTR.5.1).

### Common Misconceptions or Errors

• Students can be confused that when numerators are the same in fractions, larger denominators represent smaller pieces, and smaller denominators represent larger pieces.
• When fraction comparisons are made using area models, students may be confused that the size of the whole for each model must be the same size.

### Strategies to Support Tiered Instruction

• Instruction includes opportunities to use concrete models and drawing of number lines to connect learning with fraction understanding.
• For example, students plot fourths on the number line. Utilizing fraction strips or tiles, students can connect fractional parts to the measurement on a number line.

• Conversation includes what students notice about the fraction on the number line. “How many fourths are in three-fourths? What do we notice about the size of $\frac{\text{1}}{\text{4}}$ compared to $\frac{\text{3}}{\text{4}}$?” Students have opportunities to describe the distance from the 0 as well as the distance from other benchmark fractions.
• Instruction includes opportunities to use fraction manipulatives, concrete models, and drawings. The teacher begins instruction by modeling fractional pieces with their fraction name. It is important that students see that the fractions that they are building and comparing refer to the same size whole.
• For example, students build fractions tiles or models to equal the same size one whole like below.

Clara says that $\frac{\text{5}}{\text{4}}$ is greater than $\frac{\text{5}}{\text{2}}$ because 4 is greater than 2. Prove why she is incorrect using the number line below.

### Instructional Items

Instructional Item 1

Order the fractions below from least to greatest.
$\frac{\text{8}}{\text{5}}$, $\frac{\text{8}}{\text{3}}$, $\frac{\text{8}}{\text{10}}$, $\frac{\text{8}}{\text{1}}$

Instructional Item 2

Compare 7 fourths and 3 fourths using <, =, or >.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
5012050: Grade Three Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7712040: Access Mathematics Grade 3 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))
5012055: Grade 3 Accelerated Mathematics (Specifically in versions: 2019 - 2022, 2022 and beyond (current))
5012015: Foundational Skills in Mathematics 3-5 (Specifically in versions: 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.3.FR.2.AP.1: Compare fractional numbers with the same denominator. Denominators are limited to 2, 3 and 4.

## Related Resources

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

## Formative Assessments

Comparing Fractions:

Students compare two pairs of fractions and record their comparisons using the less than or greater than symbols.

Type: Formative Assessment

The Cake Problem:

Students compare two fractional parts of two different wholes.

Type: Formative Assessment

Four-Sixths on the Number Line:

Students are asked to use a number line that includes the location of zero and one-sixth to find the location of four-sixths.

Type: Formative Assessment

One-Third on the Number Line:

Students are given four number line diagrams and asked to choose the one that correctly shows the location of one-third.

Type: Formative Assessment

Five-Eighths on the Number Line:

Students are asked to locate five-eighths on a number line that has been anchored by zero and one, but that has not yet been scaled.

Type: Formative Assessment

Three-Fourths on the Number Line:

Students are asked to scale a number line from zero to one so that they can find the location of three-fourths.

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

Perfect Pool Plans:

In this Model Eliciting Activity, MEA, students will create a procedure for ranking pool construction companies based on the number of years in business, customer satisfaction, and available pool dimensions. In a “twist,” students will be given information about discounts available by each company. Students will evaluate their procedure for ranking and change it if necessary.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

Type: Lesson Plan

Would You Rather?:

This lesson is designed to help students generate rules for comparing fractions. The students will use fraction tiles to discover ways to compare fractions with the same denominator or fractions with the same numerator. They will also begin to use benchmark fractions to help make comparisons and complete inequalities.

Type: Lesson Plan

Who has more? Using the size of the fractional part to compare.:

Students explore how they can compare fractions by considering the denominator. Students use real world examples to create models and demonstrate that the size of the piece decreases as the denominator increases.

Type: Lesson Plan

Interactive Fraction Number Lines:

In this lesson students make number lines out of sentence strips to plot, order, and compare fractions with the same denominators.

Type: Lesson Plan

Comparing Fractions:

In this lesson, students will demonstrate their understanding of comparing fractions with the same denominator through engaging problem solving. Students will plot fractions on a numberline, play fraction war and complete a worksheet.

Type: Lesson Plan

Comparing Fractions with Brownies:

Students will demonstrate their understanding of comparing fractions with the same numerator through engaging problem solving using real-world application with brownies as a model. Students will be actively engaged in a fraction war game and "would you rather have" statements to solidify their understanding of comparing fractions with the same numerator.

Type: Lesson Plan

Fractions on a Number line:

In this lesson, students will place fractions on a number line and identify equivalent fractions. Students will explain the definition of equivalent fractions.

Type: Lesson Plan

Comparing and Placing Unit Fractions on a Number Line:

In this lesson, 3rd grade students will compare fractions which have the same numerator and explain their reasoning. The students will be able to compare the fractions by correctly placing them on a number line.

Type: Lesson Plan

Magnified Inches:

This lesson provides a parallel between fraction strips (something students should be familiar with) and measuring length with a ruler past one inch including quarters. This lesson is the follow-up to The Magnified Inch, Resource ID 46593.

Type: Lesson Plan

The Fraction String:

In this lesson students create a model of a number line using string and adding machine tape. Students discover how to partition the string into equal sections, and name the fractional pieces, including fractions greater than 1.

Type: Lesson Plan

The Magnified Inch:

This lesson provides a parallel between fraction strips (something students should be familiar with) and measuring with a ruler up to an inch including quarters.

Type: Lesson Plan

## Original Student Tutorials

Fraction Camp! Fractions Greater Than 1 on a Number Line:

Joey uses his knowledge of fractions to win games at camp by knowing where fractions greater than one are located on number lines, in this interactive tutorial.

Type: Original Student Tutorial

Nature Walk: Fractions Less Than 1 on a Number Line:

Learn to use number lines to represent fractions as Emmy explores nature in this interactive tutorial.

Type: Original Student Tutorial

Finding Fractions at Camp: Fractions on a Number Line:

Joey learns about the location of unit fractions on a number line while at camp in this interactive tutorial.

Type: Original Student Tutorial

Comparing Fractions with a Different Whole:

This task is meant to address a common error that students make, namely, that they represent fractions with different wholes when they need to compare them. This task is meant to generate classroom discussion related to comparing fractions.

Comparing Fractions:

The purpose of this task is for students to compare fractions using common numerators and common denominators and to recognize equivalent fractions.

Closest to 1/2:

How students tackle the problem and the amount of work they show on the number line can provide insight into the sophistication of their thinking. As students partition the interval between 0 and 1 into eighths, they will need to recognize that 1/2=4/8. Students who systematically plot every point, even 9/8, which is larger even than 1 may still be coming to grips with the relative size of fractions.

Locating Fractions Greater than One on the Number Line:

The goal of this task is to help students gain a better understanding of fractions and their place on the number line.

Find 2/3:

This simple-looking problem reveals much about how well students understand unit fractions as well as representing fractions on a number line.

Find 1:

This task includes the seeds of several important ideas. Part a presents the student with the opportunity to use a unit fraction to find 1 on the number line. Part b helps reinforce the notion that when a fraction has a numerator that is larger than the denominator, it has a value greater than 1 on the number line.

Ordering Fractions:

The purpose of this task is to extend students' understanding of fraction comparison and is intended for an instructional setting.

Locating Fractions Less than One on the Number Line:

In every part of this task, students must treat the interval from 0 to 1 as a whole, partition the whole into the appropriate number of equal sized parts, and then locate the fraction(s).

## STEM Lessons - Model Eliciting Activity

Perfect Pool Plans:

In this Model Eliciting Activity, MEA, students will create a procedure for ranking pool construction companies based on the number of years in business, customer satisfaction, and available pool dimensions. In a “twist,” students will be given information about discounts available by each company. Students will evaluate their procedure for ranking and change it if necessary.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

## MFAS Formative Assessments

Comparing Fractions:

Students compare two pairs of fractions and record their comparisons using the less than or greater than symbols.

Five-Eighths on the Number Line:

Students are asked to locate five-eighths on a number line that has been anchored by zero and one, but that has not yet been scaled.

Four-Sixths on the Number Line:

Students are asked to use a number line that includes the location of zero and one-sixth to find the location of four-sixths.

One-Third on the Number Line:

Students are given four number line diagrams and asked to choose the one that correctly shows the location of one-third.

The Cake Problem:

Students compare two fractional parts of two different wholes.

Three-Fourths on the Number Line:

Students are asked to scale a number line from zero to one so that they can find the location of three-fourths.

## Original Student Tutorials Mathematics - Grades K-5

Finding Fractions at Camp: Fractions on a Number Line:

Joey learns about the location of unit fractions on a number line while at camp in this interactive tutorial.

Fraction Camp! Fractions Greater Than 1 on a Number Line:

Joey uses his knowledge of fractions to win games at camp by knowing where fractions greater than one are located on number lines, in this interactive tutorial.

Nature Walk: Fractions Less Than 1 on a Number Line:

Learn to use number lines to represent fractions as Emmy explores nature in this interactive tutorial.

## Student Resources

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

## Original Student Tutorials

Fraction Camp! Fractions Greater Than 1 on a Number Line:

Joey uses his knowledge of fractions to win games at camp by knowing where fractions greater than one are located on number lines, in this interactive tutorial.

Type: Original Student Tutorial

Nature Walk: Fractions Less Than 1 on a Number Line:

Learn to use number lines to represent fractions as Emmy explores nature in this interactive tutorial.

Type: Original Student Tutorial

Finding Fractions at Camp: Fractions on a Number Line:

Joey learns about the location of unit fractions on a number line while at camp in this interactive tutorial.

Type: Original Student Tutorial

Comparing Fractions with a Different Whole:

This task is meant to address a common error that students make, namely, that they represent fractions with different wholes when they need to compare them. This task is meant to generate classroom discussion related to comparing fractions.

Comparing Fractions:

The purpose of this task is for students to compare fractions using common numerators and common denominators and to recognize equivalent fractions.

Closest to 1/2:

How students tackle the problem and the amount of work they show on the number line can provide insight into the sophistication of their thinking. As students partition the interval between 0 and 1 into eighths, they will need to recognize that 1/2=4/8. Students who systematically plot every point, even 9/8, which is larger even than 1 may still be coming to grips with the relative size of fractions.

Locating Fractions Greater than One on the Number Line:

The goal of this task is to help students gain a better understanding of fractions and their place on the number line.

Find 2/3:

This simple-looking problem reveals much about how well students understand unit fractions as well as representing fractions on a number line.

Find 1:

This task includes the seeds of several important ideas. Part a presents the student with the opportunity to use a unit fraction to find 1 on the number line. Part b helps reinforce the notion that when a fraction has a numerator that is larger than the denominator, it has a value greater than 1 on the number line.

Ordering Fractions:

The purpose of this task is to extend students' understanding of fraction comparison and is intended for an instructional setting.

Locating Fractions Less than One on the Number Line:

In every part of this task, students must treat the interval from 0 to 1 as a whole, partition the whole into the appropriate number of equal sized parts, and then locate the fraction(s).

## Parent Resources

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

## 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

Comparing Fractions with a Different Whole:

This task is meant to address a common error that students make, namely, that they represent fractions with different wholes when they need to compare them. This task is meant to generate classroom discussion related to comparing fractions.

Comparing Fractions:

The purpose of this task is for students to compare fractions using common numerators and common denominators and to recognize equivalent fractions.

Closest to 1/2:

How students tackle the problem and the amount of work they show on the number line can provide insight into the sophistication of their thinking. As students partition the interval between 0 and 1 into eighths, they will need to recognize that 1/2=4/8. Students who systematically plot every point, even 9/8, which is larger even than 1 may still be coming to grips with the relative size of fractions.

Locating Fractions Greater than One on the Number Line:

The goal of this task is to help students gain a better understanding of fractions and their place on the number line.

Find 2/3:

This simple-looking problem reveals much about how well students understand unit fractions as well as representing fractions on a number line.

Find 1:

This task includes the seeds of several important ideas. Part a presents the student with the opportunity to use a unit fraction to find 1 on the number line. Part b helps reinforce the notion that when a fraction has a numerator that is larger than the denominator, it has a value greater than 1 on the number line.