# Standard 2: Perform operations with fractions.

General Information
Number: MA.5.FR.2
Title: Perform operations with fractions.
Type: Standard
Subject: Mathematics (B.E.S.T.)
Grade: 5
Strand: Fractions

## Related Benchmarks

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MA.5.FR.2.AP.1a
Explore adding and subtracting mixed numbers and fractions greater than 1 with like denominators.
MA.5.FR.2.AP.1b
Explore adding and subtracting fractions less than one with unlike denominators. where one denominator is a multiple of the other (e.g., + , ).
MA.5.FR.2.AP.2
Explore multiplying a unit fraction by a unit fraction.
MA.5.FR.2.AP.3
Explore the impact on the size of the product when multiplying a given number by a fraction less than 1 or by a whole number.
MA.5.FR.2.AP.4
Explore the division of a one-digit whole number by a unit fraction. Denominators are limited to 2, 3 or 4.

## Related Resources

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

## Educational Games

Ice Ice Maybe: An Operations Estimation Game:

This fun and interactive game helps practice estimation skills, using various operations of choice, including addition, subtraction, multiplication, division, using decimals, fractions, and percents.

Various levels of difficulty make this game appropriate for multiple age and ability levels.

Addition/Subtraction: The addition and subtraction of whole numbers, the addition and subtraction of decimals.

Multiplication/Division: The multiplication and addition of whole numbers.

Percentages: Identify the percentage of a whole number.

Fractions: Multiply and divide a whole number by a fraction, as well as apply properties of operations.

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

Multiplying by a Fraction Greater Than One:

Students are asked to describe the size of a product of a fraction greater than one and a whole number without multiplying.

Type: Formative Assessment

Multiplying by a Fraction Less than One:

Students are asked to describe the size of a product of a fraction less than one and a whole number without multiplying.

Type: Formative Assessment

More Than or Less Than Two Miles:

Students are asked to reason about the size of the product of fractions and whole numbers presented in context.

Type: Formative Assessment

Estimating Products:

Students are given three products, each involving a whole number and a fraction, and are asked to estimate the size of the product and explain their reasoning.

Type: Formative Assessment

Fractions Divided by Whole Numbers:

Students are given a division expression and asked to write a story context to match the expression and use a visual fraction model to solve the problem.

Type: Formative Assessment

Bags of Fudge:

Students are asked to solve a word problem involving division of a whole number by a fraction.

Type: Formative Assessment

Relay Race:

Students are asked to solve a word problem involving division of a fraction by a whole number.

Type: Formative Assessment

Whole Numbers Divided by Fractions:

Students are given a division expression and asked to write a story context to match the expression and use a visual fraction model to solve the problem.

Type: Formative Assessment

Using Visual Fraction Models:

Students interpret a visual fraction model showing multiplication of two fractions less than one.

Type: Formative Assessment

Multiplying Fractions by Fractions:

Students are asked to consider an equation involving multiplication of fractions, then create a visual fraction model, and write a story context to match.

Type: Formative Assessment

Adding More Fractions with Unlike Denominators:

Students are asked to add pairs of fractions with unlike denominators.

Type: Formative Assessment

Adding Fractions with Unlike Denominators:

Students are asked to add two pairs of fractions with unlike denominators.

Type: Formative Assessment

Subtracting More Fractions:

Students are asked to subtract improper fractions and mixed numbers with unlike denominators.

Type: Formative Assessment

Subtracting Fractions:

Students are asked to subtract fractions with unlike denominators.

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

Let's Have a Fraction Party!:

In this lesson, students will use addition and subtraction of fractions with unlike denominators to solve word problems involving situations that arise with the children who were invited to a party. They will use fraction strips as number models and connect the algorithm with these real-life word problems.

Type: Lesson Plan

Fractions make the real WORLD problems go round:

In this lesson students will use a graphic organizer to to solve addition and subtraction word problems. Students will create their own word problems in PowerPoint, by using pen and paper, or dry erase boards to help them to connect to and understand the structure of word problems.

Type: Lesson Plan

Aaron and Anya's Discovery: Adding Fractions with Unlike Denominators:

In this situational story, Aaron and Anya find several pieces of ribbon/cord of varying fractional lengths. They decide to choose 3 pieces and make a belt. All of the fractions have different denominators; students have to determine common denominators in order to add the fractional pieces. After students successfully add three fractional pieces, they make a belt and label it with their fractional pieces.

Type: Lesson Plan

Real-World Fractions:

This lesson focuses on providing students with real-world experiences where they will be required to multiply fractions. Students will be required to use visual fraction models or equations to represent the problem.  This is a practice and application lesson, not an introductory lesson.

Type: Lesson Plan

Using Models to Add Fractions with Unlike Denominators:

This lesson is specific to adding fractions with unlike denominators. It requires students to already have a working knowledge of adding fractions with common denominators, and equivalent fractions. Subtracting fractions with unlike denominators will follow in a subsequent lesson.

Type: Lesson Plan

Multiplying Fractions With GeoGebra Using An Area Model:

In this lesson, students will derive an algorithm for multiplying fractions by using area models. They will use a GeoGebra applet to visualize fraction multiplication. They will also translate between pictorial and symbolic representations of fraction multiplication.

Type: Lesson Plan

Multiplying a Fraction by a Fraction:

In this lesson, students will solve problems related to training for a marathon to apply and make sense of multiplying fractions. The student will complete a function table to help illustrate patterns in the numerator/denominator relationships. This lesson utilizes the linear model as a concrete representation and moves towards the standard algorithm (a/b) x (c/d) = ac/bd.

Type: Lesson Plan

Area Models: Multiplying Fractions:

In this lesson students will investigate the relationship between area models and the concept of multiplying fractions. Students will use area models to develop understanding of the concept of multiplying fractions as well as to find the product of two common fractions. The teacher will use the free application GeoGebra (see download link under Suggested Technology) to provide students with a visual representation of how area models can be used at the time of multiplying fractions.

Type: Lesson Plan

Multiplying a Fraction by a Fraction:

Students will multiply a fraction times a fraction. The students will section off a square through rows and columns that will represent the strategy of multiplying numerators and then denominators.

Type: Lesson Plan

Garden Variety Fractions:

Students explore the multiplication of a fraction times a fraction through story problems about a garden using models on Geoboards and pictorial representations on grid paper. Students make a connection between their models and the numerical representation of the equation.

Type: Lesson Plan

Using Models to Subtract Fractions with Unlike Denominators:

This lesson is specific to subtracting fractions with unlike denominators. It requires students to already have a working knowledge of subtracting fractions with common denominators and equivalent fractions.

Type: Lesson Plan

Modeling Fraction Multiplication:

This lesson involves students modeling fraction multiplication with rectangular arrays in order to discover the rule for multiplication of fractions.

Type: Lesson Plan

Adding and Subtracting Mixed Numbers with Unlike Denominators:

This lesson helps fifth graders combine their understanding of adding and subtracting fractions with unlike denominators, finding equivalent fractions, and adding and subtracting mixed numbers with like denominators to move on to adding and subtracting mixed numbers with unlike denominators.

Type: Lesson Plan

Discovering Common Denominators:

Students use pattern blocks to represent fractions with unlike denominators. Students discover that they need to convert the pattern blocks to the same size in order to add them. Therefore, they find and use common denominators for the addition of fractions.

Type: Lesson Plan

Looking for Patterns in a Sequence of Fractions:

Students generate and describe a numerical pattern using the multiplication and subtraction of fractions.

Type: Lesson Plan

It's My Party and I'll Make Dividing by Fractions Easier if I Want to!:

During this lesson students will relate their understanding of whole number division situations to help them interpret situations involving dividing by unit fractions. They will then develop models and strategies for representing the division of a whole number by a unit fraction.

Type: Lesson Plan

Natural Disaster Dividing Fractions:

In this lesson, students will extend learning of dividing unit fractions and whole numbers within the context of governmental response to an emergency situation.

Type: Lesson Plan

## Original Student Tutorials

Carnival Craziness!:

Learn to divide whole numbers by unit fractions as you help Allie and Cameron create equal shares of candy and prizes for guests at a carnival in this interactive tutorial.

Type: Original Student Tutorial

Chocolate Shop Challenge Part 2: Dividing Unit Fractions and Whole Numbers Using Number Lines:

Solve real-world word problems involving dividing a unit fraction by a whole number and dividing a whole number by a unit fraction using number lines in this chocolate-themed, interactive tutorial.

This is part 2 of a 2-part series. Click HERE to open "Chocolate Shop Challenge Part 1: Dividing Unit Fractions and Whole Numbers Using Fraction Bar Models"

Click HERE to open the related tutorial, "David Divides Desserts: Divide a Unit Fraction by a Whole Number"

Type: Original Student Tutorial

David Divides Desserts: Divide a Unit Fraction by a Whole Number:

Learn to solve word problems involving division of a unit fraction by a whole number by using models, expressions, equations, and strategic thinking in this interactive, dessert-themed tutorial.

Type: Original Student Tutorial

Share and Share Alike:

Learn how to divide a unit fraction by a whole number to share yummy picnic goodies equally in this interactive tutorial.

Type: Original Student Tutorial

Adding Potions with Unlike Fractions Part 1:

Explore how to add fractions less than one with unlike denominators in this magical, interactive tutorial.

Type: Original Student Tutorial

Scaling Up to Escape:

Try to escape from this room using multiplication as scaling in this interactive tutorial.

Note: this tutorial is an introductory lesson on multiplying a given number without calculating before working with fractions.

Type: Original Student Tutorial

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

How many servings of oatmeal?:

This task provides a context for performing division of a whole number by a unit fraction. This problem is a "How many groups?'' example of division: the "groups'' in this case are the servings of oatmeal and the question is asking how many servings (or groups) there are in the package.

Type: Problem-Solving Task

Painting a room:

The purpose of this task is to provide students with a situation in which it is natural for them to divide a unit fraction by a non-zero whole number. Determining the amount of paint that Kulani needs for each wall illustrates an understanding of the meaning of dividing a unit fraction by a non-zero whole number.

Type: Problem-Solving Task

Painting a Wall:

The purpose of this task is for students to find the answer to a question in context that can be represented by fraction multiplication. This task is appropriate for either instruction or assessment depending on how it is used and where students are in their understanding of fraction multiplication.

Type: Problem-Solving Task

Origami Stars:

The purpose of this task is to present students with a situation in which they need to divide a whole number by a unit fraction in order to find a solution. Calculating the number of origami stars that Avery and Megan can make illustrates student understanding of the process of dividing a whole number by a unit fraction.

Type: Problem-Solving Task

Mixed Numbers with Unlike Denominators:

The purpose of this task is to help students realize there are different ways to add mixed numbers and is most appropriate for use in an instructional setting. The two primary ways one can expect students to add are converting the mixed numbers to fractions greater than 1 or adding the whole numbers and fractional parts separately. It is good for students to develop a sense of which approach would be better in a particular context.

Type: Problem-Solving Task

Making S'Mores:

The purpose of this instructional task is to motivate a discussion about adding fractions and the meaning of the common denominator. The different parts of the task have students moving back and forth between the abstract representation of the fractions and the meaning of the fractions in the context.

Type: Problem-Solving Task

Making Cookies:

This tasks lends itself very well to multiple solution methods. Students may learn a lot by comparing different methods. Students who are already comfortable with fraction multiplication can go straight to the numeric solutions given below. Students who are still unsure of the meanings of these operations can draw pictures or diagrams.

Type: Problem-Solving Task

Jog-A-Thon:

The purpose of this task is to present students with a situation where it is natural to add fractions with unlike denominators; it can be used for either assessment or instructional purposes. Teachers should anticipate two types of solutions: one where students calculate the distance Alex ran to determine an answer, and one where students compare the two parts of his run to benchmark fractions.

Type: Problem-Solving Task

Running to School:

The task could be one of the first activities for introducing the multiplication of fractions.  The task has fractions which are easy to draw and provides a linear situation.  Students benefit from reasoning through the solution to such word problems before they are told that they can be solved by multiplying the fractions; this helps them develop meaning for fraction multiplication.

Type: Problem-Solving Task

Running a Mile:

The solution uses the idea that multiplying by a fraction less than 1 results in a smaller value. The students need to explain why that is so.

Type: Problem-Solving Task

Reasoning about Multiplication:

This is a good task to work with kids to try to explain their thinking clearly and precisely, although teachers should be willing to work with many different ways of explaining the relationship between the magnitude of the factors and the magnitude of the product.

Type: Problem-Solving Task

Half of a Recipe:

This is the third problem in a series of three tasks involving fraction multiplication that can be solved with pictures or number lines. The first, Running to school, does not require that the unit fractions that comprise 3/4 be subdivided in order to find 1/3 of 3/4. The second task, Drinking Juice, does require students to subdivide the unit fractions that comprise 1/2 in order to find 3/4 of 1/2. This task also requires subdivision and involves multiplying a fraction and a mixed number.

Type: Problem-Solving Task

Grass Seedlings:

The purpose of this task is to gain a better understanding of multiplying with fractions. Students should use the diagram provided to support their findings.

Type: Problem-Solving Task

Fundraising:

This problem helps students gain a better understanding of multiplying with fractions.

Type: Problem-Solving Task

Folding Strips of Paper:

The purpose of this task is to provide students with a concrete experience they can relate to fraction multiplication. Perhaps more importantly, the task also purposefully relates length and locations of points on a number line, a common trouble spot for students. This task is meant for instruction and would be a useful as part of an introductory unit on fraction multiplication.

Type: Problem-Solving Task

Finding Common Denominators to Subtract:

Part (a) of this task asks students to use two different denominators to subtract fractions. The purpose of this is to help students realize that any common denominator will work, not just the least common denominator. Part (b) does not ask students to do it in more than one way; the purpose is to give them an opportunity to choose a denominator and possibly compare with another student who chose a different denominator. The purpose of part (c) is to help students move away from a reliance on drawing pictures. Students can draw a picture if they want, but this subtraction problem is easier to do symbolically, which helps students appreciate the power of symbolic notation.

Type: Problem-Solving Task

Finding Common Denominators to Add:

Part (a) of this task asks students to find and use two different common denominators to add the given fractions. The purpose of this question is to help students realize that they can use any common denominator to find a solution, not just the least common denominator. Part (b) does not ask students to solve the given addition problem in more than one way. Instead, the purpose of this question is to give students an opportunity to choose a denominator and possibly to compare their solution method with another student who chose a different denominator.  The purpose of part (c) is to give students who are ready to work symbolically a chance to work more efficiently.

Type: Problem-Solving Task

How many marbles?:

This task is intended to complement "How many servings of oatmeal?" and "Molly's run.'' All three tasks address the division problem 4÷1/3 but from different points of view. This task provides a how many in each group version of 4÷1/3. This task should be done together with the "How many servings of oatmeal" task with specific attention paid to the very different pictures representing the two situations.

Type: Problem-Solving Task

Egyptian Fractions:

One goal of this task is to help students develop comfort and ease with adding fractions with unlike denominators. Another goal is to help them develop fraction number sense by having students decompose fractions.

Type: Problem-Solving Task

Drinking Juice:

This is the second problem in a series of three tasks involving fraction multiplication that can be solved with pictures or number lines. This task does require students to subdivide the unit fractions that comprise 1/2 in order to find 3/4 of 1/2.

Type: Problem-Solving Task

Do These Add Up?:

This task addresses common errors that students make when interpreting adding fractions word problems. It is very important for students to recognize that they only add fractions when the fractions refer to the same whole, and also when the fractions of the whole being added do not overlap. This set of questions is designed to enhance a student's understanding of when it is and is not appropriate to add fractions.

Type: Problem-Solving Task

Dividing by One-Half:

This task requires students to recognize both "number of groups unknown" (part (a)) and "group size unknown" (part (d)) division problems in the context of a whole number divided by a unit fraction. It also addresses a common misconception that students have where they confuse dividing by 2 or multiplying by 1/2 with dividing by 1/2.

Type: Problem-Solving Task

Comparing a Number and a Product:

The purpose of this task is for students to compare a number and its product with other numbers that are greater than and less than one. As written, this task could be used in a summative assessment context, but it might be more useful in an instructional setting where students are asked to explain their answers either to a partner or in a whole class discussion.

Type: Problem-Solving Task

Calculator Trouble:

This particular problem deals with multiplication. Even though students can solve this problem by multiplying, it is unlikely they will. Here it is much easier to answer the question if you can think of multiplying a number by a factor as scaling the number.

Type: Problem-Solving Task

Banana Pudding:

The purpose of this task is to provide students with a concrete situation they can model by dividing a whole number by a unit fraction. For students who are just beginning to think about the meaning of division by a unit fraction (or students who have never cooked), the teacher can bring in a 1/4 cup measuring cup so that students can act it out. If students can reason through parts (a) and (b) successfully, they will be well-situated to think about part (c) which could yield different solution methods.

Type: Problem-Solving Task

## Professional Development

Fractions, Percents, and Ratios, Part A: Models for Multiplication and Division of Fractions:

This professional development module shows teachers how to use area models to understand multiplication and division of fractions.

Type: Professional Development

## Student Center Activity

Fractions Jigsaw:

This problem provides students an opportunity to find equivalent fractions and carry out some simple additions and subtractions of fractions in a context that may challenge and motivate students. Users need to download, print, and cut-out the fraction jigsaw. Then, they must arrange the square pieces right-side up so that the edges that touch contain equivalent fractions. The Teachers' Notes page offers rationale, suggestions for implementation, discussion questions, and ideas for extension and support.

Type: Student Center Activity

## Teaching Ideas

Adding Fractions with Unlike Denominators:

Kahn Academy video - How to add fractions with unlike denominators.

Type: Teaching Idea

Let's Make Fudge:

"Students work in small groups to read a recipe involving fractions, halve the recipe's values, and make a batch of fudge. The importance of selection of appropriate measuring tools is taught. Abbreviations for capacity measuring tools are reviewed"( from the Beacon Lesson Plan Library).

Type: Teaching Idea

Divide Fractions:

This interactive resource provides three activities which model the concept of dividing fractions, as well as mixed numbers, by using number lines or circle graphs.  It includes the equation showing the standard algorithm.

Type: Teaching Idea

## Tutorials

Arithmetic Operations with Fractions:

In this tutorial, the four operations  are applied to fractions with the visualization of the number line. This tutorial starts by adding fractions with the same denominators and explains the logic behind multiplication of fractions.  This tutorial also highlights the application and extension of previous understandings of mulitplication to multiply a fraction or whole number by a fraction.

a.  Interpret the product (a/b) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x qb.  In general, (a/b) x (c/d) = ac/bd.

Type: Tutorial

Creating Common Denominators:

This tutorial explores the addition and subtraction of fractions with unlike denominators. Using the number line, this mathematical process can be easily visualized and connected to the final strategy of multiplying the denominators (a/b + c/d = ad +bc/bd).  The video number line does show negative numbers which goes beyond elementary standards so an elementary teacher would need to reflect on whether this video will enrich student knowledge or cause confusion.

Type: Tutorial

Least Common Denominators:

In this tutorial, students will be exposed to the strategy of finding the least common denominator for certain cases.  Elementary teachers should note this is not a requirement for elementary standards and consider whether this video will further student knowledge or create confusion.  This chapter explains how to find the smallest possible common denominator. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12.

Type: Tutorial

Adding and Subtracting Fractions:

This tutorial for student audiences will assist learners with a further understanding of the rules for adding and subtracting fractions. Students will be able to navigate the teaching portion of the tutorial at their own pace and test their understanding after each step of the lesson with a "Try This" section. The "Try This" section will monitor students answers and self-check by a right answer turning orange and a wrong answer dissolving.

Type: Tutorial

How Do You Divide Fractions?:

This five-minute video answers the question "Must one always invert and multiply?" when dividing fractions. An alternative algorithm is presented which works well in certain cases. The video focuses on sense-making in using either method, and on judging the reasonableness of answers.

Type: Tutorial

Multiplying Fractions:

The video describes how to multiply fractions and state the answer in lowest terms.

Type: Tutorial

Adding Fractions:

In this web-based tutorial, students learn procedures for adding fractions with like and unlike denominators. The tutorial includes visual representations of the problems using pizzas, animations of the algorithm, and links to related lessons, worksheets, and practice problems.

Type: Tutorial

Subtracting Fractions:

In this web-based tutorial, students learn procedures for subtracting fractions. The tutorial includes visual representations of the problems using pizzas, animations of the algorithm, and links to related lessons, worksheets, and practice problems.

Type: Tutorial

## 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

## Student Resources

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

## Original Student Tutorials

Carnival Craziness!:

Learn to divide whole numbers by unit fractions as you help Allie and Cameron create equal shares of candy and prizes for guests at a carnival in this interactive tutorial.

Type: Original Student Tutorial

Chocolate Shop Challenge Part 2: Dividing Unit Fractions and Whole Numbers Using Number Lines:

Solve real-world word problems involving dividing a unit fraction by a whole number and dividing a whole number by a unit fraction using number lines in this chocolate-themed, interactive tutorial.

This is part 2 of a 2-part series. Click HERE to open "Chocolate Shop Challenge Part 1: Dividing Unit Fractions and Whole Numbers Using Fraction Bar Models"

Click HERE to open the related tutorial, "David Divides Desserts: Divide a Unit Fraction by a Whole Number"

Type: Original Student Tutorial

David Divides Desserts: Divide a Unit Fraction by a Whole Number:

Learn to solve word problems involving division of a unit fraction by a whole number by using models, expressions, equations, and strategic thinking in this interactive, dessert-themed tutorial.

Type: Original Student Tutorial

Share and Share Alike:

Learn how to divide a unit fraction by a whole number to share yummy picnic goodies equally in this interactive tutorial.

Type: Original Student Tutorial

Adding Potions with Unlike Fractions Part 1:

Explore how to add fractions less than one with unlike denominators in this magical, interactive tutorial.

Type: Original Student Tutorial

Scaling Up to Escape:

Try to escape from this room using multiplication as scaling in this interactive tutorial.

Note: this tutorial is an introductory lesson on multiplying a given number without calculating before working with fractions.

Type: Original Student Tutorial

## Educational Games

Ice Ice Maybe: An Operations Estimation Game:

This fun and interactive game helps practice estimation skills, using various operations of choice, including addition, subtraction, multiplication, division, using decimals, fractions, and percents.

Various levels of difficulty make this game appropriate for multiple age and ability levels.

Addition/Subtraction: The addition and subtraction of whole numbers, the addition and subtraction of decimals.

Multiplication/Division: The multiplication and addition of whole numbers.

Percentages: Identify the percentage of a whole number.

Fractions: Multiply and divide a whole number by a fraction, as well as apply properties of operations.

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

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

How many servings of oatmeal?:

This task provides a context for performing division of a whole number by a unit fraction. This problem is a "How many groups?'' example of division: the "groups'' in this case are the servings of oatmeal and the question is asking how many servings (or groups) there are in the package.

Type: Problem-Solving Task

Painting a room:

The purpose of this task is to provide students with a situation in which it is natural for them to divide a unit fraction by a non-zero whole number. Determining the amount of paint that Kulani needs for each wall illustrates an understanding of the meaning of dividing a unit fraction by a non-zero whole number.

Type: Problem-Solving Task

Painting a Wall:

The purpose of this task is for students to find the answer to a question in context that can be represented by fraction multiplication. This task is appropriate for either instruction or assessment depending on how it is used and where students are in their understanding of fraction multiplication.

Type: Problem-Solving Task

Origami Stars:

The purpose of this task is to present students with a situation in which they need to divide a whole number by a unit fraction in order to find a solution. Calculating the number of origami stars that Avery and Megan can make illustrates student understanding of the process of dividing a whole number by a unit fraction.

Type: Problem-Solving Task

Mixed Numbers with Unlike Denominators:

The purpose of this task is to help students realize there are different ways to add mixed numbers and is most appropriate for use in an instructional setting. The two primary ways one can expect students to add are converting the mixed numbers to fractions greater than 1 or adding the whole numbers and fractional parts separately. It is good for students to develop a sense of which approach would be better in a particular context.

Type: Problem-Solving Task

Making S'Mores:

The purpose of this instructional task is to motivate a discussion about adding fractions and the meaning of the common denominator. The different parts of the task have students moving back and forth between the abstract representation of the fractions and the meaning of the fractions in the context.

Type: Problem-Solving Task

Making Cookies:

This tasks lends itself very well to multiple solution methods. Students may learn a lot by comparing different methods. Students who are already comfortable with fraction multiplication can go straight to the numeric solutions given below. Students who are still unsure of the meanings of these operations can draw pictures or diagrams.

Type: Problem-Solving Task

Jog-A-Thon:

The purpose of this task is to present students with a situation where it is natural to add fractions with unlike denominators; it can be used for either assessment or instructional purposes. Teachers should anticipate two types of solutions: one where students calculate the distance Alex ran to determine an answer, and one where students compare the two parts of his run to benchmark fractions.

Type: Problem-Solving Task

Running to School:

The task could be one of the first activities for introducing the multiplication of fractions.  The task has fractions which are easy to draw and provides a linear situation.  Students benefit from reasoning through the solution to such word problems before they are told that they can be solved by multiplying the fractions; this helps them develop meaning for fraction multiplication.

Type: Problem-Solving Task

Running a Mile:

The solution uses the idea that multiplying by a fraction less than 1 results in a smaller value. The students need to explain why that is so.

Type: Problem-Solving Task

Reasoning about Multiplication:

This is a good task to work with kids to try to explain their thinking clearly and precisely, although teachers should be willing to work with many different ways of explaining the relationship between the magnitude of the factors and the magnitude of the product.

Type: Problem-Solving Task

Half of a Recipe:

This is the third problem in a series of three tasks involving fraction multiplication that can be solved with pictures or number lines. The first, Running to school, does not require that the unit fractions that comprise 3/4 be subdivided in order to find 1/3 of 3/4. The second task, Drinking Juice, does require students to subdivide the unit fractions that comprise 1/2 in order to find 3/4 of 1/2. This task also requires subdivision and involves multiplying a fraction and a mixed number.

Type: Problem-Solving Task

Grass Seedlings:

The purpose of this task is to gain a better understanding of multiplying with fractions. Students should use the diagram provided to support their findings.

Type: Problem-Solving Task

Fundraising:

This problem helps students gain a better understanding of multiplying with fractions.

Type: Problem-Solving Task

Folding Strips of Paper:

The purpose of this task is to provide students with a concrete experience they can relate to fraction multiplication. Perhaps more importantly, the task also purposefully relates length and locations of points on a number line, a common trouble spot for students. This task is meant for instruction and would be a useful as part of an introductory unit on fraction multiplication.

Type: Problem-Solving Task

Finding Common Denominators to Subtract:

Part (a) of this task asks students to use two different denominators to subtract fractions. The purpose of this is to help students realize that any common denominator will work, not just the least common denominator. Part (b) does not ask students to do it in more than one way; the purpose is to give them an opportunity to choose a denominator and possibly compare with another student who chose a different denominator. The purpose of part (c) is to help students move away from a reliance on drawing pictures. Students can draw a picture if they want, but this subtraction problem is easier to do symbolically, which helps students appreciate the power of symbolic notation.

Type: Problem-Solving Task

Finding Common Denominators to Add:

Part (a) of this task asks students to find and use two different common denominators to add the given fractions. The purpose of this question is to help students realize that they can use any common denominator to find a solution, not just the least common denominator. Part (b) does not ask students to solve the given addition problem in more than one way. Instead, the purpose of this question is to give students an opportunity to choose a denominator and possibly to compare their solution method with another student who chose a different denominator.  The purpose of part (c) is to give students who are ready to work symbolically a chance to work more efficiently.

Type: Problem-Solving Task

How many marbles?:

This task is intended to complement "How many servings of oatmeal?" and "Molly's run.'' All three tasks address the division problem 4÷1/3 but from different points of view. This task provides a how many in each group version of 4÷1/3. This task should be done together with the "How many servings of oatmeal" task with specific attention paid to the very different pictures representing the two situations.

Type: Problem-Solving Task

Egyptian Fractions:

One goal of this task is to help students develop comfort and ease with adding fractions with unlike denominators. Another goal is to help them develop fraction number sense by having students decompose fractions.

Type: Problem-Solving Task

Drinking Juice:

This is the second problem in a series of three tasks involving fraction multiplication that can be solved with pictures or number lines. This task does require students to subdivide the unit fractions that comprise 1/2 in order to find 3/4 of 1/2.

Type: Problem-Solving Task

Do These Add Up?:

This task addresses common errors that students make when interpreting adding fractions word problems. It is very important for students to recognize that they only add fractions when the fractions refer to the same whole, and also when the fractions of the whole being added do not overlap. This set of questions is designed to enhance a student's understanding of when it is and is not appropriate to add fractions.

Type: Problem-Solving Task

Dividing by One-Half:

This task requires students to recognize both "number of groups unknown" (part (a)) and "group size unknown" (part (d)) division problems in the context of a whole number divided by a unit fraction. It also addresses a common misconception that students have where they confuse dividing by 2 or multiplying by 1/2 with dividing by 1/2.

Type: Problem-Solving Task

Comparing a Number and a Product:

The purpose of this task is for students to compare a number and its product with other numbers that are greater than and less than one. As written, this task could be used in a summative assessment context, but it might be more useful in an instructional setting where students are asked to explain their answers either to a partner or in a whole class discussion.

Type: Problem-Solving Task

Calculator Trouble:

This particular problem deals with multiplication. Even though students can solve this problem by multiplying, it is unlikely they will. Here it is much easier to answer the question if you can think of multiplying a number by a factor as scaling the number.

Type: Problem-Solving Task

Banana Pudding:

The purpose of this task is to provide students with a concrete situation they can model by dividing a whole number by a unit fraction. For students who are just beginning to think about the meaning of division by a unit fraction (or students who have never cooked), the teacher can bring in a 1/4 cup measuring cup so that students can act it out. If students can reason through parts (a) and (b) successfully, they will be well-situated to think about part (c) which could yield different solution methods.

Type: Problem-Solving Task

## Tutorials

Arithmetic Operations with Fractions:

In this tutorial, the four operations  are applied to fractions with the visualization of the number line. This tutorial starts by adding fractions with the same denominators and explains the logic behind multiplication of fractions.  This tutorial also highlights the application and extension of previous understandings of mulitplication to multiply a fraction or whole number by a fraction.

a.  Interpret the product (a/b) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x qb.  In general, (a/b) x (c/d) = ac/bd.

Type: Tutorial

Creating Common Denominators:

This tutorial explores the addition and subtraction of fractions with unlike denominators. Using the number line, this mathematical process can be easily visualized and connected to the final strategy of multiplying the denominators (a/b + c/d = ad +bc/bd).  The video number line does show negative numbers which goes beyond elementary standards so an elementary teacher would need to reflect on whether this video will enrich student knowledge or cause confusion.

Type: Tutorial

Least Common Denominators:

In this tutorial, students will be exposed to the strategy of finding the least common denominator for certain cases.  Elementary teachers should note this is not a requirement for elementary standards and consider whether this video will further student knowledge or create confusion.  This chapter explains how to find the smallest possible common denominator. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12.

Type: Tutorial

Adding and Subtracting Fractions:

This tutorial for student audiences will assist learners with a further understanding of the rules for adding and subtracting fractions. Students will be able to navigate the teaching portion of the tutorial at their own pace and test their understanding after each step of the lesson with a "Try This" section. The "Try This" section will monitor students answers and self-check by a right answer turning orange and a wrong answer dissolving.

Type: Tutorial

Multiplying Fractions:

The video describes how to multiply fractions and state the answer in lowest terms.

Type: Tutorial

## 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

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

How many servings of oatmeal?:

This task provides a context for performing division of a whole number by a unit fraction. This problem is a "How many groups?'' example of division: the "groups'' in this case are the servings of oatmeal and the question is asking how many servings (or groups) there are in the package.

Type: Problem-Solving Task

Painting a room:

The purpose of this task is to provide students with a situation in which it is natural for them to divide a unit fraction by a non-zero whole number. Determining the amount of paint that Kulani needs for each wall illustrates an understanding of the meaning of dividing a unit fraction by a non-zero whole number.

Type: Problem-Solving Task

Painting a Wall:

The purpose of this task is for students to find the answer to a question in context that can be represented by fraction multiplication. This task is appropriate for either instruction or assessment depending on how it is used and where students are in their understanding of fraction multiplication.

Type: Problem-Solving Task

Origami Stars:

The purpose of this task is to present students with a situation in which they need to divide a whole number by a unit fraction in order to find a solution. Calculating the number of origami stars that Avery and Megan can make illustrates student understanding of the process of dividing a whole number by a unit fraction.

Type: Problem-Solving Task

Mixed Numbers with Unlike Denominators:

The purpose of this task is to help students realize there are different ways to add mixed numbers and is most appropriate for use in an instructional setting. The two primary ways one can expect students to add are converting the mixed numbers to fractions greater than 1 or adding the whole numbers and fractional parts separately. It is good for students to develop a sense of which approach would be better in a particular context.

Type: Problem-Solving Task

Making S'Mores:

The purpose of this instructional task is to motivate a discussion about adding fractions and the meaning of the common denominator. The different parts of the task have students moving back and forth between the abstract representation of the fractions and the meaning of the fractions in the context.

Type: Problem-Solving Task

Making Cookies:

This tasks lends itself very well to multiple solution methods. Students may learn a lot by comparing different methods. Students who are already comfortable with fraction multiplication can go straight to the numeric solutions given below. Students who are still unsure of the meanings of these operations can draw pictures or diagrams.

Type: Problem-Solving Task

Jog-A-Thon:

The purpose of this task is to present students with a situation where it is natural to add fractions with unlike denominators; it can be used for either assessment or instructional purposes. Teachers should anticipate two types of solutions: one where students calculate the distance Alex ran to determine an answer, and one where students compare the two parts of his run to benchmark fractions.

Type: Problem-Solving Task

Running to School:

The task could be one of the first activities for introducing the multiplication of fractions.  The task has fractions which are easy to draw and provides a linear situation.  Students benefit from reasoning through the solution to such word problems before they are told that they can be solved by multiplying the fractions; this helps them develop meaning for fraction multiplication.

Type: Problem-Solving Task

Running a Mile:

The solution uses the idea that multiplying by a fraction less than 1 results in a smaller value. The students need to explain why that is so.

Type: Problem-Solving Task

Reasoning about Multiplication:

This is a good task to work with kids to try to explain their thinking clearly and precisely, although teachers should be willing to work with many different ways of explaining the relationship between the magnitude of the factors and the magnitude of the product.

Type: Problem-Solving Task

Half of a Recipe:

This is the third problem in a series of three tasks involving fraction multiplication that can be solved with pictures or number lines. The first, Running to school, does not require that the unit fractions that comprise 3/4 be subdivided in order to find 1/3 of 3/4. The second task, Drinking Juice, does require students to subdivide the unit fractions that comprise 1/2 in order to find 3/4 of 1/2. This task also requires subdivision and involves multiplying a fraction and a mixed number.

Type: Problem-Solving Task

Grass Seedlings:

The purpose of this task is to gain a better understanding of multiplying with fractions. Students should use the diagram provided to support their findings.

Type: Problem-Solving Task

Fundraising:

This problem helps students gain a better understanding of multiplying with fractions.

Type: Problem-Solving Task

Folding Strips of Paper:

The purpose of this task is to provide students with a concrete experience they can relate to fraction multiplication. Perhaps more importantly, the task also purposefully relates length and locations of points on a number line, a common trouble spot for students. This task is meant for instruction and would be a useful as part of an introductory unit on fraction multiplication.

Type: Problem-Solving Task

Finding Common Denominators to Subtract:

Part (a) of this task asks students to use two different denominators to subtract fractions. The purpose of this is to help students realize that any common denominator will work, not just the least common denominator. Part (b) does not ask students to do it in more than one way; the purpose is to give them an opportunity to choose a denominator and possibly compare with another student who chose a different denominator. The purpose of part (c) is to help students move away from a reliance on drawing pictures. Students can draw a picture if they want, but this subtraction problem is easier to do symbolically, which helps students appreciate the power of symbolic notation.

Type: Problem-Solving Task

Finding Common Denominators to Add:

Part (a) of this task asks students to find and use two different common denominators to add the given fractions. The purpose of this question is to help students realize that they can use any common denominator to find a solution, not just the least common denominator. Part (b) does not ask students to solve the given addition problem in more than one way. Instead, the purpose of this question is to give students an opportunity to choose a denominator and possibly to compare their solution method with another student who chose a different denominator.  The purpose of part (c) is to give students who are ready to work symbolically a chance to work more efficiently.

Type: Problem-Solving Task

How many marbles?:

This task is intended to complement "How many servings of oatmeal?" and "Molly's run.'' All three tasks address the division problem 4÷1/3 but from different points of view. This task provides a how many in each group version of 4÷1/3. This task should be done together with the "How many servings of oatmeal" task with specific attention paid to the very different pictures representing the two situations.

Type: Problem-Solving Task

Egyptian Fractions:

One goal of this task is to help students develop comfort and ease with adding fractions with unlike denominators. Another goal is to help them develop fraction number sense by having students decompose fractions.

Type: Problem-Solving Task

Drinking Juice:

This is the second problem in a series of three tasks involving fraction multiplication that can be solved with pictures or number lines. This task does require students to subdivide the unit fractions that comprise 1/2 in order to find 3/4 of 1/2.

Type: Problem-Solving Task

Do These Add Up?:

This task addresses common errors that students make when interpreting adding fractions word problems. It is very important for students to recognize that they only add fractions when the fractions refer to the same whole, and also when the fractions of the whole being added do not overlap. This set of questions is designed to enhance a student's understanding of when it is and is not appropriate to add fractions.

Type: Problem-Solving Task

Dividing by One-Half:

This task requires students to recognize both "number of groups unknown" (part (a)) and "group size unknown" (part (d)) division problems in the context of a whole number divided by a unit fraction. It also addresses a common misconception that students have where they confuse dividing by 2 or multiplying by 1/2 with dividing by 1/2.

Type: Problem-Solving Task

Comparing a Number and a Product:

The purpose of this task is for students to compare a number and its product with other numbers that are greater than and less than one. As written, this task could be used in a summative assessment context, but it might be more useful in an instructional setting where students are asked to explain their answers either to a partner or in a whole class discussion.

Type: Problem-Solving Task

Calculator Trouble:

This particular problem deals with multiplication. Even though students can solve this problem by multiplying, it is unlikely they will. Here it is much easier to answer the question if you can think of multiplying a number by a factor as scaling the number.

Type: Problem-Solving Task

Banana Pudding:

The purpose of this task is to provide students with a concrete situation they can model by dividing a whole number by a unit fraction. For students who are just beginning to think about the meaning of division by a unit fraction (or students who have never cooked), the teacher can bring in a 1/4 cup measuring cup so that students can act it out. If students can reason through parts (a) and (b) successfully, they will be well-situated to think about part (c) which could yield different solution methods.

Type: Problem-Solving Task

## Tutorials

Adding and Subtracting Fractions:

This tutorial for student audiences will assist learners with a further understanding of the rules for adding and subtracting fractions. Students will be able to navigate the teaching portion of the tutorial at their own pace and test their understanding after each step of the lesson with a "Try This" section. The "Try This" section will monitor students answers and self-check by a right answer turning orange and a wrong answer dissolving.

Type: Tutorial

How Do You Divide Fractions?:

This five-minute video answers the question "Must one always invert and multiply?" when dividing fractions. An alternative algorithm is presented which works well in certain cases. The video focuses on sense-making in using either method, and on judging the reasonableness of answers.

Type: Tutorial

Multiplying Fractions:

The video describes how to multiply fractions and state the answer in lowest terms.

Type: Tutorial

Subtracting Fractions:

In this web-based tutorial, students learn procedures for subtracting fractions. The tutorial includes visual representations of the problems using pizzas, animations of the algorithm, and links to related lessons, worksheets, and practice problems.

Type: Tutorial