# MA.5.FR.2.2

Extend previous understanding of multiplication to multiply a fraction by a fraction, including mixed numbers and fractions greater than 1, with procedural reliability.

### Clarifications

Clarification 1: Instruction includes the use of manipulatives, drawings or the properties of operations.

Clarification 2: Denominators limited to whole numbers up to 20.

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

## Benchmark Instructional Guide

• NA

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

The purpose of this benchmark is for students to learn strategies to multiply two fractions. This continues the work from grade 4 where students multiplied a whole number times a fraction and a fraction times a whole number (MA.4.FR.2.4). Procedural fluency will be achieved in grade 6 (MA.6.NSO.2.2).
• During instruction, students are expected to multiply fractions including proper fractions, improper fractions (fractions greater than 1), and mixed numbers efficiently and accurately.
• Visual fraction models (area models, tape diagrams, number lines) should be used and created by students during their work with this benchmark (MTR.2.1). Visual fraction models should show how a fraction is partitioned into parts that are the same as the product of the denominators.

• When exploring an algorithm to multiply fractions ($\frac{\text{a}}{\text{b}}$ × $\frac{\text{c}}{\text{d}}$= $\frac{\text{a x c}}{\text{b x d}}$ ), make connections to an accompanying area model. This will help students understand the algorithm conceptually and use it more accurately.
• Instruction includes students using equivalent fractions to simplify answers; however, putting answers in simplest form is not a priority.

### Common Misconceptions or Errors

• Students may believe that multiplication always results in a larger number. Using models when multiplying with fractions will enable students to generalize about multiplication algorithms that are based on conceptual understanding (MTR.5.1).
• Students can have difficulty with word problems when determining which operation to use, and the stress of working with fractions makes this happen more often.
• For example, “Mark has $\frac{\text{3}}{\text{4}}$ yards of rope and he gives a third of the rope to a friend. How much rope does Mark have left?” expects students to first find $\frac{\text{1}}{\text{3}}$ of $\frac{\text{3}}{\text{4}}$, or multiply $\frac{\text{1}}{\text{3}}$ x $\frac{\text{3}}{\text{4}}$, and then to find the difference to find how much Mark has left. On the other hand, "Mark has $\frac{\text{3}}{\text{4}}$ yards of rope to a friend. How much rope does Mark have left?” only requires finding the difference $\frac{\text{3}}{\text{4}}$ - $\frac{\text{1}}{\text{3}}$.

### Strategies to Support Tiered Instruction

• Instruction involves real-world examples and models which allow students to see that multiplication does not always result in a larger number.
• For example, the teacher provides the problem: “Tau has $\frac{\text{1}}{\text{4}}$ of the lasagna pan leftover from the party in the refrigerator. He eats one half of the leftovers for dinner. How much of the lasagna did he eat for dinner?” This can be written as $\frac{\text{1}}{\text{2}}$ x $\frac{\text{1}}{\text{4}}$ or $\frac{\text{1}}{\text{2}}$ "of" $\frac{\text{1}}{\text{4}}$

• When students think about what is left in the refrigerator now, they must think in terms of the whole pan of lasagna. Tau didn’t eat half the pan; he ate half of the portion that was left. So how much of the whole pan did he eat?

• Part A. Martiza has 4$\frac{\text{1}}{\text{2}}$ cups of cream cheese. She uses $\frac{\text{3}}{\text{4}}$ of the cream cheese for a banana pudding recipe. After she uses it for the recipe, how much cream cheese will Maritza have left?
• Part B. To find out how much cream cheese she used, Maritza multiplied 4 $\frac{\text{1}}{\text{2}}$ × $\frac{\text{3}}{\text{4}}$ as (4 × $\frac{\text{3}}{\text{4}}$) + ($\frac{\text{1}}{\text{2}}$ × $\frac{\text{3}}{\text{4}}$). Will this method work? Why or why not?
• Part C. What additional step is required to find how much cream cheese she has left?

### Instructional Items

Instructional Item 1

What is the product of $\frac{\text{1}}{\text{5}}$ x 6 $\frac{\text{1}}{\text{2}}$?
• a. $\frac{\text{6}}{\text{10}}$
• b. $\frac{\text{12}}{\text{5}}$
• c. 6 $\frac{\text{7}}{\text{10}}$
• d. 1 $\frac{\text{3}}{\text{10}}$

*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.
5012070: Grade Five Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7712060: Access Mathematics Grade 5 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))
5012065: Grade 4 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.5.FR.2.AP.2: Explore multiplying a unit fraction by a unit fraction.

## Related Resources

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

## 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.

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

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

## Lesson Plans

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.Â  This is a practice and application lesson, not an introductory 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

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

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

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

## Perspectives Video: Expert

B.E.S.T. Journey:

What roles do exploration, procedural reliability, automaticity, and procedural fluency play in a student's journey through the B.E.S.T. benchmarks? Dr. Lawrence Gray explains the path through the B.E.S.T. maththematics benchmarks in this Expert Perspectives video.

Type: Perspectives Video: Expert

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.

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.

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.

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.

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.

Fundraising:

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

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.

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.

## 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

## 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

## 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

Multiplying Fractions:

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

Type: Tutorial

## MFAS Formative Assessments

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.

Using Visual Fraction Models:

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

## Student Resources

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

## 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.

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

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.

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.

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.

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.

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.

Fundraising:

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

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.

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.

## 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

Multiplying Fractions:

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

Type: Tutorial

## Parent Resources

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

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.

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.

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.

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.

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.

Fundraising:

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

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.

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.