# MA.6.NSO.2.2

Extend previous understanding of multiplication and division to compute products and quotients of positive fractions by positive fractions, including mixed numbers, with procedural fluency.

### Clarifications

Clarification 1: Instruction focuses on making connections between visual models, the relationship between multiplication and division, reciprocals and algorithms.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Number Sense and Operations
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Area Model
• Commutative Property
• Dividend
• Divisor
• Expression

### Vertical Alignment

Previous Benchmarks

http://flbt5.floridaearlylearning.com/standards.html

Next Benchmarks

### Purpose and Instructional Strategies

In grade 5, students multiplied fractions by fractions with procedural reliability and explored how to divide a unit fraction by a whole number and a whole number by a unit fraction. In grade 6, students become procedurally fluent with multiplication and division of positive fractions. The expectation is to utilize skills from the procedural reliability stage to become fluent with an efficient and accurate procedure, including a standard algorithm. In grade 7, students will become fluent in all operations with positive and negative rational numbers.
• Instruction includes making connections to the distributive property when multiplying fractions.
• For example, when multiplying 1$\frac{\text{1}}{\text{2}}$  by $\frac{\text{3}}{\text{4}}$, it can be written as (1+ $\frac{\text{1}}{\text{2}}$$\frac{\text{3}}{\text{4}}$ to determine $\frac{\text{9}}{\text{8}}$ as the product.
• Instruction includes making connections to inverse operations when multiplying or dividing fractions.
• For example, when determining $\frac{\text{3}}{\text{4}}$ ÷ $\frac{\text{5}}{\text{8}}$, students can write the equation $\mathrm{x\left(}$$\frac{\text{5}}{\text{8}}$) = $\frac{\text{3}}{\text{4}}$ and then solve for x.
• Instruction focuses on appropriate academic vocabulary, such as reciprocal. Avoid focusing on tricks such as “keep-change-flip.” Using academic language and procedures allow for students to connect to future mathematics (MTR.5.1).
• For example, $\frac{\text{3}}{\text{4}}$ ÷ $\frac{\text{5}}{\text{8}}$ can be read as “How many five-eighths are in three-fourths?”
• Instruction includes using concrete and pictorial models, writing a numerical sentence that relates to the model and discovering the pattern or rules for multiplying and dividing fractions by fractions (MTR.2.1, MTR.3.1, MTR.5.1).
• Area Model

• Linear Model
• Bar Model

• Instruction includes providing opportunities for students to analyze their own and others’ calculation methods and discuss multiple strategies or ways of understanding with others (MTR.4.1).
• Students should develop fluency with and without the use of a calculator when performing operations with positive fractions.

### Common Misconceptions or Errors

• Students may forget that common denominators are not necessary for multiplying or dividing fractions.
• Students may have incorrectly assumed that multiplication results in a product that is larger than the two factors. Instruction continues with students assessing the reasonableness of their answers by determining if the product will be greater or less than the factors within the given context.
• Students may have incorrectly assumed that division results in a quotient that is smaller than the dividend. Instruction continues with students assessing the reasonableness of their answers by determining if the quotient will be greater or less than the dividend within the given context.

### Strategies to Support Tiered Instruction

• Teacher encourages and allows for students who have a firm understanding of multiplying and dividing decimals to convert the provided fractional values to their equivalent decimal form before performing the desired operation and converting the solution back to fractional form.
• Instruction includes the use of fraction tiles, fraction towers, or similar manipulatives to make connections between physical representations and algebraic methods.
• Instruction includes the co-creation of a graphic organizer utilizing the mnemonic device Same, Inverse Operation, Reciprocal (S.I.R.) for dividing fractions, which encourages the use of correct mathematical terminology, and including examples of applying the mnemonic device when dividing fractions, whole numbers, and mixed numbers.
• Teacher provides students with flash cards to practice and reinforce academic vocabulary.
• Instead of multiplying by the reciprocal to divide fractions, an alternative method could include rewriting the fractions with a common denominator and then dividing the numerators and the denominators.
• For example,  $\frac{\text{5}}{\text{6}}$÷ $\frac{\text{3}}{\text{2}}$ is equivalent to  $\frac{\text{5}}{\text{6}}$÷ $\frac{\text{9}}{\text{6}}$ which is equivalent to  $\frac{\text{5/9}}{\text{1}}$ which is equivalent to  $\frac{\text{5}}{\text{9}}$.
• Instruction provides opportunities to assess the reasonableness of answers by determining if the product will be greater or less than the factors within the given context.
• Instruction provides opportunities to assess the reasonableness of answers by determining if the quotient will be greater or less than the dividend within the given context.

Jasmine wants to build a 2$\frac{\text{5}}{\text{6}}$ meters long garden path paved with square stones that measure $\frac{\text{1}}{\text{4}}$ meter on each side. There will be no spaces between the stones.
• Part A. Create a model that could be used to answer the following question: How many stones are needed for the path?
• Part B. How many stones are needed for the path?

A container at a juicing plant holds 6$\frac{\text{2}}{\text{3}}$ tons of oranges. The plant can juice 1$\frac{\text{1}}{\text{2}}$ tons of oranges per day. At this rate, how long will it take to empty the container?

Explain using visual models why  $\frac{\text{4}}{\text{5}}$× $\frac{\text{2}}{\text{3}}$=  $\frac{\text{8}}{\text{15}}$ .$\frac{\text{}}{}$

### Instructional Items

Instructional Item 1
What is the value of the expression $\frac{\text{3}}{\text{5}}$ ÷$\frac{\text{5}}{\text{8}}$ ?

Instructional Item 2
What is the value of the expression 8$\frac{\text{1}}{\text{10}}$÷$\frac{\text{5}}{\text{8}}$ ?

*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.
1205010: M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205020: M/J Accelerated Mathematics Grade 6 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812015: Access M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.6.NSO.2.AP.2: Use tools to calculate the product and quotient of positive fractions by positive fractions, including mixed numbers, using the standard algorithms.

## Related Resources

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

## 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

Models of Fraction Division:

Students are asked to explain the relationship between a fraction division word problem and either a visual model or an equation.

Type: Formative Assessment

Fraction Division:

Students are asked to complete two fraction division problems – one with fractions and one with mixed numbers.

Type: Formative Assessment

Juicing Fractions:

Students are asked to write and evaluate a numerical expression involving division of fractions and mixed numbers to model and solve a word problem.

Type: Formative Assessment

## Lesson Plans

Students will be able to match a 3-D shape with its net, then using the net, they will find the surface area of the shape. They will then be able to apply this knowledge to solve real world application problems, finishing up with a design contest.

Type: Lesson Plan

Extending the Distributive Property:

In this lesson, students will build upon their arithmetic experiences with the distributive property to equate algebraic expressions through a series of questions related to real world situations and the use of manipulatives. Activities include the use of Algebra Tiles for moving the concrete learner to the abstract level and the use of matching cards.
This is an introductory lesson that only includes producing equivalent expressions such as 3(2 + x) = 6 + 3x.

Type: Lesson Plan

Using Nets to Find the Surface Area of Pyramids:

In this lesson, students will explore and apply the use of nets to find the surface area of pyramids.

Type: Lesson Plan

Wrapping Up Geometry (Lesson 1 of 2):

This lesson is the first of two in a unit on surface area. This lesson provides a foundation for understanding the concept of surface area by introducing nets of right rectangular prisms.

Type: Lesson Plan

What's on the Surface?:

In this activity, students will work in groups to evaluate the measurements of shapes that form three-dimensional composite shapes to compute the surface area.

Type: Lesson Plan

Can You Find the Relationship?:

In this lesson students will first define in their own words what the greatest common factor (GCF) and least common multiple (LCM) mean. They will take this understanding and apply it to solving GCF and LCM word problems. Students will then illustrate their understanding by creating posters based on their word problems. There are examples of different types of methods, online games, a rubric, and a power point to summarize this two-day lesson.

Type: Lesson Plan

How Many Rubik's Cubes Can You Pack?:

This two-day lesson uses a hands-on problem-solving approach to find the volume of a right rectangular prism with positive rational number edge lengths. Students first design boxes and fill with Rubik's Cubes. They create a formula from the patterns they find. Using cubes with fractional edges requires students to apply fractional units to their formulas.

Type: Lesson Plan

Students will be able to match a 3-D shape with its net, then using the net, they will find the surface area of the shape. They will then be able to apply this knowledge to solve real world application problems, finishing up with a design contest.

Type: Lesson Plan

Fill to Believe!:

In this lesson, students work cooperatively to find the volume of a right rectangular prisms, using whole and fraction units of measurement, using the volume formula, and using manipulatives to count the number of units necessary to fill the prisms, and compare it with the formula results.

Type: Lesson Plan

Can you say that another way?:

Students will model how to express an addition problem using the distributive property.

Type: Lesson Plan

Surface Area of Prisms and Pyramids:

In this lesson students will find the surface area of three-dimensional figures. Students will use nets to calculate the surface area of right rectangular prisms and right rectangular pyramids.

Type: Lesson Plan

The Price is Right:

In this activity the students will apply their knowledge of mathematical calculations to solve a real-world problem. They will analyze a collection of shipping boxes to determine which box will ship the most for the \$100 allowed.

Type: Lesson Plan

How Much Paint Will It Take?:

This is a guided inquiry lesson to help students gain greater understanding of the relationship between 2-dimensional and 3-dimensional shapes. Students create right rectangular prisms and problem-solve how to find the flat 2-dimensional surface area. Students are asked to figure out how many party favors (prisms) can be painted with a quart of glow-in-the-dark paint.

Type: Lesson Plan

Box It Up, Wrap It Up (Surface Area of Rectangular Prisms):

In this introductory lesson to surface area, students will make connections between area of two-dimensional figures and calculating the surface area of rectangular prisms using nets, within the context of wrapping birthday presents! Math is Fun :)

Type: Lesson Plan

You Can Never Have Too Many Shoes!:

This lesson teaches Least Common Multiples.

Type: Lesson Plan

How Many Small Boxes?:

In this lesson students will extend their knowledge of volume from using whole numbers to using fractional units. Students will work with adding, multiplying, and dividing fractions to find the volume of right rectangular prisms, as well as, determining the number of fractional unit cubes in a rectangular prism.

Type: Lesson Plan

Finding the Greatest Crush Factor:

This lesson uses a real-life approach to exploring the use of Greatest Common Factors (GCF). The students will utilize math practice standards as they analyze math solutions and explain their own solutions.

Type: Lesson Plan

Factoring out the Greatest:

This lesson teaches students how to find the GCF and LCM by factoring. This is a different method than is normally seen in textbooks. This method easily leads to solving GCF word problems and using the distributive property to express a sum of two whole numbers.

Type: Lesson Plan

Dividing Fractions:

In this lesson students will explore the different methods available for dividing fractions through a student-based investigation. The teacher will facilitate the discussion, but the students will discover the different methods on their own or with a partner as they work through the different steps.

Type: Lesson Plan

Wrapping Up Geometry (Lesson 2 of 2):

This lesson is 2 of 2 and is primarily formative in nature, but includes a summative assessment for students to take during the following class period.

During the lesson, students will be reviewing for their assessment on the surface area formula for a right rectangular prism.

Type: Lesson Plan

Dividing Fractions (Part 1) - Tackling Word Problems:

This lesson allows the students to explore the foundation for dividing fractions as well as correctly solving word problems involving division of fractions. It includes the use of the Philosophical Chairs activity and numerical solutions. Group activities are included to foster cooperative learning.

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

Dividing by Fractions Discovery:

This lesson allows students to derive an algorithm for dividing fractions using visual fraction models and equations to represent the problem.

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

Formula Detective: Finding the Surface Area of a 3D Figure:

This lesson allows students to derive the formulas for 3D figures by having them build models for nets.

Type: Lesson Plan

## Perspectives Video: Experts

Fluency vs. Automaticity:

How are fluency and automaticity defined? Dr. Lawrence Gray explains fluency and automaticity in the B.E.S.T. mathematics benchmarks in this Expert Perspectives video.

Type: 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

What is Fluency?:

What is fluency? What are the ingredients required to become procedurally fluent in mathematics? Dr. Lawrence Gray explores what it means for students to be fluent in mathematics in this Expert Perspectives video.

Type: Perspectives Video: Expert

Why Isn't Getting the "Right" Answer Good Enough?:

Why is it important to look beyond whether a student gets the right answer? Dr. Lawrence Gray explores the importance of understanding why we perform certain steps or what those steps mean, and the impact this understanding can have on our ability to solve more complex problems and address them in the context of real life in this Expert Perspectives video.

Type: Perspectives Video: Expert

## 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 Ideas

Space Math - Big Moons and Small Planets:

Students use a scale representation of the top 26 small planets and large moons in the solar system to compare their relative sizes to Earth. Students will use simple fractions to solve real world problems.

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

Dividing Mixed Numbers:

In this tutorial, you will see how mixed numbers can be divided.

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

## Video/Audio/Animation

Reciprocals and Divisions of Fractions:

When working with fractions, divisions can be converted to multiplication by the divisor's reciprocal. This chapter explains why.

Type: Video/Audio/Animation

## MFAS Formative Assessments

Fraction Division:

Students are asked to complete two fraction division problems – one with fractions and one with mixed numbers.

Juicing Fractions:

Students are asked to write and evaluate a numerical expression involving division of fractions and mixed numbers to model and solve a word problem.

Models of Fraction Division:

Students are asked to explain the relationship between a fraction division word problem and either a visual model or an equation.

## Student Resources

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

## 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

## Tutorials

Dividing Mixed Numbers:

In this tutorial, you will see how mixed numbers can be divided.

Type: Tutorial

Multiplying Fractions:

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

Type: Tutorial

## Video/Audio/Animation

Reciprocals and Divisions of Fractions:

When working with fractions, divisions can be converted to multiplication by the divisor's reciprocal. This chapter explains why.

Type: Video/Audio/Animation

## Parent Resources

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

## Tutorials

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