Strand: THE NUMBER SYSTEM (6.NS)

Apply and extend previous understandings of multiplication and division of whole numbers to divide fractions by fractions (Standard 6.NS.1). Compute (add, subtract, multiply and divide) fluently with multi-digit numbers and decimals and find common factors and multiples (Standards 6.NS.2–4). Apply and extend previous understandings of numbers to the system of rational numbers (Standards 6.NS.5–8).

Standard 6.NS.1

Interpret and compute quotients of fractions.

This task requires students to complete a series of steps in order to find a solution, and because they need to analyze constraints, it addresses some aspects of mathematical modeling. Students must first add fractions with familiar but unlike denominators, which is a skill developed in the 5th grade. Then students need to divide fractions by fractions.
• Cup of Rice
Students are given a word problem "One serving of rice is 23 of a cup. I ate 1 cup of rice. How many servings of rice did I eat?" They must choose between 2 possible solutions and explain their reasoning.
• Dan's Division Strategy
The purpose of this task is to help students explore the meaning of fraction division and to connect it to what they know about whole-number division.
• Drinking Juice, Variation 2
This task builds on a fifth grade fraction multiplication task, 5.NF Drinking Juice. This task uses the identical context, but asks the corresponding Number of Groups Unknown division problem. See Drinking Juice, Variation 3 for the Group Size Unknown version.
• Drinking Juice, Variation 3
This task builds on a fifth grade fraction multiplication task, 5.NF Drinking Juice. This task uses the identical context, but asks the corresponding Group Size Unknown division problem. See Drinking Juice, Variation 2 for the Number of Groups Unknown version.
• Grade 6 Math Module 2: Arithmetic Operations Including Division of Fractions (EngageNY)
In Module 2, students complete their understanding of the four operations as they study division of whole numbers, division by a fraction and operations on multi-digit decimals. This expanded understanding serves to complete their study of the four operations with positive rational numbers, thereby preparing students for understanding, locating, and ordering negative rational numbers (Module 3) and algebraic expressions (Module 4).
• Grade 6 Unit 1: Number System Fluency (Georgia Standards)
In this unit students will find the greatest common factor of two whole numbers less than or equal to 100. Find the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. Interpret and compute quotients of fractions. Solve word problems involving division of fractions by fractions using visual fraction models and equations to represent the problem. Fluently divide multi-digit numbers using the standard algorithm. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
• How many _______ are in. . . ?
This task provides a list of problems. They require that the students model each problem with some type of fractions manipulatives or drawings. The problems are meant to be a progression which require more sophisticated understandings of the meaning of fractions as students progress through them. If the task is used to help students see the connections to the invert-and-multiply rule for fraction division (as described in the solution) then they should already be familiar with and comfortable solving Number of Groups Unknown (a.k.a. "How many groups?") division problems with visual models.
• How Many Batches/What Fraction of a Batch?
The purpose of this task is to help students extend their understanding of multiplication and division of whole numbers to multiplication and division of fractions. The task does not ask students to find the product or quotient since the task is more about learning how to represent the situation, but teachers might choose to ask students to find or estimate the answers, if desired.
• How Many Containers in One Cup / Cups in One Container?
These two fraction division tasks use the same context and ask "How much in one group?" but require students to divide the fractions in the opposite order.
• How Much in One Batch?
The purpose of this task is to help students extend their understanding of multiplication and division of whole numbers to multiplication and division of fractions.
• Keep, Change, Flip
Students are taught the "Keep, Change, Flip" rule for dividing fractions by viewing this clever Flocabulary rap song. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Making Hot Cocoa, Variation 1
This is the first of two fraction division tasks that use similar contexts to highlight the difference between the "Number of Groups Unknown" a.k.a. "How many groups?" when the quotient is a fraction (or mixed number) greater than 1 (Variation 1) and when the quotient is a fraction that is less than 1 (Variation 2).
• Making Hot Cocoa, Variation 2
This is the second of two fraction division tasks that use similar contexts to highlight the difference between the "Number of Groups Unknown" a.k.a. "How many groups?" when the quotient is a fraction (or mixed number) greater than 1 (Variation 1) and when the quotient is a fraction that is less than 1 (Variation 2).
• Modeling Fraction and Mixed Number Division Using Arrays
Students will learn how to solve word problems that involve dividing fractions and mixed numbers by using a visual model. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Modeling Fraction Division Using Comparison, Group Number Unknown
In this lesson students will learn how to solve a word problem involving the division of fractions by viewing an animation about a hedgehog's hibernation. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Modeling Fraction Division, Equal Groups, Group Size Unknown
The skill of dividing two fractions by groups of unknown size is the focus of this video. Students will learn how to solve a word problem using this process. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Modeling Fraction Division, Equal Groups, Number of Groups Unknown
This animated video shows students a model they can use to solve word problems involving the division of fractions. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Reciprocity
The purpose of this task is to help students understand why dividing by a fraction gives the same result as multiplying by its reciprocal. This is accomplished by writing the division equation along with related multiplication equations and diagrams showing the situation for several different contexts.
• Running to School, Variation 2
This task builds on a fifth grade fraction multiplication task, "5.NF Running to School, Variation 1." This task uses the identical context, but asks the corresponding "Number of Groups Unknown" division problem. See "6.NS Running to School, Variation 3" for the "Group Size Unknown" version.
• Running to School, Variation 3
This task builds on a fifth grade fraction multiplication task, "5.NF Running to School, Variation 1." "6.NS Running to School, Variation 3" uses the identical context, but asks the corresponding "Group Size Unknown" division problem. See "6.NS Running to School, Variation 2" for the "Number of Groups Unknown" version.
• Standing in Line
The purpose of this task is for students to solve a problem in context that can be solved in different ways, but in particular by dividing a whole number by a unit fraction.
• The Number System (6.NS) - 6th Grade Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - The Number System.
• Traffic Jam
This task posits this word problem to students: "You are stuck in a big traffic jam on the freeway and you are wondering how long it will take to get to the next exit, which is 1 1/2 miles away. You are timing your progress and find that you can travel 2/3 of a mile in one hour. If you continue to make progress at this rate, how long will it be until you reach the exit? Solve the problem with a diagram and explain your answer."
• Video Game Credits
"It requires 1/4 of a credit to play a video game for one minute." Given this information, students are asked to answer questions about how long a student can play given a specific number of credits.

http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer Throndsen .

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.