Gr. 6 Unit: 6.NS.A.1: Apply and Extend Previous Understandings of Multiplication and Division to Divide Fractions

Unit Overview

This unit extends students’ work with division of fractions. They will use models and equations to represent problems. Students will be given a division of fraction problem and they must create a story about the problem and solve it.

Teacher Notes:

Enduring Understandings:

At the completion of the unit on division of fractions the student will understand that:

Focus Standards (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document):

6.NS.A.1 This is a culminating standard for extending multiplication and division to fractions.

Possible Student Outcomes:

The student will be able to:

Evidence of Student Learning:

The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.

Fluency Expectations and Examples of Culminating Standards:

6.NS.1 Students interpret and compute quotients of fractions and solve word problems involving division of fractions by fractions. This completes the extension of operations to fractions.

Common Misconceptions:

Students may

Interdisciplinary Connections:

Interdisciplinary connections fall into a number of related categories:

Literacy standards within the Maryland Common Core State Curriculum

Science, Technology, Engineering, and Mathematics standards

Instructional connections to mathematics that will be established by local school systems, and will reflect their specific grade-level coursework in other content areas, such as English language arts, reading, science, social studies, world languages, physical education, and fine arts, among others.

Sample Assessment Items: The items included in this component will be aligned to the standards in the unit and will include:

Items purchased from vendors

PARCC prototype items

PARCC public release items

Maryland Public release items

Interventions/Enrichments/PD: (Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) producing the modules.)

Vocabulary/Terminology/Concepts: This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster, which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.

Part I – Focus Cluster
The Number System – Apply and Extend previous Understandings of Multiplication and Division to Divide Fractions by Fractions

measurement concept: : This is referring to division of fractions. Reviewing, 4 ÷ 3 with this concept means, “How many sets of 3 are in 4?” If you have 4 pints of ice cream to divide among 3 people, how much does each person receive?

Therefore each person gets 1 ¹/₃ pints of ice cream.

partition concept:: Modeling a quotient, using the partitive concept, requires that only the dividend be modeled. The divisor represents the number of equal parts into which the dividend is to be partitioned. Thus, the modeling materials representing the dividend are rearranged, partitioned, or sub-divided into equal groups. The quotient is the number shown in each of the equal groups. Due to the very nature of the partitive concept, the divisor of a quotient must be a whole number ≥ 2.

So 1 ¹/₂ can be divided into 6 equal groups by dividing each part in 6 equal pieces. Take 1/6 of each part and add those together.

If, after some of the materials are rearranged into equal groups, there are materials remaining, the remaining materials should be traded for equivalent smaller pieces and the partitioning continued. If a number, less than the divisor, of the smallest pieces in your model remain after the partitioning has been completed, a fraction may be expressed where the remainder (the remaining number of smallest pieces) is the numerator and the divisor is the denominator. The quotient is the number in each equal set plus this fraction

So each group of 4 contains a part equal to ⁵/₈

common denominator algorithm: : The common-denominator algorithm is repeated subtraction concept of division.

Part II – Instructional Connections outside the Focus Cluster

properties of equality: Rules which allow you to balance, manipulate, and solve equations

properties of equality: Rules which allow you to balance, manipulate, and solve equations

Reflexive property: x = x
Example: 2 = 2 or I am equal to myself

Symetric property: If x = y, then y = x
Example: Suppose fish = tuna, then tuna = fish

Transitive property: If x = y and y = z, then x = z
Example: Suppose John's height = Mary's height and Mary's height = Peter's height, then John's height = Peter's height

Addition property: If x = y, then x + z = y + z
Example: Suppose John's height = Mary's height, then John's height + 2 = Mary's height + 2
Or suppose 5 = 5, then 5 + 3 = 5 + 3

Subtraction property: If x = y, then x − z = y – z
Example: Suppose John's height = Mary's height, then John's height − 5 = Mary's height − 5
Or suppose 8 = 8, then 8 − 3 = 8 − 3

Multiplication property: If x = y, then x • z = y • z
Example: Suppose Jetser's weight = Darline's weight, then Jetser's weight • 4 = Darline's weight • 4
Or suppose 10 = 10, then 10 × 10 = 10 × 10

Division property: If x = y, then x ÷ z = y ÷ z
Example: Suppose Jetser's weight = Darline's weight, then Jetser's weight ÷ 4 = Darline's weight ÷ 4
Or suppose 20 = 20, then 20 ÷ 10 = 20 ÷ 10

Substitution property: If x = y, then y can be substituted for x in any expression
Example: x = 2 and x + 5 = 7, then 2 can be substituted in x + 5 = 7 to obtain 2 + 5 = 7

Additional Resources:

http://ime.math.arizona.edu/progressions/#products