Division with Fractions – Part 3 “Partitive Division Tools” Core Mathematics Partnership Building Mathematical Knowledge and High-Leverage Instruction for Student Success Friday July 31, 2015

48 ÷ 4 = 12 48 ÷ 4 = 12 Partitive Division Measurement Division 48 ÷ 4 = 12 48 ÷ 4 = 12 total amount number of groups (shares) size of each group (share) total amount size of each group (share) number of groups (shares) Partitive Division Measurement Division “know the number of complete equal groups that can be made from the total amount” “know the size of a group to measure out repeatedly from the total amount” Partition to make a specific number of complete equal groups or to make “part of one group” or both. How much stuff is in one whole (complete) group? or “know how much of a partial group can be made from the total amount” Find: The number of equal groups. Find: The size of one complete group (share).

We are learning to: Develop our conceptual understanding of partitive division with fractions through contexts, visual models, oral language, and symbols. Identify and describe the underlying mathematical structure of operations with fractions in partitive division situations.

CCSSM Standards

Standard 5.NF.7 5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. b. Interpret division of a whole number by a unit fraction, and compute such quotients. c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem.

Standard 6.NS.1 Cluster: Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 6.NS.1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

Ratio Tables and Double Number Lines

Another Representation: Ratio Tables A recipe for one batch of cookies uses ¾ of a cup of butter. How much butter will be needed for... Number of Batches of Cookies 1 batch 2 batches 5 batches 1/3 of a batch Number of Cups of Butter 3/4 of a cup Ask who is familiar with each of these representations. Possibly ask someone who is familiar to lead us through a solution?

Another Visual Model: Double Number Line A recipe for one batch of cookies uses ¾ of a cup of butter. How much butter will be needed for... Number of Batches of Cookies 1/3 1 2 3 4 5 Number of Cups of Butter Ask who is familiar with each of these representations. Possibly ask someone who is familiar to lead us through a solution? 3/4 ?

S t r e t c h Break

Partitive Division with Ratio Tables and Double Number Lines

of a pound of coffee is enough to fill of a bag of a pound of coffee is enough to fill of a bag. How much does a full bag of coffee weigh? Work in pairs to solve this problem using A tape diagram A ratio table A double number line Chart equation with context and structure. Total amount: 3/4 of a pound # groups: 2/3 of a bag size of one share (group): 9/8 or 1 1/8 pounds per bag or for one “complete” bag. Separate 3/4 into 2 parts = 3/8. This gives 2/3 of the bag. To find the weight of the whole complete filled bag, multiply by the number of partitions in one bag, 3 x 3/8 = 9/8 pounds.

Ratio Table Number of Bags 2/3 bag 1/3 bag 1 bag Pounds of Coffee 3/4 of a pound 3/8 of a pound 3 x 3/8 = 9/8 pounds Double Number Line 1/3 2/3 1 Number of Bags Pounds of Coffee 3/8 3/4 ?

Ratio Table Number of Bags 2/3 bag 1/3 bag 1 bag Pounds of Coffee 3/4 of a pound 3/8 of a pound 3 x 3/8 = 9/8 pounds Double Number Line 1/3 2/3 1 Number of Bags Pounds of Coffee 3/4 ?

pounds of coffee is enough to fill of a bag pounds of coffee is enough to fill of a bag. How much does one full bag of coffee weigh? Estimate: More or less than 6 ¾ pounds of coffee in one full bag? Work together to solve the problem with a: Tape Diagram Ratio Table Double Number Line Probably one problem too many. We should do something other than content in the last afternoon.

The rectangle represents one full bag of coffee. of a pound pounds pounds pounds pounds pounds of a bag of coffee

Ratio Table Number of Bags 3/5 bag 1/5 bag 1 bag Pounds of Coffee 6 ¾ 6 ¾ pounds 2 ¼ 5 x 2 ¼ = 11 ¼ lb Double Number Line 1/5 3/5 1 Number of Bags Pounds of Coffee 2 ¼ 6 ¾ ? 11 ¼

pounds of coffee fills of a bag pounds of coffee fills of a bag. How much does a full bag of coffee weigh? Now try writing a series of equations (like Keisha) to find the solution. Label the numbers with descriptive phrases for context and structure. Now study this student work. How does it compare to your sequence of equations?

Three Key Learning Targets for Your Students “Partitive Division with Fractions”

I think I’m starting to understand! Partitive Division Three Key Learning Targets 1. Meaning of the Numbers and Operation: Context & Structure— Partition to make a specific number of complete equal groups or to make “part of one group” or both. 2. Effect of the Operation: Estimate whether the answer is... more or less than 2 ¼? 3. Understanding the Solution: How much stuff is in one whole (complete) group? more or less than 1? Chart equation with context and structure. This could be a recap on Friday. I think I’m starting to understand!

Partitive Division Modified Standard Approach Explain what is happening with this sequence of equations by contextualizing the numbers and operations (SMP 2) and by considering the underlying mathematical structure of the operations (SMP 7).

Summarize a new understanding and discuss what still puzzles you. Revisiting our learning intentions: Summarize a new understanding and discuss what still puzzles you. Deepen our conceptual understanding of partitive division with fractions through contexts, visual models, oral language, and symbols. Identify and describe the underlying mathematical structure of operations with fractions in partitive division situations.

PRR: Huinker Article: Huinker “Letting Fraction Algorithms Emerge through Problem Solving Read pp. 176-179, Section on “Multiplication and Division” In what ways did the examples and discussion in this reading relate to your own reasoning and problem solving this week? Stand up, move, discuss with another person.....

Day 10 Morning Reflections (Log) Summarize some key points you are taking away about partitive division with fractions.

Disclaimer Core Mathematics Partnership Project University of Wisconsin-Milwaukee, 2013-2016   This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors. This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.