Multiplication of Fractions
December MTL 09
Multiplying Fractions:
The Challenge of Computation vs. Conceptualization
December MTL Meeting
December 8, 10, & 14, 2009
Marshall Complex
Core Content Development Team
DeAnn Huinker
Connie Laughlin
Kevin McLeod
Mary Mooney
Beth Schefelker
Melissa Hedges
Assisting MTLs
Frelesha LeFlore
Cecile Labecki
Session Goal
· To make sense of the operation of multiplication regardless of the numbers.
Learning Intention
· To use real-life problem-solving situations to surface the meaning of multiplication of fractions.
Success Criteria
l At the end of today's session you will be able to analyze a word problem and understand that it involves multiplication
Exploring Multiplication Situations
With Fractions
Consider ⅔ × ¾.
Describe a real-world situation that can be modeled by this equation.
· What could the ⅔ represent?
· What could the ¾ represent?
Turn and Share
As you crafted a real-world situation, what struggles emerged?
Back to the basics...
Consider for a moment 2×3.
· What could the 2 represent?
· What could the 3 represent?
What insights surfaced?
In what ways are 2×3 and ⅔ × ¾ similar?
In what ways are they different?
Does the meaning of multiplication change when moving from whole numbers to fractions?
“Algorithms for multiplication of common fractions are easy for teachers to teach and students to use, but their meanings are elusive.”
-- Zhijun Wu
Rational Number
Interpretations
□ Part-Whole Comparisons
□ Quotients
□ Operators
□ Measures
□ Rates and Ratios
Making Spaghetti
Julie is making the family dinner. She buys 4 packages of meat for the spaghetti. Each package weighs 5/8 lb. How many pounds of meat does she buy?
Turn and talk –
· What does the 4 represent?
· What does the 5/8 represent?
Individually –
1. Work through this problem.
2. Record your thinking on note cards.
3. Represent your thinking using numbers, pictures and words.
4. Place face down in the middle of your table when done.
Making Spaghetti
Debriefing Strategies
1. Each person picks a card to study.
2. After 30 seconds pass the card to the right.
3. Study the strategy on each card.
4. After all cards have been passed, share comments, questions or ah-ha’s with the table group.
Table Discussion
l How do these strategies demonstrate an understanding of multiplication?
Taking a Look at Student Thinking
Julie is making the family dinner. She buys 4 packages
of meat for the spaghetti. Each package weighs 5/8 lb. How many pounds of meat does she buy?
In what ways do the students show understanding of the situation as multiplication?
In what ways do the students use equations that demonstrate that they understand this was a multiplication situation?
In what ways do the students use labels noting the number of groups and size of each group or portion?
“These types of situations can be modeled by the repeated-addition interpretation. The link between multiplication and addition is clearly seen here. The repeated-addition model offers a satisfying interpretation in this case.”
-- Zhijun Wu
How does your thinking change when you consider this next problem...
Taking a Run
I wanted to run 4 miles. I ran 5/8 of the distance before I stopped for water. How many miles did I run before I had to stop for water?
Put your pencils down.
Turn and talk –
· What does the 4 represent?
· What does the 5/8 represent?
Individually –
1. Work through this problem.
2. Represent your thinking using numbers, pictures and words.
3. Record your thinking on note a card.
4. Place face down in the middle of your table when done.
Taking a Run
Strategy Debrief
1. Each person picks a card to study.
2. After 30 seconds pass the card to the right.
3. Study the strategy on each card.
4. After all cards have been passed, share comments, questions or ah-ha’s with the table group.
Table Discussion
How do these strategies demonstrate an understanding of multiplication?
How did repeated addition play a role in solving this problem, if at all?
Thinking About Parts and Wholes
Combining parts:
Problem 1
4 = pkgs of meat
5/8 = weight per package (quantity)
“4 parts of 5/8 lbs each”
Finding part of a group:
Problem 2
4 = miles OR the whole run
5/8 = part of the 4 miles (5/8 is now the operator – we do not need a complete whole but we need a part of that whole.)
“5/8 parts of 4 miles”
How does the procedure connect?
In many classrooms we might see this:
5 × 4 = 20 = 20 ÷ 8 = 2 1
8 8 2
(Talk through each expression between the equal sign.)
In what way does this algorithm connect to the context of this story?
Does identifying units strengthen the connection?
5 × 4 miles= 20 miles = (20 ÷ 8) miles = 2 1 miles
8 8 2
Taking a Run – Looking at Student Work
In what ways do the students show understanding of the situation as multiplication?
What does multiplication of fractions encompass?
Multiplication of fractions involves:
· Combining equal parts
· Finding a part of a whole or part of a group
ie: Mariah's Kittens
l Doing both – combining equal parts and finding part of a whole.
Problem Sort
Read through each problem.
As a table, decide which problem type is represented in each context.
Big Ideas – MKT
The meaning of multiplication must be expanded to include rational numbers.
What experiences do students need to extend the meaning of multiplication from whole numbers to fractions?
□ Experience with real-life problem solving situations.
□ Pictorial representations support students as they reason.
□ Opportunities to explore the meaning of multiplication through a variety of problem formats involving fractions.
“Multiplying fractions challenges students to examine many of the ideas they have developed about multiplication from their work with whole numbers. The challenge is not one of computation, but rather is one of conceptualization.”
-- Zhijun Wu
Problems for Card Sort
Mrs. Smith has 120 books in her fourth grade classroom. 4/5 of the books are fiction. How many books are fiction?
All notebooks at the local store are discounted by ¼ A notebook originally cost $0.96. How much do you save on one notebook if you buy it today?
Julie bought 4/5 of a yard of fabric for her class project. Later she found that she needed only ¾ of the material. How much material did Julie use for her project?
At the supermarket potatoes are bagged in ¾ pound bags. Mom bought 3 bags of potatoes. How many pounds of potatoes did mom buy?
Red cabbage cost $0.39 a pound. Julie bought 3 1/3 pounds of red cabbage to prepare her dish. How much did she pay for the red cabbage?
I put a container holding a half gallon of ice cream into the freezer. Two days later the ice cream container is 2/3 full. How much ice cream is in the container?
Melissa is planning on making several batches of cookies. She needs 2/3 cup of sugar for every batch she makes. She plans on making 2 ½ batches. How much sugar will she need to make these batches.
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The Milwaukee Mathematics Partnership (MMP), an initiative of the Milwaukee Partnership Academy (MPA), is supported with funding from the National Science Foundation.