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Fractions as NumbersNCTM Interactive Institute, 2016
Angela WaltrupJulie McNamara
Welcome
Decorate your name tent with the following:
• Name/Position
• Where you are from
• Represent “personal” fraction numbers. These fractions have meaning and connections to your life. – (Expressed as a fraction)
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Disclaimer
The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.
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Fractions as NumbersFractions on a Number Line
During this session we will:
• Examine and define fractions as numbers emphasizing magnitude and equivalence
• Enhance our ability to generate a variety of representations and use reasoning strategies to compare and order fractions.
• Explain key mathematical ideas such as equivalence
• Solve problems using a variety of representations
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Doing What Works: Learning Together About Building on Informal Understandings of Fractions
Work Through the Problem Set
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Dr. Thomas Carpenter
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Doing What Works: Learning Together About Building on Informal Understandings of
Fractions
• Table Discussion
– Have you used similar problems with your students?
– What have students found difficult, or what do assume will be difficult for students?
– What do you need more of to support students with their initial understandings of fraction concepts?
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Examining mathematics, student thinking, and teaching practices
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The Brown Rectangle Problem
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Brown Rectangle Problem Video
• What mathematical issues do you see arising?
• How do students think about the problem?
• What do you notice the teacher doing or saying?
• As you watch the video:
– Attend to talk, student thinking, and teacher’s moves and comments
– Note detail and evidence for your observations10
Brown Rectangle Video
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Grade 4 student - Hannah
Grade 4 student - Jose
Why Fractions Matter
• “Crucial for students to learn but challenging for teachers to teach” (Barnett-Clark, Fisher, Marks, and Ross 2010)
• Understanding fractions is a “foundational skill essential to success with algebra” (U.S. Department of Education 2008)
• Large-scale assessment data confirms that students often do not become proficient with fraction concepts and procedures
• Shift in demands on Grade 3-5 teachers and students
• Algebra (and mathematics in general) is a civil right (Moses and Cobb 2001)
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The Fraction Kit
• The fraction kit introduces students to fractions as parts of a whole.
• With your fraction kit:
– Explore at your table.
• Fraction Kit Activities
– Cover Up
– Uncover
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Representations of three-fourthsUse the available manipulatives to represent
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34
BREAK
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Use the available manipulatives to represent 34
Defining a fraction and using a definition of a fraction
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CCSS definition of a fraction
• Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
OR
• Understand a fraction 1/d as the quantity formed by 1 part when a whole is partitioned into d equal parts; understand a fraction n/d as the quantity formed by n parts of size 1/d.
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Using CCSS definition of a fraction
• With a partner, take turns using this definition to explain your representations of
– When explaining: Use definition to talk as you reason and make sense of your representation.
– When listening: Attend to how your partner is making use of the definition to reason and make sense of ¾.
• If time permits, try:
– A different fraction (a fraction greater than 1?)
– A different representation (a set model? a number line?)
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Part To Whole Whole To Part
• Use the definition to reason and make sense of your representation.
• How might you connect the language of numerator and denominator to the definition?
• Can the working definition be used for different representations of fractions? (area model, fraction of a set, number line)?
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Fractions as NumbersComparing and Ordering
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Partitioning the Number Line
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0 1 2
Partitioning the Number Line
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0 1 2
Label ‘s on the Line
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1
2
0
2
1
2
2
2
0 1 2
3
2
4
2
What do you notice about…..
• Fractions that are equal to ?
• Fractions that are close to 0?
• Unit fractions that have numerators and
denominators that are close together?
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Ordering Fractions
• What strategies can you use to compare these fractions?
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Ordering Fractions
• What strategy can you use to compare these fractions?
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Ordering Fractions
• What do these fractions have in common?
• What strategy can you use to compare these fractions?
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Ordering Fractions
• What strategy can you use to compare these fractions?
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Reasoning about 12
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Locating Fractions on theNumber Line
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Estimate the location of each number on the number line:
Reflection
• What are the key take-aways, points for application to your school/classroom?
• What are some ideas for follow up/follow through?
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Fractions as NumbersNCTM Interactive Institute, 2016
Julie McNamaraAngie Waltrup
Operations with Fractions
During this session we will:
• Identify challenges students have with fraction computation
• Identify characteristics of problems that can be solved by addition, subtraction, multiplication, and division of fractions
• Identify contexts that can help students make sense of operations with fractions
• Solve problems using a variety of representations
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How would you answer this question?
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Share your strategies with others at your table
How would your students answer this question?
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Reasoning about 1112
+15
mathreasoninginventory.com
Watch video of students reasoning about this problem at
https://mathreasoninginventory.com/Home/VideoLibrary
What understandings does Alberto’s response indicate?
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Addition of Fractions
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Which problems would be solved by adding ?
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12
+ 13
Which problems would be solved by adding ?
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12
+ 13
Which problems would be solved by adding ?
12
+ 13
Which problems would be solved by adding ?
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12
+ 13
Representing Fraction Addition
• converting the fractions to fractions with common denominators
• drawing a diagram (tape, area)
• using a number line
• converting the fractions to decimals
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Subtraction of Fractions
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Subtraction of Fractions
• converting the fractions to fractions with common denominators
• drawing a diagram
• using a number line
• converting the fractions to decimals 47
Understanding Fraction Subtraction
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Whole Number Addition and Subtraction Strategies
• Decomposing/recomposing• Associative property• Commutative property• Renaming (equivalence)
Get to the Whole!
Decomposing and recomposing fractions to “get to the whole” when adding and subtracting.
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4
3
4+
: Will’s Strategy
Watch Will at
https://mathsolutions.wistia.com/medias/ct9q
xko5n3
3
4
3
4+
Beyond Invert and Multiply: Making Sense of Fraction Computation.
Math Solutions, 2015.
: Belen’s Strategy3
4
3
4+
Beyond Invert and Multiply: Making Sense of Fraction Computation.
Math Solutions, 2015.
Watch Belen at
https://mathsolutions.wistia.com/medias/
m3oc5e92qi
: Malia’s Strategy3
5
4
5+
Beyond Invert and Multiply: Making Sense of Fraction Computation.
Math Solutions, 2015.
Watch Malia at
https://mathsolutions.wistia.com/medias/plerkbj369
Student work
Student work
Student work
Reflecting on Adding and Subtracting Fractions
• What are the key take-aways, points for application to your school/classroom?
• What are some ideas for follow up/follow through?
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Multiplication and Division of Fractions
What challenges do students typically have with multiplying and dividing fractions?
58
Brendan, Grade 4
Multiplication of Fractions
Write three different word problems that illustrate the following:
1. A whole number times a fraction. (Front tables)
2. A fraction times a whole number. (Middle tables)
3. A fraction times a fraction. (Back tables)
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Using an Area Model: An Example with Whole Numbers
4
3
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3 x 4 = 12
Using an Area Model: Multiplying a Whole Number by a Fraction
4
1
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Using an Area Model: Multiplying a Whole Number by a Fraction
412
12
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Using an Area Model: Multiplying a Whole Number by a Fraction
412
12
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Farmer Liz’s Field
Fruit Vegetables
Farmer Liz
Fruit Vegetables
Farmer Liz
Farmer Liz wants to further partition the two halves of her field such that --
• half of the fruit section will be planted with fruit trees and half with fruit bushes
• half of the vegetable section will be planted with vegetables that grow above ground and half with vegetables that grow below ground
Farmer Liz
Fruit trees
Fruit bushes
Above
ground
vegetables
Below
ground
vegetables
Farmer Liz
• What fraction of Farmer Liz’s field is planted with fruit trees?
• What fraction of Farmer Liz’s field is planted with fruit bushes?
• What fraction of Farmer Liz’s field is planted with above ground vegetables?
• What fraction of Farmer Liz’s field is planted with below ground vegetables?
Multiplying fractions
• How does the “field” show that
=1
2
1
2x 1
4?
Farmer Bruce
Farmer Bruce is going to plant half of his field with flowers and leave the other half unplanted to use as a pasture.
Farmer Bruce
One more problem
• Pick one problem from the ones your group wrote initially
• Solve with patty paper
• Compare
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Have you ever heard….
Yours is not to reason why,
Just invert and multiply!
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Division with Fractions: Building on Division with Whole Numbers
You have 6 feet of ribbon and want to cut it into pieces that are 2 feet long. How many pieces can you make?
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6 feet
6 ÷ 2 = 3
Division with Fractions: Building on Division with Whole Numbers
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6 feet
Inverting and Multiplying
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6 feet
Division with Fractions: Building on Division with Whole Numbers
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1 mile
Types of Division Situations:4 ÷
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13
Reflecting on Multiplying and Dividing Fractions
• What are the key take-aways, points for application to your school/classroom?
• What are some ideas for follow up/follow through?
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mathsolutions.com/presentations
81
Disclaimer
The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.
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Thank you!!