Fractions as Numbers NCTM Interactive Institute, 2016 Angela Waltrup Julie 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) 2
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. 3
Fractions as Numbers Fractions 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 4
Doing What Works: Learning Together About Building on Informal Understandings of Fractions Work Through the Problem Set 5
Dr. Thomas Carpenter 6
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? 7
Examining mathematics, student thinking, and teaching practices 8
The Brown Rectangle Problem 9
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 observations 10
Brown Rectangle Video 11
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) 14
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 15
Representations of three-fourths 3 Use the available manipulatives to represent 4 16
BREAK Use the available manipulatives to represent 3 4 17
Defining a fraction and using a definition of a fraction 18
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 . 19
Using CCSS definition of a fraction • With a partner, take turns using this definition to 3 explain your representations of 4 – 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?) 20
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)? 21
Fractions as Numbers Comparing and Ordering 22
Partitioning the Number Line 2 0 1 23
Partitioning the Number Line 2 0 1 24
1 Label ‘s on the Line 2 2 0 1 0 1 2 3 4 2 2 2 2 2 25
What do you notice about….. 1 • Fractions that are equal to ? 2 • Fractions that are close to 0? • Unit fractions that have numerators and denominators that are close together ?
Ordering Fractions • What strategies can you use to compare these fractions? 27
Ordering Fractions • What strategy can you use to compare these fractions? 28
Ordering Fractions • What do these fractions have in common? • What strategy can you use to compare these fractions? 29
Ordering Fractions • What strategy can you use to compare these fractions? 30
Reasoning about 1 2 31
Locating Fractions on the Number Line Estimate the location of each number on the number line: 32
Reflection • What are the key take-aways, points for application to your school/classroom? • What are some ideas for follow up/follow through? 33
Fractions as Numbers NCTM Interactive Institute, 2016 Julie McNamara Angie 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 35
How would you answer this question? Share your strategies with others at your table 36
How would your students answer this question? 37
11 12 + 1 Reasoning about 5 Watch video of students reasoning about this problem at https://mathreasoninginventory.com/Home/Vid eoLibrary mathreasoninginventory.com 38
What understandings does Alberto’s response indicate? 39
Addition of Fractions 40
Which problems would be solved 1 2 + 1 by adding ? 3 41
Which problems would be solved 1 2 + 1 by adding ? 3 42
Which problems would be solved 1 2 + 1 by adding ? 3
Which problems would be solved 1 2 + 1 by adding ? 3 44
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 45
Subtraction of Fractions 46
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 48
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. 3 3 + 4 4
3 3 : Will’s Strategy + 4 4 Watch Will at https://mathsolutions.wistia.com/medias/ct9q xko5n3 Beyond Invert and Multiply: Making Sense of Fraction Computation . Math Solutions, 2015.
3 3 : Belen’s Strategy + 4 4 Watch Belen at https://mathsolutions.wistia.com/medias/ m3oc5e92qi Beyond Invert and Multiply: Making Sense of Fraction Computation . Math Solutions, 2015.
4 3 : Malia’s Strategy + 5 5 Watch Malia at https://mathsolutions.wistia.com/medias/plerk bj369 Beyond Invert and Multiply: Making Sense of Fraction Computation . Math Solutions, 2015.
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? 57
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) 60
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