Angela Waltrup Julie McNamara · 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

$Page 1: Angela Waltrup Julie McNamara · formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. OR • Understand$
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)

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 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

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 observations10

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-fourthsUse the available manipulatives to represent

16

34

BREAK

17

Use the available manipulatives to represent 34

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 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?)

20

34

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 NumbersComparing and Ordering

22

Partitioning the Number Line

23

0 1 2

Partitioning the Number Line

24

0 1 2

Label ‘s on the Line

25

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?

12

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?


29

Ordering Fractions


30

Reasoning about 12

31

Locating Fractions on theNumber Line

32

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?

33

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

35

How would you answer this question?

36

Share your strategies with others at your table

How would your students answer this question?

37

38

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?

39

Addition of Fractions

40

Which problems would be solved by adding ?

41

12

+ 13


42

12

+ 13


12

+ 13


44

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

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

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+



Watch Belen at

https://mathsolutions.wistia.com/medias/

m3oc5e92qi

: Malia’s Strategy3

5

4

5+



Watch Malia at

https://mathsolutions.wistia.com/medias/plerkbj369

Student work

Student work

Student work

Reflecting on Adding and Subtracting Fractions



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

Using an Area Model: An Example with Whole Numbers

4

3

61

3 x 4 = 12

Using an Area Model: Multiplying a Whole Number by a Fraction

4

1

62


412

12

63


412

12

64

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

73

Have you ever heard….

Yours is not to reason why,

Just invert and multiply!

74

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?

75

6 feet

6 ÷ 2 = 3


76

6 feet

Inverting and Multiplying

77

6 feet


78

1 mile

Types of Division Situations:4 ÷

79

13

Reflecting on Multiplying and Dividing Fractions



80

30% Discount code for Beyond books

mathsolutions.com

NCTMCHI30

Expires 7/31/15

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.

82

83

Thank you!!

Documents

Angela Waltrup Julie McNamara · 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