Algebra Forum IVSan Jose, CAMay 22, 2012

Compiled and Presented by

April Cherrington Joan EasterdayRegion 4 Region 1

Susie W. Hakansson, Ph.D. California Mathematics Project

Teaching and Learning Fractions with Conceptual Understanding

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Fractions from a number line approach represents a shift in thinking about fractions, moving beyond part-whole representations to thinking of a fraction as a point on the number line. Included in this session will be the following: rationale, comparing and ordering, and justification.

Description

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(25 minutes) Introduction

(105 minutes) Breakout session (includes Q&A)

(25 minutes) Summary and Reflection

Outline for Today

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CaCCSS-M Task Force Conceptual

understanding Order problems

Cognitive level

Language issues

Why number line?

Fraction progressions

Standards for Mathematical Practice

Challenges students face

Overview of break out session

Introduction

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Greisy Winicki-Landman, Chair Nadine Bezuk April Cherrington Pat Duckhorn Joan Easterday Doreen Heath Lance

Fractions Task Force

Pam Hutchison Natalie Mejia Gregorio Ponce Debbie Stetson Kathlan Latimer

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“… almost all teachers are placing a lower priority on student understanding in recent years, ….”

“… the sort of high quality PD that an really affect teachers in their ability to produce students who understand is very, very difficult to do, and very few people have much clue about how to do it.”

Scott Farrand

Demands of CaCCSS-M

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8/15 > 1/2 (?)

7/22 > 1/3 (?)

6/11 > 7/15 (?)

7/8 > 8/9 (?)

Fraction Sense: Comparing

Solve these problems mentally without using algorithms. Justify your thinking.

Cognitive Demand Spectrum

Memorization ProceduresWithoutConnectionsto understanding,meaning, or concepts

ProceduresWithConnectionsto understanding,meaning, or concepts

DoingMathematics

Tasks that require memorized procedures in routine ways

Tasks that require engagement with concepts,

and stimulate students to make connections to

meaning, representation, and other mathematical

ideas

The soldier decided to desert his dessert in the desert.

Upon seeing the tear in the painting, I shed a tear.

After a number of injections, my jaw got number.

A minute is a minute part of a day.

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Why Is English So Hard?

There is no egg in eggplant and no ham in hamburger.

How can a slim chance and a fat chance be the same, while a wise man and a wise guy are opposites?

Did you say thirty or thirteen? Did you say two hundred or two hundredths? Did you say fifty or sixty? 10

Why Is English So Hard?

Math Word Meaning 1 (Math) Meaning 2Solution The answer Two or more substances

mixed togetherExercise Math problems to solve Physical movement to stay fit

Product The answer when you multiply two or more numbers

Something made by humans or machines

Expression Math statement with numbers and/or variables

To show emotion

Dual Meaning Words

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9th and 10th graders’ responses Tom had 5 apples. He ate 2 of them. How many

apples were left? A. 10 B. 7 C. 5 D. 3 (100%)

Guinevere had 5 pomegranates. She ate 2 of them. How many pomegranates were left? A. 10 (22%) B. 7 (24%) C. 5 (23%) D. 3 (31%)

The Guinevere Effect

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Access prior knowledge Frontload language Build on background knowledge Extend language Be aware of multiple meanings of words Have students Think, Ink, Pair, Share (TIPS)

Key Strategies for English Learners

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Teachers learn to amplify and enrich--rather than simplify--the language of the classroom, giving students more opportunities to learn the concepts involved.

Aída Walqui, Teacher Quality Initiative

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“Hung-Hsi Wu attempts to bring coherence to the teaching and learning of fractions by beginning with the definition of a fraction as the length on the number line (1998).  This approach eliminates the ‘conceptual discontinuity’ (2002) encountered moving from work with whole numbers to fractions; it also brings coherence to the various meanings of fractions and allows for both conceptual work to operations on fractions (2008). Wu asserted that ‘The number line is to fractions what one’s fingers are to whole numbers ...”

Why Number Line?

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Basic Assumptions about the Number Line and Its Use

Using the number line, there are basically two types of tasks:

Given a point on the number line, assign a number to it (its coordinate)

Given a number, place it as a point on the number line

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WHY THE Number Line?

It serves as a visual/physical model to represent the counting numbers and constitutes an effective tool to develop estimation techniques, as well as a helping instrument when solving word problems.

It constitutes a unifying and coherent representation for the different sets of numbers (N, Z, Q, R), which the other models cannot do.

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WHY THE Number Line?

It is an appropriate model to make sense of each set of numbers as an expansion of other and to build the operations in a coherent mathematical way.

It enables to present the fractions as numbers and to explore the notion of equivalent fractions in a meaningful way.

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WHY THE Number Line?

The number line, in some way, looks like a ruler, fostering the use of the metric system and the decimal numbers.

It fosters the discovery of the density property of rational numbers.

It provides an opportunity to consider numbers that are not fractions.

Grades 3, 4 and 5

Number and Operations - Fractions

Common Core Standards Mathematics

Grade 3• Develop understanding of

fractions as numbers

Grade 4• Extend understanding of

fraction equivalence and ordering

• Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

• Understand decimal notation for fractions, and compare decimal fractions.

Grade 5• Use equivalent fractions as a

strategy to add and subtract fractions.

• Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Grades 6 and 7

Number and Operations - Fractions

Common Core Standards Mathematics

Grade 6• Apply and extend previous

understandings of multiplication and division to divide fractions by fractions.

Grade 7• Apply and extend previous

understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

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We will focus two of the Standards for Mathematical Practice: Reason abstractly and quantitatively

Construct viable arguments and critique the reasoning of others

CaCCSS-M: Mathematical Practice

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Reason abstractly and quantitatively

DO STUDENTS: Make sense of quantities and their relationships in problem

situations? Decontextualize a problem? Contextualize a problem? Create a coherent representation of the problem, consider the

units involved, and attend to the meaning of quantities?

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Construct viable arguments and critique the reasoning of others

DO STUDENTS: Justify their conclusions, communicate them to others, and

respond to arguments of others? Hear or read arguments of others and decide whether they

make sense, and ask useful questions to clarify or improve the argument?

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How does this activity promote

our goals for students?

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How do I think about a task that opens up

opportunities to implement the Standards for

Mathematical Practice?

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What are some of the challenges that students

have with fractions?

Question

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Overview of Breakout Sessions

Appropriate grade level problem Twelve (12) cards Videos of students working with 12 cards Human Number Line activity Reflection

Break Out Rooms Elementary-Oak Grove Room

2nd Floor South Middle Grades-SJES Room

Here High School-Milpitas Room

2nd Floor North

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