Probably the best training that I have ever participated in. The ...Probably the best training that...

Probably the best training that I have ever participated in. The content is excel-

lent and life changing. The concept of PITFALLS is huge. – First grade teacher

This is a great way to help all my students, including the English learners. The les-

sons are short and simple to execute, and therefore it can go hand-in-hand with

my textbook. – Fourth grade teacher

Math Pathways & Pitfalls Intervention Curriculum and Professional Development

WestEd.org/mpp»

Who Is the Audience for Math Pathways & Pitfalls?

K–8 classroom, after-school, and summer school teachers who want practical, evidence-based strategies and tools that boost learning of key mathematics standards and develop academic language. Excellent for use with diverse learners, including English learners.

What Is Math Pathways & Pitfalls?

Math Pathways & Pitfalls (MPP) is an intervention curriculum that fosters correct ways for students to rep-resent and reason about mathematical concepts and ex-

plicitly calls their attention to common misconceptions or pitfalls. MPP instructional materials are designed to jump-start effective discussion-based instruction with di-verse learners, including English learners. At each grade level, MPP addresses 10 to 12 of the toughest concepts in mathematics topics such as place value, subtraction, fractions, and percent. These lessons support increasing-ly sophisticated reasoning and academic language use —two competencies that are fundamental to mathemat-ics learning and college success.

Math Pathways & Pitfalls provides resources to improve instruction of key mathematical standards, regardless of the core instructional materials used. The materi-als and strategies easily adapt to a variety of teaching and learning goals and, for example, may be used as an intervention to the core curriculum or as part of an after-school or summer school program. Math Pathways & Pitfalls has the following components:

Student Lessons and Teaching Guides (K-1, 2-3, 4-5, 6-7) »

Discussion Builders Posters (K-1, 2-3, 4-8) »

DVDs for students and teacher professional development »

CDs with black line masters »

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10

20

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40

50

60

70

80

Grade 6Grade 4Grade 2

Significant Positive Impact for Students in All Grades

MPPComparison MPP

ContACt InformAtIon >> Alma Ramírez >> 510.302.4249 >> [email protected] >> WestEd.org/mpp

What makes Math Pathways & Pitfalls Easy to Use Successfully?

Math Pathways & Pitfalls lessons reinforce key math-ematics concepts through lessons that use an easy-to-follow format with a core lesson and two follow-up mini lessons, each accompanied by a teaching guide. Available in English or Spanish, the lessons offer:

Topics that align with state and local standards »

Instructional strategies that add value to any state- or »district-mandated curriculum

Lessons that provide students with guided and indepen- »dent practice

Mini lessons that help students avoid pitfalls on stan- »dardized assessment items

Mini lessons that help students solidify conceptual »understanding by writing explanations

Lesson-specific support for academic language devel- »opment and mathematical discourse

Field-tested teaching guides for teachers with diverse »teaching philosophies and styles

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Grade 4ELL

Grade 2Not ELL

Grade 2ELL

Grade 6Not ELL

Grade 6ELL

Grade 4Not ELL

Proven Results for Both Native English Speakers and English Learners

MPPComparison

national Science foundation Experimental Study

Findings from a large-scale experimental study demon-strated that teaching only seven Math Pathways & Pitfalls lessons in conjunction with the regular mathematics curriculum boosts the achievement of all students. The graphs above show students’ adjusted posttest mean scores on mathematics quizzes that assess the toughest standards using items similar to those found on state and national assessments such as NAEP.

Cost

MPP instructional materials will be available in spring 2009 from WestEd.

For information on cost, sample materials, and profession-al development workshops, please contact Alma Ramirez at 510.302.4249 or [email protected].

MPP

2

Math Pathways & Pitfalls™

How Does MPP Use Research Findings to Boost Learning?Turns Pitfalls into Pathways for Learning

As one of our students has pointed out, “My father says you can learn from your mistakes.” Research supports the idea that pitfalls serve as strong motiva-tors for inquiry and sense-making (Festinger, 1957; Borasi, 1994). Pitfalls may also prompt metacogni-tion as students learn to pay attention to their own thinking to spot potential pitfalls (Donovan & Brans-ford, 2005). The pitfalls highlighted in MPP lessons address some of the most common misconceptions cited in the research literature on student thinking and used as distractors on state and national assess-ments. During each lesson, students contrast correct and incorrect ways to solve a problem. They talk explicitly about why a particular pitfall occurs, how to avoid the pitfall, and how to think correctly about the mathematics in the problem.

Addresses Key Mathematics Standards in Number and Algebraic Reasoning

The mathematical content of MPP focuses on key concepts and skills that are found in the National Council of Teachers of Mathematics (NCTM) Cur-riculum Focal Points and in state and local standards. The primary grade lessons focus on building concepts related to number sense, operations, and equality. The upper grade lessons target concepts related to fractions, decimals, ratios, proportions, percents, algebraic expres-sions, and equations. Refer to page 17 in this book and to the MPP website (www.WestEd.org/mpp) for help select-ing lessons that support standards in which students need stronger development or extra reinforcement.

Provides Tools That Develop a Community of Learners

MPP draws from research about how to build com-munities in which students not only learn mathematics but also learn how to participate in the discourse prac-tices of mathematics (Lampert, 1990; Ball, 1997; Cobb,

Wood, & Yackel, 1993). MPP provides opportunities for students to participate safely while also inviting intel-lectual risk-taking (Beghetto, 2004). Each MPP lesson includes opportunities for individual, partner, and whole class participation, allowing students to take on more risky roles as they feel ready. The accompanying Discussion Builders poster contains sentence stems that embed the academic language used in mathemati-cal discourse and supports risk-taking and respect-ful discussion. The Video for Students also models students taking risks by presenting their ideas publicly and agreeing and disagreeing respectfully.

Addresses Multiple Modalities

MPP tasks mediate learning in multiple ways, using oral, print, and other visual modes of communication. For example, mathematical concepts in MPP lesson are initially developed by discussing the thinking of two fictional students. Their thinking, presented in print and supported by labeled drawings, purpose-fully embeds important conceptual ideas to prime the discussion. Having both text and drawings to refer to allows students to examine — and reexamine — the mathematical ideas. After the beginning of the MPP lesson lays a conceptual foundation, students’ own solutions serve as the basis for discussion and learning. The lesson encourages students to use multiple modali-ties to explain their solutions, such as explaining their thinking orally and using drawings, symbols, and text labels to make their explanations clear. Learning mediated through multiple modalities is particularly important in providing access for visual learners and English learners (Echevarria, Vogt, & Short, 2004).

Provides Word Problems and Symbolic Problems

Word problems provide a meaningful context for developing number concepts and the meanings of operations (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989). This idea is supported by a theory that describes the situated nature of learning (Brown, Collins, & Duguid, 1989; Lave & Wenger, 1991). Ulti-mately, students must also learn to work meaningfully

Linking Research, Standards, and Practice

3

and skillfully with mathematics in its abstract, or symbolic, form, especially if they are to be success-ful in higher mathematics (Sfard, 2000). MPP helps students make sense of — and solve — both word problems and problems in purely symbolic form. Also, MPP encourages mental, visual, and paper-and-pencil solution methods. Regardless of whether the problem is in context or not, students are expected to explain why their solution process makes sense.

Develops Mathematical Language

The meaning students develop for a mathematical idea is closely entwined with the way they use language to reason about the idea and learn to translate among words, symbols, and meanings of that idea (Vygotsky, 1962; Schleppegrell, 2004; Cummins, 1980; Pimm, 1995; Solomon & Rhodes, 1995). Discussion-based lessons, such as MPP, support language development. As Khisty (1992, 1995) points out, students develop language by talking. MPP teaching guides help teachers anticipate language confusion and support communication to ensure that English learners have access to the discussion. In addition to fostering oral communication, MPP lessons assist students in learning to comprehend complex mathematical text and write mathematical explanations. To support vocabulary development, lessons open with a quick review of important mathematical words, including examples of how they are used. Finally, the Discus-sion Builders elicit increasingly more sophisticated use of academic language and reasoning as students progress through the grades.

Jumpstarts Effective and Equitable Instruction Even in Non-MPP Lessons

A classroom observation study of MPP found evidence to support the idea that MPP practices transfer from MPP lessons to non-MPP lessons (Heller Research Associates, 2008). This may help explain why incorpo-rating only 15 hours of MPP lessons into the regular mathematics program has the potential to amplify mathematical learning during all lessons. This study

found that after experiencing MPP lessons, students were more likely to notice pitfalls, were more likely to explain and justify their reasoning, and had a greater level of engagement in the classroom overall. Also, a greater variety and percentage of students partici-pated in class discussions, and students with diverse ethnic and linguistic backgrounds were more likely to present their ideas.

Prompts a Metacognitive and Proactive Stance Toward Learning

Instructional approaches that encourage self-moni-toring — or metacognition — have been shown to support learning with understanding (Donovan & Bransford, 2005; National Research Council, 1999). MPP lessons include several structures and reminders for students to become more proactively aware of their thinking and learning processes. For example, MPP lessons encourage students to consider where someone might make a pitfall (or use incorrect reasoning) as they solve a problem. Students learn to ask themselves whether a solution makes sense or not. These oppor-tunities for reflection help students become more con-scious of how they learn and ways they can monitor their own learning and problem solving.

Incorporates Tools for Professional Learning and Lesson Study

MPP provides practical professional development options that respond to different needs, budgetary constraints, and time allocations. Teachers can learn how to use MPP on their own or with colleagues by viewing the MPP Video for Teachers (on the DVD included with this book) while completing the Profes-sional Development Tasks found on pages 20 –26. The MPP teaching guides, which provide mathematical insights and teaching tips for each lesson, have served as a resource for lesson study groups. Finally, each MPP lesson embeds structures and prompts that support the development of effective and equitable teaching habits. Further professional development opportunities are listed online at www.WestEd.org/mpp.


5

CD

DVD

What Tools and Resources Are Included with MPP?

Each MPP book contains two units, totaling 20 to

22 lessons, and together the four books cover key

mathematics topics for kindergarten through grade 7.

To maximize effectiveness, each book comes with a

DVD, CD, and Discussion Builders poster. This set of materials is all that is needed for successful imple-mentation for most teachers. Additional professional development is available to provide teachers with a practicum for implementing MPP and opportunities to further develop their expertise with the mathematical content and pedagogy of MPP.

Book with Lessons and Teaching Manual

Each book contains two grade levels of lessons, including engaging teaching guides and black line masters

for student pages.

DVD

Video for Students: An MPP lesson models respectful and productive mathematics discussion.

Video for Teachers: A lesson-in-action demonstrates effective facilitation, bolstered by helpful tips from

MPP teachers.

CD with Black Line Masters and Resources

The student pages and professional development tasks in this book, as well as quiz items, can be printed

out directly from the CD.

Poster

A poster of Discussion Builders sentence stems helps students engage proactively, politely, and productively

in discussion.

Professional Development

Tasks for teachers to do individually or with colleagues are included in each book and are accompanied by

black line masters found on the CD. Additional offerings are available online at www.WestEd.org/mpp.


7

Decimal Topics Unit 5 Lessons

Understand decimal forms of 1 1, 2

Understand decimal place value 3, 5

Relate fractions and decimals 2, 4

Understand and represent decimal amounts 1

Identify and find equivalent decimals 1, 2, 6, 10

Read and write decimal amounts 3

Locate decimal amounts on a number line 5

Compare and order decimal amounts 6

Add and subtract decimals with mental regrouping 7

Add and subtract decimals 8

Multiply decimals 9

Divide a decimal by a whole number 10

Understand and solve one-step equations with decimals 11ALGEBRA READINESS

What Is in Each MPP Lesson?

MPP lessons use a consistent, easy-to-follow format with a core lesson and two follow-up mini lessons, each accompanied by a teaching guide. Teaching guides provide sample prompts and solution keys for each part of the lesson. Lessons rely on drawings for visuals and suggest but do not require manipulatives,

allowing lessons to be used in schools where such materials are not available.

The four elements of each MPP lesson are: Opener, Discussion, Review and Practice, and Assess and Reinforce. Together they provide multiple opportuni-ties for students to develop, assess, and solidify their mathematical understandings.

Writing Task Mini Lesson

Assess and Reinforce

a

1/2 student page

a

2–3 days later

a

25 minutes

Multiple Choice Mini Lesson


a

1/2 student page

a

2–3 days later

a

20 minutes

Core Lesson

Opener, Discussion, Review and Practice

a

4 student pages

a

2 consecutive days

a

45 minutes each day

What are the Mathematical Topics of MPP?

MPP is designed to boost learning of key mathematical concepts and standards when used in conjunction with any adopted textbook. MPP includes a set of four books on the following mathematical topics.

Units K-1 Unit 2-3 Units 4-5 Units 6-7

Early math concepts Place value Fractions Percents

Whole number concepts Whole number operations Decimals Ratios and proportions

Algebra readiness Algebra readiness Algebra readiness Algebra concepts

Student PagesSample Lesson: Naming Equivalent Fractions (For Grades 4 and 5)

Excerpted from:

Title: Math Pathways & Pitfalls Lessons and Teaching Manual, Fractions and Decimals with Algebra Readiness, Units 4 and 5

Authors: Carne Barnett-Clarke, Alma B. Ramírez, with Debra Coggins

Copyright: Available Spring 2009

Publisher: WestEd

ISBN: 978-0-914409-60-1

222 © 2009 WestEd. All rights reserved.

STUDENT PAGE 1

NAME:

PurposeTo understand that a fraction tells how much a part is compared to the whole

Math Words

equivalent fractions

Two fractions are equivalent if they are the same amount.

halves We say one half for and three halves for .

thirds We say one third for and two thirds for .

fourths We say one fourth for and three fourths for .

Starter Problem

Write two equivalent fractions that tell what part of the picture is shaded. Write each fraction in symbols and words.

Naming Equivalent Fractions

Math Pathways & Pitfalls™ a Unit 4 a Lesson 1

32

12

24

=

12

23

13

34

14

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© 2009 WestEd. All rights reserved. 223

STUDENT PAGE 2

NAME:

Things to Remembera

a

Pitfall

OK

Student Thinking

Starter ProblemWrite two equivalent fractions that tell what part of the picture is shaded. Write each fraction in symbols and words.

Juan

Teresa

I can see it’s 2 equal parts out of 6 equal parts. That’s . If I slide

the 2 shaded pieces together, I can see that two sixths is equal to

one third of the whole rectangle.

It’s 2 parts shaded and 4 parts not shaded, so it’s .

That’s the same as .



26

241

2S A M

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STUDENT PAGE 3

NAME:

Our TurnWrite two fractions for the shaded part of the picture. Write each fraction

in symbols and in words.

1.

2.

3.



Symbols =

Words

Symbols =

Words

Symbols =

Words

S A M

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STUDENT PAGE 4

NAME:



My TurnWrite two fractions for the shaded part of the picture. Write each fraction


1.

2.

3.

Symbols =

Words

Symbols =

Words

Symbols =

Words

S A M

P L E


STUDENT PAGE 5

NAME:

STUDENT PAGE 5

NAME:





Fill in the circle next to the answer you choose.

1. What fraction of the picture is shaded?

not given


two sixths one fourth one fifth not given

25

46

64


Fill in the circle next to the answer you choose.


not given


two sixths one fourth one fifth not given

25

46

64

S A M

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STUDENT PAGE 6

NAME:

STUDENT PAGE 6

NAME:


Explain how you know and are correct fraction names for the shaded part of

this picture. You may draw a picture on the back to help you explain.




26

13




26

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Teaching GuideSample Lesson: Naming Equivalent Fractions (For Grades 4 and 5)

Excerpted from:

Title: Math Pathways & Pitfalls Lessons and Teaching Manual, Fractions and Decimals with Algebra Readiness, Units 4 and 5

Authors: Carne Barnett-Clarke, Alma B. Ramírez, with Debra Coggins

Copyright: Available Spring 2009

Publisher: WestEd

ISBN: 978-0-914409-60-1

Math Pathways & Pitfalls™ 43

Lesson at a Glance

MATERIALSLESSON ROADMAP

Mathematical goals

Understand that a fraction tells how much Aa part is compared to the whole

Identify equivalent fractions A

Mathematical language and reasoning goals

Write fraction amounts in symbols and words A

Use reasoning to verify different ways to name a given fractional part A

Prior Learning Needed

Understand fractions and •the concept of a unit

Read and write symbols for •basic fractions

Lesson Preparation

Study Lesson Foundation•

Review Teaching Guide and •Student Pages

Prepare stapled packet of •Student Pages 1–4 for each student

Copy and cut in half •Student Pages 5 and 6

Post • Discussion Builders poster

C O R E L E S S O N : D Ay 1 G r o U p I n G T I m e

Opener

Discussion Builders

Purpose

Math Words

Starter Problem

Discussion

Student Thinking

Things to Remember

Reflection

C O R E L E S S O N : D Ay 2

Review and Practice

Review Day 1 Lesson

Our Turn

My Turn

M I N I L E S S O N S : 2 – 3 D Ay S L AT E R




Discussion Builders ❍ poster

projector (optional) ❍

Student page 1 ❍

Student page 2 ❍

Teaching Guide ❍

Fraction pieces or kits ❍

Naming Equivalent FractionsTEACHING GUIDE a UNIT 4 a LESSON 1

Clipboard prompts, ❍page 37

Student page 2 ❍(completed day 1)

Student pages 3 and 4 ❍

Teaching Guide ❍


Student pages 5 and 6 ❍

Teaching Guide ❍


S A M

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Lesson at a Glance

44 ©2009 WestEd. All rights reserved.


Lesson Foundation

LESSON SNAPSHOT

MAtheMAticAL iNsights & teAchiNg tiPs

Naming Fractions by Thinking About the Whole

Teresa thought of the shaded amount as 2 equal parts “out of” 6 equal parts, but she could also think of it as 2 equal parts “com-pared to” 6 equal parts. She moved one of the shaded parts to another part of the rectangle to better visualize the shaded amount compared to the whole. That helped her see that one of the shaded parts was 1/3 of the whole rectangle. Visualizing more or fewer subdivisions when naming fraction amounts is a valuable tool that helps students think about equivalent fractions.

Provide opportunities to divide whole regions into more or fewer equal parts and to name the shaded amount. Chalk drawings allow students to easily add or erase lines.

© 2009 WestEd. All rights reserved.

STUDENT PAGE 2

NAME:

a

a

Pitfall

OKI can see it’s 2 equal parts out of 6 equal parts. That’s . If I slide






a a

26

241

2

PitfallJuan incorrectly named

the fraction by comparing

the part shaded with the

part not shaded instead of compar-

ing a part to the whole.

S A M

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Lesson at a Glance


TEACHING GUIDE a UNIT 4 a LESSON 1

Lesson Foundation

(continued)

MAtheMAticAL iNsights & teAchiNg tiPs (cONtiNueD)

A Fraction Represents a Comparison

Juan had likely learned that you need a numerator and a denominator to make a fraction. He looked for two numbers for his fraction and fell into the common pitfall of counting the num-ber of parts shaded and the number not shaded to get 2/4. Students who make this pitfall may not realize that a fraction represents a relationship between a part and the whole amount, not two parts (i.e., shaded and not shaded). The relationship between the numerator and denomi-nator communicates the amount that is being represented. Specifically, the number 2/6 is the amount represented by the part compared to the whole. Later, students will learn that a ratio can compare two parts, two wholes, or a whole and a part.

Students should use a variety of fraction models that require more than simple counting to name the fraction shown. For example, ask them to show 3/8 on a model divided into fourths or to show approximately 2/3 of a model divided into halves.

Infinite Number of Names for the Equivalent Fraction Amounts

This lesson is an opportunity to help students realize that a region can be divided into equal parts in many different ways and that an infinite number of fractions can name the shaded amount. To find other fraction names, Teresa could divide the same region into more or fewer equal parts, as long as the ratio of the shaded amount to the whole amount remains the same.

MAtheMAticAL DiscussiON suPPORt

model the correct way to say and write fraction names and words and give students ample opportunities to say and write them. encourage students to make drawings on the board to show their thinking. Drawings should be roughly accurate so that they convey the correct mean-ing, but can be sketched without a ruler. Another option to support explanations includes cutting or folding paper to show fraction equivalency.

Point out the similarities and

differences between whole

number words and fraction

words such as “four” and

“fourth” and “eight” and

“eighth.” Also point out peculiarities such as “1/2

is one half not one twoth” and “1/3 is one third

not one threeth.”

En

glish Learner

A c c e s sS A M

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Lesson at a Glance



Core Lesson Day 1

Opener

Review Discussion Builders Read the poster. suggest a section to focus on today: Presenting Alternative Ideas, Expanding on Others’ Ideas, or Posing Additional Questions.

PurposeDistribute stapled packets of Student pages 1–4. Project an image of page 1 (optional).

call on a student to read the purpose.

Math Words Point to and say the first math words. Ask students to repeat them aloud or silently.

Read the sentence containing the words.

give an example using objects or drawings.

Repeat for the other math words.

starter ProblemRead the Starter problem. call on a stu-dent to restate it in his/her own words.

Think about what the Starter Problem means. Try to use what you understand to solve the problem on your own.

I’ll walk around and write notes about things we need to discuss. Look out for pitfalls!

Look at your work. It’s easy to have a pitfall in this type of problem. You might also have made a pitfall if one of your fractions has a denominator of 4.

Don’t worry. Next we’ll discuss how two imaginary students solved this problem. One has a pitfall! You may keep your solution private, but bring up your ideas in the discussion.

STUDENT PAGE 1

S A M

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Lesson at a Glance



Core Lesson Day 1

(continued)

STUDENT PAGE 2

Discussion

student thinking

Ask students to refer to page 2. Read the statement marked oK.

explain that this statement is about the same problem students worked on earlier.

We can learn a lot about the math by studying what this student did.

Read each sentence silently and look at the drawing. Think about what they mean.

Now talk with a partner about what each sentence and each part of the drawing means.

Listen in, ask questions, and observe. Note potential contributions for the discussion.

Who can come up to show what Teresa meant when she said she could see 2 equal parts out of 6 equal parts? What fraction did she write? How do we write 2/6 in words?

Talk to your neighbor about how Teresa decided that another name for the shaded amount is 1/3.

Who can show how Teresa slid the pieces together? She wrote 1/3, 1/3, and 1/3 on top of the rectangle. Who can explain why?

Explain why 2/6 and 1/3 are equal fractions. What equation did Theresa write to show they are equal? What is another fraction equal to 2/6 and 1/3? Who can write another equation? How do you write it in words?

call on students to state things to remember about solving problems like this.

start a Things to remember list on the board. MORE DAY 1


STUDENT PAGE 2

NAME:

a

a

Pitfall







a a

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2

S A M

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Lesson at a Glance



Core Lesson Day 1

(continued)

STUDENT PAGE 2

Discussion

student thinking, continued

Read the statement marked pitfall. Remind students that this is a common pitfall.

Juan made a pitfall when he made a fraction by counting the shaded parts separately from the parts not shaded. Talk with your neighbor about why his answer does not make sense and is incorrect.

Who would like to show why it doesn’t make sense to say 1/2 of the picture is shaded?

Why is this a pitfall?

Draw a rectangle on the board and divide it into eight equal parts. Shade two parts. Ask students to talk with a neighbor about ways to write fractions for the shaded part of the picture. Remind them to look out for pitfalls. call on students to explain their reasoning.

things to Remembercall on students to add to the Things to remember list on the board. Read the list.

help students summarize and record two important Things to remember.

ReflectionAsk students to reflect on the discussion process using one of the sample prompts.

Things to Remember List (sample)A fraction can be used to tell how much a part is compared 1. to the whole. You can divide the whole into more or fewer equal parts to 2. find equivalent fractions.


STUDENT PAGE 2

NAME:

a

a

Pitfall







a a

26

241

2

Reflection Prompts (sample)Name a • Discussion Builder that we used today. How did it help the discussion? What • Discussion Builder could we use next time to make the discussion even better?What did someone do or say today that helped you • understand the math?

S A M

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Lesson at a Glance



Core Lesson Day 2

Review and Practice

ReviewAsk students to review page 2 to jog their memory.

Read the statement marked oK. call on a student to explain how the problem was solved.

Read the statement marked pitfall. call on a student to explain why it is incorrect.

call on two or three students to read an item on their Things to remember list.

Our turnAsk students to refer to page 3.

use the procedure below and the Clipboard prompts to discuss students’ solutions. Discuss the problems one at a time.

Read the problem.

Ask students to work with a neighbor to solve it.

Discuss one or two students’ solutions.

My turnAsk students to solve the problems on page 4. Remind them to watch out for pitfalls!

After allowing time to work, read the answers. have students mark and revise their papers using ink or crayon.

STUDENT PAGE 3

STUDENT PAGE 4

STUDENT PAGE 2

Oops!


STUDENT PAGE 2

NAME:

a

a

Pitfall







a a

26

241

2


STUDENT PAGE 3

NAME:

Write two fractions for the shaded part of the picture. Write each fraction


1.

2.

3.

a a


Symbols =

Words

Symbols =

Words

Symbols =

Words

Answer Key

1. 2/5, 4/10 (two fifths, four tenths), or other equivalent fractions2. 1/2, 4/8 (one half, four eighths), or other equivalent fractions3. 2/5, 4/10 (two fifths, four tenths), or other equivalent fractions

Answer Key

1. 1/2, 5/10 (one half, five tenths), or other equivalent fractions2. 1/3, 3/9 (one third, three ninths), or other equivalent fractions3. 2/3, 4/6 (two thirds, four sixths), or other equivalent fractions

S A M

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Lesson at a Glance


Mini Lessons(2–3 Days Later)


Multiple choice Mini LessonDistribute Student page 5.

problem 1

Please read problem 1.

Talk with your neighbor about which choices don’t make sense.

Who can explain why it is a pitfall to choose 4/6?

What is the correct choice?

The correct fraction compares the number of shaded parts to the total number of parts in the whole, rather than comparing the number of shaded parts to the number not shaded.

problem 2

Read the problem and find the correct choice.

Which response is correct? Explain why.

Is there more than one fraction name for the amount shaded? Explain.

Writing task Mini LessonDistribute Student page 6.

Ask a student to read the task. call on students to respond with their ideas.

Jot the ideas on the board.

Write an explanation together using their ideas. Read it aloud.

Ask students to write an explanation on their page.

STUDENT PAGE 6

STUDENT PAGE 5

Sample Explanation: If I draw an extra line, I can cut the larger piece in half. That makes 6 equal pieces. There are 2 parts out of 6 or 2/6 that are shaded. I can also slide the 2 small shaded pieces together and make it 1 large piece. Then I have 3 equal pieces with 1 piece shaded. So, 2/6 or 1/3 of the picture is shaded.

En

glish Learner

A c c e s s

Mathematical Discussion Support

When generating ideas, encourage students to mentally manipulate the pieces. For example, they might

mentally slide two small pieces together to make a large piece or cut a large piece in half to make two

smaller pieces. They can add or erase lines to illustrate the relationships among the pieces. Students could

also cut out the pieces to verify the equivalencies. Ask students to explain their thinking along the way.

° ° ° °

° ° ° °

° ° ° °

° ° ° °


STUDENT PAGE 6

NAME:

STUDENT PAGE 6

NAME:






a a

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S A M

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When one of my fifth grade students was at the board showing they really didn’t

have a solid understanding, another would say, ‘I’m confused about what you put

up there, can you prove your thinking?’ Then as a class they would work through

those mistakes until they had a solid understanding. – Teacher

Discussion Builders Workshop WestEd.org/discussionbuilders»

Who Should Attend

K–8 classroom teachers who want practical, evidence-based teaching strategies that foster oral and written academic language development. Excellent for teachers with English learners in their classroom.

What Are Discussion Builders?

Originally created for teaching math, Discussion Build-ers is a suite of classroom posters, teaching guides, and classroom strategies, developed and field-tested with funding from the National Science Foundation. Discussion Builders promotes academic language by helping students build reasoning and communications skills. Sentence stems on the posters provide students with a scaffold for voicing their ideas and questions and valuing others’ con-tributions. The three levels of posters and teaching guides scaffold progressively more sophisticated language and reasoning across the grades. The teaching guide includes tasks and blackline masters for introducing and sustaining the use of Discussion Builders in any subject area.

What teachers Learn

Powerful, easy-to-implement strategies using National »Science Foundation-funded, field-tested Discussion Builders classroom posters and teaching guide

Proven techniques for building students’ communica- »tion and reasoning skills

Practical ways to help » all students overcome academic, linguistic, and social barriers in classroom discussions

Overview of current research on the role of academic »language development in academic success

Practice facilitating productive discussions in »mathematics, literature, social studies, and other content areas

High comfort level in leading effective discussions that »boost collaborative and respectful critical thinking among students

format and Cost

The one-day workshop, poster, and teaching guide cost only $350 per participant. Teachers participating in teams of three or more are eligible to attend the work-shop for $300 each.

ContACt InformAtIon >> Kate D. Darling >> 510.302.4253 >> [email protected] >> WestEd.org/discussionbuilders

By talking to learn, students also learn how to think.

The sentence stems on these colorful posters provide

students with a scaffold for voicing their ideas and ques-

tions, valuing others’ contributions, and incorporating

increasingly sophisticated thinking strategies. Using

Discussion Builders, students learn through active

participation in classroom discussions. Accompanying

quick-guides for teachers explain how to get students

talking — and thinking — more conceptually, in any

subject (originally developed for mathematics).

Proven powerful for English language learners and

students of all achievement levels, Discussion Builders

scaffold progressively more complex reasoning across

the grades. In K–1 the focus is on helping students pres-

ent, expand, and reflect on important ideas. In grades

2–3, Discussion Builders prompt students to use these

I like Discussion Builders because they help us get to the point.

– Third grade student

skills at more sophisticated levels. The 4–8 poster strengthens students’ complex reasoning, including their abilities to consider counter-examples and conjec-tures and to justify options.

Set: $38.00 (Save 15% off the individual price by ordering the set.) • 2005 • WestEd • MATH-05-04M

Individual posters/teaching guides:

Discussion Builders Poster & Teaching Guide: K-1 • $14.95 • 2005 • WestEd • MATH-05-01M

Discussion Builders Poster & Teaching Guide: 2-3 • $14.95 • 2005 • WestEd • MATH-05-02M

Discussion Builders Poster & Teaching Guide: 4-8 • $14.95• 2005 • WestEd • MATH-05-03M

(posters’ dimensions are 24 x 36)

Discussion Builders: Posters and Teaching Guides K-1, 2-3, and 4-8

CARNE BARNETT-CLARKE AND ALMA RAMIREz

K-7 Intervention Curriculum and Professional Development - Coming Soon!

Math Pathways & Pitfalls is a research-based intervention curriculum that focuses on the toughest mathematics standards while fostering academic language development. It has been shown to boost student achievement for both native English speakers and English language learners.

To inquire about pre-ordering your copies at the prepublication discount, please email WestEd: [email protected].

Each Math Pathways & Pitfalls book contains two grade levels of lessons, including engaging teaching guides and black line masters for student pages; a DVD containing a video for students and video for teachers; a CD with black line masters and resources for all student materials; a classroom Discussion Builders poster; and teacher professional development tasks to do individually or with colleagues.

Math Pathways & Pitfalls Lessons and Teaching Manual Number Sense and Operations with Algebra ReadinessUnits K and 1ISBN: 978-0-914409-58-8

Math Pathways & Pitfalls Lessons and Teaching Manual Number Sense and Operations with Algebra ReadinessUnits 2 and 3ISBN: 978-0-914409-59-5

Math Pathways & Pitfalls Lessons and Teaching Manual Fractions and Decimals with Algebra ReadinessUnits 4 and 5ISBN: 978-0-914409-60-1

Math Pathways & Pitfalls Lessons and Teaching Manual Percents, Ratios, and Proportions with Algebra ReadinessUnits 6 and 7ISBN: 978-0-914409-61-8

For more information about Math Pathways & Pitfalls, visit WestEd.org/mpp.

About WestEd

WestEd, a national nonpartisan, nonprofit research, development, and service agency, works with education and other communities to promote excellence, achieve equity, and improve learning for children, youth, and adults. WestEd has 16 offices nationwide, from Washington and Boston to Arizona and California. Its corporate head-quarters are in San Francisco. More information about WestEd is available at WestEd.org.

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Probably the best training that I have ever participated in. The ...Probably the best training that...

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