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VMPs Ongoing Assessment Project(OGAP) was created as a part of the Vermont Mathematics Partnership funded by the US DOE

(S366A020002) and the National Science Foundation (EHR-0227057)1

Vermont Mathematics Partnership Ongoing

Assessment Project (OGAP)

Fractions at the Middle Grades

Session 1 A

Vermont Mathematics Partnership

www.vermontmathematics.org

Marge Petit, Marge Petit Consulting, MPC

June 2009

Middle School Fraction Dilemma


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What this workshop cannot do

Provide an instant fix

What this workshop is designed to do

Provide you with knowledge and tools to better recognize

what students do understand and what research says about

how to move students to a new level of understanding.

Provide a problem solving environment in which to

collectively think about developing strategies to deal with

the Middle School Fraction Dilemma.

Goals

To map how fraction conceptual understanding and procedural


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Major Research Considerations at Middle

School Whole number reasoning may interfere with development of

fraction concepts and procedural fluency (e.g., Post, Behr, Lesh &Wachsmuth, 1986; VMP OGAP, 2005)

Fraction order and equivalence form the framework forunderstanding fractions as quantities that can be operated on(e.g., Post, Cramer, Behr, Lesh & Harel, 1993)

Students may struggle with the use and understanding offormal algorithms when their knowledge is dependentprimarily on memory, rather than anchored with a deeperunderstanding of the foundational concepts. Understandingand procedural fluency should be built in a way that brings

meaning to both. (e.g., Behr et al., 1984; Behr & Post, 1992; Wong & Evans, 2007; Payne,1976; Lesh, Landau, & Hamilton, 1983 Kieren, as cited in Huinker, 2002).

One way is by developing the connections between the conceptand procedure is by moving back and forth between differentrepresentations of fractions(e.g., Behr et al., 1984; Behr & Post, 1992; Wong & Evans, 2007; Payne,1976; Lesh, Landau, & Hamilton, 1983 ).


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Therefore - Workshop Foci

Developing understanding and procedural fluency

how they represented in student work and

implications for instruction

Day 1:

Mapping fraction demand at middle grade

Equivalence, magnitude, and density

Day 2: Operations

Fraction Concept

Mapping

Fraction


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Mapping Fraction Demand Identify the fraction concepts and skills new to the grade level.

Identify applications of fraction concepts and skills at the grade level.

Identify fraction knowledge that is assumed in order for students to be

successful learning new concepts and applying fraction concepts and

skills at the grade level

Assumed

knowledge

Applied at

grade level

New to grade

level

Grade8

Grade7

Grade6

OGAP Fraction Framework

The first step to helping students is understanding what they

understand and can do.

St t f P bl


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Session 1 B Mapping Fraction Demand at Middle School

OGAP was developed as a part of the Vermont Mathematics Partnership funded by the US Department of Education (Award Number S366A020002) and the National Science Foundation (AwardNumber EHR-0227057) v 1 June 3, 2009

Concepts and Skills Grade 6 Grade 7 Grade 8

New to grade level

Applied at grade

level

Assumed knowledge


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Session 2B Fraction warm-up

1) Solve the following problem using three different strategies.

2) Id tif th diff t t d t i ht k i d t di th t t d t


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OGAP Fraction Item Analysis Sheets Operations

The Vermont Mathematics Partnership is funded by a grant provided by the US Department of Education(Award Number S366A020002) and the National Science Foundation (Award Number EHR-

0227057)June 2, 2009

Operations

Question # (s) ____

Fractional Strategies Transitional Fractional Strategies

Fractional Strategy with an Error

or Misconception Non-Fractional Strategy

Efficient or generalizable strategy:

Number sense

Estimation (when appropriate)

Efficient algorithm

Other:

Successful student generated model

Strategy not efficient or

generalizable (e.g., building up or

down)

Other:

Student generated model witherror:

Wholes different sizes

Parts obviously not equallypartitioned

Other

Appropriate operation or strategywith an error

Calculation

Other

Whole number reasoning, not

fractional reasoning

when adding fractions

when estimating sum as closestto a number

Inappropriate model, operation,

or strategy given the problemsituation

Model used inappropriate forgiven situation

Selected the wrong operationgiven problem situation

Other

Instructional Notes:


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Sessions 2 10 Telling your Classrooms Story

Based on analysis of student work and research answer the following questions to

help develop your classrooms story

1) What are some patterns in your class of developing understandings that can be built

upon?

2) What are some patterns of errors/misconceptions across your class?

3) Wh t th ti l t /id h ld b h i d d h i d?


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The Vermont Mathematics Partnership funded by The National Science Foundation (Award Number EHR-0227057) and the US Department of

Education (S366A0200002))Version 6.0 October 2, 20061

How do students see this number?

Session 2A

Researchers say


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Because many students do not

see a fraction as a single value

#$%&'#()*+,&)%-..-/$0#1)/234+5-'6.5+/#-2'(7

4&5+#-'6)9-#*).5+/#-2'(7

%&'#-.1-'6)4+5#)#2)9*20&)5&0+#-2'(*-4(;

Example 1


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Example 1

Example 3


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Example 4

Evidence in Your Students Work


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Mathematicians and Educators

working together to help all

Vermont children

succeed in mathematics

VMP is a targeted

Math & Science

Partnership funded by

the National Science

Foundation

&

the U.S. Department of

Education


National Science Foundation,

grant award number EHR 0227057

and U.S. Department of Education,

grant award number S366A020002


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Compare and Order Student Work Sort

Session 3 C


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Research -

Comparing and

Ordering Fractions

Vermont Mathematics Partnership


Session 3 B

Comparing FractionsDirections: Work with a partner to compare the fraction pairs below. Discuss your thinking

with your partner and record the strategies you used to make your comparisons.


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Created by Vermont Mathematics Partnership is funded by NSF (EHR-0227057) and US DOE (S366A020002)

3

Students should understand

and use flexibly the different

classes of fractions:

Different Numerators, Same

Denominators;

Same Numerators, Different

Denominators;

Different Numerators,

Different Denominators.(Behr, M.J., Lesh, R, and Post

(1981)

Identify examples

of different classes

of fractions.

Researchers found that students effectively used five types of

reasoning when solving problems involving fractions:(Behr, M., & Lesh,R. (1992)))

U i l i hi b h b f i h h l d h i


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Created by Vermont Mathematics Partnership is funded by NSF (EHR-0227057) and US DOE (S366A020002)

7

Looking at Your Own Mathematics Program

Comparing and Ordering Fractions Scan your program and makes notes

about opportunities students are provided to develop a variety of reasoning

strategies (i.e., modeling, unit fraction reasoning, extended unit fraction

reasoning, benchmarks, and equivalence/common denominators).

Cognitive Research and Mathematics Program

Based on your scan are there any modifications that you should make

to lessons on comparing and ordering to assure that students have the

opportunity to develop a range of reasoning strategies?

Summary


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3A Fraction Pairs

Directions: Work with a partner. Identify which fraction in each of the pairs

is largest. Discuss your thinking with your partner and record the strategies

you used to make your comparisons.

1)3 5

6 6

2)11 9

13 11

3)

7 7

9 11

4)3 7

6 15

5)1 1

7 5


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OGAP Fraction Item Analysis Sheets Equivalence and Magnitude

The Vermont Mathematics Partnership is funded by a grant provided by the US Department of Education(Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)June 2, 2009

Equivalence and Magnitude

Question # (s) ____




Efficient or generalizable

Strategy:

Number sense

Estimation

Benchmark

Unit fraction reasoning

Extended unit fractionreasoning

Equivalence/commondenominator

Other


successfully

Other

Student generated model with

error:



OtherAppropriate operation or strategy

with an error

Ignored whole number, justcompared the fraction

Fraction located on number linein the correct relative position,

but not accurately

Other




or strategy given the problem

situation

Identifies fractions asequivalent when they are the

same number of parts from

whole (e.g., 2/3 and !).

Does not recognize non-congruent parts as equivalent

Did not place on number linerelative to the defined units

Other



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OGAP was developed as a part of the Vermont Matheamtics Partnership funded by the US DOE (S366A020002) and NSF (EHR-

0227057)1

Density of

Fractions

Marge Petit, Marge Petit Consulting, MPC

June 2009

Density of Fractions


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0227057)3

For each fraction pair:

a) Find 3 fractions using at least 2 different

strategies.

b) Identify difficulties students might encounter.

Identifying fractions between fractions

Students have difficulty with the concept of the

density of rationale numbers (Tirosh, Fischbein, Graeber, & Wilson, 1998; Orten,


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0227057)7

Name two fractions that are between 1/3 and 3/4.

Do you think there are any other fractions besides the ones you

identified that are between 1/3 and !?

Misconception/error: There are more fractions between but

identified equivalent fractions (26% (9/35)) (Petit, Laird, Marsden, in press 2010)

Researchers indicate thatusing number lines have the

potential to help build understanding of the density of

rational number concept


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0227057)9

How could a number line be used to extend

this students understanding?

Name two fractions that are between 1/3 and 3/4.

Do you think there are any other fractions besides the ones you

identified that are between 1/3 and !?


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Question # (s) ____





Strategy:

Number sense

Estimation

Benchmark




Other


successfully

Other

Student generated model with

error:



OtherAppropriate operation or strategywith an error



but not accurately

Other





situation






Other



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Session 4 A - The Density of Fractions

1) For each fraction pair:a) Find three different fractions between the fraction pairs (using two different

strategies for each fraction pair). Then answer questions b, c, and d.b) What difficulties do you think students might encounter as they solve these

problems?c) What kinds of errors might result from these difficulties?d) As a set of questions, what information can the student work provide that the

evidence from a single question might not provide?

1)4 7

and10 10

2)1 1

and8 4

3)1 1and

5 4

4)

5 6and

11 11


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OGAP was developed as a part of the Vermont Matheamtics Partnership funded by US DOE ( S366A020002) and NSF (EHR-0227057) 1

Number Lines

Marge Petit Consulting, MPC

June 2009

Session 5 A

Number lines can help build understanding of

equivalence, magnitude, and the density of


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1) Make a list of everything you notice about the number line. (The

teacher uses the lists to guide a whole-class discussion.)

1) Identify where the number 4 is on this number line. What defined

where the number 4 is located? What whole numbers are represented

on this number line?

1) What do the tick marks between the number 0 and 1 indicate?

2) Name the numbers represented on the number line that are below 0

and 1)

.

1) Are there other numbers between the tick marks on the number line?

How could you determine what those numbers are?(Adapted from Petit, Laird, Marsden, in press 2010)

Features of Number Lines

Identify errors/misconceptions for each response

Response 1 Response 2


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Anticipate Errors Misconceptions and

Effective Strategies

Place the following fractions in the correct location of the number line.

1) Sample - Student Work Sort

2) Your class

Building an understanding ofequivalence


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Building an understanding of

density of fractions

Building an understanding ofaddition and subtraction of fractions


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Session 5 B: Number Lines Student Work Sort


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OGAP F ti It A l i Sh t E i l d M it d


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Question # (s) ____





Strategy:

Number sense

Estimation

Benchmark




Other


successfully

Other

Student generated model witherror:



OtherAppropriate operation or strategywith an error



but not accurately

Other

Whole number reasoning, notfractional reasoning



situation






Other



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Equivalence

Marge Petit Consulting, MPC

June 2009

Equivalence


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Conceptual understanding of equivalent fractions involvesmore than remembering a fact or applying a procedure(Wong & Evans, 2007, p. 826). -

That is, understanding equivalence as well as procedures

for finding equivalents fractions, so important for thedevelopment of other concepts, should be built in a waythat brings meaning to both. ----

---- Researchers suggest developing theconnections between the concept and procedure throughinteraction with models and manipulatives (Behr et al., 1984; Behr &Post, 1992; Wong & Evans, 2007; Payne, 1976 all cited in Petit, Laird, Marsden, in press 2010).

What patterns and relationships do younotice?


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What patterns and relationships do you

notice?

What questions can you ask that will start to

build an understanding of addition and

subtraction of fractions?

Equivalence and ProceduralFluency


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SORT your student responses

Middle School Math Program


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There was other information that surprised the teachers. Some of the students who were

able to correctly add fractions did not seem to understand the magnitude of theirresponses. Richards response makes this case.

Richards Response

This response really shocked the fifth grade teachers. Everybody agreed that there was a

problem. These kinds of responses prompted the teachers to attend the work shop on

Mathematical Proficiencies.

What they learned at the workshop surprised them. It wasnt a question of teaching

procedures OR teaching the concept. They appear to be interconnected. An expert panelindicated that understanding makes learning skills easier, less susceptible to common

errors, and less prone to forgetting as well as a certain level of skill is necessary tolearn many mathematical concepts

1This posed a new way of thinking for this team of

teachers.


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They then found several things that could help them think about how to build concepts

for operating with fractions. They were centered on three big ideas.

! Use models to build understanding of operations:o Concept learning is maximized when concepts are presented with a

variety of physical contexts. (Post, T.R., & Reys, R.E. (1979))

o Researchers have found thatstudents who can use and move betweenmodels when operating with fractions are more likely to reason with

fractions as quantities rather than apply whole number reasoning to theirsolutions. (Towsey, A. (1989)).

! Build a sense of the magnitude of the fractions: Students who have a feeling for

the bigness of fractions are able to compare fractions, place fractions onnumber lines, and operate with fractions more effectively. (Bezuk, N. S., and

Bieck, M. (1993))

! Provide a variety of contexts for students to solve problems: Students shouldexperience a variety of situations (contexts) in which they need to recognize theappropriate fraction operation and then solve problems accordingly. (Huinker, D.

(2002))

Using knowledge from readings and professional development they received through

OGAP the teachers were committed to building understanding of operations using

models, partitioning understanding, and reasoning derived from the impact of partitioningmodels.

They all agreed that the goal was for their students to attain procedural fluency when

operating with fractions. That is, students should have knowledge of procedures,

knowledge of when and how to use them appropriately, and skill in performing themflexibly, accurately, and efficiently (Adding it Up: Helping Children Learn Mathematics

2001).


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Mrs. Plum: Mrs. Plum brought Robertos work to the team meeting. She was both

excited by the work and concerned. Mrs. Plum said, Roberto obviously understood thecontext of the problem by selecting the correct operations to solve the problem. This is

great news. I am also assuming that we want students to add and subtract using efficient

strategies like Roberto did below, but I wonder if Roberto understood the algorithm he

used to add the fractions. How do I know? At what point do I need to stop worryingabout the understanding part and just accept the efficient application of an algorithm

To help Mrs. Plum please answer the following:1) Describe evidence(s) in Robertos response that show evidence of understanding of

the context of the problem and related fraction concepts.2) What questions might you ask Mrs. Plum about her instruction to assure that Robertohad a foundation for understanding?

3) If Mrs. Plum wanted to be sure that Roberto understood the algorithm, what elsecould she ask Roberto? What other OGAP questions (or questions of her own) couldMrs. Plum ask Roberto to assure that he understood the concept?

Robertos Response


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Mr. Laird: Mr. Laird brought this student response to the team meeting. Mr. Laird felt

that Mathews response provides evidence that he has a strong conceptualization when

comparing 10

3to5

2using both an area model and a number line. Many of his students are

using models and have similar understandings evidenced in their work. He thought this

might be a good opportunity to bridge equivalence to addition and subtraction offractions and wanted the groups help.

To help Mr. Laird please answer the following:1) What understandings are evidenced in Mathews work? Describe.

2) How could these evidences be capitalized on to build understanding aboutequivalence and common denominators when comparing fractions, or adding and

subtracting fractions? (e.g., What questions could be asked?) With each question

explain how it might help Mathew move to an understanding of equivalence and

common denominators when comparing fractions, or adding and subtractingfractions?


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Tina ate3

2of the candy in the candy jar. Her sister ate

4

1of the candy in the

candy jar. What is the fractional part of the candy left in the candy jar?

Explain your answer using words or diagrams.

Ms. Cunningham: Ms. Cunningham shared Kims student work found below. Even

though students have been modeling fractions when asked questions like show me

2

3

using an area model when asked the candy jar problem below, many students (like Kim)

were unable to use models to solve the problems. She is asking for advice on how totransition students to using models accurately to solve problems involving addition and

subtraction.

To help Ms. Cunningham please address the following:

a) What did Kim model correctly? What is the evidence?b) Kims model leads to an incorrect response. What errors did Kim make in her

modeling? What is the evidence?c) What questions might you ask, or activities might you do, to help Kim

understand how to use models to solve addition and subtraction problems?

Kims Response


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Mr. Hill: Mr. Hill has been spending a lot of time working on estimating sums and

differences. He decided to give the problem below to his students and bring a sample oftypical responses to the team meeting. The thing that pleased him the most was that all

the students recognized the sum was closest to 1 and had evidence that supported thisunderstanding. However, he felt the goal was for students to have a mental picture of

the magnitude of these fractions like Leslie appears to have. He wonders what he shoulddo next for students like Cody and Oscar to help them have a better sense of the

magnitude of these fractions.

To help Mr. Hill answer the following:

1) What did Cody and Oscar understand? How do you know?2) What questions might you ask Cody so that he would not have to add the fractions

before making the estimate? How would these questions help Cody?

3) What questions (or activities) would you ask Oscar to help him move from usinga model to a mental picture of the relative magnitude of the fractions beingadded? How would these help Oscar?

Cod s


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Ms. Petit: Ms. Petit was working with a student helping him to understand addition of

proper fractions. She quickly realized that Emmanuel could add/subtract fractions withcommon denominators, but not with unlike denominators, and he could draw models for

almost any reasonable fraction. For example, when asked to solve the problem (1 3

8 8! =)

Emmanuel used a model to add the fractions correctly.

Solution 1:

However, when he was asked to solve these problems (1 3

2 8! " and

1 1

2 3! " ) he incorrectly

tried to apply the same thinking he used in Solution 1.

Solution 2 Solution 3

7 B R i i h F i W


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7 B Reasoning with Fractions Warm-up

1) What happens to the relative value of a positive fraction when:

a. The numerator and the denominator are increasing at the same rate.

b. The numerator is increased and the denominator stays the same.

c. The numerator and the denominator are both increasing but the denominator isincreasing at a faster rate.

d. The denominator is increased and the numerator stays the same.

e. The numerator and the denominator are both multiplied by the same number.

7 B O ti W D 2


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7 B Operations Warm-up Day 2

c. The least possible product

d. The greatest possible product

e. The least possible quotient

f. The greatest possible quotient


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Session 8 B Multiplication Student Work Sort

Show your work.


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Show your work.


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Show your work.


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Show your work.


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Show your work.


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Show your work.


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Show your work.


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Show your work.

8 A Case Study finding the fractional part of a whole


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Mr. Smith is a sixth grade teacher. Among other questions he asked his students to solve

the following problem.

1) Shade 58

of the following figure.

Students, for the most part, found 58

of the figure as Thomas and Dyson did in the

responses below.

Shade5

8of the figure

Thomass Response Dysons Response



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8 Case Study d g t e act o a pa t o a w o e

2) Based on the evidence in the student work, what understanding of finding 58

of

the figure does Dyson have? What is the evidence?

3) Which strategy (Dysons or Thomass) would lend itself better to:a) Solving problems with larger wholes or more complex fractions (e.g., 5

8of $

243 =)? Explain your choice using examples.

b) Building equivalence and magnitude understandings? Explain your choiceusing examples

c) Understanding why 13 3

xx =


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Session 9 B Division Student Work Sort


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Session 9 C Partitive Division Student Work Sort


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S i 9 C P i i Di i i S d W k S


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S i 9 C P titi Di i i St d t W k S t


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8



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9A Division Problems


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Part I: Solve each of the following problems using 2 different strategies.(Problems 1 and 2 Source: Van de Walle (2004) Elementary and Middle School

Mathematics, Pearson Education, Inc., 5th addition, pp. 276)

1) Cassie has 154

yards of ribbon to make 3 bows for birthday packages. How much

ribbon should she use for each bow if she wants to use the same amount of ribbon

for each bow?

2) Linda has 243

yards of materials. She is making baby clothes for the bazaar. Each

dress pattern requires 11 yards of material How many dresses will she be able

10 A - Transitioning from elementary school fraction demand to middle school demand


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GOAL: To develop a strategy that builds upon students pre-existing knowledge as theyengage in mathematics topics in middle school that assume prior fraction understanding

and procedural fluency.

STEP 1: Math topics

Choose one fraction related topic new to your grade (e.g., ordering negativefractions) and one that requires application (e.g., solving proportions) at your grade

level.

Topic new to grade:

Topic applied at grade:

STEP 2: Foundational knowledge

Identify foundational knowledge students would need to have to be successful with

the new topic and with the applied topic.

Topic new to grade:



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10 A - Transitioning from elementary school fraction demand to middle school demand


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Making the Connections

STEP 6:

Based on the evidence in the pre-assessment, your answers to question # 3, and our

discussions over the last two days, develop an activity that provides your students withthe connection between what they know and understand (or dont understand) and the

new topic and to the applied topic.

Topic new to grade:




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3) Mrs. Jenkins has 324

gallons of juice for the class picnic.

The juice if divided equally into four pitchers.How much Juice is in each pitcher?

4A)

Jim is making decorations for a school dance.

He has 144

yards of wire. Each decoration needs 34

of a yard if wire.

How many full decorations can Jim make from the1

4 4



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Study each of the problems below. Describe the type of unit (e.g., rate) found in thedivisor, dividend, and quotient.

Cassie has 154

yards of ribbon

to make 3 bows for birthdaypackages. How much ribbon

should she use for each bow ifshe wants to use the same

amount of ribbon for each bow?

Linda has 243

yards of materials. She

is making baby clothes for the bazaar.

Each dress pattern requires 116

yards of

material. How many dresses will she

be able to make from the material thatshe has?

Dividend 154

yards 243

yards

Divisor 3 bows 116

yards/dress

Quotient

Which problem have a structure similar to:

Cassie -

Linda -


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Major Research Considerations at Middle School

Whole number reasoning may interfere with development of fractionconcepts and procedural fluency (e.g., Post, Behr, Lesh & Wachsmuth,1986; VMP OGAP, 2005)

Fraction order and equivalence form the framework for understandingfractions as quantities that can be operated on (e.g., Post, Cramer, Behr,Lesh & Harel, 1993)

Students may struggle with the use and understanding of formalalgorithms when their knowledge is dependent primarily on memory,

rather than anchored with a deeper understanding of the foundational

concepts. Understanding and procedural fluency should be built in a

way that brings meaning to both.(e.g., Behr et al., 1984; Behr & Post,1992; Wong & Evans, 2007; Payne, 1976; Lesh, Landau, & Hamilton, 1983Kieren, as cited in Huinker, 2002).

Model for making connections (e.g., Behr et al., 1984; Behr & Post, 1992; Wong &Evans, 2007; Payne, 1976; Lesh, Landau, & Hamilton, 1983 ).

ModelModel

OGAP Fraction Framework (June 2009)


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