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Overview of SessionOverview of Session

Introducing OGAP Proportionality

Framework

Developing an understanding of

the structure of proportional

situations

Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)

Goals of the SessionGoals of the Session

• To become familiar with findings from research on

formative assessment

• To understand the meaning of and purpose for

formative assessment


OGAP Proportionality OGAP Proportionality FrameworkFramework

Mathematical Topics And

Contexts

Structures of Problems

Other Structures

Evidence in Student Work to Inform Instruction

Proportional Strategies

Transitional Proportional

Strategies

Non-proportionalReasoning

Underlying Issues, Errors,

Misconceptions


Structures of Proportionality Structures of Proportionality Problems – How the Problems Problems – How the Problems are Builtare Built

• Multiplicative relationships in a problem (Karplus,

Polus, & Stage, 1983; VMP OGAP Pilots, 2006)

• Context (Heller, Post, & Behr, 1985; Karpus, Polus, &

Stage, 1983)

• Types of problems (Lamon, 1993)

• Complexity of the numbers (Harel & Behr, 1993)

• Meaning of quantities as defined by the context and

the units (Silver, 2006 Vermont meeting; VMP OGAP

Pilots, 2006)


• When the multiplicative relationships in a proportional

situation are integral, it is easier for students to solve

than when they are non-integral

(Cramer, Post, & Currier, 1993; Karplus,

Polus,

& Stage, 1983; VMP OGAP Pilots, 2006)

A Research FindingA Research Finding

OGAP Proportionality Framework


Multiplicative RelationshipsMultiplicative Relationships

Carrie is packing apples. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 8 bushels of apples?

3 boxes

2 bushels 8 bushels

x boxes

=

Integral multiplicative relationship

Non-integral multiplicative relationship


Multiplicative RelationshipsMultiplicative Relationships


3 boxes

x boxes 8 bushels

2 bushels

=

Non-integral multiplicative relationship

Integral multiplicative relationship


RelationshipsRelationships

Carrie is packing apples for an orchard’s mail order business. It takes 3 boxes to pack 2 bushels of apples.

How many boxes will she need to pack 7 bushels of apples?

What are the multiplicative relationships in this proportional

situation?



• When the multiplicative relationships in a proportional

situation are both non-integral then students have

more difficulty and often revert back to non-

proportional reasoning and strategies.

(Cramer, Post, & Currier, 1993; Karplus,

Polus, & Stage, 1983; VMP OGAP Pilots,

2006)



Structures of Proportionality Structures of Proportionality ProblemsProblems




Stage, 1983)



• Meaning of quantities as defined by the

context and the units (Silver, 2006 Vermont meeting;

VMP OGAP Pilots, 2006)


Case Study: Multiplicative Case Study: Multiplicative RelationshipsRelationships(VMP Pilot Study, Grade 7 Students, (VMP Pilot Study, Grade 7 Students, n=153)n=153)

• Three similar problems administered across a one week

period

• Main difference between the problems is the

multiplicative relationship within and between figures

PILOT 1: A school is enlarging its playground. The dimensions of the new playground are proportional to the dimensions of the old playground. What is the length of the new playground?

40 ft.

80 ft.120 ft.


Student Work Analysis Student Work Analysis (n=6 students)(n=6 students)

• Part 1. Solve each problem.

• Identify the multiplicative relationship within and

between the figures.

• Anticipate difficulties that students might have when

solving each problem.

• Part 2. Discussion with a partner:

• Identify the multiplicative or additive relationship

evidenced in the student response (e.g., x 3,

between figures; + 6, within figures).

• Place your analysis in the cell that corresponds with

the student number and pilot number in the table on

page 3.

• Complete Discussion Questions on page 3.


Multiplicative Relationships Multiplicative Relationships Study: Discussion QuestionsStudy: Discussion Questions

• What did you see that you expected?

• What surprised you?

• What are the implications for instruction and

assessment?


OGAP Study FindingsOGAP Study Findings(2006 Pilot, n=153)(2006 Pilot, n=153)

Multiplicative Relationships within and between figures

Percent of Correct Responses

Pilot 1 Both integral 80%

Pilot 2 One integral, one non-integral 65%

Pilot 3 Both non-integral 35.5%



• Multiplicative relationships in a problem (Karplus, Polus,

& Stage, 1983; VMP OGAP Pilots, 2006)


Stage, 1983)







Context MattersContext Matters

• More familiar contexts tend to be easier for students

than unfamiliar contexts. (Cramer, Post, & Currier,

1993)

• How proportionality shows up in different contexts

impacts difficulty. (Harel, & Behr, 1993)


Context MattersContext Matters

• Which contexts might be more familiar to students?

• How does proportionality show up in these different

contexts?

The scale factor relating two similar rectangles is 1.5. One side of the larger rectangle is 18 inches. How long is the corresponding side of the smaller rectangle?

Nate’s shower uses 4 gallons of water per minute. How much water does Nate use when he takes a 15 minute shower?

A 20-ounce box of Toasty Oats costs $3.00. A 15-ounce box of Toasty Oats costs $2.10. Which box costs less per ounce? Explain your reasoning.






Stage, 1983)







• Ratio

• Rate

• Rate and ratio comparisons

• Missing value

• Scale factor

• Qualitative questions

• Non- proportional

Types of ProblemsTypes of Problems



• Ratio – is a comparison of any two like quantities

(same unit).

The ratio of boys to girls is 1:2. The ratio of people with brown eyes to blue eyes is 1:4.

• Rate – A rate is a special ratio. Its denominator is always 1.

$5.00 per hour

$3.00 per pound

25 horses per acre

Types of ProblemsTypes of Problems


• Relationships - Part : Part or Part : Whole

• Referents - Implied or Explicit

Dana and Jamie ran for student council president at Midvale Middle School. The data below represents the voting results for grade 7.

John says that the ratio of the 7th grade boys who voted for Jamie to the 7th grade students who voted for Jamie is about 1:2. Mary disagreed. Mary says it is about 1:3. Who is correct? Explain your answer.

Types of Problems: Ratio Types of Problems: Ratio

7th Grade Votes

Jamie Dana

Boys 24 40

Girls 49 20


Types of Problems: Ratio Missing Types of Problems: Ratio Missing ValueValue

• Types of Problems: Ratio Missing Value

• Relationships - Part : Part or Part : Whole

• Referents - Implied or Explicit

There are red and blue marbles in a bag. The ratio of red marbles to blue marbles is 1:2. If there are 10 blue marbles in the bag, how many red marbles are in the bag?


Types of Problems: Rate Types of Problems: Rate Missing ValueMissing Value

• What are the meanings of the quantities in this

problem?

• What is the meaning of the answer?

Leslie drove at an average speed of 55 mph for 4 hours. How far did Leslie drive?

Start 1 hour 2 hours 3 hours 4 hours

55 miles


Types of Problems: Rate Types of Problems: Rate ComparisonComparison

• What is the general structure of rate comparison

problems?

A 20-ounce box of Toasty Oats costs $3.00. A 15-ounce box of Toasty Oats costs $2.10. Which box costs less per ounce? Explain your reasoning.

Big Horn Ranch raises 100 horses on 150 acres of pasture. Jefferson Ranch raises 75 horses on 125 acres of pasture. Which ranch has more acres of pasture per horse? Explain your answer using words, pictures, or diagrams.


Case Study - Meaning of the Case Study - Meaning of the QuantitiesQuantities

• In Part I of this case study, you will analyze 4 student

solutions to Ranch problem. The solutions represent

the kinds of “quantity interpretation” errors that

students make when they solve rate comparison

Big Horn Ranch raises 100 horses on 150 acres of pasture. Jefferson Ranch raises 75 horses on 125 acres of pasture. Which ranch has more acres of pasture per horse? Explain your answer using words, pictures, or diagrams.



• In pairs, analyze the student solutions and then

respond to the following.

• What is the evidence that the student may not be

interpreting the meaning of the quantities in the

problem?

• Suggest some questions you might ask each student or

activities you might do to help them understand the

meaning of the quantities in the problem and the

solution.



• What evidence is there

of the student’s

understanding of both

the meaning of the

quantities in the

problem

and in the solution?


Types of Problems: Missing Types of Problems: Missing ValueValue


What is the general structure of a missing

value problem?


A Research FindingA Research Finding(Harel, & Behr,1993)(Harel, & Behr,1993)

• The location of the missing value may affect

performance.

Carrie is packing apples for an orchard’s mail order business. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 7 bushels of apples?

Carrie is packing apples for an orchard’s mail order business. It takes 3 boxes to pack 2 bushels of apples. She needs 7 bushels of apples packed. How many boxes will she need?



Research ApplicationsResearch Applications

Paul’s dog eats 15 pounds of food in 18 days.How long will it take Paul’s dog to eat 45 pound bag of food? Explain your thinking.

Change this problem to make it easier, and then

harder.



Structures of The ProblemsStructures of The Problems

What type of problem is this similarity problem?

Old Playground

90 ft.

630 ft.

New Playground

110 ft.

A school is enlarging its playground. The dimensions of the new playground are proportional to the old playground. What is the measurement of the missing length of the new playground? Show your work.

What type of problem is this similarity problem?




The dimension of 4 rectangles are given below. Which two rectangles are similar?

2” x 8”4” x 10”6” x 12”6” x 15”



What is the general structure of scale factor

problems?


Jack built a scale model of the John Hancock Center. His model was 2.25 feet tall. The John Hancock Center in Chicago is 1476 feet tall.

How many feet of the real building does one foot on the scale model represent? Be sure to show all of your work.

The scale factor relating two similar rectangles is 1.5. One side of the largerrectangle is 18 inches. How long is the corresponding side of the smaller rectangle?


Structures of the ProblemsStructures of the Problems

The scale factor relating two similar rectangles is 1.5. One side of the larger rectangle is 18 inches. How long is the corresponding side of the smaller rectangle?

If a student was unable to solve this problem successfully, what variables would you change to

make it more accessible? Why?


• Students should interact with qualitative predictive and

comparison questions as they are developing their

proportional reasoning….

(Lamon,1993)




Types of Problems: QualitativeTypes of Problems: Qualitative

Kim ran more laps than Bob. Kim ran her laps in less time than Bob ran his laps. Who ran faster?

If Kim ran fewer laps in more time than she did yesterday, would her running speed be:

a) faster; b) slower; c) exactly the same; d) not enough information.

Why do you think researchers suggest these types of

problems as important stepping stones?


• Students need to see examples of proportional and

non-proportional situations so they can determine

when it is appropriate to use a multiplicative solution

strategy.

(Cramer, Post, & Currier,

1993)




Solve these problemsSolve these problems(Cramer, Post, & Currier, (Cramer, Post, & Currier, 1993)1993)

Sue and Julie were running equally fast around a track. Sue started first. When she had run 9 laps, Julie had run 3 laps.When Julie completed 15 laps, how many laps had Sue run?

3 U.S. dollars can be exchanged for 2 British pounds.How many pounds for $21 U.S. dollars?


A Research FindingA Research FindingA Classic Non-proportional A Classic Non-proportional Example* Example*

Sue and Julie were running equally fast around a track. Sue Started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?

•22 out of 33 undergraduate students treated this as a proportional relationship. (Cramer, Post, & Currier, 1993)


A Contrasting Research A Contrasting Research FindingFinding

Three U.S. dollars can be exchanged for 2 British pounds. How many pounds for 21 U.S. dollars?

•Same group – 100% solved it correctly using traditional proportional algorithm.

•No one in the same group could explain why this is a

proportional relationship while the “running laps” is not.


Case Study - Proportional and Case Study - Proportional and Non-proportional?? (VMP Pilot Non-proportional?? (VMP Pilot Study, ???)Study, ???)

Kim and Bob were running equally fast around a track. Kim started first. When she had run 9 laps, Bob had run 3 laps. When Bob completed 15 laps, how many laps had Kim run?

Do student work sort!


Vermont Version Grade 6(n= Vermont Version Grade 6(n= 82)82)

Kim and Bob were running equally fast around a track. Kim started first. When she had run 9 laps, Bob had run 3 laps. When Bob completed 15 laps, how many laps had Kim run?

• 39/82 (48%) solved as a proportion

• 33/82 (40%) solved as an additive situation

• 10/82 (12%) non-starters

What are the instructional implications?


Elements of a Proportional Elements of a Proportional Structure That Affect Structure That Affect PerformancePerformance

• Problem types (comparison, missing value, etc.)

• Mathematical topics/contexts (scaling, similarity, etc.)

• Multiplicative relationships (integral or non-integral)

• Meaning of quantities (ratio relationships and ratio

referents)

• Type of numbers used (integer vs. non-integer)

No wonder proportions are tough to teach and

learn.


What Are the Hallmarks ofWhat Are the Hallmarks ofa Proportional Reasoner?a Proportional Reasoner?

• Recognizes the nature of proportional relationships,

• Finds an efficient method based on multiplicative

reasoning to solve problems,

• Represents the quantities in the solution with units that

reflect the meaning of the quantities for the problem

situation.

• Ultimately, a proportional reasoner should not be

deterred by structures, such as context, problem types,

the quantities in the problems. (Cramer, Post, &

Currier, 1993; Silver, 2006)


Activity: Activity: Administering the OGAP Administering the OGAP Pre-assessmentPre-assessment

• Analyze each of the tasks for:

• Problem types

• Mathematical topics/contexts (scaling, similarity, etc.)

• Multiplicative Relationships (integral or non-integral)

• Ratio Relationships (part:whole or part:part) and

referents (implied or implicit - if applicable)

• Type of numbers used (integer or non-integer)

• Internal Structure (parallel or non-parallel)


General Directions: General Directions: Administering the OGAP Administering the OGAP Pre-assessmentPre-assessment

• Administer the pre-assessment and bring a set of 20 to

25 to our next session

• Calculators are not allowed

• Tips for students

• Time

• Level of teacher assistance

• Do not analyze student work before our next meeting

ses0b o g a p 062209 formatted

Technology

nsf ehr

bushels of apples

bushels xboxes

boxes xboxes

additive relationship

similar problems

figures pilot

numbersharel behr