Ses3 Structures Ppt 0709

Structures of Proportionality Problems

Krisan Stone, VMP Leslie Ercole, VMP Marge Petit, Marge Petit Consulting (MPC)

Modified October 2008 Original materials created as a part of the Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)

OGAP Proportionality Framework Structures of Problems Mathematical Topics And Contexts

Other Structures

Evidence in Student Work to Inform Instruction Proportional Strategies

Transitional Proportional Strategies

Non-proportional Reasoning

Underlying Issues, Errors, Misconceptions

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Structure of the problems that students solve Structure refers to – how the problems are built

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Structures of Proportionality Problems • 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)

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A Research Finding 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)

OGAP Proportionality Framework

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Multiplicative 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? Non-integral multiplicative relationship

3 boxes 2 bushels

=

x boxes

Integral multiplicative relationship

8 bushels

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Multiplicative 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? Non-integral multiplicative relationship

Integral multiplicative relationship

3 boxes x boxes

=

2 bushels 8 bushels

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Multiplicative Relationships What are the multiplicative relationships in this proportional situation?

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?

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A Research Finding 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)

OGAP Proportionality Framework

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Structures of Proportionality Problems • 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)

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Case Study - Multiplicative Relationships (VMP Pilot Study, Grade 7 Students, n=153)

• Three similar problems administered across a one week period (Monday, pilot 1; Wednesday, pilot 2; and Friday, pilot 3)

• 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.

120 ft. 80 ft.

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Student Work Analysis (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. October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)

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Multiplicative Relationships Study: Discussion Questions •

What did you see that you expected?

•

What surprised you?

•

What are the implications for instruction and assessment?

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OGAP Study Findings (2006 Pilot, n=153)

Multiplicative Relationships Percent of Correct within and between figures Responses Pilot 1

Both integral

80%

Pilot 2

One integral, one non-integral

65%

Pilot 3

Both non-integral

35.5%

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Structures of Proportionality Problems • 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)

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Context 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)

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Context 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.

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Structures of Proportionality Problems • 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)

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Types of Problems • • • • • • •

Ratio Rate Rate and ratio comparisons Missing value Scale factor Qualitative questions Non- proportional OGAP Proportionality Framework

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Types 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

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Types of Problems: Ratio Relationships - Part : Part or Part : Whole Referents - Implied or Explicit

OGAP Proportionality Framework

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

Dana

Boys

24

40

Girls

49

20

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. October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)

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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?

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Types of Problems: Rate Missing 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?

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Types of Problems: Rate Comparison 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.

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Case Study - Meaning of the Quantities 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 problems.

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. October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)

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Case Study - Meaning of the Quantities 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. October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)

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Case Study - Meaning of the Quantities What evidence is there of the student’s understanding of both the meaning of the quantities in the problem and in the solution?

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Types of Problems: Missing Value What is the general structure of a missing value problem?

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?

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A Research Finding The location of the missing value may affect performance. (Harel, & Behr,1993)

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?

Internal Structure

OGAP Proportionality Framework

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Research 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.

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Structures of The Problems 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. New Playground

Old Playground 90 ft.

110 ft. 630 ft.

What type of problem is this similarity problem?

OGAP Proportionality Framework

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Structures of The Problems 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” OGAP Proportionality Framework October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)

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Structures of The Problems 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 larger rectangle is 18 inches. How long is the corresponding side of the smaller rectangle? OGAP Proportionality Framework October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)

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Structures 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? October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)

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A Research Finding Students should interact with qualitative predictive and comparison questions as they are developing their proportional reasoning…. (Lamon,1993)

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Types of Problems: Qualitative Why do you think researchers suggest these types of problems as important stepping stones?

• 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.

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A Research Finding 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)

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Solve these problems (Cramer, Post, & Currier, 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?

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A Research Finding A Classic Non-proportional Example (Cramer, Post, & Currier, 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? 22 out of 33 undergraduate students treated this as a proportional relationship. October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)

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A Contrasting Research Finding 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.

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Case Study - Proportional and Non-proportional?? (VMP Pilot 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! October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)

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Vermont Version Grade 6(n= 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? October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)

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Elements of a Proportional Structure That Affect Performance • 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.

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What Are the Hallmarks of 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)

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Activity: Analyzing Pre-Assessment Tasks 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)

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General Directions: Administering the OGAP Pre-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

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