Ses3 Structures 0609 V2 Formatted

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)

Structures of Proportionality Problems

Krisan Stone, VMPLeslie Ercole, VMPMarge Petit, Marge Petit Consulting (MPC)

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

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


• Structure refers to – how the problems are built

Structure of the problems that students solve


Structures of Proportionality Structures of Proportionality Problems 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)


• 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




3 boxes

x boxes 8 bushels

2 bushels=

Non-integral multiplicative relationship

Integral multiplicative relationship



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





• 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%










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.










• 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 pound25 horses per acre

Types of ProblemsTypes of Problems


• Relationships - Part : Part or Part : Whole• Referents - Implied or ExplicitDana 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 VotesJamie Dana

Boys 24 40Girls 49 20


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

• Relationships - Part : Part or Part : Whole

• Referents - Implied or ExplicitThere 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?




• 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?



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-startersWhat 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 What Are the Hallmarks of a Proportional Reasoner?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)


Activity: Analyzing Pre-Activity: Analyzing Pre-assessment Tasksassessment 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)


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

Ses3 Structures 0609 V2 Formatted

Business

vmp ogap pilots

john hancock

running equally

scale factor

proportionality

important

vmp pilot

7th grade