Ses3 Structures PPT 0709

8/14/2019 Ses3 Structures PPT 0709

http://slidepdf.com/reader/full/ses3-structures-ppt-0709 1/23

1

Structures of ProportionalityProblems

Modified October 2008

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

Krisan Stone, VMP

Leslie Ercole, VMP

Marge Petit, Marge Petit Consulting(MPC)

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

2

Mathematical TopicsAnd Contexts

Structures of Problems

Other Structures

OGAP Proportionality Framework

Proportional

Strategies

Transitional

ProportionalStrategies

Non-proportional

Reasoning

Underlying

Issues, Errors,

Misconceptions

Evidence in Student Work to Inform Instruction



2


3

Structure of the problemsthat students solve

Structure refers to –how the problems arebuilt


4

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)



3


5

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 Finding



6

Non-integralmultiplicativerelationship

Integralmultiplicativerelationship

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?

Multiplicative Relationships

3 boxes

2 bushels 8 bushels

x boxes=



5


9

A Research Finding

When the multiplicative relationships in aproportional situation are both non-integral thenstudents have more difficulty and often revert

back to non-proportional reasoning andstrategies. (Cramer, Post, & Currier, 1993; Karplus, Polus, & Stage, 1983; VMP OGAP Pilots, 2006)



10











6


11

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 multiplicativerelationship 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 oldplayground. What is the length of the new playground?

40 ft.

80 ft.

120 ft.


12

Student Work Analysis

(n=6 students)Part 1

• Solve each problem.

• Identify the multiplicative relationship within and betweenthe figures.

• Anticipate difficulties that students might have when solvingeach problem.

Part 2

Discussion with a partner:

• Identify the multiplicative or additive relationship evidencedin the student response (e.g., x 3, between figures; + 6, within figures).

• Place your analysis in the cell that corresponds with thestudent number and pilot number in the table on page 3.

• Complete Discussion Questions on page 3.



8


15










16

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)



9


17

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

• Nate’s shower uses 4 gallons of water per minute. How muchwater 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.

Which contexts might be more familiar to students?

How does proportionality show up in these different contexts?

Context Matters


18











10


19

Types of Problems

• Ratio

• Rate

• Rate and ratio comparisons

• Missing value

• Scale factor

• Qualitative questions

• Non- proportional



20

Ratio – is a comparisonof any two like quantities(same unit).

Rate – A rate is a specialratio. Its denominator isalways 1.

$5.00 per hour $3.00 per pound25 horses per acre

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

Types of Problems



11


21

Relationships - Part : Part or Part : WholeReferents - Implied or Explicit

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

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

Types of Problems: Ratio


2049Girls

4024Boys

DanaJamie

7th Grade Votes


22

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?

Relationships - Part : Part or Part : Whole

Referents - Implied or Explicit

Types of Problems: Ratio Missing Value



12


23

Leslie drove at an average speed of 55 mph for 4 hours.

How far did Leslie drive?

Types of Problems: Rate Missing Value

What are the meanings of the quantities in this problem?

What is the meaning of the answer?


24

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.



13


25

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 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.


26


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.



14


27

What evidence is there of

the student’s understanding

of both the meaning of the

quantities in the problem

and in the solution?



28

What is the general structureof 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?

Types of Problems: Missing Value



15


29

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

A Research Finding



30

Change this problem to make iteasier, and then harder.

Paul’s dog eats 15 pounds of food in18 days. How long will it take Paul’sdog to eat 45 pound bag of food?

Explain your thinking.

Research Applications



16


31

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 isthis similarity problem?


Old Playground New Playground

90 ft.

630 ft.

110 ft.

Structures of The Problems


32

The dimension of 4 rectangles are givenbelow. Which two rectangles are similar?• 2” x 8”• 4” x 10”• 6” x 12”

• 6” x 15”

What type of problem isthis similarity problem?





17


33

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?

What is the general structure of scale factor problems?



Jack built a scale model of the John Hancock Center. His model was2.25 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 modelrepresent? Be sure to show all of your work.


34

If a student was unable to solve this

problem successfully, what variableswould you change to make it moreaccessible? Why?

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?




18


35

Students should interact withqualitative predictive and comparisonquestions as they are developing their

proportional reasoning…. (Lamon,1993)


A Research Finding


36

Why do you think researchers suggest

these types of problems as important

stepping stones?

Types of Problems: Qualitative

• Kim ran more laps than Bob. Kim ran her laps inless time than Bob ran his laps. Who ran faster?

• If Kim ran fewer laps in more time than she didyesterday, would her running speed be: A) faster;

B) slower; C) exactly the same; D) not enoughinformation.



19


37

Students need to see examples of proportional and non-proportionalsituations so they can determine when itis appropriate to use a multiplicativesolution strategy. (Cramer, Post, & Currier, 1993)

A Research Finding



38

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

Sue and Julie wererunning equally fastaround a track. Suestarted first. When shehad run 9 laps, Julie hadrun 3 laps. When Juliecompleted 15 laps, how

many laps had Sue run?

3 U.S. dollars can beexchanged for 2 Britishpounds. How manypounds for $21 U.S.dollars?



20


39

22 out of 33 undergraduate studentstreated this as a proportional relationship.

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

A Research Finding A Classic Non-proportional Example

(Cramer, Post, & Currier, 1993)


40

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

• No one in the same group could explain

why this is a proportional relationshipwhile the “running laps” is not.

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

A Contrasting Research Finding



21


41

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!

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


42

Vermont Version Grade 6(n= 82)

• 39/82 (48%) solved as a proportion

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

• 10/82 (12%) non-starters

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?

What are the instructional implications?



22


43

No wonder proportions aretough to teach and learn.

• 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)

Elements of a Proportional Structure

That Affect Performance


44

• 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 theproblems. (Cramer, Post, & Currier, 1993; Silver, 2006)

What Are the Hallmarks of a

Proportional Reasoner?




45

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)


46

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

Ses3 Structures PPT 0709

Documents

Transcript of Ses3 Structures PPT 0709