Formulating measuresFormulating measures

toward modeling in the K-12 curriculumtoward modeling in the K-12 curriculum

Judah L. SchwartzJudah L. Schwartz

Department of Education &Department of Education &

Department of Physics & AstronomyDepartment of Physics & Astronomy

Tufts UniversityTufts University

Some vocabularySome vocabulary

ModelsModels are ways of relating the are ways of relating the behavior of different measures to behavior of different measures to one another –one another –

Some vocabularySome vocabulary

ModelsModels are ways of relating the are ways of relating the behavior of different measures to behavior of different measures to one another –one another –

For example:For example:

force needed to stretch a spring is directly proportional to the force needed to stretch a spring is directly proportional to the

amount the spring is stretchedamount the spring is stretched

Some vocabularySome vocabulary

ModelsModels are ways of relating the are ways of relating the behavior of different measures to behavior of different measures to one another –one another –

For example:For example: force needed to stretch a spring is directly proportional to force needed to stretch a spring is directly proportional to

the amount the spring is stretchedthe amount the spring is stretched

density of a block of matter is inversely proportional to its density of a block of matter is inversely proportional to its temperaturetemperature

Some vocabularySome vocabulary

ModelsModels are ways of relating the behavior are ways of relating the behavior of different measures to one another –of different measures to one another –

For example:For example:

force needed to stretch a spring is directly proportional to the amount the force needed to stretch a spring is directly proportional to the amount the

spring is stretchedspring is stretched

density of a block of matter is inversely proportional to its temperaturedensity of a block of matter is inversely proportional to its temperature

population of a preyed-upon species depends on birth rates, death rates population of a preyed-upon species depends on birth rates, death rates and reproduction rates of both predator and preyand reproduction rates of both predator and prey

MeasuresMeasures are constructs that we are constructs that we devise to quantify the amount or devise to quantify the amount or degree of a property of interestdegree of a property of interest

MeasuresMeasures are constructs that we are constructs that we devise to quantify the amount or devise to quantify the amount or degree of a property of interestdegree of a property of interest

For example:For example:

forceforce needed to stretch a spring is directly proportional to the needed to stretch a spring is directly proportional to the

amount theamount the spring is stretchedspring is stretched

MeasuresMeasures are constructs that we are constructs that we devise to quantify the amount or devise to quantify the amount or degree of a property of interestdegree of a property of interest

For example:For example:

forceforce needed to stretch a spring is directly proportional to the needed to stretch a spring is directly proportional to the

amount theamount the spring is stretchedspring is stretched

densitydensity of a block of matter is inversely proportional to its of a block of matter is inversely proportional to its temperaturetemperature

MeasuresMeasures are constructs that we are constructs that we devise to quantify the amount or devise to quantify the amount or degree of a property of interestdegree of a property of interest

For example:For example:

forceforce needed to stretch a spring is directly proportional to the needed to stretch a spring is directly proportional to the

amount theamount the spring is stretchedspring is stretched

densitydensity of a block of matter is inversely proportional to its of a block of matter is inversely proportional to its temperaturetemperature

populationpopulation of a preyed-upon species depends on of a preyed-upon species depends on birth ratesbirth rates, , death ratesdeath rates and and reproduction ratesreproduction rates of both predator and prey of both predator and prey

The problem of modeling…The problem of modeling…

……is two-fold (at least!)is two-fold (at least!)

decide on measures that one believes are relevant to the decide on measures that one believes are relevant to the description and/or explanation of the phenomenon in description and/or explanation of the phenomenon in questionquestion

The problem of modeling…The problem of modeling…

……is two-fold (at least!)is two-fold (at least!)

decide on measures that one believes are relevant to the decide on measures that one believes are relevant to the description and/or explanation of the phenomenon in description and/or explanation of the phenomenon in questionquestion

thenthen

propose a theory for how those measures might be related to propose a theory for how those measures might be related to one anotherone another

Modeling in the K-12 Modeling in the K-12 curriculumcurriculum

tends to focus on relations tends to focus on relations among already given measuresamong already given measures

N.B. measures may be given by N.B. measures may be given by namename or or name & valuename & value

What is the What is the massmass of of 150 cc150 cc of aluminum of aluminum [[densitydensity 2.3 gm/cc2.3 gm/cc]?]?

What is the What is the massmass of of 150 cc150 cc of aluminum of aluminum [[densitydensity 2.3 gm/cc2.3 gm/cc]?]?

What is the What is the average speedaverage speed of a car that of a car that goes goes 60 mi/hr60 mi/hr for for 30 miles30 miles and then and then 30 mi/hr30 mi/hr for the next for the next 30 miles30 miles??

What is the What is the massmass of of 150 cc150 cc of aluminum of aluminum [[densitydensity 2.3 gm/cc2.3 gm/cc]?]?

What is the What is the average speedaverage speed of a car that of a car that goes goes 60 mi/hr60 mi/hr for for 30 miles30 miles and then and then 30 mi/hr30 mi/hr for the next for the next 30 miles30 miles??

Share Share ½ a pizza½ a pizza among among 3 people3 people

What is the What is the massmass of of 150 cc150 cc of aluminum of aluminum [[densitydensity 2.3 gm/cc2.3 gm/cc]?]?

What is the What is the average speedaverage speed of a car that of a car that goes goes 60 mi/hr60 mi/hr for for 30 miles30 miles and then and then 30 30 mi/hrmi/hr for the next for the next 30 miles30 miles??

Share Share ½ a pizza½ a pizza among among 3 people3 people

John’s ageJohn’s age is twice is twice Mary’s ageMary’s age six years six years agoago

formulating measures…formulating measures…

……a step on the road to proposing a step on the road to proposing modelsmodels

Where do measures come Where do measures come from?from?

observingobserving

Where do measures come Where do measures come from?from?

observingobserving comparingcomparing

Where do measures come Where do measures come from?from?

observingobserving comparingcomparing orderingordering

the compulsion to the compulsion to orderorder things…things…

See, for exampleSee, for example

http://www.time.com/time/specials/2007/top10/0,30576,1686204,00.htmlhttp://www.time.com/time/specials/2007/top10/0,30576,1686204,00.html

Does ordering imply a metric ? Does ordering imply a metric ? ??

No !No !

– – e.g., Apgar score for newbornse.g., Apgar score for newborns

but if there is a metric…but if there is a metric…

Where do measures come Where do measures come from?from?

observingobserving comparingcomparing orderingordering making measurementsmaking measurements

Where do measures come Where do measures come from?from?

observingobserving comparingcomparing orderingordering making measurementsmaking measurements analyzing dataanalyzing data

Kinds of measuresKinds of measures

Quotient measuresQuotient measures

Kinds of measuresKinds of measures

Quotient measuresQuotient measures Product measuresProduct measures

Kinds of measuresKinds of measures

Quotient measuresQuotient measures Product measuresProduct measures Additive measuresAdditive measures

Quotient measuresQuotient measures

A composite measure C that depends on A composite measure C that depends on two component measures A and B such two component measures A and B such thatthat

if A if A then C then C - - moremore A A moremore C C if B if B then C then C - - moremore B B lessless C C

Examples : density, speed, concentrationExamples : density, speed, concentration

Could Could

C = A – BC = A – B

be a good measure ?be a good measure ?

Quotient measures areQuotient measures are

usually intensive quantitiesusually intensive quantities

often dimensionlessoften dimensionless

messing around with 3 kinds of messing around with 3 kinds of stuff…stuff…

aluminumaluminum pinepine polystyrenepolystyrene

volume(ccvolume(cc))

mass(gm)mass(gm) volume(ccvolume(cc))

mass(gmmass(gm))

volume(cvolume(cc)c)

mass(gmmass(gm))

00 00 00 00 00 00

120120 324324 1515 8.38.3 4141 43.143.1

7373 197197 6262 34.134.1 103103 108.2108.2

3131 83.783.7 127127 69.969.9 7979 8383

110110 297297 9090 49.549.5 6767 70.470.4

……and plotting those dataand plotting those data

mass vs. volume

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120 140

volume (cc)

ma

ss

(g

m)

aluminum

pine

polystyrene

a measure seems to a measure seems to emerge!emerge!

The data for a given material seem to lie along an The data for a given material seem to lie along an undrawnundrawn straight line – different materials, different straight line – different materials, different undrawn straight linesundrawn straight lines

a measure seems to a measure seems to emerge!emerge!

The data for a given material seem to lie along an The data for a given material seem to lie along an undrawnundrawn straight line – different materials, different straight line – different materials, different undrawn straight linesundrawn straight lines

Can we assume that an as yet unmeasured piece of Can we assume that an as yet unmeasured piece of aluminum will have a mass and volume that lie on the aluminum will have a mass and volume that lie on the undrawn aluminum line? undrawn aluminum line?

Why or why not Why or why not ??

Product measuresProduct measures A composite measure C that depends on A composite measure C that depends on

two component measures A and B such two component measures A and B such thatthat

if A if A then C then C - - moremore A A moremore C C if B if B then C then C - - moremore B B moremore C C

Examples : momentum, Cartesian products Examples : momentum, Cartesian products [passenger-miles, kilowatt-hours, etc.][passenger-miles, kilowatt-hours, etc.]

Could Could

C = A + BC = A + B

be a good measure ?be a good measure ?

Product measures areProduct measures are

usually extensive quantitiesusually extensive quantities

rarely dimensionlessrarely dimensionless

How “big” a job?How “big” a job?Name of Name of

jobjob# people # people

availableavailableNature of Nature of

jobjobEstimated Estimated

time - time - hourshours

Other Other needed needed

infoinfo

Joe’s yardJoe’s yard 11 Remove debrisRemove debris 3030

Sally’s atticSally’s attic 22 Haul out boxesHaul out boxes 1818

Sam’s basementSam’s basement 33 Remove debris Remove debris & clean& clean

1111

Rose’s Rose’s storeroomstoreroom

44 Remove boxes – Remove boxes – rearrange rearrange contentscontents

88

School locker School locker roomsrooms

55 Clean out & Clean out & paint lockerspaint lockers

77

Town archivesTown archives 66 Sort-keep-Sort-keep-discard old discard old

recordsrecords

66

……one measure of “size” of job (in person-hrs)one measure of “size” of job (in person-hrs)

JoeJoe 30 person-hrs30 person-hrs Sally Sally 36 person-hrs36 person-hrs Sam Sam 33 person-hrs33 person-hrs Rose Rose 32 person-hrs32 person-hrs SchoolSchool 35 person-hrs35 person-hrs Town Town 36 person-hrs36 person-hrs

……which @ $10./person-hr “size” of job (in $) iswhich @ $10./person-hr “size” of job (in $) is

JoeJoe $ 300.$ 300. Sally Sally $ 360.$ 360. Sam Sam $ 330.$ 330. Rose Rose $ 320.$ 320. SchoolSchool $ 350.$ 350. Town Town $ 360.$ 360.

Town = Sally > School > Sam > Rose > JoeTown = Sally > School > Sam > Rose > Joe

Additive measuresAdditive measures

If the company charged If the company charged

$ 25 / person sent to the job site$ 25 / person sent to the job site

&&

$ 20 / hour the job takes$ 20 / hour the job takes

““Size” of job in $ Size” of job in $

= $ 25/person * P persons + $20 / hour * T hours= $ 25/person * P persons + $20 / hour * T hours

JoeJoe $ 625.$ 625. SallySally $ 410.$ 410. SamSam $ 295.$ 295. RoseRose $ 260.$ 260. SchoolSchool $ 265.$ 265. TownTown $ 270.$ 270.

Joe > Sally > Sam > Town > School > RoseJoe > Sally > Sam > Town > School > Rose

some interesting problems some interesting problems

[[that turn out to be mostly quotient measuresthat turn out to be mostly quotient measures]]

Square-NessSquare-Ness Below is a collection of rectangles. Below is a collection of rectangles.

1. Which of the rectangles is the “squarest”? 1. Which of the rectangles is the “squarest”? 2. Arrange the rectangles in order of square-ness from 2. Arrange the rectangles in order of square-ness from

most to least square. most to least square. 3. Devise a measure of square-ness, that allows you to 3. Devise a measure of square-ness, that allows you to

order any collection of rectangles in order of square-ness. order any collection of rectangles in order of square-ness. 4. Devise a second measure of square-ness and discuss 4. Devise a second measure of square-ness and discuss

the advantages and disadvantages of each of your the advantages and disadvantages of each of your measures. measures.

Which is more crowded?Which is more crowded?

How much more crowded?How much more crowded?

Smooth-ness of spheresSmooth-ness of spheres

A chestnut is roughly spherical A chestnut is roughly spherical

So is a tennis ball, a ping-pong ball, So is a tennis ball, a ping-pong ball, as well as an orange and as well as an orange and the Earth. the Earth.

Devise a measure of smooth-ness of Devise a measure of smooth-ness of spheres that allows you to assign a smooth-spheres that allows you to assign a smooth-ness value to spherical objects.ness value to spherical objects.

Devise a measure of smooth-ness of Devise a measure of smooth-ness of spheres that allows you to assign a smooth-spheres that allows you to assign a smooth-ness value to spherical objects.ness value to spherical objects.

Plot a graph of your measure of smooth-Plot a graph of your measure of smooth-ness showing how the value of smooth-ness ness showing how the value of smooth-ness varies from least smooth to most smooth. varies from least smooth to most smooth.

Devise a measure of smooth-ness of Devise a measure of smooth-ness of spheres that allows you to assign a smooth-spheres that allows you to assign a smooth-ness value to spherical objects.ness value to spherical objects.

Plot a graph of your measure of smooth-Plot a graph of your measure of smooth-ness showing how the value of smooth-ness ness showing how the value of smooth-ness varies from least smooth to most smooth. varies from least smooth to most smooth.

Is there a minimum smooth-ness? A Is there a minimum smooth-ness? A maximum smooth-ness?maximum smooth-ness?

Devise a measure of smooth-ness of Devise a measure of smooth-ness of spheres that allows you to assign a smooth-spheres that allows you to assign a smooth-ness value to spherical objects.ness value to spherical objects.

Plot a graph of your measure of smooth-Plot a graph of your measure of smooth-ness showing how the value of smooth-ness ness showing how the value of smooth-ness varies from least smooth to most smooth. varies from least smooth to most smooth.

Is there a minimum smooth-ness? A Is there a minimum smooth-ness? A maximum smooth-ness?maximum smooth-ness?

What would happen to the value of the What would happen to the value of the smooth-ness of the Earth if its radius were smooth-ness of the Earth if its radius were twice as large but the mountains remained twice as large but the mountains remained the same height?the same height?

……and some othersand some others

sharp-ness of bends in roadssharp-ness of bends in roads

……and some othersand some others

sharp-ness of bends in roadssharp-ness of bends in roads runny-ness of liquids runny-ness of liquids

……and some othersand some others

sharp-ness of bends in roadssharp-ness of bends in roads runny-ness of liquids runny-ness of liquids equilateral-ness of trianglesequilateral-ness of triangles

……and some othersand some others

sharp-ness of bends in roadssharp-ness of bends in roads runny-ness of liquids runny-ness of liquids equilateral-ness of trianglesequilateral-ness of triangles rough-ness of sandpaper rough-ness of sandpaper

……and some othersand some others

sharp-ness of bends in roadssharp-ness of bends in roads runny-ness of liquids runny-ness of liquids equilateral-ness of trianglesequilateral-ness of triangles rough-ness of sandpaper rough-ness of sandpaper square-ness of parallograms square-ness of parallograms

……and some othersand some others

sharp-ness of bends in roadssharp-ness of bends in roads runny-ness of liquids runny-ness of liquids equilateral-ness of trianglesequilateral-ness of triangles rough-ness of sandpaper rough-ness of sandpaper square-ness of parallograms square-ness of parallograms sticky-ness of tapesticky-ness of tape

……and some othersand some others

sharp-ness of bends in roadssharp-ness of bends in roads runny-ness of liquids runny-ness of liquids equilateral-ness of trianglesequilateral-ness of triangles rough-ness of sandpaper rough-ness of sandpaper square-ness of parallograms square-ness of parallograms sticky-ness of tapesticky-ness of tape smooth-ness of plotted datasmooth-ness of plotted data

……and finallyand finally

[[formulating a product measureformulating a product measure]]

Suppose you know for a variety of kinds of Suppose you know for a variety of kinds of airplanesairplanes

Suppose you know for a variety of kinds of Suppose you know for a variety of kinds of airplanesairplanes

the number of passengers that can be the number of passengers that can be carriedcarried

Suppose you know for a variety of kinds of Suppose you know for a variety of kinds of airplanesairplanes

the number of passengers that can be the number of passengers that can be carriedcarried

average speed in mi/hraverage speed in mi/hr

Suppose you know for a variety of kinds of Suppose you know for a variety of kinds of airplanesairplanes

the number of passengers that can be the number of passengers that can be carriedcarried

average speed in mi/hraverage speed in mi/hr average flight length in miaverage flight length in mi

Suppose you know for a variety of kinds of Suppose you know for a variety of kinds of airplanesairplanes

the number of passengers that can be the number of passengers that can be carriedcarried

average speed in mi/hraverage speed in mi/hr average flight length in miaverage flight length in mi average fuel consumption in gallons/hraverage fuel consumption in gallons/hr

Suppose you know for a variety of kinds of Suppose you know for a variety of kinds of airplanesairplanes

the number of passengers that can be the number of passengers that can be carriedcarried

average speed in mi/hraverage speed in mi/hr average flight length in miaverage flight length in mi average fuel consumption in gallons/hraverage fuel consumption in gallons/hr operating cost in $/hroperating cost in $/hr

define a measure of “efficiency” of airplanesdefine a measure of “efficiency” of airplanes