Formulating measures

1. Formulating measures toward modeling in the K-12 curriculum Judah L. Schwartz Department of Education & Department of Physics & Astronomy Tufts University

2. Some vocabulary • Models are ways of relating the behavior of different measures to one another –

3. Some vocabulary • Models are ways of relating the behavior of different measures to one another – For example: • force needed to stretch a spring is directly proportional to the amount the spring is stretched

4. Some vocabulary • Models are ways of relating the behavior of different measures to one another – For example: • force needed to stretch a spring is directly proportional to the amount the spring is stretched • density of a block of matter is inversely proportional to its temperature

5. Some vocabulary • Models are ways of relating the behavior of different measures to one another – For example: • force needed to stretch a spring is directly proportional to the amount the spring is stretched • density of a block of matter is inversely proportional to its temperature • population of a preyed-upon species depends on birth rates, death rates and reproduction rates of both predator and prey

6. Measures are constructs that we devise to quantify the amount or degree of a property of interest

7. Measures are constructs that we devise to quantify the amount or degree of a property of interest For example: • force needed to stretch a spring is directly proportional to the amount thespring is stretched

8. Measures are constructs that we devise to quantify the amount or degree of a property of interest For example: • forceneeded to stretch a spring is directly proportional to the amount thespring is stretched • density of a block of matter is inversely proportional to its temperature

9. Measures are constructs that we devise to quantify the amount or degree of a property of interest For example: • force needed to stretch a spring is directly proportional to the amount thespring is stretched • density of a block of matter is inversely proportional to its temperature • population of a preyed-upon species depends on birth rates, death rates and reproduction rates of both predator and prey

10. The problem of modeling… …is two-fold (at least!) decide on measures that one believes are relevant to the description and/or explanation of the phenomenon in question

11. The problem of modeling… …is two-fold (at least!) decide on measures that one believes are relevant to the description and/or explanation of the phenomenon in question then propose a theory for how those measures might be related to one another

12. Modeling in the K-12 curriculum • tends to focus on relations among already given measures • N.B. measures may be given by name or name & value

13. What is the mass of 150 cc of aluminum [density2.3 gm/cc]?

14. What is the mass of 150 cc of aluminum [density2.3 gm/cc]? What is the average speed of a car that goes 60 mi/hr for 30 miles and then 30 mi/hr for the next 30 miles?

15. What is the mass of 150 cc of aluminum [density2.3 gm/cc]? What is the average speed of a car that goes 60 mi/hr for 30 miles and then 30 mi/hr for the next 30 miles? Share ½ a pizza among 3 people

16. What is the mass of 150 cc of aluminum [density2.3 gm/cc]? What is the average speed of a car that goes 60 mi/hr for 30 miles and then 30 mi/hr for the next 30 miles? Share ½ a pizza among 3 people John’s age is twice Mary’s age six years ago

17. formulating measures… …a step on the road to proposing models

18. Where do measures come from? • observing

19. Where do measures come from? • observing • comparing

20. Where do measures come from? • observing • comparing • ordering

21. the compulsion to order things… See, for example http://www.time.com/time/specials/2007/top10/0,30576,1686204,00.html

22. Does ordering imply a metric ? ?

23. No ! – e.g., Apgar score for newborns

24. but if there is a metric…

25. Where do measures come from? • observing • comparing • ordering • making measurements

26. Where do measures come from? • observing • comparing • ordering • making measurements • analyzing data

27. Kinds of measures • Quotient measures

28. Kinds of measures • Quotient measures • Product measures

29. Kinds of measures • Quotient measures • Product measures • Additive measures

30. Quotient measures • A composite measure C that depends on two component measures A and B such that • if A  then C  - more A  more C • if B  then C  - more B  less C • Examples : density, speed, concentration

31. Could C = A – B be a good measure ?

32. Quotient measures are • usually intensive quantities • often dimensionless

33. messing around with 3 kinds of stuff…

34. …and plotting those data

35. a measure seems to emerge! • The data for a given material seem to lie along an undrawn straight line – different materials, different undrawn straight lines

36. a measure seems to emerge! • The data for a given material seem to lie along an undrawn straight line – different materials, different undrawn straight lines • Can we assume that an as yet unmeasured piece of aluminum will have a mass and volume that lie on the undrawn aluminum line? Why or why not ?

37. Product measures • A composite measure C that depends on two component measures A and B such that • if A  then C  - more A  more C • if B  then C  - more B  more C • Examples : momentum, Cartesian products [passenger-miles, kilowatt-hours, etc.]

38. Could C = A + B be a good measure ?

39. Product measures are • usually extensive quantities • rarely dimensionless

40. How “big” a job?

41. …one measure of “size” of job (in person-hrs) • Joe 30 person-hrs • Sally 36 person-hrs • Sam 33 person-hrs • Rose 32 person-hrs • School 35 person-hrs • Town 36 person-hrs

42. …which @ $10./person-hr “size” of job (in $) is • Joe $ 300. • Sally $ 360. • Sam $ 330. • Rose $ 320. • School $ 350. • Town $ 360. Town = Sally > School > Sam > Rose > Joe

43. Additive measures • If the company charged $ 25 / person sent to the job site & $ 20 / hour the job takes

44. “Size” of job in $ = $ 25/person * P persons + $20 / hour * T hours • Joe $ 625. • Sally $ 410. • Sam $ 295. • Rose $ 260. • School $ 265. • Town $ 270. Joe > Sally > Sam > Town > School > Rose

45. some interesting problems [that turn out to be mostly quotient measures]

46. Square-Ness • Below is a collection of rectangles. • 1. Which of the rectangles is the “squarest”? • 2. Arrange the rectangles in order of square-ness from most to least square. • 3. Devise a measure of square-ness, that allows you to order any collection of rectangles in order of square-ness. • 4. Devise a second measure of square-ness and discuss the advantages and disadvantages of each of your measures.

47. Which is more crowded?How much more crowded?

48. Smooth-ness of spheres • A chestnut is roughly spherical • So is a tennis ball, a ping-pong ball, as well as an orange and the Earth.

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

50. Devise a measure of smooth-ness of spheres that allows you to assign a smooth-ness value to spherical objects. • Plot a graph of your measure of smooth-ness showing how the value of smooth-ness varies from least smooth to most smooth.