curves of best fit. using real data to illustrate the behaviour of functions, notably exponential...

CURVES OF BEST FIT

USING REAL DATA TO ILLUSTRATE

THE BEHAVIOUR OF FUNCTIONS,

NOTABLY EXPONENTIAL GROWTH AND

DECAY, AND TO CREATE

MATHEMATICAL MODELS FOR THE

PURPOSES OF INTERPOLATION

AND EXTRAPOLATION.

THE TOOLS:

• Plotting data manually and drawing a line of best fit or sketching a curve of best fit.

• Graphing Calculator to plot and use regression tools to create mathematical models.

• Graphing/spreadsheet programs which can provide regression analysis.

INTEL MICROPROCESSOR HISTORY

y = 0.5117e0.2361x

0

20

40

60

80

100

120

140

0 5 10 15 20 25

YEARS AFTER 1971

CL

OC

K S

PE

ED

MH

z

Series1

Expon. (Series1)

WHERE CAN WE USE THIS IN THE CURRICULUM?

MATH 8 and 9

The Mathematics 8 and 9 curriculum is meant to reinforce the main goals of mathematics education:

• using mathematics confidently to solve problems

• using mathematics to better understand the world around us

• communicating and reasoning mathematically

• appreciating and valuing mathematics

• making connections between mathematics and its

applications

• committing themselves to lifelong learning

• becoming mathematically literate and using mathematics to

participate in, and contribute to, society

MATH 8• Patterns

B1 graph and analyse two-variable linear relations

MATH 9

• PATTERNSB2 graph linear relations, analyse the graph, and interpolate or extrapolate to solve problems

• DATA ANALYSISD3 develop and implement a project plan for the collection, display, and analysis of data by

- formulating a question for investigation- selecting a population or a sample- collecting the data- displaying the collected data in an appropriate manner- drawing conclusions to answer the question

COMING RIGHT AWAY…

PRECALCULUS 12RELATIONS AND FUNCTIONS

12. Graph and analyze polynomial functions (limited to polynomial functions of degree 5).

• 12.7 Solve a problem by modelling a given situation with a polynomial function and analyzing the graph of the function.

Foundations of Mathematics and Pre-calculus (Grade 10)

Relations and Functions Specific Outcomes1. Interpret and explain the relationships among data, graphs and situations.

1.1 Graph, with or without technology, a set of data, and determine the restrictions on the domain and

range.1.2 Explain why data points should or should not be connected on the graph for a situation.1.3 Describe a possible situation for a given graph.1.4 Sketch a possible graph for a given situation.1.5 Determine, and express in a variety of ways, the domain and range of a graph, a set of ordered pairs or a table of values.

Apprenticeship and Workplace Mathematics (Grade 11)

Statistics

Develop statistical reasoning.1. Solve problems that involve creating and interpreting graphs, including:

bar graphs, histograms, line graphs, circle graphs.

1.1 Determine the possible graphs that can be used to represent a given data set, and explain the advantages and disadvantages of each.

1.2 Create, with and without technology, a graph to represent a given

data set.

1.3 Describe the trends in the graph of a given data set.

1.4 Interpolate and extrapolate values from a given graph.

1.7 Solve a contextual problem that involves the interpretation of a

graph.

FOUNDATIONS OF MATHEMATICS 12

1. Represent data, using polynomial functions

1.4 Graph data and determine the polynomial function that best approximates the data.1.5 Interpret the graph of a polynomial function that models a situation, and explain the reasoning.1.6 Solve, using technology, a contextual problem that involves data that is best represented by graphs of polynomial functions, and explain the reasoning.

FOUNDATIONS OF MATHEMATICS 12

2. Represent data, using exponential and logarithmic functions, to solve problems.

2.4 Graph data and determine the exponential or logarithmic function that best approximates the data.2.5 Interpret the graph of an exponential or logarithmic function that models a situation, and explain the reasoning.2.6 Solve, using technology, a contextual problem that involves data that is best represented by graphs of exponential or logarithmic functions, and explain the reasoning.

FOUNDATIONS OF MATHEMATICS 12Mathematics Research Project

1. Research and give a presentation on a current event or an area of interest that involves mathematics.1.1 Collect primary or secondary data (statistical or informational) related to the topic.1.2 Assess the accuracy, reliability and relevance of the primary or secondary data collected by:

-identifying examples of bias and points of view-identifying and describing the data collection methods-determining if the data is relevant-determining if the data is consistent with information

obtained from other sources on the same topic.

1.3 Interpret data, using statistical methods if applicable.1.5 Organize and present the research project, with or without technology.

PRECALCULUS 12RELATIONS AND FUNCTIONS

10. Solve problems that involve exponential and logarithmic equations.

10.5 Solve a problem that involves exponential growth or decay.

10.7 Solve a problem that involves logarithmic scales, such as the Richter scale and the pH scale.

10.8 Solve a problem by modelling a situation with an exponential or a logarithmic equation.

CALCULUS 12AP CALCULUS

BETTER QUESTION:

HOW CAN WE PROVIDE INTERESTING AND MEANINGFUL EXTENSIONS, ESPECIALLY IN THE AREA OF FUNCTIONS AND THEIR APPLICATIONS?

“An increasing emphasis on visualization, primarily in the area of the graphical representation of functions, is an important aspect of Grade 12 mathematics. My experiences indicate various levels of student reluctance to accept and therefore appreciate the utility of these visualizations in understanding course content and in problem-solving.”

GRAPHING DATA AND DRAWING CURVES OF BEST FIT

http://ptaff.ca/soleil

World Population Growth History Charthttp://www.vaughns-1-pagers.com/history/world-population-growth.htm

Africaplus

Madagascar

Asiaplus

USSR /Mideast Europe

NorthAmericaCanada

USMexicoCarrib.

SouthAmerica

plusCentralAmerica

Oceaniaplus

AustraliaNew

ZealandPhilippines

Total(millions)

10000 B.C. 5,000 year

increments

            1

5000 B.C.             5

2000 B.C. 1,000 year

increments

            27

1000 B.C.             50

0 A.D.

500 yearincrements

            200

500 A.D.             300

1000 A.D.             400

1500 A.D.             500

1650 A.D.

 50 yearincrements

  327 103 0.5 12 2 600

1750 A.D.   475 144 3 11 2 750

1800 A.D.   597 192 5.3 19 2 900

1810

10 year

increments  

      7.2     1,000

1820       9.6      

1830       13      

1840       17      

x

PO

PU

LAT

ION

(B

ILLI

ON

S)

YEAR

WORLD POPULATION GROWTH

FROM 1950 TO 2007

THE GRAPHING CALCULATOR

CLOCK SPEED

TRANSISTORS

MIPS

Year MHz Transistors

1971 0.108 2,300

1972 0.2 3,500

1974 2 6,000

1976 2 6,500

1978 10 29,000

1979 8 29,000

1982 12 134,000

1985 33 275,000

1988 33 275,000

1989 50 1200000

1990 25 855,000

1991 33 1200000

1992 33 1400000

1992 66 1200000

1993 66 3.10E+06

1994 200 3.30E+06

1994 100 3.30E+06

1995 133 3.30E+06

1996 233 4.50E+06

Using Graphmatica

TR

AN

SIS

TO

RS

(m

illio

ns)

YEAR

CURVE OF BEST FIT, 1971-1999

CURVE OF BEST FIT, 1971-1999

YEAR

TR

AN

SIS

TO

RS

(m

illio

ns)

Moore’s Law:

• predicted in 1965 that the number of transistors on a microprocessor would double every two years. Starting with 2300 transistors in 1971 on the Intel 4004 chip, …

MICROPROCESSOR TRANSISTOR GROWTH OVER TIME

y = exp(0.35x - 696.13)

TR

AN

SIS

TO

RS

(m

illio

ns

YEAR

MOORE’S LAW – LOGARITHMIC VIEW

Y=0.0023(2^((x-1971)/2))

TR

AN

SIS

TO

RS

(m

illio

ns)

YEAR

PLOT OF COOLING DATA

Using ExcelUsing Excel

INTEL MICROPROCESSOR HISTORY

0

20

40

60

80

100

120

1965 1970 1975 1980 1985 1990 1995 2000

YEAR

CLO

CK S

PEED

MHz

Series1

INTEL MICROPROCESSOR HISTORY

y = 4E-203e0.2361x

0

20

40

60

80

100

120

140

1965 1970 1975 1980 1985 1990 1995 2000

YEAR

CLO

CK S

PEED

MHz

Series1

Expon. (Series1)

INTEL MICROPROCESSOR HISTORY

y = 4E-203e0.2361x

0

200

400

600

800

1000

1200

1400

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

YEAR

CLO

CK S

PEED

MHz

Series1

Expon. (Series1)

INTEL MICROPROCESSOR HISTORY

y = 0.5117e0.2361x

0

20

40

60

80

100

120

140

0 5 10 15 20 25

YEARS AFTER 1971

CLO

CK S

PEED

MHz

Series1

Expon. (Series1)

INTEL MICROPROCESSOR HISTORY

y = 0.5085e0.2352x

0

500

1000

1500

2000

2500

3000

0 10 20 30 40

YEARS AFTER 1971

CLO

CK

SPE

ED M

Hz

Series1

Expon. (Series1)

Microwave Popcorn Microwave Popcorn Popping DensityPopping Density

How about modelling real tidal behaviour…

http://tbone.biol.sc.edu/tide/tideshow.cgi

“One factor contributing to the misuse of regression is that it can take considerably more skill to critique a model than to fit a model.”

R. Dennis Cook; Sanford Weisberg "Criticism and Influence Analysis in Regression", Sociological Methodology, Vol. 13. (1982), pp. 313-

361.

dkamin@vsb.bc.ca