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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.
