Transcript Curves of Best Fit

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.
INTEL MICROPROCESSOR HISTORY
140
CLOCK SPEED MHz
120
y = 0.5117e0.2361x
100
80
Series1
Expon. (Series1)
60
40
20
0
0
5
10
15
YEARS AFTER 1971
20
25
• Graphing/spreadsheet programs
which can provide regression
analysis.
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
• PATTERNS
B2 graph linear relations, analyse the graph, and
interpolate or extrapolate to solve problems
• DATA ANALYSIS
D3 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 12
RELATIONS 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 Outcomes
1. 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 12
Mathematics 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 12
RELATIONS 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 12
AP 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 Chart
http://www.vaughns-1-pagers.com/history/world-population-growth.htm
Africa
plus
Madagascar
10000
B.C.
5000
B.C.
2000
B.C.
1000
B.C.
Asia
plus
USSR /
Mideast
Europe
North
America
Canada
US
Mexico
Carrib.
South
America
plus
Central
America
Oceania
plus
Australia
New
Zealand
Philippines
1
5,000 year
increments
5
27
1,000 year
increments
50
200
0 A.D.
500
A.D.
1000
A.D.
300
500 year
increments
400
1500
A.D.
500
1650
A.D.
1750
A.D.
50 year
increments
1800
A.D.
1810
1820
Total
(millions)
327
103
0.5
12
2
600
475
144
3
11
2
750
597
192
5.3
19
2
900
7.2
10 year
increments
9.6
1830
13
1840
17
1,000

WORLD POPULATION GROWTH

FROM 1950 TO 2007


POPULATION
(BILLIONS)















YEAR




x
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
TRANSISTORS (millions)
CURVE OF BEST FIT, 1971-1999
YEAR
TRANSISTORS (millions)
CURVE OF BEST FIT, 1971-1999
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, …
YEAR
TRANSISTORS (millions
MICROPROCESSOR TRANSISTOR
GROWTH OVER TIME
y = exp(0.35x - 696.13)
YEAR
TRANSISTORS (millions)
MOORE’S LAW – LOGARITHMIC VIEW
Y=0.0023(2^((x-1971)/2))
YEAR
PLOT OF COOLING DATA
Using Excel
INTEL MICROPROCESSOR HISTORY
120
CLOCK SPEED MHz
100
80
60
Series1
40
20
0
1965
1970
1975
1980
1985
YEAR
1990
1995
2000
INTEL MICROPROCESSOR HISTORY
140
CLOCK SPEED MHz
120
y = 4E-203e0.2361x
100
80
Series1
Expon. (Series1)
60
40
20
0
1965
1970
1975
1980
1985
YEAR
1990
1995
2000
INTEL MICROPROCESSOR HISTORY
1400
y = 4E-203e0.2361x
CLOCK SPEED MHz
1200
1000
800
Series1
Expon. (Series1)
600
400
200
0
1965
1970
1975
1980
1985
1990
YEAR
1995
2000
2005
2010
INTEL MICROPROCESSOR HISTORY
140
CLOCK SPEED MHz
120
y = 0.5117e0.2361x
100
80
Series1
Expon. (Series1)
60
40
20
0
0
5
10
15
YEARS AFTER 1971
20
25
INTEL MICROPROCESSOR HISTORY
3000
CLOCK SPEED MHz
2500
y = 0.5085e0.2352x
2000
Series1
Expon. (Series1)
1500
1000
500
0
0
10
20
YEARS AFTER 1971
30
40
Microwave Popcorn
Popping 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.
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