Download modelling as real world problem solving: matching rhetoric

MODELLING AS REAL WORLD PROBLEM
SOLVING: MATCHING RHETORIC WITH
ACTION (PART A)
PETER GALBRAITH
The University of Queensland
[email protected]
Curricular Goal (Australian Curriculum)
“mathematics aims to ensure that students are
confident, creative users and communicators of
mathematics, able to investigate, represent and
interpret situations in their personal and work lives
and as active citizens”.
http://www.australiancurriculum.edu.au/Mathematics/Aims
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●
Such abilities can only develop if mathematical
experiences are drawn genuinely from these same
areas of personal, vocational, and civic
contexts.
●
Whatever other useful purposes they serve, text book
problems cannot meet this need.
Purpose: The intention is to highlight components of
modelling activity essential to the above curricular goal
using illustrative examples - including how some students
have approached a modelling task.
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Challenge: Much work carried out under the banner of
mathematical modelling, skates over the aspect most
crucial to developing expertise – formulating a
mathematical problem from a messy real world
context.
This essential component is also involved elsewhere
in the modelling process
– justifying assumptions
– selecting the mathematics
– interpreting mathematical results
– evaluating whether an alleged solution fits the
needs of the original problem.
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Modelling as real world problem solving
●
perspective and authority derives from the use
of mathematics to model problems in fields
outside education. ( e.g. Pollak
(telecommunications), Burkhardt (physics))
Modelling process (after Pedley, 2005)
1. Understand real problem situation
2. Frame appropriate math question
3. Formulate a model (assumptions etc)
4. Analyse the model
5. Compare math outcomes with reality
6. Modify and repeat until adequate soln found.
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Example 1.
Geoffrey Taylor’s analysis of the atomic bomb
test in New Mexico in 1945 followed the
publication in Life magazine in 1947, of photos
of the expanding blast wave taken over a
succession of small time intervals.
Blast wave photo.
http://en.wikipedia.org/wiki/Nuclear_weapon_yield
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Approach:
From the photographs Taylor estimated the energy of the
blast, using mathematics accessible to senior high school
mathematics students.
Fundamental assumption is that the radius of the blast
wave (R) depends on the time elapsed since the explosion
(t), the instantaneous energy released (E), and the density
of air (ρ).
Thus R = C (ta Eb ρc ), where C is a dimensionless
constant .
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● R = C (ta Eb ρc ), where C is a dimensionless constant.
●
Now dim R = [R] = L, [t] = T, [E] = ML2 T-2, and [ρ] = ML-3
Thus dimensionally we need: L1 = M(b+c) L(2b-3c) T(a-2b).
●
Equating dimensions: b + c = 0; 2b - 3c = 1; a – 2b = 0;
and hence a = 2/5, b = 1/5, and c = -1/5. So R = C (Et2/ρ)1/5,
where C ≈ 1 from known data.
●
Photo contains a length scale, and label showing time at
t = 0.025 (s). Expand the photo from the link, take
measurements, and use the given scale to estimate the
radius - gives R ≈ 132 m.
●
Given that the density of air is 1.2 kg/m3 obtain a value for E
of about 7.7x1013 joule – an energy equivalent of about 16.7
kilotonnes of TNT.
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Aftermath:
Americans:
Taylor’s
subsequent
letter
to
the
“I see that the atomic bomb you detonated had a
power equivalent of about 17 kilotons of TNT”,
Outcome: great consternation at the revelation of
supposedly classified information – and all from
photos in a popular magazine!
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Reflection on modelling matters
● striking example of a modelling problem, identified
and developed from an article in a magazine.
● reinforces that the popular press is a fertile source
of problems, many of which are suitable for school
students.
● mathematics needed is no more than secondary
level, yet the problem was a very significant one.
● Again we focus attention on the formulation stage;
for it was there that deep understanding of the
context was essential;
- knowing which three input variables were sufficient,
- the assumption C ≈1 was based on knowledge of a
combination of factors previously assessed
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Joe
Hugh
Bob
Rudi
F
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Real World Problem
• Climate change is the term used to describe the
changing nature of the world's weather patterns.
Many meteorologists have stated that climate
change could potentially result in detrimental
impacts upon our current way of life, through
rising global temperatures and sea levels. The
increasing temperatures due to climate change
have been reported to cause rising sea levels
due to the expansion of water around the world
due to heat. The concept that the climate is
changing is strongly supported by evidence
obtained by experts.
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Mathematical Question
• At what point in time will the Q1 lobby,
which is 3m above sea level, be
submerged due to sea level rises, and
what will the mean maximum temperature
of that month in Surfers Paradise be at
that point in time?
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Defining the x-variable
• The independent variable, x, is the time in months,
in one month intervals, since January 1938.
Therefore: x = 0 = January 1938 the first month of
temperature being recorded.
x = 1 = February 1938 the second month
of the temperature being recorded.
x = 2 = March 1938, the third month of the
temperature being recorded.
The range of values for the number of months since
January 1938 in the data set is 0<x<635.
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Defining the y-variable
• The dependent variable, y, is the mean
maximum temperature of Surfers
Paradise over one month measured in
degrees Celsius. The range of the y
values for the mean maximum
temperature of Surfers paradise is
19.1<y<30.6.
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Assumptions
•
•
•
•
•
It is assumed that the influence of rainfall and evaporation in the ocean
is negligible due to the fact that this would only contribute to the water
cycle which would in turn feed back into the oceans. If this is not the
case, the process used to determine the increase in sea level from the
temperature increase would not be able to be used to produce accurate
results.
It is assumed that the whole surface of Surfer's Paradise is a flat plane
3 metres above sea level that contains no obstructions to the path of
the ocean as it rises. If this was not the case, the calculations for the
rise in sea level would be too simplistic to be valid, and there could be a
significant difference between them and the actual sea level increase
necessary to reach the lobby of the Q1.
It is assumed that the trend demonstrated in the data set used
continues into the future. If the trend does not continue into the future, it
would not be possible to accurately predict data points.
It is assumed that there is a correlation between temperature and rising
sea levels; if there is no correlation, it would be illogical to attempt to
predict the change in sea level using the temperature of Surfer’s
Paradise.
It is assumed that the melting of the polar ice caps do not contribute to
the rise of the sea level and that rising sea levels are entirely caused by
expansion of water due to heat. If this assumption proves invalid, the
predicted point in time at which the Q1 lobby would be at sea level
would be too far in the future, as melting polar ice could increase the
rate at which the sea level rises; making the predicted time frame for
the Q1 lobby to be at sea level inaccurate.
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Building a model
• The general form for a periodic function is , y =a
sin (2πb (x + c)) + d, where a is the amplitude of
the periodic function, b is the period of the
periodic function, c is the phase shift of the
periodic function and d is the vertical translation
of the periodic function. A periodic function
does, however, only predict a wave of constant
vertical translation if a constant d value is used.
Because of this, it would not be possible to
predict a change in the mean maximum
temperature of Surfer’s Paradise unless an
equation representing the general change in the
climate was used in place of the d value.
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Building a model cont:
• In order to obtain the equation for d, one must
determine the average rate at which the
temperature increases; the equation essentially
generates an equilibrium line for the periodic
function that is gradually sloping upwards.
• A linear regression performed upon the data gave
an equation of y = 0.001x + 24.67; an equation that
will return a positive value when x>0. This
equation is very important; it essentially proves
that the equilibrium line of the periodic function
for the mean maximum temperature over time
graph is sloping upwards; meaning that climate
change is causing an increase in temperature.
• This linear equation can be substituted into the
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general periodic equation instead of d.
Building a model cont:
The a value in a periodic equation is the amplitude of the
wave; essentially half the range of values for the y axis.
 The b value in a periodic function is the period of the
function being the average distance between two crests
on a wave. The period is simply the x value of one crest
minus the x value of its preceding crest.

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Building a model cont:
• Calculating the period as 12, the b value for the
periodic equation can be calculated to be
0.5235. This value can be substituted into the
periodic equation as the b-value.
• To determine the value of c, the equation can
temporarily exclude the c value and be used to
predict values for y. Because these values will
have been translated incorrectly due to the lack
of a c value; they can be matched up with
corresponding y values and the x distance
between these selected corresponding points
can be used to determine the value of c.
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Building a model cont:
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Building a model cont:

As shown on the previous slide, one pair of crests and one pair of troughs
have been identified for analysis. In the pair of troughs, the first has an x
value of 605, and the second has an x value of 609. This is indicative of a
horizontal phase shift of 4 months. Of the pair of crests, the first has an x
value of 623, and the second has an x value of 627; these two also have a
horizontal phase shift of 4. This would indicate that the predicted y values
have been shifted 4 months forward, and thus the c value is -4. Using a c
value of -4, the equation to predict the mean maximum temperature over a
month in Surfer’s Paradise becomes .
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Evaluating the model

Now that the model has been completed, it is necessary to validate it
by comparing the actual values from the data set with those
predicted by the model. This can be done by graphing the actual
values from the data set against the predicted ones, as done below.
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Verification of the model

This perception of the model being fairly accurate can be
verified by independently comparing a single predicted data
point with a single actual data point. Below, the model is
used to generate the mean maximum temperature of
Surfer’s Paradise over the 80th month from January 1938.
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Verifying the Model
• This calculation, however, produces a value of
24.702291 when performed automatically by a
calculator due to the fact that it has not compounded
errors due to rounding. The actual value of the mean
maximum temperature in the 80th month from January
1938 was 24.8 degrees. Thus, the calculated value was
off by approximately 0.0977 degrees, while the value
calculated by hand with rounding errors using the model
was off by 3.94 degrees. The fact that the model, when
used with a computer, produced a result that was
incorrect by less than one tenth of a degree in this
particular case would indicate that it is quite accurate.
Based upon this, it would be reasonable to assume that
the model could accurately predict the future trend of the
mean maximum temperature of Surfer’s Paradise.
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Solving the Mathematical problem

According to Baltimore City College, the volume of water will
increase by 0.0088% when heated 10 degrees. The Q1 lobby
is only approximately 3 metres above sea level, and it is
unlikely that the world temperature will rise by 10 degrees due
to global warming, so this value must be made smaller. To
determine the expansion of water at temperatures lower than
10 degrees, a ratio must be used as seen on the poster.
therefore for each degree of temperature increase, the world’s
water will expand by 0.00088%. The volume of the Earth’s
water is approximately 1.3 billion cubic kilometers, and its
surface area is 361 million square kilometers [Cook, D, 2004].
Because the volume of a rectangular prism is , and the world’s
oceans have already been put in terms of a rectangular prism,
it is possible to calculate the increase in the height of the
ocean by using the formula on the next slide.
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Solving the Mathematical problem cont:
Therefore, one can calculate the increase in sea level in kilometers when given the
increase in temperature in degrees Celsius. Assuming a height increase of at least 3
metres to reach the lobby of the Q1, the equation becomes
Therefore, the temperature increase necessary to create a rise in sea level of
0.003 kilometers, or 3 metres, is 0.946671 degrees Celsius. Of course, for this
change to actually take effect the change in temperature must be substantial and
sustained; it should be the mean of the mean maximum temperatures generated
by the model. Due to the periodic nature of the model, the equation should have
a linear regression performed upon it to determine an equation for its equilibrium
line, thereby allowing one to determine the point in time at which the mean of the
mean maximum temperatures will have increased to the point at which the sea27
level will have risen by 3 metres.
Solving the Mathematical problem cont:
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Improving the Mathematical Problem

After reading over the process previously applied to determine the
amount of time taken for the lobby of the Q1 to be at sea level, it
became apparent that the equation to calculate the sea level increase
when given the temperature increase was incorrect, as the percentage
of water expansion was not correctly placed inside the equation; it
should have been decreased by a factor of 10 to allow one to multiply
the percentage by the temperature increase. This is taken into account
in the revised calculations below.
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Improving the Mathematical Problem
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Verifying our Predictions
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Comment.
The assumptions were careful and relevant, containing additional justifications,
foreshadowing potential limitations as well as necessary simplifications.
Innovative thinking was evident – for example the replacement of ‘d’, by a
linear expression to capture the slowly increasing base temperature. Their
incorporation of the translation constant ‘c’ was clever, although they inferred a
value of 4 from the graph – the actual value is 3 (1/4 of a period).
Conceptually they should have regressed actual values onto model generated
values, which also gives a positive y-intercept. The purpose was to show that
the model was globally doing the right thing, so the subsequent use of the
wave formula was justified. This was also tested, although a range of points
rather than a single point should have been used. The students picked up an
arithmetic error, caused by misreading expansion data, by realising their
predicted date was not feasible, and subsequently corrected the error. They
wrongly attributed a mode error (degrees instead of radians) to rounding, but
checked the result using an independent method. Some issues remained
regarding the outcome but time intervened. The students revisited their original
caveat concerning the possible impact of melting ice.
A feature was the way the students continually spun between phases three
and six of the modelling process – testing, evaluating, and revisiting was a way
of life for them. Apart from the modelling, the students deepened their
understanding of mathematics topics they invoked as part of the solution
process – sometimes extraordinarily so.
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Acknowledgement: Thanks to Trevor Redmond and the ABP students.
Summary
Two types of problem are addressed
•A specific real world problem (e.g. the atomic test)
•A genuine real world context and problem area is selected, within
which a variety of specific problems can be formulated (e.g. climate
change and sea level)
Two purposes motivate this approach to modelling
•To solve a particular problem at hand
•Over time to build skills that enable students to apply their
mathematical knowledge to real world problems – by fostering the
development of a mental modelling infrastructure through cumulative
structured experience.
Thorough familiarity with the problem context is a key
•Acting out (goal shooting in football, hockey etc)
•Site visits (adding road lane in a cutting)
•Simulating (driving a car prior to addressing traffic related problems)
•Applying modelling skills to hobbies (hydroponic cultures).
•Contacting direct sources. (speed bump specifications)
•Immersion in internet material (climate change, dietary issues)
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