Mathematics SL
Internal Assessment
IA Portfolio
Type II Task
Mathematical Modeling
Mr. Wai 2012
Population Trends in China
• Aim: In this task, you will investigate different
functions that best model the population of
China from 1950 to 1995.
• The following table shows the population of
China from 1950 to 1995.
Year
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
Population
554.8 609.0 657.5 729.2 830.7 927.8 998.9 1070.0 1155.3 1220.5
(millions)
• Define all relevant variables and parameters clearly. Use
technology to plot the data points from the above table on
a graph.
• You should define elapsed time t, in years, since 1950 as
the independent variable, and population P, in millions of
habitants, as the dependent variable.
• Things to note here:
– You should identify the dependent and independent variables
with a variable and the unit that it is measured in clearly and
correctly.
– “The independent variable is elapsed time t, in years, since
1950” is better than “the independent variable is time”
– “The independent variable is population P, in millions of
habitants” is better than “the independent variable is
population”
– ALL variables representing a quantity, such as t and P, are
italicized.
• Parameters will be provided when a mathematical model is
proposed.
• Comment on any apparent trends shown in the graph.
What type of functions could model the behaviour of the
graph? Explain your choices.
• Create a scatter plot based on your selected dependent and
independent variable.
Graph 1. Population in China from 1950 to 1995
• Comment on any apparent trends shown in the graph.
What type of functions could model the behaviour of the
graph? Explain your choices.
• You should explain the behaviour and graphical pattern
observed.
– It is increasing
– rate of change is increasing for the first half, then it seems to
decrease
• A linear function, a cubic function, or an exponential
function seem reasonable based on the apparent trends
shown.
• Things to note in this section:
– There is no correct answer here, as long as your choice is
accompanied with a reasonable justification.
– The choice of mathematical model made here is only
preliminary, there will be room for improvement later.
– Make sure the graph takes up at least half a page.
– Make sure all Graphs and Tables are titled and numbered.
(Graph 1. Population in China from 1950 to 1995) This makes it
easier to refer to them.
• Analytically develop one model function that fits the
data points on your graph.
• At this stage of the course, you have learned about
many different types of functions.
• Suppose I have chosen cubic function as my model.
• However, you have not learned exactly HOW and WHY
regression works, thus the “baby way” to analytically
develop a cubic function to model population as a
function of time,
P(t) = at3 + bt2 + ct + d, where a, b, c, d, are the
parameters of the cubic model, and P and t are the
independent and dependent variables respectively, is
to work out a system of equations where four of the
data points are the solutions to my cubic model.
• Analytically develop one model function that fits the
data points on your graph.
• I have chosen P5(5, 609.0), P15(15, 729.2),
P30(30, 998.9), P40(40, 1155.3) to be the four points.
• Why do you think I chose those four points?
• Using the four chosen data points, we can
establish a system of equations, using our
cubic model P(t) = at3 + bt2 + ct + d
P(5) = (5)3a + (5)2b + (5)3c + d = 609.0
P(15) = (15)3a + (15)2b + (15)3c + d = 729.2
P(30) = (30)3a + (30)2b + (30)3c + d = 998.9
P(40) = (40)3a + (40)2b + (40)3c + d = 1155.3
• Thus we have the matrix system AX = B, where
where A is the coefficient matrix, B column
vector on the right side, X is the column vector
(a b c d)T.
• The system is solved by applying matrix algebra: X
= A–1B.
• Thus the cubic mathematical model obtained
analytically is
P(t) = –0.009486t3 + 0.7127t2 + 0.8491t + 588.1
• I chose 4 significant digits for my parameters
because the data is given to 4 significant digits.
• Analytically develop one model function that fits the
data points on your graph.
• Suppose I chose an exponential function instead,
because I think population growth exponentially.
• P(t) = abt, where a, and b are the parameters of the
model.
• Using the two chosen data points, we can establish a
system of equations, with our exponential model
P(t) = abt
P(5) = ab5 = 609.0
P(40) = ab40 = 1155.3
• Solving the system of equation below will give the
solution a ≈ 555.8 and b ≈ 1.018
• Thus the exponential model obtained analytically using
this “baby way” is P(t) = 555.8(1.018)t
• On a new set of axes, plot your model and the original data.
Comment on how well your model fits the original data.
Revise your model if necessary.
• Graph 2 shows the original data, the cubic model (red
curve) and the exponential model (green curve) both of
which are created analytically.
Graph 2. Mathematical models for the population of China from 1950 to 1995
• We can observe that both model fits the data
fairly well.
• The cubic functions pass through more data
points. (at least 4, because that is how we
analytically arrived at the four parameters of the
cubic function). However it is reasonable to say
that the cubic model will underestimate the
population of China based on the behaviour of
the data.
• The exponential function passes through less
data points, and they appear to be further away,
especially in 1950, 1970 and 1975. Also it is
reasonable to say that the exponential model will
overestimating the population of China based on
the behaviour of the data.
• Use technology to find another function that
models the data. On a new set of axes, draw
both your model functions. Comment on any
differences.
• In the task of Population Trends in China,
students are instructed to use logistic
regression on a Graphic Display Calculator to
model the data. Doing so on the calculator
you will arrive at a logistic function as follow:
• On a new set of axes, plot the logistic model
and the original data. Comment on how well
this model fits the original data.
Graph 3. Logistic Regression model created on a GDC
• Keys things to note:
– The logistic function eliminates the main short coming in
both the cubic function and exponential function, which is
that both cubic function and exponential function does not
appear to be able to predict the population beyond 1995
satisfyingly since they contradict each other, thus both of
them cannot be correct at the same time.
– We should still discuss the implication of each of the three
models we examined in terms of population growth for
China in the future.
• In the exponential model, one implication is that the population in
China will continue to grow indiscriminately, which is not probable,
since resources are limited.
• In the cubic model, one implication is that the population will start
to decline beyond 1955, which is not an expected behaviour of any
population.
• In the logistic model, without future data it is hard to conclude
quantitatively that it is the superior model. However, it does
eliminate the immediate short coming of both the cubic and
logistic model.
• Here are additional data on population trends in China
from 2008 World Economic Outlook published by the
International Monetary Fund (IMF)
Year
1983
1992
1997
2000
2003
2005
2008
Population
1030.1 1171.7 1236.3 1267.4 1292.3 1307.6 1327.7
(millions)
• Comment on how well each of the models above fit the
IMF data for the years 1983-2008
• Create a graph using the original data and the
additional data on population in China along with the
three models that we have came up with. Then
comment on how well each models fit the IMF data
and the original data based on the graph.
Graph 4. Comparing all three models for population in China from 1950 to 2008
• Discuss the quality of fit for each of the functions with
the new data points specifically, then in general (total
data pool), and lastly if any of them addresses the
concerns we had before.
• For an example, you can discuss how the two data
points that are within the original time range how they
still fit very well with all three functions.
• However beyond 1995, the additional data did show
that our original suspicion that the exponential
function will overestimate the population where the
cubic function will underestimate the population.
• Even though the logistic overestimates the population
beyond year 2000, it is still the best model out of the
three that was being considered. Thus…
• Thus… We should use a logistic model to find the curve that
best model our data set.
• To improve our current model, we must include the
additional data from IMF to make adjustment to the
parameters in the logistic model.
• Entering the new data, the final logistic model will be as
follows:
• However, you must note that there is still limitations to our
refined model with the modified parameters. Since this
model is created using a finite number of data (58 years), it
is not reasonable to use the model to predict the
population of China in 30 years. (maybe quickly explain why
30 years)
To Sum Up
• Remember that IB is looking for the process, on how you
came to conclude that which ever model you selected in
the end is reasonable and justifiable.
• Emphasize at each stage how your current model fit the
data, and how it can be improved.
• As an activity, you should go through the slides and locate
where I have included items that satisfies all the
requirements for the following:
– Criterion C: correctly defined variables, parameters, and
constraints, and analyze them to enable the formulation of
mathematical model so that the model can be applied to
additional data while taking into consideration on how well it
fits the data.
– Criterion D: correctly and critically interpreted the
reasonableness of the results of the model in the context of the
task, to include possible limitations and modifications of the
results, to the appropriate degree of accuracy.
Special Mentions
• Professor Carlos Abel Eslava Carrillo
from Tecnológico de Monterrey – Campus
Eugenio Garza Lagüera