Introduction to Systems
Modeling
Why Systems Modeling?
Demonstrates aggregate change over time
resulting in behaviors such as:
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Linear Growth and Decline
Quadratic Motion
Exponential Growth and Decay
Bounded Growth
Periodic Behavior
Systems Models in Mathematics
Address Common Core Standards in Mathematics
Standards for Mathematical Practices
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Model with mathematics
Reason abstractly and quantitatively
Use appropriate tools strategically
Construct viable arguments and critique the reasoning of others
Make sense of problems and persevere in solving them
Standards for Mathematical Content
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Build a function that models a relationship between two quantities.
Construct and compare linear, quadratic, and exponential models and
solve problems.
Understand that different situations may be modeled by the same
structure.
Implement the modeling cycle in constructing a model.
Systems Models in Science
Support teaching science as inquiry by providing:
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Opportunities for careful observation and analysis of real
world data and relationships.
The ability to test hypotheses, analyze results, form
explanations, judge the logic and consistency of conclusions,
and predict future outcomes.
Support Academic Standards for Science by providing:
• Examples of dynamic changes in stable systems.
• Examples showing the effects of limiting factors on population
dynamics.
Goals
Overarching Goal: Understand how to use a systems
model as a tool for learning in science and mathematics.
1.
2.
3.
4.
5.
Identify problems that have similar model structures.
Identify the measurable variables and rates of change in a
systems model.
Represent the relationships between variables in a systems
model mathematically.
Validate a systems model through comparison to real-world
data.
Develop an inquiry-based investigation that uses a systems
model.
How are these problems similar?
1. Graph the speed of a car that accelerates
from rest at a constant acceleration.
2. Graph the amount of money in my son’s
savings account if his weekly allowance is
constant and he spends none of his
allowance.
3. Graph the amount of water in a bathtub if
the faucet is running at a constant rate and
the drain is closed.
A Linear Graph
A Linear Model
Problem
Change in Y
Y
1
acceleration
speed
2
weekly allowance
savings
3
faucet output in gal/min
volume of water in
bathtub in gal
How are these problems similar?
4. Graph the amount of radioactive material as
it decays over time.
5. Graph the amount of money left in my son’s
savings from his summer job if his weekly
spending is a fixed percentage of the money
still in his savings.
6. Graph the temperature of a cup of coffee as
it cools to room temperature.
Exponential Decay
A Decay Model
Problem
Change in Y
Y
4
A fraction of the isotope
radioactive isotope
5
A fraction of savings
savings
6
A fraction of the difference
between coffee and room
temperatures
coffee temperature
How are these problems similar?
7. Graph the number of burnt trees in a forest
fire as trees are transformed from living to
burning to burnt.
8. Graph the number of immune people as
people progress from being healthy to being
sick to recovering to become immune to the
disease.
Bounded Growth
Transformation to Bounded Growth
Problem
X
Change in X
Y
Change in Y
Z
7
Green trees
Catch fire rate
Burning
trees
Burnt out
rate
Burnt trees
8
Healthy
people
Get sick rate
Sick people
Recovery
rate
Immune
people
How are these problems similar?
9. Graph two populations, the predator and its
prey, for a period of many years.
10.Graph glucose and insulin levels during a day
in which three meals are eaten at regular
intervals.
11.Graph the motion of a frictionless vertical
spring.
Periodic Behavior
Interdependence
What have we learned?
Problems from different disciplines can be represented by similar
model structures.
(Goal 1)
Graphing the expected output for a model can show the
expected model structure, including the variables and the
rates of change between variables.
(Goal 2)
Each model structure has particular mathematical relationships
between its variables and their rates of change.
(Goal 3)
Systems Models in Inquiry-based
Investigations
1. Select the problem you desire to investigate.
2. Find a model representing this problem.
3. Validate the model by comparing its output
to real-world data.
4. Experiment with the model by setting the
parameters to see their effects on the output.
Coffee-Cooling
1. The Problem
– How long will it take my cup of coffee to cool to
room temperature?
2. The Model
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http://mvhs.shodor.org/activities/physical/cooling.html
Software: http://www.vensim.com/venple.html
3. Validating the Model
– Compare to experimental or theoretical data
4. Inquiry with the Model
– Changing parameters
Do one of your own
1. The Problem
– Design a problem based on one of the pre-built
models at mvhs.shodor.org
2. The Model
– Select from those on the mvhs.shodor.org website.
3. Validating the Model
– Compare to experimental or theoretical data
4. Inquiry with the Model
– Changing parameters
The Modeling Cycle
Overarching Goal: Understand how to apply the
modeling cycle to dynamic systems in science and
mathematics.
Image courtesy of Common Core State Standards Initiative
Downloaded from www.corestandards.org
The Money Game
• How much money do I need to put aside for
retirement?
• Once I retire, how much of my savings can I
spend annually without running out?
• What assumptions can I make about the
growth rate of my investments?
• What assumptions can I make about my life
expectancy?
Formulating the Model
1. What is the main topic of this model?
Note: This must be a measurable variable.
2. What actions cause this variable to change?
What makes this variable increase?
What makes this variable decrease?
3. What parameters affect those actions?
4. How can the relationships between the
parameters, the actions and the main topic be
represented mathematically?
The Answers in Vensim symbols
1. The main topic of our model is money.
This is called a box variable in Vensim.
Money
2. Interest and Deposits add to the Money.
Spending subtracts from the Money.
These actions are called flows or rates in Vensim.
Interest
Money
Spending
Deposits
More answers in Vensim symbols
3. Interest is based on the amount of money currently saved and the
interest rate paid by the financial institution.
Money is already in our model as a box variable (our main topic).
The interest rate is a parameter (called a variable – auxiliary/constant).
The blue arrows show that the mathematical formula for interest uses
the variables Money and interest rate.
Interest
interest rate
Money
Spending
Deposits
4. What is the math behind the model?
Flows are quantities that are added to or subtracted from box variables.
Constants or formulas may be stored in flows.
The blue arrows pointing toward Interest mean that Interest equals a formula
that uses interest rate and Money.
The lack of blue arrows pointing toward Spending and Deposits means those
rates are probably constants.
Interest
interest rate
Money
Spending
Deposits
Doing the Math
Let’s suppose we start with $1000 in our Money account which earns a 5%
annual interest rate. Assume there are no deposits and no spending. How
fast will our money grow?
Year
Money
Interest
1
1000
.05*1000 = 50
2
1050
.05*1050 = 52.50
3
1102.50
.05*1102.50 = 55.12
4
1157.62
.05*1157.62 = 57.88
What’s the Rule?
Year
Money
Interest
1
1000
.05*1000 = 50
2
1050
.05*1050 = 52.50
3
1102.50
.05*1102.50 = 55.12
4
1157.62
.05*1157.62 = 57.88
The money we have at the start of Year 2 is the
money we had in Year 1 + the interest earned.
Have = Had + Change
Model Computation and Validation
• Open the model money.mdl
• Explore the model to see the underlying math.
• Run the model and verify that the Money in
the table agrees with the calculations done by
hand.
• With zero deposits and zero spending, what
kind of growth does the Money graph show?
Inquiry-based Investigations
• How much money do I need to put aside for
retirement?
• Once I retire, how much of my savings can I
spend annually without running out?
• What assumptions can I make about the
growth rate of my investments?
• What assumptions can I make about my life
expectancy?
Reflections on this model
1. If deposits and spending are set to 0, what
kind of growth does the money graph show?
2. What are some other real-world problems
that undergo exponential growth?
3. How would you change this model to
represent those problems?
What have we learned?
Complex problems can be broken down into simple
parts involving simple arithmetic.
(Goals 2 & 3)
Model output must be compared to real world data
to validate the model.
(Goal 4)
Models can be used to experiment with parameters
to see the differences in outcomes.
(Goal 5)