Introduction to Systems Modeling

Demonstrates aggregate change over time resulting in behaviors such as:– Linear Growth and Decline– Quadratic Motion– Exponential Growth and Decay– Bounded Growth– Periodic Behavior

Why Systems Modeling?

Address Common Core Standards in Mathematics

Standards for Mathematical Practices•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• 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 Mathematics

Support teaching science as inquiry by providing:• 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.

Systems Models in Science

Overarching Goal: Understand how to use a systems model as a tool for learning in science and mathematics.

1. Identify problems that have similar model structures.2. Identify the measurable variables and rates of change in a

systems model.3. Represent the relationships between variables in a systems

model mathematically.4. Validate a systems model through comparison to real-world

data.5. Develop an inquiry-based investigation that uses a systems

model.

Goals

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.

How are these problems similar?

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

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.

How are these problems similar?

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

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.

How are these problems similar?

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

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.

How are these problems similar?

Periodic Behavior

Interdependence

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)

What have we learned?

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.

Systems Models in Inquiry-based Investigations

1. The Problem– How long will it take my cup of coffee to cool to room

temperature?2. The Model– 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

Coffee-Cooling

1. The Problem– Design a problem based on one of the pre-built

models at mvhs.shodor.org2. 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

Do one of your own

Overarching Goal: Understand how to apply the modeling cycle to dynamic systems in science and mathematics.

The Modeling Cycle

Image courtesy of Common Core State Standards InitiativeDownloaded from www.corestandards.org

• 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?

The Money Game

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?

Formulating the Model

Money

The Answers in Vensim symbols1. The main topic of our model is money. This is called a box variable in Vensim.

2. Interest and Deposits add to the Money. Spending subtracts from the Money. These actions are called flows or rates in Vensim.

MoneyInterest Spending

Deposits

More answers in Vensim symbols3. 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.

MoneyInterest Spending

Depositsinterest rate

4. What is the math behind the model?

MoneyInterest Spending

Depositsinterest rate

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 formulathat uses interest rate and Money.The lack of blue arrows pointing toward Spending and Deposits means thoserates are probably constants.

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?

Doing the Math

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

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

• 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?

Model Computation and Validation

• 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?

Inquiry-based Investigations

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?

Reflections on this model

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)

What have we learned?