1. Introduction to Systems Modeling
Resources needed:
For instructor
Computer with projection capability
Internet access
PowerPoint and Vensim software
Web browser with Java installed
Chart paper for posting goals
Square Post-it Notes for participant use
Large Post-it Notes for participant reflection
Moose Model Presenter Guide
Overview
For participants
One computer per 1 or 2 participants
Vensim software
Moose Model handout
Introduction to Systems Modeling handout
moose.mdl file downloadable from a website or distributed via flash drives
Predprey, projectilemotion, freefall, verticalspring, boundedgrowth2, expdecay3, linear3 .mdl files downloadable

2. What is a Systems Model? Systems models:
Help us gain a deeper understanding of real world problems by providing us with graphical output based on the interactions of the factors inherent to the problem.
Show us the patterns of behavior that are common to diverse problems.
Express change through mathematical expressions representing the rate of change over time.
In the first part of the workshop, we will talk about systems modeling, what it is and how it really just takes what is happening in the real world (and by extension in science… science class) and shows it mathematically and visually, thereby organizing many different phenomena into patterns that most things follow. That lens (mathematically, visually, i.e. modeling) provides us and our students with a deeper understanding of the science.

3. Today’s Goals
Overarching Goal: Understand how to use a systems model as a tool for learning in science and mathematics.
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.
Have the goals written on a chart and posted where all can see and refer to as they move through the session.
Discuss what each goal means as it flies in - don’t necessarily read them word-for-word.
Foreshadow how each goal will be accomplished.

4. Why 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
Explain that aggregate change over time means that we are describing the behavior of the group with mathematical equations. Describe each graph type as you fly in each type of behavior.
Ask the participants to work in small groups to list problems that behave in the ways listed above. Hand out square post-it notes for them to record their answers.
On chart paper, make 5 columns, one for each behavior. Ask participants to post their problems in the columns.
Accept all problems.
Save this list to come back to later for discussion.
Explain that we will now look at models that illustrate each behavior.

5. How are these problems similar?
Graph the speed of a car that accelerates from rest at a constant acceleration.
Graph the amount of money in my son’s savings account if his weekly allowance is constant and he spends none of his allowance.
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? In each case, the rate of change (the slope) is constant.
Use a whiteboard or chart to sketch the graph as you elicit details from the group. Ask questions about what we could determine from the graph.
Some participants may argue that the money example would be better represented by a step-function.

6. A Linear Graph
Linear systems are simple, but important to understand. They are building blocks for more complex systems.
Point out that the units on the x-axis are always some measure of time – weeks, seconds, minutes, years, etc.
Point out that the units on the y-axis will depend on the problem being modeled. This graph was deliberately made unit-less to allow it to represent a variety of situations.

7. 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
Here’s the model for the three problems. In each case, the change in Y is added to Y to compute the next value.
In systems modeling, measurable variables are computed in a step-by-step fashion. In the bathtub model, the tub starts out empty. Each minute, a fixed number of gallons of water is added to the tub. The volume increases linearly.
Point out that you could change the setting on the faucet to change the rate at which the water flows into the tub. How would this change the graph? Point out that it would change the slope.
Box is the variable you are measuring. Flow is adding “stuff” to the box.
Look at participant examples of linear models that were posted on the chart from Slide 8. For any new problem, discuss the meaning of the box and flow variables for that problem.

8. How are these problems similar?
Graph the amount of radioactive material as it decays over time.
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.
Graph the temperature of a cup of coffee as it cools to room temperature.
Point out that in each problem, the measurable variable (radioactive isotope or savings or temperature of coffee) decreases quickly at first and then more slowly. The less there is, the less there is to take away since a constant percentage is removed.
Coffee cooling is slightly different as the coffee cools to room temperature (not zero as the others).

9. Exponential Decay
Note that the curve is steepest at the beginning and nearly flat at the end. Why does that happen?

10. 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
Here’s part of the model for the three problems. In each case, the change in Y is subtracted from Y to compute the next value and the change in Y depends on the current value of Y. Therefore, the change in Y is not constant.
What variable needs to be added to number 6? Room temperature.
Remember - Coffee cooling is slightly different as the coffee cools to room temperature (not zero as the others).
Exponential Growth Model is different – flow points into box not out as above. What would the model and graph look like? Tell them that we will be building an exponential growth model later in this session.
Look at participant suggestions from Slide 8 for this type of graph. For any new problem, discuss the meaning of the box and flow variables for that problem.
May need to bring up HAVE = HAD + CHANGE to help participants understand how the model produces exponential growth or decay.

11. How are these problems similar?
Graph the number of burnt trees in a forest fire as trees are transformed from living to burning to burnt.
Graph the number of immune people as people progress from being healthy to being sick to recovering to become immune to the disease.
Point out that in each problem, the measurable variable (burnt trees or immune people) experiences bounded growth. It increases slowly at first, then more rapidly, and then levels off as it nears the maximum number possible. Caution them that while several variables are mentioned in the problems, they are considering the graph for only one of them.

12. Bounded Growth
Here we see each stage of the process. The blue curve is the bounded growth curve.

13. 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
Here’s the model for the two problems. In each case, the amount subtracted from X depends on both X and Y. In other words, to burn, the green tree needs to encounter a burning tree. To get sick, a healthy person has to encounter a sick person. The amount subtracted from Y depends on Y only. In other words, a fraction of the burning trees will finish burning each day. A fraction of the sick people will recover from the illness and gain immunity.
Again, talk them through the model – what happens in each box as the pipes (or flows) add and subtract amounts.
When things are added to Y they are subtracted from X. When they are added to Z, they are subtracted from Y.
Revisit participant ideas from the chart made on slide 8.

14. How are these problems similar?
Graph two populations, the predator and its prey, for a period of many years.
Graph glucose and insulin levels during a day in which three meals are eaten at regular intervals.
Graph the motion of a frictionless vertical spring.
Point out that in each problem, the measurable variables oscillate in value.
This time, we want two graphs, especially for 9 and Invite a participant to sketch the expected graph for 9 on a whiteboard or chart paper.

15. Periodic Behavior
These two graphs could be moose and wolf populations or glucose and insulin levels. This particular graph is from a wolf-moose model. The real population graph is not this smooth. There are 50 years of data for an Isle Royale study with multiple factors affecting the populations, including ticks that weaken the moose population and a genetic deformity appearing in the wolf population due to in-breeding.
Ideal spring motion would be perfectly sinusoidal.

16. Interdependence
Discuss the meanings of the flows for a predator-prey model: that is, births and deaths. Talk about the arrows showing that the number of prey (X) affects the birth rate of the predator (Y). The number of predator (Y) affects the deaths of the prey (X).
Do the same for glucose-insulin. The amount of glucose (X) increases due to food eaten. It affects the amount of insulin (Y) produced. In turn, the insulin (Y) causes the glucose (X) to be converted into energy.
Some arrows found in the predator-prey model would not exist in the glucose-insulin model.
Again, revisit participant ideas from Slide 8.

17. 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)
Elicit responses from participants and chart.
Note that we haven’t looked at a quadratic example, but we will see some in other sessions.
Fly in each statement on the slide and lead a discussion on how it connects to participant responses and to the goals. Indicate on the chart using check marks for statements that relate to statements on the slide and write the number(s) of the goal(s) that relate beside each statement. (Refer to posted goals chart)
About 80 minutes to this point.
Fly in the goal(s) connections for each statement.
Discuss the following prompt:
How does what you have learned today relate to the overarching goal?

18. Systems Models in Science
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.
Briefly go over these standards. Point out that we are going to see these standards in action as we build a model.

19. Systems Models in Mathematics
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.
Briefly go over these standards.

20. Model Exploration Open the predprey.mdl from the CAST website
What could your students learn from using this model?
Describe the effect of modifying a parameter.
What is the math behind the model?
Now, since we understand a little about what systems modeling is and how it supports student learning of mathematics and science, we will explore a few already assembled models.
First, let’s all look at the Predator Prey model based on the interaction of moose and wolves on the Isle Royale island in Lake Superior. Thanks to 50 years of data collected about the populations, we can learn a lot about the factors important to the survival of a species.
Show the participants how to run the model with the Run a Simulation button, how to vary parameters with the Automatically Simulate on Change button, and how to see the equations behind the scenes with the Y=x2 button.

21. Hands-On Practice Select another CAST model to explore.
What could your students learn from using this model?
Describe the effect of modifying a parameter.
What is the math behind the model?
Ask the participants to select a downloaded CAST model to explore. Give them 15 minutes to think about how they could use the model with their students. Ask them to make notes answering these questions for group sharing at the end.

22. Group Sharing The Models: The Questions:
Linear Free Fall Projectile Motion Vertical Spring
Exponential Decay Bounded Growth
The Questions:
What could your students learn from using this model?
Describe the effect of modifying a parameter.
What is the math behind the model?
Ask groups to share out the models they explored and the ways they could use the models with their students.
In the next part of the session, we will learn how to build a model. Participants may choose to follow along or they may continue exploring the pre-built models downloaded from the CAST website.

23. Reflection I used to think ….. But now I know…..
On a large post-it note, complete this thought:
I used to think …..
But now I know…..
Reflect on the ideas you have learned so far in this session. Tell us how your thinking about systems modeling, graphs, mathematical relationships, and/or problem-solving have changed.
Please print neatly so we can read and share your thoughts.
This would be a good time for a 10 minute break. While participants take their break, read and organize the reflections, posting them on the wall for review after the break.

24. The Modeling Cycle
Overarching Goal: Understand how to apply the modeling cycle to dynamic systems in science and mathematics.
Now, since we understand a little about what systems modeling is and how it supports student learning of mathematics and science, we can learn more deeply by building a simple model ourselves, which will deepen our understanding of the modeling process and how the real world data end up producing the graph of the model, in turn deepen our understanding about how carefully we need to think about which data we are putting into our model.
For Model Builders:
In the next stage of this session, we will pose a problem and design, build and test a model.
Go through the modeling cycle diagram, describing each step. We will see examples of each step as we build a model.
Image courtesy of Common Core State Standards Initiative
Downloaded from

25. Building a Simple Model – Step 1
Start with a question
How long will it take for the moose on Isle Royale to over-populate the island?
Identify the measurable variables in the question.
Identify the major actions that cause those variables to increase or decrease.
Identify factors that affect those actions.
Explain that now that we have learned about different model structures and their graphs, we are going to build a simple model that will help us answer a question.
This problem was selected because the model structure is applicable to many different problems, as we will see later.
Ask the participants to work in groups to identify the measurable variables, the actions that cause those variables to increase or decrease, and the factors that could affect those actions. Explain that there are many unknowns in the question as it is currently stated. Encourage them to think broadly.
On chart paper, make three columns, one for Measurable Variables, another for Actions, and a third for Factors. Ask the groups to share out one variable with corresponding actions and factors, one variable per group. Record their answers on the chart paper. Accept all responses.
Expected Responses:
Measurable Variables: Time, Moose, Island Size, Island Resources
Actions: Births and Deaths are actions for the Moose Population
Factors affecting Births and Deaths: Predators on the island, Vegetation for the moose, Disease, Weather conditions (favorable or harsh)

26. Building a Simple Model – Step 2
Figure out a way to validate your model
How long will it take for the moose on Isle Royale to over-populate the island?
Can I collect data in the classroom with a wet-lab experiment?
Can I find real world data to use for comparison?
Can I calculate numerically what I expect to happen?
Explain that it’s always wise to consider how you will validate your model before you start building it. If you can’t validate the model of a complex problem, perhaps a simpler problem could be validated. From that simpler problem, you may gain some insight into the more complex problem.

27. Building a Simple Model – Step 3
Figure out the relationships between the pieces of the model
Major measurable variable: moose
Major action for increasing population: births
Major action for decreasing population: deaths
Are births and deaths dependent on the number of moose in the population in any way?
Before building the model, we need to ask ourselves about the dependencies in the model.
Pose this questions:
Are more births possible when there are more moose? What other factors affect the number of births?
Expected answers:
If the moose are all healthy, then the number of births will be some fraction of the number of mature female moose.
Moose health would also affect the number of births since litter size and live births will decrease in times of stress.
Pose this question:
How are deaths related to the number of moose?
If there are no predators and the moose are healthy, deaths will be due to natural causes and will depend on the number of elderly moose.
The presence of predators and moose illness will increase the number of moose deaths.
Explain that we will be begin with the simplest situation – no predators and healthy, well-fed moose.

28. Building a Simple Model – Step 4
Figure out how to validate an idealized moose population model
Can we set up an experiment in our classroom?
Can we find real world data?
Can we verify population growth mathematically?
Ask the participants which method of validation they would use for an idealized population model. Since a classroom experiment with moose is impractical and real world data would not be under ideal conditions, we must rely on mathematical calculations.
Suppose we start with 100 moose, 50 male and 50 female. Assume each female has one calf per year and calves mature in one year. To simplify even more, let’s ignore deaths.
How many moose will there be a year after the first set of calves have been born? How many will there be the next year? How about the third year?
Set up a table on chart paper and record answers there. Explain that these numbers will be needed during the model building activity. Let the participants decide how they want to handle fractional moose (112.5, for example).
Expected Table:
Year Moose Population Moose Births
or 113
or or 169

29. Building a Simple Model – Step 5
Open Vensim and build this model
Explain that participants will now learn how to build this model in Vensim. Distribute the Moose Model handout and have the group follow along as you build the model with them.
The first 4 pages of the handout explain how to build the model shown above. If you have adequate time, you may have the group continue with pages 5-8 where they will learn how to create custom graphs and tables. Or you may leave those pages for independent work after the workshop.
Use the Presenter’s Guide for the Moose Model as a reference as you lead the group through the building process. Notice the reflection questions sprinkled throughout the handout.
To see a pre-built model of the moose population, open the moose.mdl file.

30. 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)
Elicit responses from participants and chart.
Fly in each statement on the slide and lead a discussion on how it connects to participant responses and to the goals.
Indicate on the chart using check marks for statements that relate to statements on slide and write the number(s) of the goal(s) that relate beside each statement.
Fly in the goal(s) connections for each statement.
Discuss the following prompt:
How does this learning relate to the overarching goal?

31. Other Models to Explore
Kinematics
Coffee Cooling
Epidemic
Predator-Prey
If time allows, visit the MVHS website to show participants other models they may explore.