1. Resources needed:
For instructor
Computer with projection capability
Internet access
PowerPoint, Excel and Vensim software
Web browser with Java installed
A handful of coins
Chart paper for posting goals
Prior to beginning the session, make sure that the PowerPoint, Excel files, the VenSim fire model and the Fire! Website are all open on the presentation computer and working. Minimize all files except the PowerPoint, so they are ready to open seamlessly during the presentation.
For participants
One computer per 1 or 2 participants
Excel and Vensim software
Flipping Pennies Excel file downloadable from a website or distributed via flash drives
Flipping Pennies Simulation handout
Forest Fire Data Analysis handout
Forest Fire blank Excel file downloadable from a website or distributed via flash drives
ForestFire.mdl file downloadable from a website or distributed via flash drives
Forest Fire Inquiry handout

2. Goals
Overarching Goal: Understand that computer models require the merging of mathematics and science.
Understand how computational reasoning can be infused into teaching.
Develop a working definition of computational reasoning.
Recognize the importance of graph interpretation skills in understanding model behavior.
Recognize that probability and random numbers are important mathematical ideas that can be modeled using tools of computational reasoning.
Understand that probability can be used to simulate real-world phenomena and make predictions.
Have 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 – don’t necessarily read them word-for-word.
Foreshadow how each goal will be accomplished.

3. Agenda Why computational reasoning?
Probability – theoretical vs. real-world behavior
Probability in an agent-based model
Probability in a systems-based model
Analysis of model output via graphs
Comparison of agent-based and systems-based models
Curriculum applications
Review the agenda for the session.

4. What is Computational Reasoning?
Understanding how to analyze, visualize and represent data using mathematical and computational tools
Using computer models to support theory and experimentation in scientific inquiry
Using models and simulations as interactive tools for understanding complex concepts in science and mathematics
Computational Reasoning (our definition – CAST/MVHS) means:
Understanding how to analyze, visualize and represent data using mathematical and computational tools
This includes graphing data, identifying trends, and recognizing error.
Using computer models to support theory and experimentation in scientific inquiry
Just as variables must be carefully defined in a scientific experiment, so must assumptions about variables be made explicit in a computer model designed to represent a real world problem.
Building a model requires a deep understanding of the problem being represented.
Many science problems are done on computers – the model is run before any testing is done in the real world
Using models and simulations as interactive tools for understanding complex scientific concepts
Interacting with a computer model by running it under various conditions is similar to conducting an experiment.
The results achieved by varying the parameters in the model help the student understand the underlying concepts.

5. Why Computational Reasoning?
Addresses Common Core Standards in Mathematics
Standards for Mathematical Practices
MODEL WITH MATHEMATICS
Reason abstractly and quantitatively
Use appropriate tools strategically
Look for and express regularity in repeated reasoning
Standards for Mathematical Content
Making Inferences and Justifying Conclusions
Understand and evaluate random processes underlying statistical experiments
Make inferences and justify conclusions from sample surveys, experiments and observational studies.
Point out how computational reasoning is relevant in mathematics.
Modeling / using equations crosses all domains.
In these models we are doing virtual experiments.
We will be using models of coin tossing and forest fire burning to illustrate these Common Core Standards.

6. Why Computational Reasoning?
Supports Practices for K-12 Classrooms
(source: A Framework for K-12 Science Education)
Developing and using models
Using mathematics, information and computer technology, and computational thinking
Supports teaching science as inquiry by providing:
Models of real world events that are difficult to demonstrate in wet lab experiments
Opportunities for careful observation and analysis of scientific investigations
The ability to test hypotheses, analyze results, form explanations, judge the logic and consistency of conclusions, and predict future outcomes.
Point out how computational reasoning is relevant in science.
In science, we can use models to represent difficult real-world events.
Have to be able to challenge models. Do the results of the model make sense? This leads to critical thinking skills that we want students to have.
We will be using models of coin tossing and forest fire burning to illustrate how computational reasoning supports inquiry-based learning.

7. Hands-on Probability Theoretical probability vs. the real world
What is the probability of getting a head when you toss a coin?
In 10 trials, will you get an equal number of heads and tails?
In 1000 trials, will you get an even split?
Will everyone in the room get the same answer?
How would you design an experiment to answer these questions?
Ask participants to work in pairs or small groups to answer these questions.
Have a handful of coins available in case a pair needs a coin.
Ask the pairs to keep track of how many heads and tails they get on 10 flips of the coin.
Ask a participant to keep track of the head and tail counts as each pair shares their head/tail counts.
Discuss the results.
Expected Answers
The first question accesses prior knowledge. Be sure to ask the participants why the probability of getting a head is ½.
The second, third and fourth question are predictions. The focus here should not be on evaluating the predictions as right or wrong. Rather, ask participants to explain their reasoning and leave the questions open. Emphasize that predictions are an important part of computational reasoning and scientific inquiry.
Use the answers to the last question to set up the computer experiment.

8. Probability via the Computer
Since probability is usually expressed as a fraction between 0 and 1, a computer uses a formula that will generate numbers between 0 and 1 in no discernible pattern – we call these random numbers.
To simulate flipping a coin, we make the rule that numbers less than ½ represent heads and numbers more than ½ represent tails.
If thousands of numbers are generated, approximately half of those numbers should be less than 0.5 and the other rest should be greater than 0.5
If only 10 numbers are generated, will half of them be less than 0.5?
Try this out using the interactive Excel spreadsheet.
Now we’ll use the computer to simulate coin-flipping.
Explain that a computer’s actions are based on code written by a programmer.
Since probability is expressed as a fraction between 0 and 1, code must be written that can generate numbers between 0 and 1.
Those numbers must not fall in any discernible pattern.
Computer scientists have figured out formulas that will generate numbers between 0 and 1 that fall in what appears to be a random order.
The programmer decides which way 0.5 falls – but since random number generators usually generate numbers with multiple decimal places, it is very rare to have exactly 0.5.
We can see this in action by opening the flipping_pennies.xls file.
Direct them to the location of the file.

9. Flipping Pennies Simulation
Open the flipping_pennies.xls spreadsheet.
Answer the questions on the handout.
Open the flipping_pennies.xls spreadsheet.
Point out the red triangle and the error calculation. Ask them to watch what happens to percent error as they work.
Distribute the one page Flipping Pennies Simulation handout.
Direct the pairs to conduct the investigation and answer the questions on the handout. As they work, continue to remind them to work together to answer the questions. Give them a 2 minute warning when you want them to finish up. Ask everyone to get through at least the 100 flips portion.
Then ask them to predict what might happen in the future if another simulation used random numbers.

10. Conclusions drawn from the Flipping Pennies Simulation
After conducting the simulation, consider these questions.
Will a simulation that uses random numbers give the same result every time it is run? Explain.
Is such a simulation a valid representation of reality? Explain.
What can you learn from a simulation if it doesn’t always give the same result? Explain.
Questions to ask to summarize findings or handout completion, before the posted questions.
How did your numbers compare to others?
What range of numbers did you see for 10 flips?
Did anyone get 10 heads or tails?
What happened as you went to 100, 1000, or 10,000 flips? Was there a trend? If so, what was it?
Why would you use the simulation instead of doing the actual flipping?
Expected Answers
No, because the actual numbers generated will vary from run to run. Each flip is independent of the others.
It depends on the situation being modeled. If, like coin tossing, individual outcomes can vary widely, then the simulation would be valid. Also, our real-world experiences validate this model.
If you run the simulation many times, you should see a trend that will indicate what that average behavior is like. However, that is no guarantee that the next run will be close to the trend.
Might want to note the difference between average behavior and individual.
Anecdote: If give students an assignment to flip a certain number of pennies for homework, you can spot the students who “faked” their data. Real data will have occasional “runs” of heads or tails and no observable patterns to the alternation. “Faked” data will be lacking long runs and often show some patterns.

11. Reach Out and Torch Someone!
Using an agent-based forest fire simulation to explore:
Probability
Random Numbers
Averages
Predictions and Hypothesis-Testing
Assumptions
Our next simulation will allow us to apply what we have just learned about probability, random numbers, and averages to predictions about the spread of a forest fire given certain assumptions. Every model has underlying assumptions which need to be explicitly stated.

12. Reach Out and Torch Someone!
Provide an overview of the simulation:
On the screen, you see a group of trees organized in neat rows and columns, like a tree farm. Each tree is an agent in the modeling world. An agent is an individual object.
When lightning strikes a tree (the one in the center in blue), its four neighbors (highlighted in pink) have a chance (probability) of catching on fire also. A tree can affect its N, E, S or W neighbor, but not the diagonal.
The simulation we are about to look at assumes that all trees are the same and there is no wind. It allows the user to set the location of the lightning strike and the burn probability.
Ask participants to discuss the questions in their small groups. May need to re-phrase question one as “What would we expect the outcome to be?”
Share out group responses and chart on a large piece of chart paper. At this time accept all responses.
Possible responses:
We could observe the pattern of burned trees that occur.
We could look for the number of trees that burn in relationship to the probability of burning.
We could observe the pattern of burned trees in relationship to the location of the initial lightning strike.
We could observe whether the same trees burn every time we run the simulation if we keep the lightning strike and burn probability constant.
We expect a uniform spread to the fire.
What could we learn by observing this simulation?
What should we look for?

13. Simulating a Forest Fire
Open
Set the burn probability in the Probability box.
Click on any tree to start the fire.
Note the percent of trees burned.
Does the percent of trees burned stay the same for a given burn probability?
Does the location of the lightning strike affect the percent of trees burned?
What does the value of a burn probability mean?
Ask everyone to follow along as you briefly demonstrate the forest fire applet at
Point out the dropdown menu in the Probability box. Hidden assumption in the previous discussion may have been that there was a 100% probability that neighbors would burn.
Select Custom Probability in order to enter your own values for the probability that a burning tree’s neighbors will catch fire. Connect to coin toss.
Show that the fire will spread as soon as the user clicks on any tree in the forest.
Note that the percent of trees burned is reported after the fire has stopped. Show them where that appears on screen.
Explain iterations.
Have participants work in pairs:
Ask participants to experiment with a variety of probabilities and a variety of lightning strike locations keeping in mind the three questions posed on the slide. Specify the amount of time or the number of runs.
Ask each pair to discuss the three questions on the slide with another group. Walk around the room and listen to the discussion to note points of understanding and points of confusion.
Share out responses to the questions. Consider what you learned from listening to the small group discussion to determine the order in which you call on groups to share their responses.
Chart responses and facilitate discussion.
If it does not come out in the discussion:
To help participants clarify answers, you can ask how the random number generator affects the spread of the fire and the percent of trees burned.
Point out that setting the burn probability to a value such as ¼ does not mean that ¼ of the trees burn down. Each tree next to a burning tree has a 1/4th chance of burning. Sometimes none of the neighbors burn, and other times several neighbors burn. Relate this to the coin tossing experiment.
Point out that in the case of a burn probability of ¼, when the random number is generated, if it is less than ¼, the tree will catch on fire. On the other hand, if the random number is greater than ¼, the tree will not catch on fire.

14. Reach Out and Torch Someone!
Conclude the large group discussion by bringing back the previous slide.
Reference the chart participants created
Is there anything we want to add, emphasize, anything we feel differently about?
Mark up the chart with participants’ suggestions, for example you could star statements that participants still agree with, or put an exclamation mark besides statements that participants feel were really important things one could learn by observing the simulation.
If it does not come out in the discussion:
Emphasize the phrase ‘learn by observing’. Scientists observe the real world to identify patterns and to generate hypotheses.
What could we learn by observing this simulation?
What should we look for?

15. Simulating a Forest Fire
Question:
How do you think the percent of trees burned is related to the burn probability?
Experimental Design:
Get in groups of four to design an experiment.
Share your experiment with the class.
Ask the participants how they would set up an experiment to determine the relationship between the percent of trees burned and the burn probability.
Expected responses:
We need to test burn probabilities from 0 to 1 in regular increments (by tenths, for example).
We need to agree on the location of the lightning strike.
We need to do multiple runs with each probability since the results will vary.
We need to agree on the number of runs to do.
We could average the percent of trees burned for the multiple runs on a single probability to determine the value to use in our data table.
We could collect data from each participant or participant pair and plot the values on a graph.
Distribute the Forest Fire Data Analysis handout.
Point out that already know the answer to 0% probability and 100% probability. Set up graph with probability on the x-axis and % burned on the y-axis. Fill in these known answers for 0 and Ask the participants to predict the shape of the graph in between on their handout. Then have the group share possible shapes and describe what each shape means.

16. Forest Fire Data Collection
Question:
How do you think the percent of trees burned is related to the burn probability?
Procedure:
Using the burn probability assigned to you, run the simulation 10 times and average your results.
Share your average with the class to create a comprehensive data set.
Sketch a graph of percent burned vs. burn probability using the Fire_graphing.xls spreadsheet.
Facilitate the activity described in the Forest Fire Data Analysis handout:
Ask the participants to discuss in their small groups the predictions described in the first part of the handout. Put the predicted graphs on chart paper and ask groups to explain their thinking. Accept all responses.
For Running the Experiment, assign each group a burn probability between 0 and 1. Move around the room as the groups collect the data for their 10 runs.
Open the Fire_graphing.xls spreadsheet on the projecting computer for participant groups to enter their results.
(Optional) Ask the group if their 10 runs varied widely in percent burned or whether the values were all close to the average. Based on their answers, mark their values with a v (for vary widely) or c (for close together).
Discuss the shape of the curve – it’s a logistics curve. Point out that the percent burned does not increase until the burn probability is greater than 0.4. The percent burned levels off near 100% when the burn probability is greater than If an anomaly appears in the graph, lead a discussion of the factors that contribute to this. Would more trials “improve” the shape of the graph?

17. How is the percent of trees burned related to the burn probability?
Compare the graph on the chart paper to this graph collected from a previous workshop.
Ask the participants to discuss:
What is similar between the 2 graphs?
What is different?
What might be the cause of the differences?
Expected responses:
Both graphs are in the shape of a logistics curve.
The percent burned values are not identical on the two graphs.
Variations in results can be explained by the random numbers generated by the model.
Note: The simulation results indicate that as burn probabilities increase from 0.4 to 0.6, the percent of trees burned increases dramatically. Anecdotal evidence from state forest managers shows that there is a narrow window separating a fire that burns out from a fire that burns wildly. One research article verifying this may be found at

18. Evaluating the Simulation
Questions:
How realistic is this simulation?
What are its limitations?
Name some other factors that influence the spread of a forest fire.
Lead a discussion focused on these questions and others suggested by the participants.
Expected responses
The simulation assumes that all trees have the same probability of catching fire when, in reality, forests may consist of a variety of types of trees, each with its own burn probability.
Weather conditions can affect the spread of a fire - humidity and wind, among others.
Trees in a forest are not equally spaced unless the forest was planted for harvesting.

19. Connecting Flipping Pennies and Forest Fires
What is similar about the two simulations we have run today?
What is different about the two simulations we have run today?
How might this impact your teaching about the concept of probability?
Lead a large group discussion about the similarities and differences seen in the flipping pennies and forest fire simulations. Might want to chart.
Expected responses:
Both simulations
Use probability and random numbers.
Get different results even when they are run with the same conditions.
Demonstrate that real world outcomes cannot be expected to match theoretical probabilities perfectly.
Differences:
The forest fire simulation is more complex than the flipping pennies simulation since the outcome of the forest fire depends on the burn probability being applied to successive groups of trees. Therefore, if several trees catch on fire at the beginning, more trees can burn later. But, if few or no trees burn at the beginning, fewer trees are likely to burn down over all.
On the other hand, the result of one coin flip has no effect on later coin flips. Each coin flip is an independent event.
Lead a discussion of the changes this will make in their teaching of probability.
What science concepts are related? (Predator-prey, peppered moth, spread of disease, genetics)

20. What have we learned?
Uncertainty in the real world can be modeled using random number generators. (Goals 2, 4 & 5) Model outcomes will vary when random numbers are used to model probabilities, but trends can be observed through graphs of data collected with multiple runs. (Goals 1 - 5) The assumptions behind a computer model must be made explicit to understand the model. (Goal 2) Computer models can be used to challenge your preconceptions. (Goals 1 & 2)
Ask participants what specific ideas / content they have learned and chart. Then relate to the statements above.
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. (Refer to posted goals chart)
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?

21. Focus on the Forest
Using a systems-based model of a forest fire to explore:
Probability
Graph Interpretation
Patterns in Model Behavior
Predictions and Hypothesis-Testing
Assumptions
Now we will take a different approach to modeling the spread of a forest fire. In this version, we will look at the forest as a whole – as a collection or aggregate of individual trees. In the previous model, our focus was on the behavior of each tree – did it catch on fire or not? In this version, we will focus on how many trees catch on fire, not which ones.

22. Telling the Story of a Forest Fire
Lightning strikes a tree in the forest.
Other trees, depending on their location and their condition, can catch fire from that tree.
The number of newly burning trees depends on
the burn probability
the number of burning trees
the number of non-burning trees that come in contact with the burning trees
Burning trees eventually cease burning and can no longer spread the fire.
Ask the participants to imagine a forest fire started by a lightning strike. Tell the story with a focus on the trees as a group. Explain that we are going to look at a model that deals with the number of trees, not their locations on a grid.
Record the story on chart paper so it can be seen when the next slide is shown.

23. Reading the Model
Point out that this model is similar to a concept map. The boxes and arrows are used to show relationships between the variables.
This is a systems model, which we talked about is different from an agent model. “What in this diagram could be a hint for you that we are dealing with a systems model here?”
Anticipated responses:
Trees (plural) instead of Tree (singular)
Arrows, indicating what is happening to the forest as trees convert from non-burning to burning to burnt
Feed-back arrow indicating that e.g. the catch on fire rate depends on the number of trees and the number of burning trees.
Ask the participants to give their interpretation of the model.
Prompts if they struggle:
Try to make sense of the diagram.
What does rate mean?
What factors affect the rates?
Compare the model to the story on chart paper.
Review these points with the participants after receiving their input.
Trees catch on fire to become burning trees based on a burn probability, the number of burning trees, and the number of green trees.
Burning trees eventually burn out depending on the number of days it takes to burn out.
It will take at least one burning tree to start this fire.
Ask the participants to consider the real world conditions that would affect burn probability and days to burn.
When are trees more likely to catch fire? When are they less likely to burn?
Do all trees take the same amount of time to burn out?
Point out that in this model the flows (thick arrows in the middle) act as a timeline.

24. Analyzing the Output
The output for this model is expressed as a graph. Let’s look at this graph in the context of the forest fire.
Draw the graph on chart paper.
Point out that the y-axis is number of trees and the key for the curves is below the graph.
Encourage them to use the scales on the graph to tell the story of the forest fire.
Have participants come to the chart paper and mark the graph with comments.
Expected responses:
The green curve shows that the forest had 400 trees, most of which burned by the 5th day of the fire.
The blue curve shows that the number of burnt trees increases rapidly through day 3, but then increases more slowly, leveling off after day 6. The curve is a logistics curve.
The red curve shows the number of trees actively burning each day. The fire is most active near the end of the 3rd day of the fire. After that, there are fewer trees left to catch fire and most trees have burnt out.
Alternate strategy:
Project graph on white board or chart paper
Assign each curve to a different group. What is the story of your graph?
Give each group 2 large Post-Its
On one post-it: What is the shape of the graph? What do you see?
On the other: What does it mean?
Have participants place post-its on the projected graph.
Discuss green, then red, then blue graphs and the participant answers for each
Another question to ask: Which day is critical?

25. Making Predictions Open the ForestFire.mdl model. Run the model.
Predict how the graph would change if
you increased the burn probability.
you increased the days to burn.
Distribute the ForestFireInquiry handout.
Open the ForestFire.mdl model. Show participants where to find the model on their computers and follow along with them on the demo computer.
Point out the Run and Run AutoSim buttons on the model screen.
Direct the participants to work in pairs to follow the directions for Part 1 and make their predictions.
Share out group responses for questions 1 and 2 and write on chart paper.

26. Increasing the Burn Probability
Run the model In AutoSim mode.
How does the forest fire change as the burn probability is changed?
If you run the model three times with the same settings, does the graph change?
Is there any evidence of random numbers in this model?
In Part 2, we will see the effects of modifying the two parameters, burn probability and days to burn.
Follow the steps in Part 2 of the ForestFireInquiry handout.
Expected responses for questions 1-3 in Part 2:
Smaller burn probabilities result in fewer trees burning. Larger burn probabilities result in the forest burning down faster.
No, we get the same output as long as we use the same input. This is a deterministic model. Results are based on equations.
There is no uncertainty built into this model. Given the same inputs, the same outputs occur. Therefore, there is no evidence of the use of random numbers.
Note: the sensitivity to probability is much higher in this model than in the Agent-based Fire! Model.

27. Double the Burn Probability
Here are two graphs from the model. The one on the left used a burn probability of 0.1. The one on the right used 0.02.
Notice that the forest completes its burn in fewer days and more trees are burning by the end of the first day.

28. Increasing Days to Burn
Run the model In AutoSim mode.
How does the graph change when days to burn is increased?
How does the number of days to burn change the behavior of the forest fire?
Expected responses for questions 4-5 in Part 2:
When trees take longer to burn out, there are a greater number of trees burning on day 3 and it takes longer for the forest fire to end.
The forest fire spreads faster and takes longer to burn out when the days to burn increases.

29. Quadruple Days to Burn
Here are two graphs from the model. The one on the left used Days to Burn = 1. The one on the left used Days to Burn = 4.
Note that more trees are burning on Day 2 and the trees take longer to burn out.

30. Comparing Agent Models to Systems Models
What is similar about the two versions of forest fire simulations we saw today?
What is different about them?
Lead a large group discussion about the similarities and differences seen in the two versions of the forest fire simulation. Consider charting as a Venn diagram.
Expected responses:
Both simulations
Show that more trees burn when the burn probability is higher.
Differences:
The agent-based model:
Gets different results even when run with the same conditions.
Demonstrates that real world outcomes cannot be expected to match theoretical probabilities perfectly.
Illustrates the variability within a group.
Can be used to determine the behavior of the group if data from several runs are analyzed.
Requires an understanding of the actions of the individual.
The aggregate-based model:
Gets the same results when run with the same conditions.
Illustrates the behavior of the group, not individuals.
Requires an understanding of the average behavior of the group.

31. Comparing Graphs from the Forest Fire Models
Ask the participants to consider the two graphs shown. Consider these questions:
What story does the blue curve on each graph tell?
What is similar about the graphs?
What is different about the two blue graphs?
Expected responses:
The graph on the left tells us that as we vary the burn probability in the forest fire model, the percent of the forest burned will change. When the burn probability is very low (less than 0.4), a low percentage of trees in the forest burn. When the burn probability is high (greater than 0.7), nearly all of the trees burn. Burn probabilities between 0.4 and 0.7 cause a nearly linear increase in the percentage of trees burned. The graph on the right tells us that as time progresses, more and more trees burn until all of the trees have burned. We have no idea what value was used for the burn probability.
The blue dots on the left-hand graph form a logistics curve as does the blue curve on the right-hand graph.
The x-axis on the left-hand graph is the burn probability while the x-axis on the right-hand graph is time in days. The y-axis on the left-hand graph is percent of trees burned while the other y-axis is the number of trees burned.
Other possible comments:
Either graph could be used for Cost-benefit analysis. When is it critical to get to the fire and make the most difference?
This is why it is important to label axes.
The graph on the right is one forest burning at one probability setting. The graph on the left is the cumulative data for many burns at many different probabilities.

32. Other Systems?
Ask participants to look at this systems model and consider other topics that might fit.
Pose these questions:
What other topics have similar behavior to a forest fire?
When does the behavior of one individual affect other individuals?
Expected responses:
Communicable diseases can spread like a forest fire.
Animal populations experience bounded growth depending on the carrying capacity of their territory.
Enzyme reactions growth in the form of a logistics curve.
Pose this question:
Could these topics also be represented by agent-based models? If so, what are the agents?
Susceptible and sick people could be the agents in a spread of disease model.
Food sources, animals and their predators could be the agents in a population model.
Enzymes and catalysts could be the agents in an enzyme model.

33. What more have we learned?
Theoretical probabilities can be used to calculate the behavior of a group of objects in a model. (Goal 5) Models can be used to test predictions about the behavior of a system under varying conditions. Graph interpretation requires an understanding of both axes, the shape of the curve, and the underlying model. (Goal 3) The same problem can often be represented in both agent-based and systems-based models. (Goals 1 & 2) Problems that seem different on the surface may have characteristics in common when looked at from a modeling perspective.
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. (Refer to posted goals chart)
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?

34. How does this apply to your curriculum?
List topics in your curriculum that involve…
Random behavior
Probability
Interactions between individual agents
Changes in aggregate behavior over time
Examples
Biology/Environmental Science – predator/prey, epidemics, genetic drift, food chains, ecosystem disturbances
Chemistry – enzyme kinetics, gas chromatography, heat, diffusion
Physics – mechanics, radioactive decay
Earth/Space – climate change, erosion, percolation
Mathematics – fractals, random walks, probability
Lead a discussion on the application of these concepts to the participants’ curriculum.