Overarching Goal: Understand that computer models
require the merging of mathematics and science.
1. Understand how computational reasoning can be infused into
teaching.
2. Develop a working definition of computational reasoning.
3. Recognize the importance of graph interpretation skills in
understanding model behavior.
4. Recognize that probability and random numbers are important
mathematical ideas that can be modeled using tools of
computational reasoning.
5. Understand that probability can be used to simulate real-world
phenomena and make predictions.
1.
2.
3.
4.
5.
6.
7.
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 systemsbased models
Curriculum applications
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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
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
o
Understand and evaluate random processes underlying statistical
experiments
o
Make inferences and justify conclusions from sample surveys,
experiments and observational studies.
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.
Theoretical probability vs. the real world
1. What is the probability of getting a head when you
toss a coin?
2. In 10 trials, will you get an equal number of heads and
tails?
3. In 1000 trials, will you get an even split?
4. Will everyone in the room get the same answer?
5. How would you design an experiment to answer these
questions?
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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.

Open the flipping_pennies.xls
spreadsheet.

Answer the questions on the handout.
After conducting the simulation, consider these questions.
1.
Will a simulation that uses random numbers give the
same result every time it is run? Explain.
2.
Is such a simulation a valid representation of reality?
Explain.
3.
What can you learn from a simulation if it doesn’t
always give the same result? Explain.
Using an agent-based forest fire simulation to
explore:





Probability
Random Numbers
Averages
Predictions and Hypothesis-Testing
Assumptions
 What could we learn by observing this simulation?
 What should we look for?
Open http://www.shodor.org/interactivate/activities/Fire
Set the burn probability in the Probability box.
Click on any tree to start the fire.
Note the percent of trees burned.
1.
2.
3.
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?
 What could we learn by observing this simulation?
 What should we look for?
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.
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.
120
100
%burned
80
60
40
20
0
0.0
0.2
0.4
0.6
burn probability
0.8
1.0
Questions:

How realistic is this simulation?

What are its limitations?

Name some other factors that influence the
spread of a forest fire.
 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?
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)
Using a systems-based model of a forest fire to
explore:
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
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
Probability
Graph Interpretation
Patterns in Model Behavior
Predictions and Hypothesis-Testing
Assumptions



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.
burn
probability
Trees
Catch on Fire
Rate
days to burn
Burning
Trees
Burnt out Rate
Burnt
Trees
Forest Fire
400
300
200
100
0
0
1
2
Burnt Trees : Current
Burning Trees : Current
Trees : Current
3
4
5
6
Time (Day)
7
8
9
10


Open the ForestFire.mdl model.
Run the model.
Predict how the graph would change if
1. you increased the burn probability.
2. you increased the days to burn.

Run the model In AutoSim mode.
1. How does the forest fire change as the burn
probability is changed?
2. Do your neighbors get the same result you do
when you all use the same burn probability?
3. Is there any evidence of random numbers in this
model?
4. How does the graph change when days to burn is
increased?
5. How does the number of days to burn change the
behavior of the forest fire?
 What is similar about the two versions of
forest fire simulations we saw today?
 What is different about them?
Forest Fire
400
120
%burned
100
300
80
200
60
100
40
20
0
0
0
0.0
0.2
0.4
0.6
burn probability
0.8
1.0
1
2
Burnt Trees : Current
Burning Trees : Current
Trees : Current
3
4
5
6
Time (Day)
7
8
9
10
change
probability
Variable
1
var1 to var2
change rate
var2 longevity
Variable
2
var2 to var3
change rate
Variable
3
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.
(Goal 5)
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 systemsbased models.
(Goals 1 & 2)
Problems that seem different on the surface may have characteristics in
common when looked at from a modeling perspective.
(Goals 1 & 2)
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