An Introduction to Computational Reasoning in

High School Science and Mathematics

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.

Goals

1. Why computational reasoning?2. Probability – theoretical vs. real-world

behavior3. Probability in an agent-based model4. Probability in a systems-based model5. Analysis of model output via graphs6. Comparison of agent-based and

systems-based models7. Curriculum applications

Agenda

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

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 Conclusionso Understand and evaluate random processes underlying

statistical experimentso Make inferences and justify conclusions from sample

surveys, experiments and observational studies.

Why Computational Reasoning?

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?

Hands-on Probability

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.

Probability via the Computer

Open the flipping_pennies.xls spreadsheet.

Answer the questions on the handout.

Flipping Pennies Simulation

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.

Conclusions drawn from theFlipping Pennies Simulation

Using an agent-based forest fire simulation to explore:

Probability Random Numbers Averages Predictions and Hypothesis-Testing Assumptions

Reach Out and Torch Someone!

Reach Out and Torch Someone!

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. Does the percent of trees burned stay the same for a given burn probability?

2. Does the location of the lightning strike affect the percent of trees burned?

3. What does the value of a burn probability mean?

Simulating a Forest Fire

Reach Out and Torch Someone!

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.

Simulating a Forest Fire

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.

Forest Fire Data Collection

How is the percent of trees burned related to the burn probability?

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

20

40

60

80

100

120

burn probability

%b

urn

ed

Questions:

How realistic is this simulation?

What are its limitations?

Name some other factors that influence the spread of a forest fire.

Evaluating the Simulation

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?

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)

What have we learned?

Using a systems-based model of a forest fire to explore:

Probability Graph Interpretation Patterns in Model Behavior Predictions and Hypothesis-Testing Assumptions

Focus on the Forest

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.

Reading the Model

Trees BurningTrees

BurntTrees

Catch on FireRate

Burnt out Rate

burnprobability

days to burn

Analyzing the Output

Forest Fire

400

300

200

100

0

0 1 2 3 4 5 6 7 8 9 10Time (Day)

Burnt Trees : CurrentBurning Trees : CurrentTrees : Current

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.

Making Predictions

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?

Experimenting with the model

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?

Comparing Graphs from the Forest Fire Models

Forest Fire

400

300

200

100

0

0 1 2 3 4 5 6 7 8 9 10Time (Day)

Burnt Trees : CurrentBurning Trees : CurrentTrees : Current

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

20

40

60

80

100

120

burn probability

%b

urn

ed

Other Systems?

Variable1

Variable2

Variable3

var1 to var2change rate

var2 to var3change rate

changeprobability var2 longevity

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

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.(Goals 1 & 2)

What more have we learned?

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

How does this apply to your curriculum?