Day 4

Context

Having modeled and written conditional arguments for several things we now turn our attention to modeling things that we can't see.

Explanation

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We will take a short break from writing conditional arguments and focus on modeling in general and specifically modeling things that we can't see. This is a common problem in science. Scientists are continually trying to figure out things that they can't actually see. They often have to rely on observations that are caused by things that they can see and indirectly build their models from that.

After this class you should be able to:

Model

Modeling

One quality of a scientist is the ability to ask and logically answer the following questions about an observed situation:

These three questions could be broken down as follows:

What is happening?
  • Observation (statement of perceived facts)
  • Description of the situation
  • Identification of important aspects (variables)
Why is it happening?
  • Modeling (interpretation of facts, theories)
  • Language model (including both written and oral)
  • Mathematical model (including graphs)
  • Physical model (including sketches)
  • Matter rearrangement model (chemical equations)
What does it mean?
  • Prediction (applying a model to a new situation)
  • Sensitivity analysis
  • Limit analysis

Modeling is central to a scientist?s activities. It might be said that a scientist's goal is to correctly model observable events. The models can take several forms. Here are a few:

1. Language models, both written and oral. These models use words to clarify observed events. Some people may consider a description of what happened to be a sufficient model, but for a scientist the description is only the framework of a good model. Often these models are called theories. These models often explain, in a logical way, why something happens. They may also show interconnections between variables and can be used to predict other events.

An example: The weight of an object depends on the mass of the object and the attraction of the object by gravity.

2. Mathematical models. These models use the language of mathematics to clarify observed events. Mathematics is a very powerful tool to simply define the connections between variables for some observed event. Mathematical models may include every variable in the system (if the model builder knows all of the variables), but is more likely to only include the most important variables. The value of the model has to do with its ability to correctly describe the observed event. Each variable retained in the model is given a symbol and the symbols are related by a mathematical equation.

The idea of using a symbol for a variable should be familiar since the words we use to communicate with every day are just symbols (cow is a symbol for a four legged animal that produces milk). It should also be clear that the context of the symbol is important and that the same symbol could mean two different things (turn right, you are right). An example of that in science is that g could mean grams (a unit of mass) or it could mean the acceleration due to gravity (a number, usually about 9.8 meters per seconds squared). The context will tell. If an object has a mass of 2.4 g, you would assume it means 2.4 grams. If, on the other hand, the symbol g is found in a mathematical equation, you would assume it to be the constant 9.8 m/s^2.

The mathematical model corresponding to the language model above is: weight = mass times the gravitational constant, or, using shorter symbols, wt = mg.

3. Graphical models. Graphical models are generated by choosing one variable to observe and then allowing a different variable to change over a range of values. For the model being used, as an example, you may wish to observe weight and see how it changes as the mass goes from zero to 100. Mathematical models and graphical models have a direct relation, knowing one determines the other. You can, however, predict graphical models from language models just as well when words like directly proportional, inversely proportional, etc. are used to describe relationships between variables.

4. Physical models (and their corresponding paper and pencil sketches). These are actual, physical systems that mimic the observed event. Some successful physical models have nothing to do with the observed event (frying an egg to model your brain on drugs), but most of the time they are systems that retain the important variables, but aren't too cluttered by incorporating all of them.

5. In chemistry and nuclear physics there are what I call matter models. These are symbolic relationships, similar to mathematical models (equations), which describe the rearrangement of matter. They are often called chemical or nuclear equations. An example is the combination of oxygen and hydrogen to form water.

2H2 + O2 → 2H2O

Good Models

What constitutes a good model? The first response of many students is that it must be a scaled down copy of the actual thing being modeled. Model airplanes, model cars, model trains could all fit into this category. But do they have to all of the parts to be a good model?

My father-in-law was a model train collector. He had one engine that cost $3,000 an had every detail of the real thing. But when he chose trains to give to my children he looked for durability and ease of use. That was a lot more helpful for us. They still looked like and acted like the real thing, but they weren't as long and had fewer moving parts. Which was the better model for my young children?

My daughter's first doll was a "Raggedy Ann" doll that was all fabric with painted on eyes, etc. It still had a head, some arms, legs and a body, but it really didn't look much like a real person. But it was the right model for her at that age. Here is a model that uses images that are nothing like what it represents.

Do you get the message? Are our brains really like eggs? Is a drug made from a frying pan? Even so I would say this was a very effective model!

So, when we say a good model I guess we need to know what we mean. In this class we will define a good model as a model that agrees with our observations. If it mimics what we see and makes predictions that are later observed it is a good model.

A good model agrees with all of our observations.

Can you come up with a good model for the Think-A-Majig?