Modeling A model is an idealized and/or simplified version of an object or process found in the real world.

Mathematical Models ● Start with of a set of variables corresponding to measurable quantities in nature or society: E for energy m for mass c for speed of light ● Once we have a set of variables, then we search for equations that relate the variables: E=mc 2

Mathematical Models ● Remember a “relation” is a precursor to a formula or function. ● If the variables DON'T relate to the real world then we are doing pure mathematics which cannot be verified experimentally.

Scientific Method There is no way to prove a model is correct; however, we can compare the model's predictions to observations in the real world and convince ourselves it is a good model over time. One example of this problem is the current debate over climate change aka global warming. Despite overwhelming observational evidence some still deny that the model of climate change is correct.

Historical Context ● One of the most famous mathematical models of all time is Sir Isaac Newton's formulas for motion and gravity, which work great at human level. ● Albert Einstein expanded upon Newton's model by showing it was not accurate for extremes of mass or speed, the very large or very fast. ● Max Planck further refined our understanding of nature by developing Quantum Mechanics which is a mathematical model of the subatomic.

Did you know? ● Anything you view on a computer screen, including smartphones and videogames, is ultimately a mathematical model. ● The only thing a digital device can store or process is numbers, even the letters on this slide are stored as numbers. ● Pictures are stored as a long series of numbers representing color or shade. So a picture is really worth a thousand numbers, not a thousand words.

Modeling or Not? ● For each situation on the worksheet, decide whether or not it is an example of mathematical modeling. ● You must explain why you think it is or is not mathematical modeling with a sentence or two. ● No answer = 0 points Wrong answer = 1 point Wrong answer with explanation = 2 points Right answer = 3 points Right answer with explanation = 4 points