Quantitative Reasoning

Activity – Analyzing the Results Why aren’t all the results the same? How do we compare results? What kind of errors occurred? Is our error small or big? Is our result precise, accurate or what? So, do our results agree?

Why aren’t all the results the same? Useful questions to ask, if results don’t agree: –Which object did you measure? –What units where you using? –What is your estimated error?

How do we compare results? Don’t compare apples with oranges! Need to use the same units 1 inch = 2.54 cm To get inches from centimeters, divide by 2.54 To get centimeters from inches, multiply by 2.54

What kind of errors occurred? There are systematic and random errors To beat down random errors, measure the same thing many times, and the errors will even out, i.e. the overall error will be smaller The systematic error can be reduced by doing a better experiment, or understanding your instruments better (miscalibrations etc.) Human error is not an acceptable error source in science! It just means you are a bad experimenter.

Is our error small or big? It depends! If you have a small error and the measured length is also small, you might have a huge error! Use percentages: –Percent error = (estimated error)/(result) x 100% –Example: 51.3 cm ± 0.2 cm gives –Percent error = (0.2 cm)/(51.3cm) x 100 % = 0.4 % (This is a pretty small error)

Is our result precise or accurate or what? Two different concepts: precision and accuracy! High precision means small error High accuracy means close to an accepted value Examples: * * * * high precision, high accuracy * * * * high precision, low accuracy * * * * low precision, high accuracy * * * * low precision, low accuracy accepted value

So, do our results agree? Results agree, if they are within the error margins of each other Examples: | O | values very different, but errors large: agreement! | O | values closer, but errors smaller: no agreement!

Quantitative Reasoning Amazingly powerful tool to understand the world around us Fundamentals: –Ratios –Graphs –Area &Volume –Scaling –Arithmetical statements

Achieving Scientific Literacy (Arons Article) Two types of knowledge –Declarative (Learned Facts, “book knowledge”) –Operative (actually knowing how to solve problems) Trouble with GenEd courses – Too much in too little time –Getting a “feeling” for the subject doesn’t work –Need to understand the underpinnings first (area, volume, scaling, energy, atoms,…)

Scaling Often one is interested in how quantities change when an object or a system is enlarged or shortened Different quantities will change by different factors! Typical example: how does the circumference, surface, volume of a sphere change when its radius changes?

How does it scale? Properties of objects scale like the perimeter, the area or the volume –Mass scales like the volume (“more of the same stuff”) –A roof will collect rain water proportional to its surface area