Generally, the things that are true are declarative statements, such as: "All polynomials have a root" or "Some whole number are the sum of two primes." The way a statement is shown to be true is by providing a PROOF, which is a very carefully done argument, one that follows agreed up conventions as determined by the community of mathematicians. This nature of this convention for a good argument has actually changed over time. The original conventions were set up by Aristotle. Once a statement has been provided with a correct argument, it is called a THEOREM. Simply stated, the goal of mathematics is the creation and proofs of theorem.

Mathematics is the language of science. Many (but not all) of the laws, theories, rules of science have a mathematical description. The advantage of this is that, so stated, one can use the vast "machinery" of mathematics to not only to elucidated these scientific statements, but often create and prove them. Generally, open any journal in science and it fi packed with diagrams and formulas. One of the very important mathematical ideas is that of the "derivative" - a term from calculus that has to do with CHANGE. Differential calculus is the study of how things change, and how things change (either over time or under the influence of something else) is clearly and important issue in science.

Whenever a measurement is made, three types of errors can arise:

(1) SYSTEMIC: [Something is wrong with the system you are using.] For example, no matter how careful you are, if your scale is not calibrated (so that no weight read 0), all measurements are systematically effected.

(2) RANDOM: These errors happen randomly, of course. Say you are using a ruler to measure something and the rule gets longer or shorter depending upon the temperature that in the lab that day. [This is need to be a very carefully taken measurement to see this issue.] Assume the ruler is calibrated to be accurate at when the room is 75 degrees, but then, because of "who knows what" some days it is warmer and some days colder.

This has been done in Unit 1. The idea is that if there is lots of data, one needs ways to get a sense of the data. This techniques give a rough and approximate view of the data, and there are easily created examples of data sets for which these notions are misleading.

Mean: This is the average value of the numbers in the set (add them all up and divide by how many)

Mode: This is the most common value (or if there is a tie, the most common values).

Standard Deviation: [Messiest to compute} The standard deviation is the square root of the average of the squares of the difference between each value of the data set and the mean of that data set. [That is: compute the mean, for each data point, subtract the value from the mean, square that result and take the average of those squares.] The standard deviation is a measure of the spread of that data. If the data is NORMALLY distributed (that is, if you draw a HISTOGRAM of the data and it is very close to the shape of a special kind of curve called a normal curve (this MAY be defined later), then approximately 68.2% of the data is within 1 standard deviation from the mean [THAT statement is a THEOREM, that is, it has a "correct" proof], 95.4% is within 2 standard deviations from the mean, and 99.74% are within three standard deviations from the mean.

Given a data set, on can construct a Histogram to better visualize the date. There are lots of ways to do this: it is not a well-defined process. One way is to break up all the values into groups of ranges [often called BINS (which depends upon the data itself and your mood at the moment): say: 0 to 10, 11 to 20, 20 to 30, ,,,,,,,,,,,70 to 80. [Assuming all the data lies from 0 to 80]. Then count how many of the data point lie in each range and then make a graph where along the base you put the various ranges and when the height of a box above each range is the number of data points that fall in that range. Different choices of ranges yield different histograms, and that is where the notion is not well-defined.

Categorical: If you just keep track of the category of the date in an experiment, say the color of the insects you see, then one says that the measurement scale is a categorical scale.

Ordinal scales have an implied order of the data points (in contrast to color, unless, of course, you think about color frequency!), so that the 'best" school is 1 and the next best is "2," etc.

Interval: This is a scale in which the difference between measurement values carries meaning: For example, if you are measuring the temperature of a room. In this case, 10 degrees warmer has meaning. [But school 1 and 3 are two apart and school 8 and 10 are also two apart, but that does not mean that the way school 1 is better than school 3 is the same as how school 8 and 10 differ.

Whenever you measure something, even after you eliminate "SYSTEMIC," and "STUPID" errors, usually "RANDOM" errors remain. Your hand still shakes as you touch something, or the wind blows one way or another, etc. One way to minimize these random errors is to take repeated measurements and then averages them. The reason is that the + and -'s tend to cancel out. The fancy reason is that (usually)m random errors are normally distributed and by taking the average the resulting distribution is also normally distributed but with a standard deviation equal to the original standard deviation divided by the square root of the number of repetitions you take. AND THEN, even as one makes a measurement, one makes a compromise: Equipment can display only so many digits, and when you round to a certain number of digits, further error is created called Round-Off error. Those, too, can be lessened by the process of averaging.

If you perform arithmetic on numbers whose true value is known only approximately, the result of the operation is generally off by even more than the original numbers. For example, say you have measured stick as 10 inches but for a variety of reasons you are only confident that the true value is somewhere in the range from 9.5 to 10.5. And say you do the same for a second stick, getting the range from 3.5 to 4.5. Then the two stick laid end to end might in fact be anywhere from 13 (9.5+3.5) to 15 (10.5+4.5), so the error range (which was equal to 1) has now DOUBLED to two! It gets worse if you multiply (as in finding the area of the rectangle whose sides are those two sticks) for now the range is from 33.25 (9.5 x 3.5) to 47.25 (10.5 x 4.5) an error range, now, of fourteen!!

re THEOREMS: Which, if any, of the following statements are THEOREMS in mathematics? Why?

(a) 2+3=5
(b) The sum of an even number and an odd number is an odd number
(c) If a fair coin is tossed, the chance of getting a HEADS is 1/2.
(d) Two points determine a line.
(e) Three points determine a circle.
(f) The mean of any three numbers is larger than their median.

re: ERRORS: Characterize the errors below as either random, systemic, or stupid. Explain.

(a) Using a cloth tape measure that has stretched
(b) Weighing yourself in your clothes
(c) Weighing yourself without your clothes
(d) Weighing yourself in the cold
(e) Weighing produce at the grocery which is inside of a plastic bag.
(f) Weighing yourself standing on one foot

re: SCALES: characterize as categorical, ordinal, interval or ratio scales

(a) The rank of a tennis player
(b) The weight of a ball
(c) The rating (X, R, PG13, etc) of a movie
(d) The characterization of land as: swamp, field, forest, etc
(e) The elevation above sea level of the top of a mountain as a measure of how far its peak is from the center of the earth.

re: ARITHMETIC of approximate numbers
If a box is measured and its sides, in inches, are: 3, 4 and 7, and if each of those measurements could be off as much as 1/2 inch (so that the "3" is really somewhere between 2.5 and 3.5), what is the range of possible values for the volume of the box?