A simple instructional activity illustrates basic principles of *data analysis:* Imagine that we have *data* from a set of related *physical experiments,* with two *variables* (x, y) that are *mathematically related* by a *theory;* all *experimental systems* are identical, except that we *control* the value of x so it's different in each system, then we *observe* the value of y. Here is one possible data set:

In experiment #1, x = 3 and y = 9. Ask students to *generate* three theories, with each proposing a simple mathematical relationship between x and y; of course, they also can propose additional theories that are more complex.

For experiment #2, with x = 5, ask them to make a *prediction* (when x = 5, y = __ __) using each theory. Then tell them that, in a physical experiment with the second system, the real observation was "x = 5, y = 15". For each of their original theories, ask them to compare their *theory-based predictions* with *reality-based observations,* in a *Reality Check* that helps them *evaluate* each theory. Then all of you can discuss their experiences, and develop the concepts of *plausible competitive theories* (after experiment #1) before a *crucial experiment* (#2) lets them distinguish between these theories, so only one theory retains a high *evaluative status.*

This simple two-experiment activity can help students understand some basic principles of Science Process, and some of its terms *(italicized),* in the context of their problem-solving experiences.

The activity could be extended by letting students generate a set of data for each of their theories, for experiments with x = 3, 5, 7, 9, 11, and put each data set into a table. Then they can graph each set of data, either manually with pencil-and-paper or (after they've made manual graphs) with a computer program, so they can see what each theory “looks like” when it's represented in graphical form.

By comparing their two approaches to analyzing data — first when they mentally searched for patterns in experiment #1, and later when they graphed each of their data sets — students can recognize that the two approaches are related strategies for achieving similar results when they are designing a theory by generating-and-evaluating theories.

The graphs they make can answer questions about some theories about mathematical theories: are x and y related directly, as in (y = mx) or (y = mx + b)? or is a "squaring" involved, as in (y = xx)? Other graphs can help answer other questions:

In addition to these graphs, they can use graphs to test a wider range of theories about mathematical relationships between x and y: (described below at *)

possible title for a section — Graphical Representations of Theory-Based Models

In the activity above, each graph-point is an experiment, so interpreting a graph is similar (in some ways) to interpreting a series of experiments, as examined in *Section 5 of my details-page*.

data sets in which variables have the relationships used in this activity (direct and squared) naturally arise from lab-experiments with physics of motion, so these could be used to make a connection with real-life phenomena;

in a computerized version of this activity (not available now) the program could show an "interesting animation" to represent the process of running a new physical experiment with a new experimental system (in which x has been changed from 3 to 5 to...) to make it more interesting for students using the computer program;

when students make tables this can help them with pattern-finding, because in a table all numbers are visible at same time;

* possible types of graphs: direct (y = mx, y = mx + b), inverse (xy = k); plus logarithmic or exponential (y = log x, y = ln x) (y = 10^{x}, y = e^{x}) after students understand logs, as explained *here**.*

teachers can show students how to make graphs — by hand or with a computer — so they get a linear line; students can do this manually (once or twice) by a teacher providing a data set to illustrate log-relationships, etc, and students do the math using a calculator and putting the results into a table; then, once they understand what a computer program is doing, they can just use a computer.

but it might be better for students if they can recognize "the shapes" for different math-relationships, and then use a computer program to distinguish between relationships that produce similar shapes; the shape can be linear (y = mx, y = mx + b); curving upward with increasing slope (for y=xx, y=e^{x}); curving upward with decreasing slope (y = square root of x, y = √ X , plus some log-functions); a different-looking shape for xy = k); and then use Excel as "tiebreaker" similar-looking shapes (like y=xx, y=e^{x}, y=10^{x}); I.O.U. - eventually I can include graphs that "show these shapes" in this section, to supplement these descriptions using words and equations.

maybe students can use log paper? log-log paper? probably not for most students, who should just (except for advanced students?) use computerized graphing for log-relationships?

an interesting type of experiment (or series of experiments) uses calibration curves; this is described *here* where Paragraph 1e describes two useful skills (over-and-down on a graph, and "y = mx + b" by substituting-and-solving), with students doing both and comparing the results; they should also be able to calculate a slope "m" manually from a graph (which they could draw, or print from a computer-generated graph) by choosing two points (initial & final) and finding values for x and y (both initial & final) and substituting numbers into the slope-equation, (Yfinal - Yinitial / Xfinal - Xinitial), and comparing their values with the value of "m" generated in the computer's equation for "y = mx + b")