FEd Summer 2002 Newsletter -Approximate Best Fit Modeling of Physics Phenomena by All High School Physics and Chemistry Students

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Summer 2002

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## Approximate Best Fit Modeling of Physics Phenomena by All High School Physics and Chemistry Students

Stewart E Brekke

Most upper grade and high school students can do "approximate best fit" modeling of physical and biological phenomena. In every lab in physics and chemistry that allowed simple modeling, I attempted to do a mathematical modeling. This modeling was done by using an "approximate best fit method" in which the student finds the "approximate best fit curve" and the "approximate best fit equation" to fit the curve. Thereby, students describe a physics phenomenon mathematically, as it should be, by finding its equation using only algebra and a calculator.

Many students were not used to this approach and often tried at first to simply describe verbally what happened in the lab experiment. But I pointed out to them that a worthy conclusion of a high school student who has done an experiment is to describe the physics event in mathematical terms. Most simple and direct experiments such as the relationship between the initial height of a tennis ball and its first bounce height can be modeled approximately by a simple equation such as y = kx, y = k/x, y = kx2 or y = kx1/2.

After I get the students started, they take the data and plot it on a rectangular coordinate system. Other coordinate systems can be used, however. I then put the curves of each of the above equations on the board: a generic line through the origin with a generic equation under it, y = kx; a generic hyperbola with y = k/x under it; a generic parabola (usually half of one) with y = kx2, and a generic square root curve, with the generic equation y = kÖ x under it. The student then tries to identify the best curve that fits the data points approximately and sketches it on the graph approximating the points. This is called "approximate best fit modeling."

The student then picks a point on the sketched curve and solves for the constant k. After solving for the constant k, the equation is completed (such as y = 0.45x or y = 1.66/x). The student then substitutes the variables used in his equation, such as y = H, initial height, and x =B, first bounce height. Then the equation describing the first bounce height of the tennis ball versus its initial height becomes, for example, B =0.45H. Only a meter stick and a tennis ball are needed for this "approximate best fit" line modeling exercise. In the linear case a ruler can be used for help with the modeling, where the student puts the ruler at the origin of the graph and tries to put half of the data points above the line and half of the data points below the line. The student then draws in the line and picks a point on the line, then solving for the constant in the model y = kx of a line through the origin.

At first the sketching of the "approximate best fit curve" is difficult for the students since they have never done this type of graphing, and I often have to help them. I also have to warn them that this type of graphing is only done in the physics class and the chemistry class since if they do an approximate best fit in a math class, they will probably not be doing their math graphing correctly. I ask the students why the curves fit so well in algebra class but not in physics or chemistry class. I explain to them that most often in math class we are dealing with ideal situations. I often refer to Plato's Theory of Ideas in which in a perfect world, an Ideal world, we make no errors in measurement. But when we take measurements in a real situation, we make errors in measuring and therefore all the points are not in a perfectly straight line, or in a perfect hyperbola. Therefore, we must make approximations in measuring and in our equations in physics class. I purposefully do not use the computer to model the data since the students can do it easily by hand and calculator.

By doing this type of modeling for all kinds of physics experiments the student can see how we get some of the formulas we do physics problem-solving with. One type of formula is made by modeling data such as the speed of sound formula v = 331.45 m/s + 0.6T, where T equals the temperature of the air. Even Ohm's Law was found in this manner, by modeling empirically.

Some of the formulas used in physics class are derived from deduction from other known formulas. For example, the relation E = hf was found by using induction with best fit modeling. Combining it with the standard wave equation v = c = fl , using deduction, gives us E = hc/l . In this manner the students can see the different ways in which physics formulas that they use in class are obtained, some by inductive best fit modeling and some by deductive methods or by a combination of both.

These "approximate best fit modeling" experiments can be used for science fair projects. The science fair project I still have is by one young boy, a basketball player, who found the "approximate best fit" equation of a line predicting the initial height of a basketball versus its first bounce height on a regulation hardwood basketball floor as B = 0.60 H and won first place in our school science fair. Another modeling experiment that often works out very well is the curve and formula relating the period of a simple pendulum to its length. The students can easily take data using a meter stick and stopwatch, and the curve is approximated by y = kÖ x where 2p /Ö g = k. Therefore, T = 2.01Ö L. The students can then find their percent error also.

Other modeling experiments are the relationship between the area of a flashlight projection and its distance from the bulb of the flashlight, the time of free rolling of a ball down an incline versus its height at the top of the incline, the relationship between the hand and the arm length, the height and the foot length, finding g, finding p , finding the number of turns of a wire on a long iron nail versus the number of paperclips it can pick up, and so on. The ability of high school students and even upper grade-school students to model using the "approximate best fit modeling" technique is well within the capability of every student in physics, chemistry, biology and earth science.

Finally, even using the periodic table, especially the noble gases, for modeling specific heat, density, and ionization potential versus atomic number or mass number provides a non-experimental academic exercise in modeling. Other experimental curves from the periodic table and physics and chemistry texts can be modeled by hand and calculator if they are smooth or linear using the "approximate best fit method."

Linear modeling can also be done by more motivated students using the standard statistical regression formula. I had a high school science fair winner, now an assistant principal in an elementary school, find the equation of the stretch of a rubber band versus the applied mass using a regression line determined by the method of least squares. This can be done easily with some time and effort using a cheap calculator by many students if they have the time and motivation. Calculators have made many time-consuming and error prone calculations much more accessible to even at risk students, although the young girl who did the equation for the stretch of a rubber band was above average in ability and motivation.

For many years, I have done these approximate best fit modeling techniques with regular chemistry and physics students, from the most at risk students to the most motivated honors students. I have had classes start out at the beginning of the year by modeling the first bounce height of a tennis ball versus its initial height to help learn the meter units as we always must. Even the stretch of a rubber band versus mass applied can be modeled mathematically to practice combining the use of meter units and kilogram units. The approximate best fit method of mathematically modeling physics and chemistry phenomena is simple and very useful in the high school physics class and can be used by all students, even those in the university freshman classes.

Stewart Brekke is a retired high school teacher. He resides in Bensenville, IL.