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 = kx^{2} or y = kx^{1/2}.
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 = kx^{2}, 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 problemsolving
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 gradeschool 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 nonexperimental 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 timeconsuming 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.
His email address is sbrekk@cs.com
