The National Student Research Center
E-Journal of Student Research: Math
Volume 4, Number 1, June, 1997
The National Student Research Center
is dedicated to promoting student research and the use of the
scientific method in all subject areas across the curriculum,
especially science and math.
For more information contact:
- John I. Swang, Ph.D.
- Founder/Director
- National Student Research Center
- 2024 Livingston Street
- Mandeville, Louisiana 70448
- U.S.A.
- E-Mail: nsrcmms@communique.net
- http://youth.net/nsrc/nsrc.html
TABLE OF CONTENTS
- Does The Divisibility Rule
For Three Work For Other Numbers?
- Measuring Fractal Dimension
of Partial Aggregates
- Is the Formula For Finding
the Area of a Triangle Always Accurate?
- Fractal Resistors
- Does The Size Of A Circle
Affect The Value Of Pi?
TITLE: Does The Divisibility Rule For Three Work For Other
Numbers?
STUDENT RESEARCHER: Amanda Senules
SCHOOL: Mandeville Middle School
Mandeville, Louisiana
GRADE: 6
TEACHER: John I. Swang, Ph.D.
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
I would like to do a scientific research project to determine
if the proven divisibility rule for three works with other
numbers, like 2, 4, 5, 6, and 7. The divisibility rule for
three is that you add the digits of a number together, and if
the sum is divisible by three, the original number is divisible
by three. My first hypothesis states that the divisibility
rule for three works. My second hypothesis states that the
divisibility rule for three does not work with the number 2.
My third hypothesis states that the divisibility rule for three
does not work with the number 4. My fourth hypothesis states
that the divisibility rule for three does not work with the
number 6. My fifth hypothesis states that the divisibility
rule for three does not work with the number 7. My sixth
hypothesis states that the divisibility rule for three does not
work with the number 5.
II. METHODOLOGY:
First, I wrote my statement of purpose and a review of
literature on divisibility rules. Next, I developed my
hypothesis. I then developed my methodology to test my
hypothesis. After that, I made my data collection sheet. I
then began my experiment.
Step 1: First, I added the digits of the dividend 1,234 until
it made one digit. 1+2+3+4=10 1+0=1
Step 2: Next, I divided the sum 1 by 3.
Step 3: It did not divide evenly. The proven divisibility
rule for 3 stated that if three doesn't go evenly into the
digit divided from, the original number (in this case 1,234) is
not divisible by 3.
Step 4: I then performed long division to check. I repeated
the process with the divisors 2, 4, 5, 6, and 7. After that, I
randomly picked three other dividends and repeated the process.
My variable held constant was the process shown above. My
manipulated variables were the 6 different divisors and 4
different dividends. My responding variables were the answers
and if the process works on all experimental numbers.
After the experiment, I completed my analysis of data, summary
and conclusions, and application. Finally, I published my
research in the journal, The Student Researcher.
III. ANALYSIS OF DATA:
For my experiment, I randomly picked 4 dividends, 1,234, 8,046,
2,469, and 4,733.
Using the divisor 2 with the divisibility rule for three, I
found that it worked for one of the dividends, but not for the
other three.
Using the divisor 3 with the divisibility rule for 3, I found
that it worked for all of the dividends I used.
Using the divisor 4 with the divisibility rule for 3, I found
that it worked for 3 of the dividends, but not for the other
one.
Using the divisor 5 with the divisibility rule for 3, I found
that it worked with all of the dividends I used.
Using the divisor 6 with the divisibility rule for 3, I found
that it worked for 3 of the dividends I used, but not for the
other 1.
Using the divisor 7 with the divisibility rule for 3, I found
that it worked with all of the dividends I used.
With the 4 randomly picked dividends, the divisibility rule for
3 worked with the odd divisors I used and not the even.
V. SUMMARY AND CONCLUSION:
After doing my experiment, I found that with the dividends I
used, the divisibility rule for three works with the divisors
3, 5, and 7, but not with the divisors 2, 4, and 6.
In conclusion, my experiment did not prove a lot. For example,
none of the dividends I used were divisible by 5, and the
chances of the digits adding up to a digit divisible by 5 are
one out of nine. If I would have used a dividend divisible by
5, it probably would have proven that the divisibility rule
does not work with the divisor 5. For example: 1,235 would
become 1+2+3+5=11 1+1=2
Five does not go into two evenly, so, according to the
divisibility rule for three, 5 won't go into the original
number evenly. Really, 5 does go into the number. I have just
proven that the divisibility rule does not work with the number
5.
To prove a divisibility rule right, you must test it with
thousands of dividends.
With the dividends I used, I accept my first hypothesis which
stated that the divisibility rule for 3 works. I accept my
second hypothesis which stated that the divisibility rule for 3
does not work with the divisor 2. I accept my third hypothesis
which stated that the divisibility rule for three does not work
with the number 4. I accept my fourth hypothesis which stated
that the divisibility rule for three does not work with the
number 6. I reject my fifth hypothesis which stated that the
divisibility rule for 3 does not work with the number 7. I
reject my sixth hypothesis which stated that the divisibility
rule for three does not work with the number 5.
V. APPLICATION:
I can apply my findings to every day life by telling people not
to use a divisibility rule they made up without thoroughly
testing it first. I can also tell people not to use a
divisibility rule for one divisor with another divisor.
TITLE: Measuring Fractal Dimension of Partial Aggregates
STUDENT RESEARCHERS: Noah Forbes, Katie Johnson, Brad Jones
SCHOOL ADDRESS: Belmont High School
221 Concord Ave.
Belmont, MA 02178
GRADE: 12
TEACHER: Paul Hickman - phickman@copernicus.bbn.com
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
We wanted to investigate how much of an aggregate can be
removed and still have the same fractal dimension. This
included a study of the limitations of the computer in
measuring the fractal dimension of objects. We thought that
the computer measurement would lose accuracy as more of the
fractal was removed.
II. METHODOLOGY:
Using the five aggregates the class had grown and scanned into
the computer, we measured their original fractal dimensions
using the program "Fractal Dimension 5.1". Then we measured
subsequent fractal dimensions as we removed parts of the
aggregate. First, we removed half of the aggregate and
measured the fractal dimension using both the box and the
circle method. The half of the aggregate we kept was
representative of the whole aggregate. Next, we cut only a
branch from the aggregate and magnified it to 200%
magnification. We then measured the fractal dimension of the
branch. In two cases, we measured the fractal dimension of the
branch at normal size to see if the magnification was what
changed the fractal dimension.
All five of the aggregates were grown for around fifteen
minutes from a .2 M CuSO4 solution. We used Electrodeposition
cells with a copper cathode and a circular copper anode about
eight cm in diameter.
III. ANALYSIS OF DATA:
Our measurements of the other groups aggregates agreed well
with their own (see Data Table fig. 1). This consistency shows
that our technique for measuring the aggregates was relatively
accurate. When measuring the half aggregates, we found that
the fractal dimension measured by the box method was very close
while the fractal dimension measured by the circle method
differed significantly. The measurements using the circle
method were above the original in some cases and below in
others. When the measured fractal dimension was greater, the
half aggregate was very full and would take up most of the
circle when it was being measured. When the resulting fractal
dimension was lower than the original, the half aggregate
tended to be very linear and would leave a lot of empty space
when being measured.
When we measured the branches at 200% magnification, the
fractal dimensions were high for the box method and very low
for the circle method. The box method was too high because
when the branch was magnified, the pixels were magnified also.
As a result, the branch lost some of its original shape. The
shape became too square along the edges making the fractal
dimension measured high. We found that it was the
magnification that made the box method too high by measuring a
few of the aggregates at original size. When the branches were
at original size the box method measured the fractal dimension
very closely to the original fractal dimension. The circle
method was very low because the branch was long and thin, and
as a result left a lot of empty space when measured. We found
that the less linear the branch was the higher the fractal
dimension using the circle method. No branch, when measured
with the circle method, had a fractal dimension very close to
the original.
Data Tables:
Group Fractal Dimension
Box Method Circle Method
Class Results Our Results Class Results Our Results
Au ------- 1.763 ------- 1.750
C&J 1.60 1.720 1.67 1.658
Honey 1.780 1.780 1.842 1.734
Mush 1.7 1.726 1.8 1.661
PaLaTo 1.610 1.636 1.735 1.616
FD at 1/2 Aggregate
Au 1.755 1.539
C&J 1.704 1.854
Honey 1.790 1.485
Mush 1.730 1.852
PaLaTo 1.622 1.643
FD at Branch
Au 1.829 1.198
C&J 1.820 1.188
Honey 1.827 1.193
Mush 1.779 1.026
PaLaTo 1.768 1.491
Mush @100 1.720 1.028
IV. SUMMARY AND CONCLUSION:
We found that, when using the computer program Fractal
Dimension 5.1, the fractal dimension of an aggregate can be
measured accurately using the box method, no matter how much of
the aggregate is present. We found the circle method to be
affected more by the shape of the aggregate. The circle method
measures the whole aggregate well, but as pieces are removed it
becomes less and less accurate. The circle method lost
accuracy as the aggregates became less and less circular.
Since the circle method measures the Fractal Dimension by
placing a circle over the object, it works best when the object
that one is measuring is a circle. Objects that are not as
circular, such as the half aggregate and the branch, leave too
much of the circle empty, so the resulting fractal dimension is
not accurate. We found that the computer was not perfect;
however, it was more accurate than we hypothesized.
V. APPLICATION:
Our research with fractals taught us that there exists
limitations in the measurement of self-similarity in accordance
to the medium in which it is measured. The same result is seen
in the magnification of a picture. One can only magnify a
picture so many times before the detail in the picture is lost.
The same loss of detail occurs as the fractal is enlarged and
as more of the original fractal is removed.
TITLE: Is the Formula For Finding the Area of a Triangle
Always Accurate?
STUDENT RESEARCHER: Michael Phillips
SCHOOL: Mandeville Middle School
Mandeville, Louisiana
GRADE: 6
TEACHER: John I. Swang, Ph.D.
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
I would like to do a scientific research project to see if the
formula for finding the area of a triangle is always accurate.
The formula for finding the area of a triangle is A = 1/2 bh.
My hypothesis states that the formula for finding the area of a
triangle will always be accurate on all triangles that I test.
II. METHODOLOGY:
First, I wrote my statement of purpose and hypothesis. Then I
reviewed the literature on triangles and area. Next, I wrote a
methodology to test my hypothesis.
My manipulated variable was the size of the triangles and the
angles of the triangles. My responding variable was the answer
to the formula. My variable held constant was the formula.
I drew six triangles of different size on 1 cm graph paper.
Then I computed the area of each triangle using the formula.
Next, I found the actual area of each triangle by counting the
number of square centimeters within it.
I then analyzed the data, wrote my summary and conclusion, and
applied my findings to the world outside of my classroom. Then
I published my findings in a national journal.
III. ANALYSIS OF DATA:
For the right triangle, the measurement of the area with the
formula was 16 sq. cm. and the actual area was 16 sq. cm. For
the acute triangle, both measurements of the area were 10 sq.
cm. For the obtuse triangle, both measurements of the area
were 10 sq. cm. For the scalene triangle, both measurements
of the area were 12 sq.cm. For the equilateral triangle, both
areas were 1.5 sq. cm. For the isosceles triangle both areas
were 12 sq. cm.
IV. SUMMARY AND CONCLUSION:
In my research, I discovered that the formula for finding the
area of a triangle worked on all kinds of triangles.
Therefore, I accept my hypothesis which stated that the formula
would always be accurate.
V. APPLICATION:
I could apply my findings to the world by using the formula for
finding the area of triangles, because it is easier and always
accurate.
TITLE: Fractal Resistors
STUDENT RESEARCHERS: Anna Hutchinson, Keith Waters
SCHOOL ADDRESS: Belmont High School
221 Concord Ave.
Belmont, MA 02178
GRADE: 12
TEACHER: Paul Hickman - phickman@copernicus.bbn.com
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
We began by looking at Sierpinski's triangle (fig 1) and carpet
(fig 2) on the computer program J. Gasket Meister. This
program allows one to generate a pattern with a few simple
rules. The gaskets were interesting to us and when our teacher
mentioned an experiment building a triangle out of resistors,
we decided to imitate the experiment with the carpet. We
wanted to find out more about the relationship between the
triangle experiment and the carpet experiment.
The theory behind the resistor network is that there is less
resistance to current with many resistors in parallel. We
tried to find a relationship between the resistance of the
square and triangle networks at iterations 1, 2, and 3, and
their Fractal Dimensions. Our hypothesis stated that the slope
of the ln-ln graph of the carpet experiment would be less than
that of the triangle experiment because the carpet has a higher
fractal dimension, and therefore there would be more parallel
resistors and less resistance.
II. METHODOLOGY:
We had the first three iterations of the triangle already built
on circuit boards with twenty-seven 1000 ohm resistors; we
built the first three iterations of the carpet on eight circuit
boards. Then we measured the resistance for the first, second,
and the third iteration of the triangle with an ohmmeter. In
order to get accurate results, we disconnected the iterations
from the rest of the network to measure them, and then
reconnected them to measure the whole network. We measured
each side of each iteration and took the average resistance to
get a better reading. We used the same procedure to measure
three iterations of the carpet.
We then made a ln-ln graph of the average resistance vs. the
number of resistors on a side (length) for the triangle and
carpet: the slope of these graphs gave us a figure related to
the fractal dimensions of the resistor networks.
III. ANALYSIS OF DATA:
Our results for the triangle matched those of previous
researchers; in fact, it matched exactly the mathematically
predicted value, which was 0.737. The slope of the graph for
the results for the carpet was 0.613.
The fact that we were able to get a straight line in the ln-ln
graph for the carpet indicates that it is possible to replicate
the triangle experiment with the carpet and get a slope. The
results show that the slope for the carpet was less than that
of the triangle; therefore the rate of increase of the
resistance was lower for the carpet.
IV. SUMMARY AND CONCLUSION:
These results agree with our hypothesis. We came to the
conclusion that as the fractal dimension goes up, the
resistance of that network goes down. This conclusion makes
sense because, if an object has a higher fractal dimension, it
would take more resistors to build a network of that object,
and therefore the resistance would be lower. If we were to
measure a solid sheet of resistor, (which would have an FD of
two) the resistance would be really low.
V. APPLICATION:
The research we did relates to resistance on the atomic and
molecular scale, which scientists are trying to learn more
about now. It is the structure of these networks that
scientists can use as a model for the imperfect patterns found
in natural electronic materials.
TITLE: Does The Size Of A Circle Affect The Value Of Pi?
STUDENT RESEARCHER: Dana Blount
SCHOOL: Mandeville Middle School
Mandeville, Louisiana
GRADE: 6
TEACHER: John I. Swang, Ph.D.
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
I am doing a mathematical research project to see if the Pi is
affected by the size of a circle. Pi is the ratio of the
circumference of a circle to its diameter. Pi is always equal
to 3.14. My null hypothesis states that the value of Pi will
not be affected by the size of the circle.
II. METHODOLOGY:
First, I wrote my statement of purpose and did my review of the
literature on Pi, circumference, diameter, and circles. Then I
developed my hypothesis. Then I wrote the following
methodology to test my hypothesis.
First, I gathered my materials, which included ten circular
objects of different sizes, paper, pencil, ruler, calculator,
and data collection form. Then I took one of my circular
objects and made a mark on the edge. I placed the mark on a
piece of paper and marked where the mark, on the circular
objects, was on the paper. I carefully rolled the circular
object on the paper until the marked point touched the paper
again. I marked this spot on the paper. Then I measured the
distance between the two marks on the paper to find the
circumference of the circular object. Next, I measured the
width of the circular object to find the diameter. Then I
divided the circumference by the diameter to compute the value
for Pi. I recorded my data on my data collection form. I
repeated this process with all of my circular objects. Next, I
listed my variables as shown below.
My variable held constant is the formula for Pi. My
manipulated variable is the size of the circular objects. My
responding variable is the value of Pi for the circles.
Then I analyzed my data using the information on my data
collection form. Next, I wrote my summary and conclusion where
I accepted or rejected my hypothesis. Then I applied my
findings to everyday life. Finally, I published my findings in
The Student Researcher.
III. ANALYSIS OF DATA:
I found that two out of the ten circular objects I used for my
experiment had a Pi value of 3.14. Two of the circular objects
had a Pi value above 3.14. Six circular objects had a Pi value
below 3.14. My average Pi value was 3.06. These differences
were due to the fact that my instrument of measurement was not
accurate enough to find the true Pi value of 3.14.
IV. SUMMARY AND CONCLUSION:
After I analyzed mt data, I found out with my instrument of
measurement that the majority of the circular objects had a Pi
value below 3.14. Two out of my ten circular objects had a Pi
value of 3.14. Two of the ten circular objects had a Pi value
above 3.14. I therefore, accept my hypothesis which stated
that the size of the circles would not affect the value of Pi.
If my instrument measurement had been more accurate the value
of Pi for the circular objects would have always been 3.14.
This research should be repeated and more exact measurements
should be taken to ensure that an accurate value of Pi is
computed.
V. APPLICATION:
I can apply my findings to everyday life by telling teachers
and students that Pi always equals 3.14 if you use the right
instruments that allow you to accurately measure a circular
object's circumference and diameter.
© 1997 John I. Swang, Ph.D.