Download Proportional Reasoning

Proportional Reasoning
Dr. Harold Williams
Montgomery College Planetarium
http://Montgomerycollege.edu/Departments/planet/
Understanding Equations without
doing Algebra (solving for
unknowns)
• Besides Arithmetic: addition, subtraction, multiplication,
and division
• You must be able to do Proportional Reasoning. In other
words it will be on the test.
• Besides if you don’t understand proportional reasoning you
are innumerate. You are also seriously handicapped,
unnecessarily. You are missing understanding of a major
reasoning tool used by human being for many
millenniums, though not be most people.
• Don’t worry it is not hard, like spelling, and it does not
take as much time to understand, as memorizing addition
or multiplication tables. But if you try to memorize it
you will fail to understanding it.
Lets Take a Break
What is this?
• Define N1=~U1 and N1+U1=1
• T(N1, U1)  (N2, U2)
where U2 > U1
• Where now N2=~U2 and N2+U2=1
are defined.
• So N1 > N2
Don’t worry if you don’t
understand yet!
After all you do not know what N and U are
and you do not know the significance of the
subscripts 1 and 2 or what T is so that it
transforms 1 to 2.
So far the past slide is so much
mathematical mummery!
End of Break
Who do we blame for this
problem of lack of proportional
reasoning?
• Two thirds (2/3) of elementary teachers can
not do proportional reasoning.
• Actually more careful studies show that
25% of preservice elementary teachers
understand proportional reasoning.
Lets compare these blame claims
• Two thirds (2/3) of
elementary teachers can
not do proportional
reasoning.
• 75% of preservice
elementary teachers do not
understand proportional
reasoning.
• So one third (1/3) of
elementary teachers can
do proportional reasoning.
•
• 25% of preservice
elementary teachers
understand proportional
reasoning.
Notice the use of unitary
• This is a two state system either you don’t
understand it or you do understand it, in this
model. No doubt, this was assessed in some
deliberate way.
• 2/3+1/3=1
• 75%+25%=100%
Changing units!
Lets compare these blame claims
• 2/3 of elementary teachers
can not do proportional
reasoning.
• 3/4 of preservice
elementary teachers do not
understand proportional
reasoning.
• 1/3 of elementary teachers
can do proportional
reasoning.
• 1/4 of preservice
elementary teachers
understand proportional
reasoning.
Let us reorder as a time sequence the blame
claims since you have to be a preservice
teacher before you are a teacher.
• 3/4 of preservice
elementary teachers do not
understand proportional
reasoning.
• 2/3 of elementary teachers
can not do proportional
reasoning.
• 3/4>2/3 the good news is
that education classes
work a little; Not
understanding is reduced.
• 1/4 of preservice
elementary teachers
understand proportional
reasoning.
• 1/3 of elementary teachers
can do proportional
reasoning.
• 1/4<1/3 the good new is
that education classes work
a little; Understanding is
increased.
What is this Now?
• Define N1=~U1 and N1+U1=1
• T(N1, U1)  (N2, U2)
where U2 > U1
• Where now N2=~U2 and
N2+U2=1 are defined.
• So N1 > N2
Maybe you still don’t understand
Don’t worry be happy!
But for how much Longer will you be
happy?
Let us try and understand
• Suppose U stood for understanding
• Then maybe N stood for not understanding
• Then N=~U if ~ means not and N+U=1 makes
sense.
• T might stand for Teaching or maybe we should
relabel T  L since we are a learning college.
• 1 must stand for state one and 2 must stand for
state two, latter state, after T or L have been
applied to N and U. Since U2 > U1 then
understanding did increase and likewise N1 > N2
so not understanding decreased.
So now maybe you know what
• Define N1=~U1 and N1+U1=1
• L(N1, U1)  (N2, U2)
where U2 > U1
• Where now N2=~U2 and N2+U2=1
are defined.
• So N1 > N2
means.
What is proportional reasoning?
• using proportions in reasoning obviously!
• But what is proportional, the symbol is .
• In the Stephan-Boltzmann equation
E/A=T4 where  is the luminosity, E is
the energy, A is the area,  is the StephanBoltzmann constant, and T is the
temperature.
Applying this to E/A=T4
• With proportional reasoning on equations
you take the equation and look at a variable
•   T4 since T can take on many different
values T is a variable.
• ~   because  is a constant and only
varies in the multiverse. What is the
multiverse?  is a constant and take a
particular value in our universe.
Some arithmetic for the test:
or for Life, proportional reasoning
is much more important than a test
or even AS101.
• Since E/A=T4 and therefore   T4
• and this law describes stars which are hot
with a temperature T and shine because of
it.
• If I have two stars which are identical
except one is twice as hot as another the
hotter one is 24 =22  2  2=16 times
brighter.
Some more arithmetic for the test
• If I have two stars which are identical except one
is three times as hot as another the the hotter one
is 34 =3333=81 times brighter.
• If I have two stars which are identical except one
is four times as hot as another the the hotter one is
44 =4444=256 times brighter.
• I hope you have had enough of calculation by
exhaustion, and see the pattern, and do not need
me to substitute all of the integers, which go on
for ever, and to then start with the real numbers,
which go on for ever even in a short number
interval.
Applying this to Newton’s
Universal Law of Gravity
• F=GM1m2/R2 where F is the force, G is the
Newtonian gravitational constant, M1 is mass 1,
and m2 mass 2 and R is the distance between the
two masses.
• F M1 or F  m2
• F 1/ R2= R-2 called inverse square we will see
this dependence again in Couloumb’s law for
charges and in light intensity fall off versus
distance for stars. This is geometry; we live in a
universe with 3 space dimensions, 3D universe.
• F ~G except in the multiverse since G is constant
in our universe and is sort of a unit conversion
term.
More in the strange Multiverse
• There is no evidence that the multiverse even
exists at the moment. It is a poetic device or idea
in physics.
• Yes physics has poetic devices or ideas and they
have some times become detected like the
neutrino. There is no evidence that the multiverse
even exists at the moment! It is a poetic device or
idea in physics There is no evidence that the
multiverse even exists at the moment. It is a
poetic device or idea in physics like neutrinos, .,
used to be.
• Which is now the e, , and  in the “Standard
model” in particle physics weak interactions.
Back to E/A=T4
• Suppose we want to include the size of the star in
calculating or computing its brightness  or its
total energy output E.
• The area A scales with size D=2R where D is the
diameter and R is the radius of the spherical star as
A=4R2.
• E/A=E/4R2 =T4 solving for E the total energy
for the star E=4R2T4.
• So now we can conclude E  T4 and E  R2.
• E  R2 tells us how size matters with spherical
stars. The bigger the brighter, but not as E  R,
but as E  R2 . This is a huge difference in
behavior. E  R2 is a direct square not an inverse
square. If you get this wrong you perish!
Latter we will see
• An inverse square relationship with r, the
distance from a star, not it size R in
brightness which is a direct square.
• One decreases with distance the other
increases with distance, but they are
different distances on vastly different scales.
• Your height is a different distance than how
far I am away from you which is also a
distance.
• Which distance matters greatly!
In support for
No Child Left Behind or
No College Student Left Behind
and particularly
No Community Left Behind and
therefore
No Community College Student Left Behind
after all community is very important!