mm-1000
===
Subject: Unsure/Stuck on a Word Problem... Please Help
IÕm unsure about how to go about solving this word problem.
The
problem: a rod, 1 mile long, is cut almost all the way
through... an
earthquake causes the cut to bend up to mimic the point of an
isosceles triangle. (This pushes the rod in 6 inches). What I
need to
find out is: can a truck (72 inches high) drive under the rod
in the
center or not.
IÕm totally stuck on how to go about this problem... 
IÕve
found the
length of a mile in inches (63,360) but after this, I have no
idea of
how (formulas, etc.) I could figure out the effect of the rod
being
pushed in...
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Subject: Re: Unsure/Stuck on a Word Problem... Please Help
You have an isosceles triangle with the measures of the
congruent sides
being half a mile. The base of the triangle is 12 inches less
than a mile
(since the rod is pushed in 6 inches, I assume, on each end).
I would draw
a picture of this to help you visualize the problem. Then all
you have to
do is determine the altitude of the triangle. Use the
Pythagorean Theorem.
Aaron
> IÕm unsure about how to go about solving this word 
problem.
The
> problem: a rod, 1 mile long, is cut almost all the way
through... an
> earthquake causes the cut to bend up to mimic the point of
an
> isosceles triangle. (This pushes the rod in 6 inches). What
I need to
> find out is: can a truck (72 inches high) drive under the
rod in the
> center or not.
> IÕm totally stuck on how to go about this problem... 
IÕve
found the
> length of a mile in inches (63,360) but after this, I have
no idea of
> how (formulas, etc.) I could figure out the effect of the
rod being
> pushed in...
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Subject: Re: Unsure/Stuck on a Word Problem... Please Help
<<
IÕm unsure about how to go about solving this word problem.
The
problem: a rod, 1 mile long, is cut almost all the way
through... an
earthquake causes the cut to bend up to mimic the point of an
isosceles triangle. (This pushes the rod in 6 inches). What I
need to
find out is: can a truck (72 inches high) drive under the rod
in the
center or not.
Draw the triangle. The base is one mile - 6 inch, the sides
are .5 miles
long.
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Subject: Re: Unsure/Stuck on a Word Problem... Please Help
> a rod, 1 mile long, is cut almost all the way through... an
> earthquake causes the cut to bend up to mimic the point of
an
> isosceles triangle. (This pushes the rod in 6 inches).
Reference: See HalmosÕs Problems for Mathematicians
Young and Old.
> What I need to find out is: can a truck (72 inches high)
drive
> under the rod in the center or not.
> IÕm totally stuck on how to go about this problem... 
IÕve
found the
> length of a mile in inches (63,360) but after this, I have
no idea...
The base of the triangle is 5279.5 feet.
Each leg is 5280/2 = 2640 feet long.
So the height of the triangle h = sqrt(2640^2 - 5279.5/2)^2)
= sqrt(1319.9375) ~ 36 miles high.
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Subject: Re: Unsure/Stuck on a Word Problem... Please Help
> a rod, 1 mile long, is cut almost all the way through... an
> earthquake causes the cut to bend up to mimic the point of
an
> isosceles triangle. (This pushes the rod in 6 inches).
> Reference: See HalmosÕs Problems for Mathematicians
> Young and Old.
> What I need to find out is: can a truck (72 inches high)
drive
> under the rod in the center or not.
> IÕm totally stuck on how to go about this problem... 
IÕve
found the
> length of a mile in inches (63,360) but after this, I have
no idea...
> The base of the triangle is 5279.5 feet.
> Each leg is 5280/2 = 2640 feet long.
> So the height of the triangle h = sqrt(2640^2 - 5279.5/2)^2)
> = sqrt(1319.9375) ~ 36 feet high.
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Subject: Re: Critical, higher-order mathematical thinking and
process
Apart from the sleight of hand which makes an algorithm the
same as a
theorem (which it isnÕt), thereÕs much which I 
can agree with
in this post.
It leaves one issue, though, and that is the introduction of
students to
formal or abstract proof. At what stage of development are
children ready
for it? The point of problem solving as far as I was ever
concerned is
to
allow children to experience theorem development often in a
concrete
context, and (I think, more powerfully than that, even), to
allow children
to discover, and prove, *their* own theorems. To experience
what it is to
make mathematical discovery. This is very powerful, and a
stage which
traditional (I donÕt like the phrase, though I use it)
approaches to
education ignore.
Manipulatives may or may not be part of this approach; I
always saw one of
their main strengths as allowing children to see the links
between algebra,
measure, space and number. As well as being concrete in
.....well..... a
concrete sense.
So, when to intoduce formal proof? And there is a difference
between
developing a proof oneself, and following/understanding
someone elsesÕs
proof. At what stage are children ready for these things?
I have neither reference nor context, but somewhere in
PiagetÕs writings is
a lovely story of a child who proved scientifically that trees
caused
the
wind, because everytime their branches waved around, the wind
blew. This is
apropos nothing really; I just liked the story when I read it
many years
ago!
M.
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Subject: Re: Critical, higher-order mathematical thinking and
process
mark.mashed.houghton@mashed.paradise.net.nz:
>At what stage of development are children ready
>for it?
>So, when to intoduce formal proof? And there is a difference
between
>developing a proof oneself, and following/understanding
someone elsesÕs
>proof. At what stage are children ready for these things?
Ask yourself, At what age was I ready....?. Your ability to
answer
that, at
least for yourself, depends on what kind of study you did,
and how old were
you
when you studied it. Some students 20 to 25 years of age will
not deal
with
formal proof; other students deal with formal proofs
reasonably well at the
age
conveniently available, but each of us at least has ourself.
G C
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Subject: Re: Critical, higher-order mathematical thinking and
process
> Ask yourself, At what age was I ready....?.
Well - statements donÕt get more teacher centred than this,
do they??
Your ability to answer that, at
> least for yourself, depends on what kind of study you did,
and how old
were you
> when you studied it. Some students 20 to 25 years of age
will not deal
with
> formal proof; other students deal with formal proofs
reasonably well at
the age
> of 14 or 15 years. We may not have psychological research
on learning
> conveniently available, but each of us at least has ourself.
I didnÕt mention age. I said (and meant) stage of
development. Which is one
of the reasons why using onself (*especially* when oneself is
a math
teacher) is somewhat problematic. And, I guess, why teaching
is so
problematic.
M.
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Subject: Re: Critical, higher-order mathematical thinking and
process
mark.mashed.houghton@.... replied after a comment:
>I didnÕt mention age. I said (and meant) stage of
development. Which is
one
>of the reasons why using onself (*especially* when oneself
is a math
>teacher) is somewhat problematic. And, I guess, why teaching
is so
>problematic.
inconvenient; referring to our own education, since at one
time we were
students but not yet teachers, is natural. We might have
learned
differently
than some of our students.
Age COULD matter. Someone at age 8 may fail to understand
some concepts
and
gain some skills which at age 14 he would find comprehensible
and
relatively
easy to perform. Some other development occurs as one grows.
For some of us, Math at some level became a language. We
could say that we
are
like our students, but this may only be a mere poetic
statement to
empathize
with those struggling with a subject which we ourselves at
one time
struggled.
We may be our students, but our students have not all become
us.
G C
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Subject: Re: Critical, higher-order mathematical thinking and
process
Mathematical algorithms may not always be put forth as
statements that
lend themselves to being proved (do this, do that), but they
are
always capable of being forth as such. IÕm using algorithm 
in
a
mathematical context more broadly than just referring to do
this, do
that. IÕm letting the term cover the necessary and 
sufficient
context: They always give the desired result, and this fact
must be
provable. So the algorithm, if not initially stated as such,
must be
stateable as a theorem, by which I mean any result that is
derivable
from prior definitions, axioms, or results.
IÕll restate the maxim: A proof is a mathematical proof or a
solution
to a problem is a mathematical solution if and only if every
step of
the proof or solution is an application of some prior
definition,
axiom, or theorem. If you canÕt identify what 
youÕve applied
for each
step in your proof or solution, then you really canÕt say
that you
have a mathematical proof or solution. (You might of course
get lucky
in that all the steps are identifiable in such a way and then
find
someone who can do this for you.) (Not teaching students this
maxim
leads to cranks thinking that proved something when they
really have
proved nothing. The web is loaded with claims that various
famous
unsolved problems have been solved. In every last case, these
individuals do not adhere to this maxim.)
ItÕs not about how soon we can introduce formal proof, but
how soon we
can introduce the idea that youÕve shown something to be 
true
or
youÕve solved a problem if and only if you are applying an
accepted
prior something at each and every step. (My thesis here in
this thread
is that this maxim doesnÕt apply to just formal proof of 
major
theorems: It applies to each step in any multi-step
mathematical
solution to any mathematical problem, no matter what the
level.)
I donÕt know how early we can start teaching this idea, but 
I
think we
can communicate some primitive version of it even in the
early grades.
But if not, certainly by the time they reach middle school.
When we
encourage students to justify or explain why, we are asking
them for a
proof or convincing argument (see below). So I see no reason
why we
canÕt start communicating at least some primitive version of
this idea
when we start asking them to explain why. And I certainly
think that
all adults, especially all math teachers no matter what the
level,
should personally know of and adhere to this maxim whenever
they claim
that they can mathematically solve a problem or otherwise
mathematically prove something.
Does all this mean that IÕm against non-mathematical proofs 
or
justifications? No. There are some authors who 
define the term
proof
such that it is a convincing argument, nothing more, nothing
less. We
can use manipulatives or visual representations to convince.
But we
ought to make sure that behind these concrete or visual
representations, there are arguments that are mathematical in
that
they adhere to the maxim. This way, when the students learn
these
mathematical arguments, there is agreement between what they
learn
before and what they learn later. (This is the job of the math
teacher, to know math well enough to make sure that the chosen
manipulatives and visual representations agree with the
theory. This
Multiplying negative numbers.)
Paul
> mark.mashed.houghton@mashed.paradise.net.nz:
>At what stage of development are children ready
>for it?
>So, when to intoduce formal proof? And there is a difference
between
>developing a proof oneself, and following/understanding
someone elsesÕs
>proof. At what stage are children ready for these things?
> Ask yourself, At what age was I ready....?. Your ability to
answer
that, at
> least for yourself, depends on what kind of study you did,
and how old
were you
> when you studied it. Some students 20 to 25 years of age
will not deal
with
> formal proof; other students deal with formal proofs
reasonably well at
the age
> conveniently available, but each of us at least has ourself.
> G C
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Subject: Re: Critical, higher-order mathematical thinking and
process
<SNIP I donÕt know how early we can start teaching this 
idea,
but I think we
> can communicate some primitive version of it even in the
early grades.
> But if not, certainly by the time they reach middle school.
When we
> encourage students to justify or explain why, we are asking
them for a
> proof or convincing argument (see below). So I see no
reason why we
> canÕt start communicating at least some primitive version
of this idea
> when we start asking them to explain why.
Right. IÕd agree with most of this (including the big snip -
though IÕm
tempted into a semantical debate about proving algorithms or
proving
theorems). But letÕs tease it out. Can you give some 
examples
of this
process in the classroom? I mean, apply your theory. (I know
that the
nature
of usenet is often antagonistic, so I should point out that
this is *not* a
challenge!! IÕm actually interested in this, not trying to
score debating
points).
If we agree that you should teach proof - and I do agree -
the more
important issue is *how* do we teach it? Otherwise,
curriculum development
becomes some sort of mathematical shopping list.
> Does all this mean that IÕm against non-mathematical 
proofs
or
> justifications? No. There are some authors who 
define the
term proof
> such that it is a convincing argument, nothing more,
nothing less. We
> can use manipulatives or visual representations to
convince. But we
> ought to make sure that behind these concrete or visual
> representations, there are arguments that are mathematical
in that
> they adhere to the maxim.
Oh - absolutely. ThereÕs an extra issue with using
manipulatives, though,
and that is that proofs with them can often add insight into
a problem.
Proofs which rely on purely written arguments often get
bogged down with
issues of formal notation, which can obscure that. BTW, IÕm
not taking a
dogmatic stance *against* such proofs - they are an important
part of
mathematics as well, as they represent the use of math as a
language of
communication.
One more question: you called your thread critical,
higher-order
mathematical thinking and process. Perhaps you like to say
where the
critical thinking enters into this? How can that aspect be
developed?
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Subject: i need a topic for a math fair....please help me!
I have a math fair due soon, and i dont have a subject to do
it on
yet, and i cant seem to think of one...can someone please
help me out
e-mail the ideas to hokiecrazy77@netscape.net
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Subject: Re: i need a topic for a math fair....please help me!
>I have a math fair due soon, and i dont have a subject to do
it on
>yet, and i cant seem to think of one...can someone please
help me out
>e-mail the ideas to hokiecrazy77@netscape.net
1. Cyclic Redundancy Check (recently discussed here)
2. Numbers to negative bases (e.g. base megative 3)
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Subject: Re: i need a topic for a math fair....please help me!
> I have a math fair due soon, and i dont have a subject to
do it on
> yet, and i cant seem to think of one...can someone please
help me out
> e-mail the ideas to hokiecrazy77@netscape.net
It would help to know what grade youÕre in and what your
interests are :-)
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Subject: Re: i need a topic for a math fair....please help me!
What level are you?
Create a report related to geometry, thatÕs my favorite.
Although at the
college level geometry is harder. :-(
Assuming you are high school, apply geometry in some way.
Maybe use trigonometry in an engineer diagram.
John
> I have a math fair due soon, and i dont have a subject to
do it on
> yet, and i cant seem to think of one...can someone please
help me out
> e-mail the ideas to hokiecrazy77@netscape.net
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Subject: web pointer for spanish algebra 1 material?
Anyone got a web pointer for spanish algebra 1 material?
Esp. the easy stuff. TIA!
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Subject: Papert on Piaget
I found the reference I (mis)quoted in another thread. This
is actually
Seymour Papert quoting Piaget in Time Magazine (as one of the
Top 100
Scientists, I think):
In one of his most famous experiments, Piaget asked children,
What makes
the wind? A typical Piaget dialogue:
Piaget: What makes the wind?
Julia: The trees.
P: How do you know?
J: I saw them waving their arms.
P: How does that make the wind?
J (waving her hand in front of his face): Like this. Only
they are bigger.
And there are lots of trees.
P: What makes the wind on the ocean?
J: It blows there from the land. No. ItÕs the waves...
Piaget recognized that five-year-old JuliaÕs 
beliefs, while
not correct by
any adult criterion, are not incorrect either. They are
entirely
sensible
and coherent within the framework of the childÕs way of
knowing.
Classifying
them as true or false misses the point and shows a lack of
respect
for
the child. What Piaget was after was a theory that could find
in the wind
dialogue coherence, ingenuity and the practice of a kind of
explanatory
principle (in this case by referring to body actions) that
stands young
children in very good stead when they donÕt know enough or
have enough
skill
to handle the kind of explanation that grownups prefer.
Piaget was not an educator and never enunciated rules about
how to
intervene
in such situations. But his work strongly suggests that the
automatic
reaction of putting the child right may well be abusive.
Practicing the art
of making theories may be more valuable for children than
achieving
meteorological orthodoxy; and if their theories are always
greeted by
Nice
try, but this is how it really is... they might give up after
a while on
making theories. As Piaget put it, Children have real
understanding only
of
that which they invent themselves, and each time that we try
to teach them
something too quickly, we keep them from reinventing it
themselves.
M.
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