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Awards announcements
PWAP CHOSEN FOR DIGITAL DOZEN IN OCTOBER
The Problems
with a Point Web site was chosen to be included in October's
Digital Dozen, a
list of exemplary web sites for educators selected by the
Eisenhower National
Clearinghouse (ENC). The list is published each month
at ENC Online
(enc.org). The October Digital Dozen can be found at
http://www.enc.org/weblinks/dd/archive/0,1577,10-2001,00.shtm
and
this month's Digital Dozen can be found at
http://enc.org/weblinks/dd/.
PWAP
INCLUDED AS NCTM ILLUMINATIONS RESOURCE
The Problems with a Point site has
also been chosen as a 'Selected Web
Resource' (SWR) on the NCTM/MarcoPolo
Illuminations web site.
http://illuminations.nctm.org/swr/list.asp?Std=5&Grd=9
Featured problem: "Inventing a formula for arithmetic sequences"
Most
often, when students first study arithmetic sequences, they get
formulas,
learn how to use them, and then practice plugging in values to
get answers.
This one-page problem sequence leads students who have just a
little bit of
algebra to develop the formulas for themselves. Students
start by using their
own sense of pattern to extend the sequence 4, 7, 10,
13, 16, ..., and then
examine how much greater the second, third, fourth,
and seventh term is from
the first term. This lets them see for themselves
that the difference from
the first term is a multiple of a particular
value, and allows them to
predict the hundredth term without a formula. In
the course of repeating this
exploration with a sequence that has "nastier
numbers" and descends, the term
"rate of change" is introduced to describe
the difference between any two
successive terms in the sequence. Students
also see how the difference of
*non*-successive terms depends on that rate
of change value by computing the
difference between the 26th and 21st term
of a sequence without knowing what
either of those terms is. The last few
problems move students from numerical
examples and patterns to develop a
formula. Starting with the abstract
sequence a, a + b, a + 2b, ...,
students first figure out the rate of change
(b), then the first through
fifth terms (they already have the first three of
them), and then the
hundredth, 6000th, and nth terms. The last of these---Tn
= a + (n - 1)b---
is the formula that they derive directly from the numerical
examples. To
clarify and extend that idea, they look again at differences of
terms,
computing T9 - T1, T100 - T1, T100 - T83, and Tn - Tk for this
abstract
sequence. For example, T9 - T1 represents eight "steps" along
the
sequence, and so is 8 times the rate of change. Similarly Tn - Tk =
(n
- k)b. Finally, students explain in words, or using their formula, how
to
find Tn of any arithmetic sequence if they know T1 and the rate
of
change.
For students to remember, let alone understand, how to find
terms in an
arithmetic sequence, it helps if the formula is not merely a
magical
incantation. By inventing the formula for themselves---which they can
do---
they not only get to understand its parts and notations better, but
also
help train their ability to invent mathematical formulas on their own.
This
is an important asset in situations where they must solve novel
problems,
or where they have forgotten or never learned the required formulas
or
techniques. Hints for five of the problems are provided, to suggest
ways
that you can help students who feel stuck, while still allowing them
to
have the experience of inventing the formula for themselves. The
underlying
mathematics -- the constant difference between successive terms in
an
arithmetic sequence -- relates closely to the theme of linear
functions.
This set, suitable for a single lesson, is designed as a very
first
experience with the ideas of arithmetic sequence, introducing the
term
"arithmetic sequence" itself, as well as the term "rate of
change."
Students in middle school, with only enough algebra to understand
the
expression a + 2b, can do the problems, though the *form* of the rule
that
they eventually create will depend on how much of algebraic convention
they
are comfortable with. This set can also be used as an opportunity to
re-
examine, with new understanding, a piece of math they once learned just
by
memorizing the rules. For this latter purpose, it might be suitable
for
tutoring situations, or, with just a bit of in-class instruction,
as
homework.
See the full problem on the Web at
http://www2.edc.org/mathproblems/getp.asp?name=pgSeqSer2
New on the PwaP Web site
Here is a sample of the new problems available
on our site:
WHAT GOES UP, MUST COME DOWN #2
http://www2.edc.org/mathproblems/getp.asp?name=sbprojectile2
Students
make predictions, then analyze data, in graphical or numerical
form, to
determine at what angle a ball should be thrown in order to
maximize the
horizontal distance traveled before the ball hits the ground.
Some students
may be surprised to learn that the answer depends on the
height from which
the ball is thrown.
TRIGONOMETRIC INVERSES
http://www2.edc.org/mathproblems/getp.asp?name=ekInverseTrigFun
Students
use their knowledge of trigonometric functions as they learn
about the
inverses of those functions. They write and solve simple
trigonometric
equations.
EXPLORING FIXED POINTS
http://www2.edc.org/mathproblems/getp.asp?name=ekFixedPoint
Students
iterate functions and consider the long-term behavior of the
iterates. They
learn to look for fixed points and consider whether they
are attracting or
repelling. Graphical interpretation of function
iteration (web diagrams) are
presented.
AREA OF REGULAR POLYGONS
http://www2.edc.org/mathproblems/getp.asp?name=ekPentForm
Students
use congruence tests, area of a triangle, and trigonometry (the
tangent
ratio) to find formulas for the areas of regular pentagons and
hexagons. They
generalize the formulas to other regular polygons.
TRIGONOMETRIC
GRAPHS
http://www2.edc.org/mathproblems/getp.asp?name=ekTrigGraph
Students
plot points to see the shapes of the graphs of the sine and
cosine functions.
They use points on the unit circle to reason about why
the graphs look as
they do.
NEIGHBORHOOD VALUES
http://www2.edc.org/mathproblems/getp.asp?name=ekComplexDist
Students
use the geometric (coordinate) interpretation of complex numbers
and the
distance formula (or Pythagorean Theorem) to find absolute values
of complex
numbers.
SIX DEGREES OF SEPARATION
http://www2.edc.org/mathproblems/getp.asp?name=ekSixDegrees
This
introduction to matrix multiplication has students look at a
matrix
describing how people in a small group know each other. The process
for
finding an entry in a product matrix is introduced and interpreted
in
terms of the situation, giving students a basis for why multiplication
is
defined as it is.
PLENTIFUL ANAGRAMS
http://www2.edc.org/mathproblems/getp.asp?name=ekAnagrams1
This
problem set uses anagrams to motivate looking at the number of
arrangements
of different letters. After finding arrangements for small
numbers of letters
(2 to 4), they look for and justify a pattern.
A WINNING CONNECTION
http://www2.edc.org/mathproblems/getp.asp?name=ekCorrelate
In
this introduction to the idea of statistical correlation, students
interpret
scatter plots of (mostly baseball) statistics to see if there is
some
connection between the variables given. They order three plots by
the
strength (or lack) of connection.
WELL, WHAT DID YOU EXPECT?
http://www2.edc.org/mathproblems/getp.asp?name=ekFairExp
This
problem set connects the ideas of fairness and expected value.
Students are
guided to calculate expected values, and they use that
information to decide
if a game is fair. Then they develop the formula for
expected value of a
single-valued event, apply it, and consider expected
value of a double-valued
event (both winning and losing
possibilities).
What people are saying about PwaP
From Elena:
W-O-W, I can't believe it! I had been looking
for the problem on cutting a
square into squares for months, and I found it
in your data base! Thank
you! A friend of mine (very briefly) told me about
it when I saw her at the
Orlando Conference, and I had been looking for a
statement of the problem
ever since...and you had it.
Have you guys
completely cornered the market on cool problems? I hope there
are some still
left at large! But, it is still nice to have all this :-)
From
Andrea:
thanks for the address. I checked it for scientific notation and
ended up
with a wonderful 1+ day lesson for my honors students. Blew them
away at
first, but once they all calmed down (8th grade, remember) it was
very fun.
Let us know what YOU think! E-mail pwap@edc.org.
About Problems with a Point
The Problems with a Point Web site is a
searchable and well-indexed
collection of problems and orchestrated problem
sets designed to help
students in grades 6 through 12 develop both deep
conceptual mathematical
understandings and technical skills. Accessible to
teachers, students, and
parents over the Web, this resource includes problems
and problem sets for
development, practice, assessment, and integration of
concepts and skills,
classified by categories such as topic, difficulty
level, and use of
technology.
http://www2.edc.org/mathproblems
You
can view each problem in two versions. The first version, HTML,
displays the
problem in Web browsers but doesn't produce clear mathematical
symbols or
artwork. The second version, PDF, produces mathematics suitable
for classroom
handouts (PDF) as well as clear mathematical display on the
Web. To print or
view the PDF version, however, you will need Adobe Acrobat
Reader, which can
be downloaded free from Adobe.
Macintosh users: http://www.adobe.com/support/downloads/acrmac.htm
Windows
users: http://www.adobe.com/support/downloads/acrwin.htm
To sign up for the Problems with a Point newsletter, or
to change
your subscription, please visit:
http://www2.edc.org/mathproblems/
and
follow the links to the newsletter.
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