December 2001 Newsletter

PwaP hits 200 problems!
Featured problem: "Checkers challenge"
New on the PwaP Web site
What people are saying about PwaP
About Problems with a Point (PwaP)

PwaP hits 200 problems!

The Problem with a Point Web site now includes over 200 problem sets. There
are lots to choose from, so please let us know which ones are your
favorites!

Featured problem: "Checkers challenge"

This problem set presents two games designed to help students see the sense
behind the rules for addition and subtraction of signed numbers. In the
first game, Checkers Challenge, players roll a die and add or remove
checkers to a common pile, depending on the roll. The player with more
checkers in the pile at the end of the game wins. After playing the game,
students answer specific questions about the game, such as how how adding
their opponent's checkers has the same effect on who's winning as removing
their own. Next, they are asked to consider the checkers to be a model for
signed numbers---black for positive numbers and red for negative numbers---
and they take another look at the game with that model in mind. Finally,
they play the second game, Numbers Challenge, which uses virtually the same
rules but using only numbers, without the physical checkers. They explain
how the two games are really the same.

Adding and subtracting signed numbers is often a mystery to students, who
memorize the rules (especially for subtraction) without understanding why
those rules exist in the first place. The problem set begins with a
concrete situation so that students can easily see how removing
(subtracting) one type is the same as adding the other type. (For example,
if there are 2 more red checkers than black, removing 1 black means there
are now 3 more red than black. If there are only 2 red checkers---so no
black checkers can be removed---adding 1 red will also give 3 more red than
black.) After they learn the rules of the game, the problems lead them to a
more abstract version. With the second game, they have an opportunity to
practice in that abstract setting with the option of converting to the
concrete, as needed, for help. Scoresheets for both games are included to
help students organize the work.

Although this problem set may be most appropriate for an introduction to
operations with signed numbers, it would also be helpful as reinforcement
or follow up after the concept has been introduced. Two players are
required for the games, but for out-of-class work, parents and tutors can
play with a student and then use the game to help them articulate the
answers to the questions.

See the full problem on the Web at
http://www2.edc.org/mathproblems/getp.asp?name=ekCheckers

New on the PwaP Web site

Here is a sample of the new problems available on our site:

POLAR CIRCLES
http://www2.edc.org/mathproblems/getp.asp?name=ekPolarCoord
Students learn about using polar coordinates to describe points in a
plane, motivated by trying to graph a circle on a graphing calculator.
They also consider the graphs of some simple curves (spirals and a circle)
using equations in polar coordinates.

EXPLORING FIXED POINTS OF QUADRATICS
http://www2.edc.org/mathproblems/getp.asp?name=ekFixedPoint2
Students explore the existence of fixed points of monic quadratic
functions---specifically, for what monic quadratics are there no, one, or
more than one fixed point? As a challenge, they can also explore what kind
of fixed points (attracting, repelling, or neither) each will be. (The
challenge will make the problem set more than one day's work.)

FINDING THE LINE OF BEST FIT
http://www2.edc.org/mathproblems/getp.asp?name=ekLeastSquares
Students compare two models for a data set (women's tennis earnings) using
the absolute value of the residuals (or error). Then they use technology
(calculator or software) to find the least squares regression line, and
they compare one of the earlier models to that line using squares of the
residuals. (Note, the problem set "Tennis, anyone?" makes an excellent
precursor to this one.)

TENNIS, ANYONE?
http://www2.edc.org/mathproblems/getp.asp?name=ekLinFit
This introduction to fitting lines to data has students fit a line using
whatever criteria for "fitting well" they wish. They then compare three
lines and consider which describes the trend in the data (women's tennis
earnings) best.

IS IT FAIR GAME?
http://www2.edc.org/mathproblems/getp.asp?name=ekFairGame
Students consider variations of a game and decide whether the game is fair
or not. At the end of the set, they modify the rules to make up their own
fair and unfair games.

BABIES AND BASKETBALLS
http://www2.edc.org/mathproblems/getp.asp?name=ekSimulation
This problem set introduces students to simulations. Students compare a
birth situation (baby genders) to flipping coins, then create and carry
out a simulation for a basketball situation (making 3 of 4 free throws).

GET OUT THE VOTE!
http://www2.edc.org/mathproblems/getp.asp?name=ekGetVote
Students take a single election (with preferences among three candidates
specified) and see how different election methods produce different
winners.

What people are saying about Pwap

I looked at the "Problems with a Point" website. I agree that it is "pretty good." I thought the problem about the area of a new triangle formed by extending the sides of the old was outstanding.
—Dr. David Wright, Professor of Mathematics, Brigham Young University
[The problem set, "What are arithmetic sequences and series for?" is a really nice problem! In fact, I think this is easily the nicest "application" of arithmetic series that I've ever seen!]

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.