January 2002 Newsletter

Featured problem: "Perspective in art"
New on the PwaP Web site
About Problems with a Point (PwaP)

Featured problem: "Perspective in art"

This problem set begins by walking students through creating a perspective drawing of a three-dimensional rectangular prism, using the projection method. Students analyze the results to find similar figures and the scale factor from the larger to the smaller. With that information, they use the method on a given polygon to create a similar polygon. Finally, the problem set gives students an open exploration of perspective drawing, suggesting looking at the effect of different scale factors and different vanishing points; they write a short report describing how these variables affect the visuals of the drawing.

Similar figures are often approached in terms of recognizing when two figures are similar, rather than creating a similar figure. This technique allows one to do so with any polygon, even if the angles involved are difficult to measure with certainty. The connection with art gives students some motivation to learn this interesting technique. The exploration (and report) at the end exercises students' skills with visualization and communication, as they describe what they see in the different drawings they create.

The problem set lends itself very well to dynamic geometry software. The technique and its results can be intriguing for students at all levels, and the exploration at the end can allow more inquisitive students to delve as deeply as they wish. The fun of drawing can help for those class days when it's difficult to get students focused—but they will learn something in the process!

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

New on the PwaP Web site

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

GUESS THE RULE
http://www2.edc.org/mathproblems/getp.asp?name=ekGuessRule
This problem set uses a spreadsheet to give students a flexible environment to play "Guess the Rule" with linear and quadratic rules. They translate their guesses into an algebraic form for the spreadsheet, in order to check the validity of their guesses. The spreadsheet must be downloaded from the PDF or HTML version of the problem. (See the Answers page or section for the link.)

POLYGONS IN POLYGONS
http://www2.edc.org/mathproblems/getp.asp?name=ekAreaInPoly
Students explore the relationship between the areas of regular polygons and the polygons formed by connecting the midpoints of the original polygons' sides. They use trigonometry and the formula for the area of regular polygons to find and prove the relationship.

THE PLATONIC SOLIDS
http://www2.edc.org/mathproblems/getp.asp?name=ekPlatonic
In this problem set, students explore creating polyhedra using regular polygons. They create the five Platonic solids (regular solids) and, in doing so, discover why there are only five.

POLAR CONVERSION
http://www2.edc.org/mathproblems/getp.asp?name=ekPolarConv
Students use triangle trigonometry to find the formula for converting polar coordinates to Cartesian coordinates. They look at each quadrant as a separate case and prove the resulting formulas are equivalent.

A POLAR MYSTERY
http://www2.edc.org/mathproblems/getp.asp?name=ekPolarMyst
Students work with polar coordinates and the connection between the polar and Cartesian coordinates of a point. They look at (and graph) an equation in polar coordinates and prove what kind of shape it is by converting the equation to Cartesian coordinates (change of variables).

CROSS CUTTING
http://www2.edc.org/mathproblems/getp.asp?name=ekCrossSection
Students learn about cross sections, and find different shape cross sections for different polyhedra. Note: This is a graphics-intensive problem set, so please be patient. (The .pdf file is 700 Kb.)

MATRIX MIRROR, ON THE WALL
http://www2.edc.org/mathproblems/getp.asp?name=ekMatrixRefl
In this problem set, students practice matrix multiplication (2-by-2 times 2-by-1, and 2-by-2 times 2-by-3) as they see how to use matrices to represent reflections of geometric objects.

$6.5 M FOR SCIENCE, $13 M FOR A BUILDING
http://www2.edc.org/mathproblems/getp.asp?name=PGSciNotation
In this problem set, students practice converting numbers to scientific notation as they develop techniques for multiplying and dividing numbers written in scientific notation.

CONDITIONAL PROBABILITIES AND BAYES' FORMULA
http://www2.edc.org/mathproblems/getp.asp?name=ekCondProb
Students work with conditional probabilities and discover that P(A|B) is not necessarily P(A)/P(B). They are given Bayes' formula and verify it for three cases, when the intersection of A and B is 1) A, 2) empty, and 3) a proper subset of both A and B. (Students do not need to use the intersection language.)

PROBABILITY DISTRIBUTIONS
http://www2.edc.org/mathproblems/getp.asp?name=ekProbDist
This introduction to probability distributions has students create tables or graphs (after the first problem, it's their choice) for different situations involving die rolls. The last question asks about the sum of the probabilities in the distribution.

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.