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| Home > Teacher Resources > Newsletter Table of Contents > August/September 2002 Newsletter |
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August/September 2002 Newsletter
Welcome Back! Many of you have just begun a new academic year and the Problems with a Point team hopes you had an enjoyable, restful break. This is the time of year that lots of folks dig into their files for their favorite problems to use during the school year. We hope you think of PwaP as a useful addition to your files. If you're looking for a problem on a given topic, with certain prerequisites, or using specific technology, go to http://www2.edc.org/mathproblems/search.asp to find what you're looking for. And if you've found some problems in which you think others would be interested, please send them to us! You can suggest a problem at http://www2.edc.org/mathproblems/suggest.asp. Featured problem: "Extending the Pythagorean Theorem" Given the lengths of two sides of a right triangle, you can use the Pythagorean theorem to solve for the length of the third side, but what can you do if the triangle doesn't contain a right angle? This problem set has students compute the length of the third side of a triangle given the lengths of the other two sides and one of the angles. Students derive the Law of Cosines by dropping a perpendicular from a vertex to its opposite side, then using the Pythagorean theorem and facts about the cosine function to solve for missing lengths.Starting with their understanding of a familiar concept, the Pythagorean theorem, students devise a way to deal with a non-familiar situation. Along the way, they develop of a method that might help them recall or rederive the Law of Cosines if they ever forget it. The Law of Cosines is an important topic, but is often the cause of student mistakes and misconceptions. "Extending the Pythagorean Theorem" can be used successfully as an introduction to the Law of Cosines or as a precursor to a discussion on the Cosine Law. It could also be used as an extension to a unit on the Pythagorean theorem or as a review of Pythagorean and trigonometric concepts since it provides practice in these areas. http://www2.edc.org/mathproblems/getp.asp?name=ekCosineLaw Here is a sample of the new problems available on our site: AN ODD DIVISIBILITY PROPERTY IN PASCAL'S TRIANGLE http://www2.edc.org/mathproblems/getp.asp?name=sbpascalprime Students investigate the odd numbered rows of Pascal's triangle, then make and confirm a conjecture about the "inside" entries in row p being divisible by the prime number p . PROOF NOT POSITIVE http://www2.edc.org/mathproblems/getp.asp?name=ekDiscGeom This problem set presents a non-Euclidean geometry, and then has students analyze a proof that works for Euclidean (plane) geometry but not for the non-Euclidean one. They are challenged to see the features of the geometries that cause the discrepancy. This set can emphasize the need to identify and perhaps challenge one's assumptions, and it can also start a discussion about the distinction between discrete and continuous. HOW TANGENT GOT ITS NAME http://www2.edc.org/mathproblems/getp.asp?name=ekTanCircle Students prove the connection between the tangent line of a circle and the tangent of an angle. They use this connection to explain some values and a property of the tangent function. SECANT TO NONE http://www2.edc.org/mathproblems/getp.asp?name=ekSecCircle Students prove a connection between the tangent line of a circle and the secant of an angle. They use this connection to explain some values of the secant function. ARCTIC STRING TRICKS http://www2.edc.org/mathproblems/getp.asp?name=ekArcticString Students analyze some fun string tricks to see why they work. The related set "Knot again!" is highly recommended as a prerequisite to give students experience drawing string manipulations. WHEN ZEROS AREN'T REAL http://www2.edc.org/mathproblems/getp.asp?name=ekComplexZeros Students practice completing the square or the quadratic formula as well as factoring principles as they find complex zeros of polynomial functions. 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. To sign up for the Problems with a Point newsletter, or
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