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| Home > Teacher Resources > Newsletter Table of Contents > October/November 2002 Newsletter |
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October/November 2002 Newsletter
PwaP is featured in ENC Focus Problems with a Point received more recognition from the Eisenhower National Clearinghouse (ENC) this fall, when PwaP was one of the mathematics resources featured in "Supporting Success in the Urban Classroom," by Carol Damian and Terese Herrera in the latest issue of ENC Focus, "Success in the Urban Classroom," (Volume 9, Number 4, 2002). The newsletter is available online at http://www.enc.org/focus/urban/ and the full list of featured mathematics resources can be found at http://www.enc.org/focus/urban/document.shtm?input=FOC-002981-index. In October, 2001, PwaP was selected as an exemplary website in ENC's "Digital Dozen." Featured problem: "What goes up must come down #2" If you want to throw a ball as far, horizontally, as possible, at what angle with the horizontal should you aim your throw? Does it depend on the velocity at which you throw or the height above ground that the ball leaves your hand? These are the questions that students investigate in this problem set.Students are given a quadratic function that provides a very good approximation for the height of a ball, t seconds after it is thrown, in terms of the initial velocity, angle, and height the ball is thrown from. Using this information, and either a graphing utility or spreadsheet software, they experiment, then develop conjectures about the optimal angle at which to throw in order for the ball to go the farthest. Many students "know" that the angle is 45 degrees, so they are surprised to learn that this is not the case. In fact, 45 degrees is the optimal angle only when the ball is thrown from ground level, which is pretty hard to do when throwing from a standing position! Even though the function used is an approximation (since it ignores wind resistance, height above sea level, and other mitigating factors), students find it to be an interesting "real-world" application. The problem could serve as a motivator for algebra or precalculus students, since calculus provides us with useful techniques for finding maximum and minimum values for functions. Working through the problem might also make calculus students better appreciate the optimization techniques they're learning (or will learn). It can be used individually or with groups or whole classes of students. http://www2.edc.org/mathproblems/getp.asp?name=sbprojectile2 Here is a sample of the new problems available on our site: TANGENT LINES AND RADII OF CIRCLES http://www2.edc.org/mathproblems/getp.asp?name=sbTanPerp Students use calculus (specifically, the relationship between the derivative and the tangent line) to prove that the line tangent to a circle and the radius at the point of tangency are perpendicular. Although the problem does not require the use of implicit differentiation, it is a useful technique to practice and apply here. DERIVING THE QUADRATIC FORMULA http://www2.edc.org/mathproblems/getp.asp?name=jaQuadFormDerivation Students learn how to derive the quadratic formula using algebra inspired by the transformation of graphs. The derivation is based on a symmetry argument (without use of the method of completing the square). MIND YOUR "IF P'S" AND "THEN Q'S" http://www2.edc.org/mathproblems/getp.asp?name=ekIfP This problem set has students think through logical if-then statements, including the connection between the original and contrapositive statements, as well as between the converse and inverse statements. CAN YOU FIND THEM ALL? http://www2.edc.org/mathproblems/getp.asp?name=sbPseudoTrace Students investigate the "pseudotraces" of two 10 by 10 arrays (square matrices), which are the sums of all possible collections of 10 entries from the array, chosen so that the collection contains exactly one entry from each row and column. Place value recognition is an important aspect of the solution. In the challenge problem, an n by n array is considered. (The solution for the challenge problem requires some algebra.) FIBONACCI'S RABBITS http://www2.edc.org/mathproblems/getp.asp?name=sbfibonacci1 Students determine the recursive formula for the Fibonacci sequence by solving a version of the famous "rabbit problem." A challenge problem asks students to write a computer or calculator program that will find any term in the Fibonacci sequence. HOW FAR IS IT? http://www2.edc.org/mathproblems/getp.asp?name=jaBinocularVision Students use trigonometric functions and triangle congruence to understand why we need two eyes to determine the distance to an object. They perform a sensitivity analysis to better understand their mathematical model. EVEN STEVEN? http://www2.edc.org/mathproblems/getp.asp?name=sbeven In the context of determining the fairness of a method for choosing who goes first in a game, students compute the probability that the product of two different numbers chosen at random from the set {1,2,...,n} is odd (for n=4, 10, and 100). A challenge problem has students consider what happens if all counting numbers can be used (rather than just a finite subset). 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. |
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