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| Home > Teacher Resources > Newsletter Table of Contents > June/July 2002 Newsletter |
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June/July 2002 Newsletter
Featured problem: "The power of trig" Featured problem: "The power of trig" Does a calculator store the trigonometric function values for every possible angle measure—or does it calculate them somehow? This problem set gives one way to calculate these values. (Calculators actually use a different method, invented in 1959. See the teacher's note in the Answers section.) Students start by approximating the sine function near 0 using a line and considering the accuracy for values up to (pi)/2. Using differentials—a ratio of the difference in function values and the difference of the independent values—students find other polynomial approximations (the Taylor polynomials) and consider their accuracy.Taylor polynomials are often introduced in calculus courses, because they rely on derivatives of the function being approximated. Of course, differentials are the precursor to derivatives. With reasonable assumptions about observed patterns (such as the differentials approach 0, 1, or -1 as the independent interval decreases), students get a preview of derivatives with a payoff—they can see how transcendental functions can be approximated with a high degree of accuracy using relatively simple polynomials. This problem set is most appropriate for pre-calculus students. It can be given to a class or to individual students, especially those with an expressed curiosity about calculators and how they work. (Research into the actual method of calculators is also recommended, but students may need help interpreting the mathematical notation used in the sources they find.) Spreadsheets are particularly helpful (but not required), and the Hints section includes some tips for setting one up. Note that the derivative-slope connection is present in the structure of the differential but not emphasized in the problem set; if students are already familiar with derivatives, this is an excellent opportunity to give the connection more impact. http://www2.edc.org/mathproblems/getp.asp?name=ekPowerTrig Want a more advanced version for Calculus students? See "Taylor series," http://www2.edc.org/mathproblems/getp.asp?name=ekPowerTrigCalc Here is a sample of the new problems available on our site: TAKE IT EASY! http://www2.edc.org/mathproblems/getp.asp?name=pgSaulAlgebra This collection of elementary algebra problems, in multiple choice format, rewards students for thinking before applying techniques. Each problem can be solved in a standard way, or more easily by seeing what is really being asked and what is special about the situation. MODELING TEMPERATURE CHANGES http://www2.edc.org/mathproblems/getp.asp?name=ekTempMod Through the context of modeling temperature changes over the course of a year or more, students work with changing amplitutde and period of the sine function as well as incorporating phase shifts. EULER PATHS AND CIRCUITS http://www2.edc.org/mathproblems/getp.asp?name=ekEulerPath Students are introduced to Euler paths and circuits, and eventually make a conjecture about when a vertex-edge graph has either. The proof of the conjecture is a challenge problem. CAVALIERI'S PRINCIPLE http://www2.edc.org/mathproblems/getp.asp?name=ekVolRev2 Students explore Cavalieri's Principle in two and three dimensions. They compare right and oblique objects of the same type (such as triangles, rectangular prisms, and cones) to see that their areas or volumes are the same. MAXIMUM AREA 3 http://www2.edc.org/mathproblems/getp.asp?name=ekMaxArea3 Students form (and perhaps prove) conjectures about the n-gons with a fixed perimeter and the maximum area. They consider triangles, quadrilaterals, and then general n-gons. POISON http://www2.edc.org/mathproblems/getp.asp?name=sbpoison Students play and analyze the game of Poison, a 2-player game in which players start with a certain number of counters and take turns removing between 1 and 3 counters. The player who is forced to take the last (poison) chip loses. This is a variant of the game Nim. FIRST TO 50 http://www2.edc.org/mathproblems/getp.asp?name=sbfirstto50 Students play and analyze the game "First to 50," a 2-player game in which players take turns choosing a number from 1 to 9, keeping a running sum of their numbers. The player whose number makes the total 50 wins. Students analyze this game in order to determine winning strategies, then test and revise their strategies on similar games. While the first three problems require no algebra, the Challenge problem might be difficult for students with no experience with algebraic symbolism. WHERE'S THE (DECIMAL) POINT - PART 2 http://www2.edc.org/mathproblems/getp.asp?name=sbdecimaldiv Students work through and explain an algorithm for dividing decimal numbers (without a calculator): slide the decimal points in both numbers to the right (an equal number of places) until the divisor is an integer and then compute the new quotient. Then, they develop a method for dividing numbers expressed in scientific notation. A LOT OF CHANCES http://www2.edc.org/mathproblems/getp.asp?name=ekFairLot Students look at probabilities and expected values for a state lottery. They consider what prize amounts will make the lottery fair. 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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