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| Home > Teacher Resources > Newsletter Table of Contents > February 2002 Newsletter |
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February 2002 Newsletter
Featured problem: "Powers of two" Featured problem: "Powers of two" This problem set—accessible to students at many levels—has students explore sums of powers of two. They begin with a short explanation of what powers of two are, then write the first eight counting numbers as powers of two. Then they are told they have a balance with a set of weights which are powers of 2—1 g, 2 g, and 4 g to begin. They show they can weigh objects whose weight is a counting number from 1 to 7 g; then they add an 8-gram weight to the set and find the weights that can now be measured with the new set. After adding a 16-g weight, they are asked to find a pattern to describe how adding the next size power of two changes the possible weights that can be measured. They are asked to find the quickest way to measure any object, then to use that idea to write 227 as a sum of powers of 2.This problem set is as much about creating an algorithm—specifically, finding an efficient way to write any number as a power of 2—as it is about powers of two. As students add weights to their set and notice how an added weight doubles the range of possible measures, they should be able to find (perhaps with some consideration) a "halving" strategy for the algorithm. The balance analogy gives a more concrete model to the process of writing a counting number as a sum of powers of two, which could be invaluable for some students. This problem set can be used as an introduction to exponents or to algorithms. It can also be used to introduce the binary (base-2) number system: Follow the problem set with a "coding" activity, in which students code the sum according to which powers are used. For example, for the set that includes (in descending order) 16, 8, 4, 2, and 1 g weights, 00101 would mean the 4 and 1 are the only weights used—giving 5. So 101 is the binary representation of 5. The problem sets "Powers of three" and "Powers of two and three" are also good activities to follow. Depending on the level of your students, you may also want to present all or part of "Again and again"—which includes binary searches and an interesting pattern in binary numbers. See the full problem on the Web at http://www2.edc.org/mathproblems/getp.asp?name=ekPower2 Here is a sample of the new problems available on our site: LEMONADE AND FRUIT PUNCH http://www2.edc.org/mathproblems/getp.asp?name=ekDrinks Students find the best combination of two types of drinks to get the most profit for a lemonade stand. They work with functions of two variables and graph inequalities, then examine the feasible region to find where the maximum profit would be. COULD BE THIS OR THAT http://www2.edc.org/mathproblems/getp.asp?name=ekAmbiguous Students learn about the ambiguous case by solving equations involving sine and considering geometric facts (sum of angles of a triangle, congruence theorems). SETTING UP PROPORTIONS http://www2.edc.org/mathproblems/getp.asp?name=ekPropSetUp Students learn how to properly set up proportions by first organizing the three given values and the unknown value. Some solving of set-up proportions is required. A MYSTERY TO SOLVE http://www2.edc.org/mathproblems/getp.asp?name=ekAssoc The problem set highlights the importance of the associative property in solving equations over the real numbers. This problem set has students work with an unfamiliar operation on a small set of numbers to solve a simple equation. However, the technique of operating by an inverse doesn't work in this case, because the set isn't associative. MUCH ADO ABOUT NOTHING http://www2.edc.org/mathproblems/getp.asp?name=pgZero In this problem set, students look at equivalent multiplication and division sentences (equations) to see why division by 0 causes problems (is undefined). THE CORRELATION AND THE CAUSE http://www2.edc.org/mathproblems/getp.asp?name=ekCorrCause Students use their understanding of correlation to consider the concept of lurking (underlying) variables, as opposed to cause-and-effect relationships. EXPECTED VALUE http://www2.edc.org/mathproblems/getp.asp?name=ekExpect Students create the general formula for expected value of a probabilistic situation with discrete outcomes. BD1, THE BINOMIAL DISTRIBUTION ROBOT http://www2.edc.org/mathproblems/getp.asp?name=ekBiDist Students think through probabilistic situations and create graphs so they can see what a binomial probability distribution looks like, and how a situation causes that kind of distribution. THE GAME SHOW PROBLEM http://www2.edc.org/mathproblems/getp.asp?name=ekGameShow Students look at the classic game show problem (also known as the Monty Hall problem) and solve it experimentally, using a simulation. They also explain why their activity is a simulation, and consider the accuracy of the results. What people are saying about Pwap This is the first math website I have found to be of any value. Thanks. This (great!) problem can be also solved with the use of the Heron's
formula. To omit simple algebraic transformations, we'll finally need to
find out when a^2-(b-c)^2 is maximum (a, b, and c are the lengths of the
sides, where a and b+c are fixed). This happens when b=c, because in that
case nothing will be subtracted from a^2. 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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