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[1] Average as a Balance Point
Average is probably the most used and probably the most important statistic
associated with a data set. People speak of slugging average in
baseball, average sales for the past month, average amount
of rainfall, average temperature, average class size,
average test score, etc. Performance in a very broad spectrum of
activities is measured in comparison to an average. That is, in how much the
performance varies from the average. In a numerical data set, the variation of
an element,
The interpretation of average as a balance point reflects the fact that the amount of negative variation in a numerical data set equals the amount of positive variation in the data set. The following analysis of telephone rates for the 10-10-345 system illustrates average as a balance point. In a holiday special, the 10-10-345 phone company offers the following rates (per minute) in addition to a thirty cent connection fee.
[2] Number Game
The Interactive Mathematics Program (IMP), a reformed secondary school
mathematics program, emphasizes solving word problems, small group activities,
discovery learning, and extended group problems. In 1999, the United States
Department of Education named IMP one of the nation's top five exemplary
mathematics programs. Creating number games, like the following one, is part
of their program.
Pick any number, multiply it by 2, add 8, divide by 2, and subtract the
original number. The answer is 4. Small Group Activity. Ask each group to explain why the sequence of steps in the preceding game always yields four. Then pair off the small groups and have each group make up an interesting challenge game for their partner group. After the groups have solved their challenge game, share a few of the games with the whole class. Post some of the number games on the department's bulletin board for others to play. [3] Average, Median, Mode of a Deck of CardsRemove the four aces from a deck of playing cards. The numerical card value for cards 2 through 10 is the face value of the card. Let the numerical value for jacks, queens, and kings be 10.
[4] Haitien Exchange Rate
The exchange rate of United States to Haitien currency was one U.S. dollar to
five Haitien dollars in January 2002. On November 27, 2002, the New York
Times reported that "Haiti's currency, the gourde, has lost 40 percent
of its value in the last year." Determine what the exchange rate should be in
January 2003. [5] Cost of ElectricityThe rate schedule for Orange County, NY of the Central Hudson Power and Light Company shows the following charges in terms of kilowatt hours (kWh):. These charges include the cost of the electricity, delivery service, meter maintenance, and New York State taxes.
1. Do the following:
2. Describe how the line in part 1b would change under each of the following scenarios. Explain your reasoning.
[6] Displaying Soup CansA display of soup cans forms a triangular shape with several rows of cans
placed on top of one another according to the restriction that each row,
counting from the top, has the same number of cans as its row number. Two
natural questions are 1) how many cans are in a display having n rows? And 2)
how large a display (number of rows) can be formed from k cans?
Experiment by counting the number of cans in a display having 1, 2, 3, rows to
gain an understanding of the problem
Consider mentally constructing a display from the top down one row at a time.
That is, begin with one row (one can), then add a second row, then a third
row, etc. Note the number of cans in a 2-row display is the number required
for a 1-row display plus 2 and the number of cans in a 3-row display is the
number required for a 2-row display plus 3 and so on. Form a 2-column table listing the number of rows and the corresponding number of cans.
Develop a recursive sequence model for the number of cans in an n-row display.
Let c(n) = number of cans needed to form an n-row display.. c(n) = c(n-1) + n
c(0) = 0 (zero row, zero sum, zero display)
Discover a formula for c(n) and then evaluate c(10).
Hint: Plot the data from the table (extend the table to n=10), recognize the
basic shape of the data, and then graphically fit a curve to the data or use
the suitable regression program in your calculator.
An Alternative Approach. Express c(n) as the sum of the seats in n rows rather
than as a recursive sequence. That is,
Karl Friedrich Gauss (1777-1855), a famous German mathematician, developed a
creative way of summing the first n positive integers when he was a nine year
old schoolboy. One day his teacher, hoping to keep the students busy for
awhile, assigned them the task of summing the first 100 positive integers.
Almost immediately Gauss wrote the correct answer (5,050) on his paper. How
did he compute the sum? Certainly not by adding the 100 numbers one at a
time. Gauss' method was to (mentally) write down the 100 integers, in increasing order, in a row. On the second row he wrote down the 100 integers in decreasing order. He then added each number in the second row to the corresponding number in the first row, getting 101 each time. 1 + 2 + 3 + . . . + 100 100 + 99 + 98 + . . . + 1
Gauss now had 100 sums of 101 or 10,100. Since each integer had been counted
twice (once in the first row and once in the second row), the 10,100
represented two times the sum of the first 100 positive integers.
Thus
Using Gauss' method, compute c(12). Generalize Gauss' method, to obtain a formula for c(n).
Superimpose the plot of c(n) on the data plot formed from the table of values.
[7] Notices
* Supported by the National Science Foundation and the U.S. Military Academy. |
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