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Goal-oriented, state-to-state search Most notes from Dr. Michael Wick Curly used a shovel to dig his own swimming pool. He figured he needed a pool because digging it was hard work and he could use it to cool off after working on it all day. He also planned to build a rectangular concrete deck around the pool that would be 6 feet wide at all points. The pool is rectangular and measures 14 feet by 40 feet. What is the area of the deck? 52 feet + 26 feet + 52 feet + 26 feet = 156 feet Counts each corner twice! 156 feet x 6 feet = 936 square feet Two lengths: 40 ft x 6 ft x 2 = 480 sq ft Two widths: 14 ft x 6 ft x 2 = 168 sq ft Four corners: 6 ft x 6 ft x 4 = 144 sq ft Total 792 sq ft 52 ft x 6 ft = 312 sq ft 312 sq ft x 2 = 624 sq ft for extended lengths 14 ft x 6 ft = 84 sq ft 84 sq ft x 2 = 168 sq ft for widths Total = 624 sq ft + 168 sq ft = 792 sq ft Area of entire figure = 52 ft x 26 ft = 1352 sq ft Area of pool alone = 40 ft x 14 ft = 560 sq ft Area of deck = 1352 – 560 = 792 sq ft The object of the game Frisbin is to throw three Frisbees at three different-sized bins that are set up on the ground about 20 feet away from the player. If a Frisbee lands in the largest bin, the player scores 1 point. If a Frisbee lands in the medium-sized bin, the player scores 5 points. If a Frisbee lands in the smallest bin, the player scores 10 points. Kirk McCoy is playing the game. If all three of his Frisbees land in bins, how many different total scores can he make? Tom, John, Fred, and Bill each brought his favorite food to a dinner. From the clues below, determine each man’s occupation and favorite food. 1. Tom is neither the nurse nor the teacher. 2. Fred and the pilot play in a jazz band together. 3. The burger lover and the teacher are not musically inclined. 4. Tom brought hot dogs. 5. Bill sat next to the burger fan and across from the steak lover. 6. The secretary does not play an instrument or sing. Tom Pilot Hdog John Scty Burg Fred Nurse Steak Bill Tchr Chkn A mixture is 25% red paint, 30% yellow paint, and 45% water. If 4 quarts of red paint are added to 20 quarts of the mixture, what is the percentage of red paint in the new mixture? 1. How many quarts of red paint are in the new mixture? 2. How many quarts of paint are in the new mixture? 3. What percentage of the new mixture is red paint? 1. How many quarts of red paint are in the new mixture? How many quarts of red paint are in the original mixture? And how many quarts of paint are in the original mixture? How many quarts of paint are in the original mixture? How many quarts of red paint are in the original mixture? How many quarts of red paint are in the new mixture? 2. How many quarts of paint are in the new mixture? 3. What percentage of the new mixture is red paint? How many quarts of paint are in the original mixture? 20 (given) How many quarts of red paint are in the original mixture? 25% of 20 = 5 quarts How many quarts of red paint are in the new mixture? 5 + 4 = 9 quarts How many quarts of paint are in the new mixture? 20 + 4 = 24 quarts What percentage of the new mixture is red paint? 9 / 24 = 0.375 = 37.5% Use a number instead of a variable Use smaller or easier numbers Do a set of specific easier examples and look for a pattern Do a specific easier example and figure out an easier process Change, fix, or get rid of some conditions Eliminate unnecessary information The average of a group of quiz scores is 31.8. There are k quiz scores in the group. The average of 10 of these quiz scores is 24.3. Find the average of the remaining quiz scores in terms of k. The average of a group of quiz scores is 31.8 30. There are k quiz scores in the group. The average of 10 of these quiz scores is 24.3 25. Find the average of the remaining quiz scores in terms of k. The average of a group of quiz scores is 30. There are k 50 quiz scores in the group. The average of 10 of these quiz scores is 25. Find the average of the remaining quiz scores in terms of k 50. Sum of all scores is 30 x 50 = 1500. Sum of 10 scores is 25 x 10 = 250. Sum of other scores is 1500 – 250 = 1250. Average of those 40 scores is 1250 31.25 40 1250 1500 250 30 50 25 10 40 50 10 50 10 30k 25 10 k 10 31.8k 243 k 10 In this election, there are 29 issues and candidates. In the last election, there were 28,311 voters, representing 18,954 households, and they voted at 14 polling places. This time there will be 34,892 voters. How many polling places will be needed? • • • • • • Polling places (last election): 15 Voters (last election): 30,000 Households: 20,000 Issues: 30 Voters (this election): 35,000 Polling places (this election): ? Polling places (last election): 15 Voters (last election): 30,000 Households: 20,000 Issues: 30 Voters (this election): 35,000 Polling places (this election): ? 30,000 2,000 voters/polling place 15 35,000 17.5 polling places (this election) 2,000 28,311 ≈ 2,022.2 voters per polling place 14 34,892 17.25 polling places (this election) 2,022.2 The divisors of 360 add up to 1170. What is the sum of the reciprocals of the divisors of 360? Use divisors of 24, which are 1, 2, 3, 4, 6, 8, 12, and 24. Their sum is 60. The sum of reciprocals is 1 1 1 1 1 1 1 1 1 2 3 4 6 8 12 24 24 12 8 6 4 3 2 1 24 24 24 24 24 24 24 24 24 12 8 6 4 3 2 1 24 60 sum of divisors 24 number The divisors of 10 are 1, 2, 5, and 10. Their sum is 18. The sum of reciprocals is 1 1 1 1 10 5 2 1 1 2 5 10 10 10 10 10 10 5 2 1 18 sum of divisors 10 10 number sum of divisors 1170 number 360 How many squares are there on a checkerboard? 1 8x8 square 4 7x7 squares 9 6x6 squares 16 5x5 squares 25 4x4 squares 36 3x3 squares 49 2x2 squares 64 1x1 squares total: 204 squares A train leaves Roseville heading east at 6:00 a.m. at 40 miles per hour. Another eastbound train leaves on a parallel track at 7:00 a.m. at 50 miles per hour. What time will it be when the two trains are the same distance away from Roseville? One possibility: Change the condition so that the first train travels for an hour and then stops, and the second travels at 10 miles per hour. Second train will “make up” 10 miles per hour. Let’s reduce the problem to finding when the second train, running at 10 miles per hour will reach the spot that the first train reaches at the one hour point. 1 hour 40 mi/hr 40 mi 40 mi 4 hr 10 mi/hr Subproblem ◦ Results from breaking a problem down into its parts ◦ Use answers to construct solution ◦ CS Terminology Divide and Conquer Wishful Thinking Easier Problem ◦ Doesn’t involve solving the original problem ◦ Use method to construct solution ◦ CS Terminology Hardcode Releases A diagonal of a polygon is a line segment that connects two nonadjacent vertices of the polygon. A certain polygon has 25 sides. How many diagonals can be drawn? A. B. C. D. 300 625 275 500 Strategy: Do a set of specific easier examples and look for a pattern. Ted has to load a truck with television sets. The cargo area of the truck is a rectangular prism that measures 8 ft by 21 ft by 11 ft. Each television set measures 1 1/2 ft by 1 2/3 ft by 1 1/3 ft. How many sets can be loaded into the truck? A. B. C. D. 528 504 625 508 Strategy: Use a small or easier number. In China each calendar year is given one of 12 names, which rotate year after year. The year 2000 was the year of the Dragon. The year 2001 was the year of the Snake. The subsequent ten years are, in order, the years of the Horse, Sheep, Monkey, Rooster, Dog, Boar, Rat, Ox, Tiger, and Rabbit. After the year of the Rabbit, the year of the Dragon will occur again, and then the whole cycle will repeat. What will the year 3000 be? A. B. C. D. Dragon Sheep Boar Monkey Strategy: Do a set of specific easier examples and look for a pattern. In the land of Kantanu, it was considered good luck to own goats. Barsanta owned some goats at the time of her death and willed them to her children. To her first born, she willed onehalf of her goats. To her second born, she willed one-third of her goats. And last she gave one-ninth of her goats to her third born. Assuming that Barsanta had 17 goats and barring a Solomonic approach, how many goats did the second child receive? A. B. C. D. 4 6 8 10 Strategy: Change, fix, or get rid of some conditions. A square has an area of S2. A regular hexagon has a perimeter of T. If p is the perimeter of the square and h is a side of the hexagon, then find h + p in terms of S and T. Strategy: Use values instead of variables; use logic A. B. C. D. h + p = 6T + S/4 h + p = 4T + 6S h + p = T/6 + 4S h + p = T/4 + 6S Knights always tell the truth. Knaves always lie. Ima says “Dewey is a knave.” Dewey says “Neither Ima nor I are knaves.” Who, if any, is a knight and who, if any, is a knave? Strategy: Use logic Summing numbers – what’s the sum of all numbers between 1 and 100? Summing digits of numbers – what’s the sum of all of the digits of all of the numbers between 1 and 100? http://mathforum.org/library/drmath/view/5 7919.html Error reads: “missing semicolon on line XY” You go to line XY, and it looks like this: ◦ System.out.println(“A perfectly valid statement”); What’s the problem? G. Polya, “How To Solve It” (2004)