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Copyright © Wright Group/McGraw-Hill Project Masters 373 Project Masters Project Masters PROJECT6 Name Date Time Ground Areas of Famous Large Buildings Copyright © Wright Group/McGraw-Hill 403 The ground areas of buildings, their footprints, are almost always given in square feet or square meters. Some buildings have very large ground areas. When their areas are given in square feet, the numbers are so large that it is hard to imagine how big the buildings really are. For large buildings, if you convert the area in square feet to an estimate in acres, you can get a better idea of the size of the building. Estimate the ground area, in acres, of each building in the table below: Example:The Colosseum, in Italy, covers an area of about 250,000 ft 2. One acre is about 50,000 ft2. So 5 acres is about 250,000 ft2. The Colosseum covers an area of about 5 acres (5 football fields). Building Country Date BuiltGround Estimated Area Area (ft 2)(in acres) Colosseum Italy 70–224 250,000 ft 2 acres Egyptc.2600 B.C. 571,500 ft 2 acres France 1194–1514 60,000 ft 2 acres Vatican City 1506–1626 392,300 ft 2 acres Taj Mahal India 1636–1653 78,000 ft 2 acres Pentagon 1941–1943 1,263,000 ft 2 acres 1936 2,800,000 ft 2 acres 5 Pyramid of Cheops Chartres Cathedral St. Peter’s Basilica Ford Parts CenterU.S. (Michigan)U.S. (Virginia) 11 1 8 1.5 25 56 Reference 1 acre 43,560 square feet For estimating, think of 1 acre as about 50,000 square feet. A football field (excluding the end zones) is approximately 1 acre. PROJECT2 A Perfect-Number Challenge Copyright © Wright Group/McGraw-Hill 382 Name Date Time Starting FactorsSum of Perfect Number Factors Number21, 2 3 648 Perfect numbers become big very quickly. The third perfect number has 3 digits, the fourth has 4 digits, the fifth has 8 digits, the sixth has 10 digits, and the thirty-second has 455,663 digits! In other words, perfect numbers are hard to find. You can find perfect numbers without having to find the sum of the proper factors of every number. Here is what you do: 1.Complete the pattern of starting numbers in the first column in the table. 2.List the factors of each starting number in the second column. 3.Write the sum of the factors of each starting number in the third column. 4.If the sum of the factors of the starting number is prime, multiply this sum by the starting number itself. The product is a perfect number. Record it in the last column. The first perfect number is 6. Try to find the next three perfect numbers. People have been fascinated by perfect numbers for centuries. The ancient Greeks knew the first four. The fifth perfect number was not found until the year 1456. The search for perfect numbers is now carried out on computers. When this book went to press, 42 perfect numbers had been identified. All the perfect numbers found so far are even numbers.

PROJECT 1 The Search for Prime Numbers 374 Name Date Time You probably know the following definitions of prime and composite numbers: A prime numberis a whole number that has exactly two factors. The factors are 1 and the number itself. For example, 7 is a prime number because its only factors are 1 and 7. A prime number is divisible by only 1 and itself. A composite numberis a whole number that has more than two factors. For example, 10 is a composite number because it has four factors: 1, 2, 5, and 10. A composite number is divisible by at least three whole numbers. The number 1 is neither prime nor composite. For centuries, mathematicians have been interested in prime and composite numbers because they are the building blocks of whole numbers. They have found that every composite number can be written as the product of prime numbers. For example, 18 can be written as 2 º 3 º 3. Around 300 B.C., the Greek mathematician Euclid (yOO´klid) proved that there is no largest prime number. No matter how large a prime number you find, there will always be larger prime numbers. Since then, people have been searching for more prime numbers. In 1893, a mathematician was able to show that there are more than 50 million prime numbers between the numbers 1 and 1 billion. The Greek mathematician Eratosthenes ( ˘er´ -t ˘os´ th -n ¯ez´), who lived around 200 B.C., devised a simple method for finding prime numbers. His strategy was based on the fact that every multiple of a numberis divisible by that number. For example, the numbers 2, 4, 6, 8, and 10 are multiples of 2, and each of these numbers is divisible by 2. Here is another way to say it: A whole number is a factor of every one of its multiples. For example, 2 is a factor of 2, 4, 6, 8, and 10. The number 2 has only one other factor, the number 1, so 2 is a prime number. All other multiples of 2 are composite numbers. Eratosthenes’ method is called the Sieve of Eratosthenes.The directions for using the sieve to find prime numbers are given on Math Masters,page 375. Since the time of Eratosthenes, mathematicians have invented more powerful methods for finding prime numbers. Some methods use formulas. Today, people use computers. The largest prime number known when this book went to press had 9,152,052 digits. If that number were printed in a book with pages the same size as this page, in the same size type, the book would be about 1,400 pages long. ee

PROJECT 1 Name Date Time The Sieve of Eratosthenes 375 Follow the directions below for Math Masters,page 376. When you have finished, you will have crossed out every number from 1 to 100 that is not a prime number. 1. Because 1 is not a prime number, cross it out. 2. Circle 2 with a colored marker or crayon. Then count by 2, crossing out all multiples of 2; that is, 4, 6, 8, 10, and so on. 3. Circle 3 with a different colored marker or crayon. Cross out every third number after 3 — 6, 9, 12, and so on. If a number is already crossed out, make a mark in a corner of the box. The numbers you have crossed out or marked are multiples of 3. 4. Skip 4, because it is already crossed out, and go on to 5. Use a new color to circle 5, and cross out multiples of 5. 5. Continue in the same pattern. Start each time by circling the next number that is not crossed out. Cross out all multiples of that number. If a number is already crossed out, make a mark in a corner of the box. Use a different color for each new set of multiples. 6. Stop when there are no more numbers to be circled or crossed out. The circled numbers are the prime numbers from 1 to 100. 7. List all the prime numbers from 1 to 100.

PROJECT 1 The Sieve of Eratosthenes continued 376 Name Date Time 12345678 910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

PROJECT 1 Name Date Time The Sieve of Eratosthenes continued 377 1. What are the crossed-out numbers greater than 1 called? 2. Notice that 6 is a multiple of both 2 and 3. Find two other numbers that are multiples of both 2 and 3. 3. Find a number that is a multiple of 2, 3, and 5. (Hint:Look at the colors.) 4. Find a number that is a multiple of 2, 3, 4, and 5. 5. Choose any crossed-out number between 50 and 60. List its factors. 6. List the crossed-out numbers that have no marks in the corners of their boxes. 7. Find a pair of consecutive prime numbers. Are there any others? If yes, list them. 8. The numbers 3 and 5 are called twin primes because they are separated by just one composite number. List all the other twin primes from 1 to 100. 9. Why do you think this grid is called a sieve?

PROJECT 1 The Sieve of Eratosthenes continued 378 Name Date Time 101 102 103 104 105 106 107 108 109 100 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200

PROJECT 2 Name Date Time Deficient, Abundant, and Perfect Numbers 379 A factorof a whole number Nis any whole number that can be multiplied by a whole number to give Nas the product. For example, 5 is a factor of 30 because 6 º 5 30. Also, 6 is a factor of 30. Every whole number has itself and 1 as factors. A proper factorof a whole number is any factor of that number except the number itself. For example, the factorsof 10 are 1, 2, 5, and 10. The proper factorsof 10 are 1, 2, and 5. A whole number is a deficient numberif the sum of its proper factors is less than the number. For example, 10 is a deficient number because the sum of its proper factors is 1 2 5 8, and 8 is less than 10. A whole number is an abundant numberif the sum of its proper factors is greater than the number. For example, 12 is an abundant number because the sum of its proper factors is 1 2 3 4 6 16, and 16 is greater than 12. A whole number is a perfect numberif the sum of its proper factors is equal to the number. For example, 6 is a perfect number because the sum of its proper factors is 1 2 3 6. Exploration List the proper factors of each number from 1 to 50 in the table on Math Masters, pages 380 and 381. Then find the sum of the proper factors of each number, and record it in the third column of the table. Finally, make a check mark in the appropriate column to show whether the number is deficient, abundant, or perfect. Divide the work with the other members of your group. Have partners use factor rainbows to check each other’s work. When you are satisfied that all the results are correct, answer the questions on page 381.

PROJECT 2 Deficient, Abundant, and Perfect Numbers cont. 380 Name Date Time Number Proper FactorsSum of Deficient Abundant Perfect Proper Factors 1 0✓ 2 3 4 5 6 1, 2, 3 6✓ 7 8 9 10 1, 2, 5 8✓ 11 12 1, 2, 3, 4, 6 16✓ 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

PROJECT 2 Name Date Time Deficient, Abundant, and Perfect Numbers cont. 381 Number Proper FactorsSum of Deficient Abundant Perfect Proper Factors 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Source: The Math Teacher’s Book of Lists.San Francisco: Jossey-Bass, 2005. Refer to the results in your table. 1. What are the perfect numbers up to 50? 2. Is there an abundant number that is not an even number? 3. Are all deficient numbers odd numbers? 4. What is the next number greater than 50 for which the sum of its proper factors is 1? 5. The sum of the proper factors of 4 is 1 less than 4. List all the numbers through 50 for which the sum of the proper factors is 1 less than the number itself. 6. What do you think is the next number greater than 50 for which the sum of its proper factors is 1 less than the number itself?

PROJECT 2 A Perfect-Number Challenge 382 Name Date Time Starting FactorsSum of Perfect Number Factors Number 2 1, 2 3 6 4 8 Perfect numbers become big very quickly. The third perfect number has 3 digits, the fourth has 4 digits, the fifth has 8 digits, the sixth has 10 digits, and the thirty-second has 455,663 digits! In other words, perfect numbers are hard to find. You can find perfect numbers without having to find the sum of the proper factors of every number. Here is what you do: 1. Complete the pattern of starting numbers in the first column of the table. 2. List the factors of each starting number in the second column. 3. Write the sum of the factors of each starting number in the third column. 4. If the sum of the factors of the starting number is prime, multiply this sum by the starting number itself. The product is a perfect number. Record it in the last column. The first perfect number is 6. Try to find the next three perfect numbers. People have been fascinated by perfect numbers for centuries. The ancient Greeks knew the first four. No one found the fifth perfect number until the year 1456. Computers now carry on the search for perfect numbers. When this book went to press, 42 perfect numbers had been identified. All the perfect numbers found so far are even numbers.

PROJECT 3 Name Date Time An Ancient Multiplication Method 383 Thousands of years ago, the Egyptians developed one of the earliest multiplication methods. Their method uses an idea from number theory. Every positive whole number can be expressed as a sum of powers of 2. Write a number sentence to show each of the numbers below as the sum of powers of 2. For example, 13 1 4 8. 1. 19  2. 67  Follow the steps below to use the Egyptian method to multiply 19 º 62. Step 1List the powers of 2 that are less than the first factor, 19. Step 2List the products of the powers of 2 and the second factor, 62. Notice that each product is double the product before it. Step 3Put a check mark next to the powers of 2 whose sum is the first factor, 19. Step 4Cross out the remaining rows. Step 5Add the partial products that are not crossed out. 62 124 992 1,178 So 19 º 62 1,178 3. Explain why you don’t have to multiply by any number other than 2 to write the list of partial products when you use the Egyptian method. 20 21 22 23 24 25 26 1248163264 19 º 62 = 162 2 124 4 248 8 496 16 992 ✓ ✓ ✓ 19 º 62 = 1,178 162 2 124 4 248 8 496 16 992

PROJECT 3 An Ancient Multiplication Method cont. 384 Name Date Time 4. Try to solve these problems using the Egyptian method. 5. Here is another ancient multiplication method, based on the Egyptian method. People living in rural areas of Russia, Ethiopia, and the Near East still use this method. See whether you can figure out how it works. Then try to complete the problem in the third box, using this method. 85 º 14 38 º 43 45 º 29  13 º 25 38 º 43 45 º 29  13 25 38 43 45 29 650 19 86 2258 3 100 9 172 11 116 1 200 4 344 5 232 325 2 688 2 464 11 ,376 1 928 1,634 1,634 325 Try This

PROJECT 3 Name Date Time Comparing Multiplication Algorithms 385 Think about the advantages and disadvantages of each multiplication method that you know. Record your thoughts in the chart below. Algorithm Advantages Disadvantages Partial Products Lattice Egyptian 43 º 62 ✓162 ✓2 124 4 248 ✓8 496 16 992 ✓32 1,984 2,666 6 43 2 4 2 66 1 8 0 80 6 6 2 1 43 º62 60 [40s] 2,400 60 [3s] 180 2 [40s] 80 2 [3s] 6 2,666

PROJECT 3 Ancient Math Symbols 386 Name Date Time 1. The ancient Egyptians used picture symbols, called hieroglyphs, to write numbers. Here is how they might have multiplied 11 º 13 using the algorithm you learned in this project. On the back of this sheet, try to multiply 21 º 16 using the Egyptian algorithm and Egyptian numerals. 2. Do you know any Roman numerals? They were used in Europe for centuries until Hindu-Arabic numerals replaced them. Today, Roman numerals appear mainly in dates on cornerstones and in copyright notices. It is sometimes said that “multiplication with Roman numerals was impossible.” Is that true? See whether you can multiply 12 º 15 using Roman numerals and the Egyptian algorithm. Use the back of this sheet. Examples of Roman Numerals: I1II2III3 IV4V5VI6 IX9X10XX20 XL40L50LX60 C100D500M1,000 (1 º 13) (2 º 13) (4 º 13) (8 º 13) = 1 = 10 = 100 = 1,000 = 10,000 = 100,000 = 1,000,000(11 º 13) ✓ ✓ ✓

PROJECT 4 Name Date Time Computation Trick #1—Super Speedy Addition 387 Set the Stage:Tell a friend that you have become a whiz at addition. To prove it, you are going to add five 3-digit numbers in your head within seconds. Props Needed:calculator Performing the Trick:Examples: Trial 1 Trial 2 Trial 3 1. Ask your friend to jot down a 3-digit number on a piece of paper. Each digit must be different. 493 261 682 2. Ask your friend to write another 3-digit number below the first number. Each digit must be different. 764 503 149 3. One more time. (This is the “notice-me number.”) 175 935 306 4. Now it is your turn. Write a number so that the sum of your number and the first number is 999. (For example, in Trial 1, 493 506 999.) 506 738 317 5. Write another number so that the sum of this number and the second number is 999. (For example, in Trial 1, 764 235 999.)235496850 6. Pause a few seconds, and then give the sum of the five numbers. Have your friend check your super speedy addition on a calculator. 2,173 2,933 2,304 Figure out how to do this trick. How does it work?

PROJECT 4 Computation Trick #2—Subtraction Surprise 388 Name Date Time Set the Stage:Tell a friend that your subtraction skills have soared. You are now able to give the answer to a subtraction problem without ever seeing the problem. Props Needed:calculator Performing the Trick:Examples: Trial 1 Trial 2 1. Ask your friend to secretly write a 3-digit number on a piece of paper. Each digit must be different. 135 562 2. Tell your friend to reverse the digits and write the new number below the first number. 531 265 3. Now have your friend use a calculator 531 562 to subtract the smaller number from 135265 the larger number. 396 297 4. Say: Tell me either the digit in the hundreds 3 in the 7 in the place or the digit in the ones place.hundreds place ones place 5. Pause a few seconds, and then give the answer. 396 297 Figure out how to do this trick. How does it work?

PROJECT 4 Name Date Time Computation Trick #3—Crazy Calendar Addition 389 Set the Stage:Tell a friend that you have become so good at addition that you can tell what an addition problem is by merely looking at the answer. Props Needed:calculator and a calendar Performing the Trick:Examples: Give your friend a calendar. 1. Ask your friend to choose a month and to secretly circle any three dates that are next to each other, either in a row or in a column. Trial 1 Trial 2 Trial 3 2. Now ask your friend to add the three dates on a calculator and to give you the calculator showing the sum in the display. 30 27 39 3. Ask: Are the three dates you circled in a row or in a column?column row column 4. Pause a few seconds, and then give the answer. 3, 10, 17 8, 9, 10 6, 13, 20 Figure out how to do this trick. How does it work? Sun. Mon. Tue. Wed. Thu. Fri. Sat. 12345 6789101112 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

PROJECT 4 12-Month Calendar 390 Name Date Time JANUARY 2 1 34567 SMTWTFS 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 MARCH 3 2 14 SMTWTFS 7 8 9 10 11 56 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 FEBRUARY 2 134 SMTWTFS 567891011 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 JUNE 3 2 1 SMTWTFS 45678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 MAY SMTWTFS JULY SMTWTFS 2345671 8 9101112131415 16 17 18 19 20 21 22 23 24 25 26 27 28 29 SEPTEMBER 12 SMTWTFS 3456789 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 AUGUST SMTWTFS DECEMBER 12 SMTWTFS 3456789 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 NOVEMBER SMTWTFS APRIL SMTWTFS 2345671 8 9101112131415 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 30 311 7 8 9 10 11 1223456 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 OCTOBER SMTWTFS12345 6789101112 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 31 2 1 34567 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 312 134 567891011 12 13 14 15 16 17 18 19 20 21 22 2923 24 25 26 27 28 30

PROJECT 4 Name Date Time Computation Trick #1—Super Speedy Addition 391 Why Does It Work? All you need to do to solve this addition problem is to look at the notice-me number. Here is why: Remember that you created two pairs of numbers — each with a sum of 999. These two pairs of numbers add up to 1,998 (999 999 1,998). This is 2 short of 2,000. The remaining number is the notice-me number. If you subtract 2 from the notice- me number and add the result to 2,000, you will always get the answer! The total will always be:Example:493 764 (notice-me number 2)175(175 2) 2,000 2,000 506 235 2,173 If you want to do more: Here are some variations you might want to try. You might use 7 or 9 numbers instead of 5. The trick is done in exactly the same way. However, think about how your formula would change if you did this. You might also try this with 6-digit numbers. Once again, the procedure is the same, but the formula would change. Record your findings below.

PROJECT 4 Computation Trick #2—Subtraction Surprise 392 Name Date Time Why Does It Work? The trick depends on the way in which you had your classmate create the subtraction problem. There are only 9 possible solutions to a subtraction problem created in that way: 99 198 297 396 495 594 693 792 891 You might have noticed that the digit in the tens place is always 9 and that the digits in the hundreds place and the ones place always add up to 9. For example, if your classmate tells you that the digit in the hundreds place is 4, then you know that the digit in the ones place must be 5, since 4 5 9. You know that the digit in the tens place is always 9. Therefore the answer is 495. What is the answer if your classmate tells you that the digit in the ones place is 9? If you want to do more: Will this trick work with a 4-digit number? With a 5-digit number? Describe your findings.

PROJECT 4 Name Date Time Computation Trick #3—Crazy Calendar Addition 393 Why Does It Work? Example: If three numbers are evenly spaced, you can find the middle number by dividing the sum of the numbers by 3. The numbers in a row and the numbers in a column of a calendar are evenly spaced. The numbers in a row are consecutive whole numbers. They are 1 apart. The numbers in a column are 7 apart. This is because there are 7 days in a week. After you find the middle number by dividing the sum of the numbers by 3, it is easy to find the other two numbers. If the three numbers are in a row, subtract 1 from the middle number to get the first number. Add 1 to the middle number to get the third number. If the three numbers are in a column, subtract 7 from the middle number to get the first number. Add 7 to the middle number to get the third number. If you want to do more: What would happen if the three dates chosen were on a diagonal? Would the trick still work? Why or why not? Sun. Mon. Tue. Wed. Thu. Fri. Sat. 12345 6789101112 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

PROJECT 5 How Would You Spend $1,000,000?—Emily’s Idea 394 Name Date Time Emily decided that if she had $1,000,000 she would spend it on a fabulous 10-day trip to Florida for her, 19 of her friends, and 4 chaperones—24 people altogether. With $1,000,000, she knew that she could make this a trip no one would ever forget. Emily began by thinking about everything she and her friends might need for their trip. She visited a local department store to find out how much different items cost. She decided to purchase a vacation wardrobe for everyone, including the chaperones, at a cost of $50,750. Her next stop was a sporting goods store for items such as snorkel gear, swimsuits, and sunglasses. The store clerk calculated that all of her purchases there would cost $24,100. Emily knew that she would need transportationto Florida and for traveling around while there. She made a few telephone calls to find out the prices for transportation. Emily found that when she politely explained her project to people, most of them were willing to help her. After doing some research, she chartered an airplane for a flight from Chicago to Orlando and back ($54,780). She purchased two stretch limos to use in Florida ($165,160 + $10,000 for gas and two around-the-clock chauffeurs). She also purchased a minivan to carry the chaperones and the luggage ($20,700) while in Florida. Lodgingwas another consideration. Emily decided that her group would stay at one of the resorts inside a theme park ($33,550). She went to a travel agency to get some information about the activitiesthat she and her friends might try while they were there. For $177,200, Emily made reservations for several special breakfasts as well as dinner shows, rented a water park for 12 hours, and purchased 10-day passes to the theme park where she and her group were staying.

PROJECT 5 Name Date Time How Would You Spend $1,000,000? cont. 395 Emily decided to keep a record of the money she was spending by listing her purchases in major categories. At the right is part of the chart that she made. Emily also decided that for each category she would keep a detailed record so that she would know exactly how she was spending the $1,000,000. Below is an example of her record for one category. These are examples of just a few of the expenses for Emily’s amazing trip. About how much money has Emily spent so far? About how much money does Emily have left to spend? Major Category—Vacation Wa rdrobe Item Quantity Unit Price Total Price Boxer shorts 100 $ 12.50 $ 1,250.00 Socks 200 $ 5.50 $ 1,100.00 Shorts 240 $ 38.00 $ 9,120.00 T-shirts 200 $ 32.00 $ 6,400.00 S wimsuits 100 $ 36.00 $ 3,600.00 Jean s 100 $ 34.00 $ 3,400.00 Shoes 60 $ 50.00 $ 3,000.00 Flip -fl ops 6 0 $ 2 4.00 $ 1,440.00 Sunglasses 20 $ 29.50 $ 590.00 Long -sleeve shirts 40 $ 57.00 $ 2,280.00 Tax $ 2,570.00 Chaperones’ $4,000.00 $ 16,000.00 Wa rdrobe A llotmentper person Total $50,750.00 Majo r Ca te g o r y Co s t Vacation Wa rdrobe $ 50, 750 Sports Equipment $ 24, 100 Tran sportation $250, 640 Lodging $ 33, 550 A ctivities $ 177,200

PROJECT 5 How Would You Spend $1,000,000? cont. 396 Name Date Time Project Guidelines Imagine that you have just inherited $1,000,000. One of the conditions for receiving the money is that you must investigate, research, and present exactly how you will spend it. You must follow the guidelines below. ThemeThe $1,000,000 must be spent carrying out one particular plan. For example:A plan that would help save the rain forest; a plan to build new parks and playgrounds in your city; a plan for a trip around the world; or a plan to open a ballet studio. GoalSpend as close to $1,000,000 as possible, but not more than $1,000,000. ResearchInclude all of the expenses involved in carrying out the details of your plan. For example:If you are opening a ballet studio, you must consider how many teachers you will need and how much you will pay them. If you are buying a car, you will need to consider the cost of gas, maintenance, and insurance for the length of time you will own the car. AccountingIn an organized way, record exactly how you will spend the $1,000,000. The purchases needed to carry out your plan should be organized in several major categories. Each major category must total at least $10,000. For example:Think about how Emily organized the purchases for her Florida vacation, as described on Math Masters,pages 394 and 395. DisplayPresent the research and accounting for your plan in a report, on a display board, on a poster, or in a portfolio. You might even make a video production. For example:Emily presented her project as a report. In addition to her calculations, she included pictures and sample receipts whenever possible.

PROJECT 5 Name Date Time How Would You Spend $1,000,000?—Totals 397 Total $1,000,000 Accounting Sheet Totals of Major Categories Major Category Cost

PROJECT 5 How Would You Spend $1,000,000?—Itemized 398 Name Date Time Total Price $ Accounting Sheet A Major Category Itemized Category: Item Quantity Unit Price Total Price

PROJECT 5 Name Date Time How Would You Spend $1,000,000?— Categories 399 In the table below, list all of your major expense categories and the total amount for each. (Refer to your accounting sheets—Math Masters,pages 397 and 398.) Write each amount as a fraction, decimal, and percent of $1,000,000. Round each decimal to the nearest hundredth. Round each percent to the nearest whole percent. Category Total $ Spent Fraction Decimal Percent 1,000 0,000   1,000 0,000   1,000 0,000   1,000 0,000   1,000 0,000   1,000 0,000   1,000 0,000   1,000 0,000   1,000 0,000   1,000 0,000   1,000 0,000   1,000 0,000   1,000 0,000 

PROJECT 5 How Would You Spend $1,000,000?— Graph 400 Name Date Time Make a circle graph of your categories for spending $1,000,000. Use your Percent Circle and the information on Math Masters,page 399. Begin by drawing the section for the smallest part of the $1,000,000. Continue with the larger parts. Mark the largest part last. Because of rounding, the percents may not add up to exactly 100%. Give the graph a title and label each section.

PROJECT 6 Name Date Time Playing Areas for Five Contact Sports 401 Use your calculator to find each playing area. Scale: 1 mm (drawing) represents 1 ft (actual) *Calculate with decimals. For example, 29 ft 6 in. is equal to 29.5 ft. Source: COMPARISONSby the Diagram Group. Reprinted by permission of St. Martin’s Press. Sport Dimensions Playing Area Boxing 20 ft by 20 ft ft 2 Karate 26 ft by 26 ft ft 2 Aikido 29 ft 6 in. by 29 ft 6 in.* ft 2 Wrestling 39 ft 3 in. by 39 ft 3 in.* ft 2 Judo 52 ft 6 in. by 52 ft 6 in.* ft 2 Judo Wrestling Aikido Karate Boxing

Sport Dimensions Playing AreaMore or Less than 1 Acre? Tennis (doubles) 78 ft by 36 ft ft 2 more less Basketball 94 ft by 50 ft ft 2 more less Water Polo 98 ft by 65 ft ft 2 more less Swimming 165 ft by 69 ft ft 2 more less Ice Hockey 200 ft by 85 ft ft 2 more less Ice Skating 200 ft by 100 ft ft 2 more less Football (U.S.) 300 ft by 160 ft* ft 2 more less Field Hockey 300 ft by 180 ft ft 2 more less Soccer 360 ft by 240 ft ft 2 more less Rugby 472 ft by 226 ft ft 2 more less PROJECT 6 Playing Areas for Other Sports 402 Name Date Time Use your calculator to find each playing area. Circle moreor lessto tell whether each area is more or less than 1 acre. 1 acre 43,560 square feet Rugby Soccer Field Hockey Football (U.S.) Ice Skating Ice Hockey Swimming Water Polo Basketball Tennis (doubles) Scale:1 mm (drawing) represents 1 yd or 3 ft (actual) *Not including end zones Source: COMPARISONSby the Diagram Group. Reprinted by permission of St. Martin’s Press.

PROJECT 6 Name Date Time Ground Areas of Famous Large Buildings 403 The ground areas of buildings, their footprints, are almost always given in square feet or square meters. Some buildings have very large ground areas. When their areas are given in square feet, the numbers are so large that it is hard to imagine how big the buildings really are. For large buildings, if you convert the area in square feet to an estimate in acres, you can get a better idea of the size of the building. Estimate the ground area, in acres, of each building in the table below: Example:The Colosseum, in Italy, covers an area of about 250,000 ft 2. One acre is about 50,000 ft 2. So 5 acres is about 250,000 ft 2. The Colosseum covers an area of about 5 acres (5 football fields). Building Country Date BuiltGround Estimated Area Area (ft 2)(in acres) Colosseum Italy 70–224 250,000 ft 2 acres Egypt c. 2600 B.C. 571,500 ft 2 acres France 1194–1514 60,000 ft 2 acres Vatican City 1506–1626 392,300 ft 2 acres Taj Mahal India 1636–1653 78,000 ft 2 acres Pentagon 1941–1943 1,263,000 ft 2 acres 1936 2,800,000 ft 2 acres 5 Pyramid of Cheops Chartres Cathedral St. Peter’s Basilica Ford Parts CenterU.S. (Michigan)U.S. (Virginia)Reference 1 acre 43,560 square feet For estimating, think of 1 acre as about 50,000 square feet. A football field (excluding the end zones) is approximately 1 acre.

PROJECT 7 Finding Areas with Standard Methods 404 Name Date Time Use any method you want to find the area of each polygon below. Record the area in the table to the right. You can use different methods with different figures. If you use any area formulas, remember that height is always measured perpendicular to the base you choose. Measure base and height very carefully. Figure Area A about cm 2 B about cm 2 C about cm 2 D about cm 2 E about cm 2 F about cm 2 1 cm 2 A B C D EF

Read the paragraphs below, and then use Pick’s Formula to find the areas of the polygons on the previous page. Record your results in the table below. Compare them to the results you recorded in the table on the previous page. You should expect some differences—measures are always estimates. Pick’s Formula for Finding Polygon Areas by Counting In 1899, Georg Pick, an Austrian mathematician, discovered a formula for finding the area of a polygon on a square grid (such as graph paper). If a polygon has its vertices at grid points, its area can be found by counting the number of grid points on the polygon (P) and the number of grid points in the interior of the polygon (I) and then by using the formula A( 1 2º P) I1. The unit of area is one square on the grid. For figure B on the previous page, the unit of area is cm 2. P4 (grid points on polygon) I12 (grid points in interior) A( 1 2º P) I1 ( 1 2º 4) 12 1 13 cm 2 Draw two polygons. Be sure that the vertices are at grid points. Use Pick’s Formula to find the areas of the polygons. PROJECT 7 Name Date Time Finding Areas with Pick’s Formula 405 FigurePIArea( 1 2ºP)I1 A cm 2 B cm 2 C cm 2 D cm 2 E cm 2 F cm 2 1 cm 2 Area: Area:

PROJECT 7 Finding Areas with Pick’s Formula cont. 406 Name Date Time You might have found the area of this shaded path in Lesson 9-6. Now use Pick’s Formula to find the area. 1. The area of the path is cm 2. 2. Do you think Pick’s Formula is a good way to find this area? Explain. 1 cm 2

PROJECT 8 Name Date Time The Swing Time of Pendulums 407 1. Your teacher will demonstrate an experiment with a pendulum that is 50 cm long. Record the results below. a. It took about seconds for 10 complete swings of the pendulum. b. About how much time did it take for one swing? Round your answer to the nearest 0.1 second. second(s) 2. Form a pendulum that is 75 cm long. Time 10 complete swings of the pendulum. Time the swings to the nearest second. Practice timing 10 complete swings several times. Then time 10 swings and record the results below. a. It took about seconds for 10 complete swings of the pendulum. b. About how much time did it take for one swing? Round your answer to the nearest 0.1 second. second(s) 3. Record the results for a 50-cm and a 75-cm pendulum in the table at the right. 4. Experiment with different lengths of pendulum string. Find the time for 10 complete swings for each of the other pendulum lengths. Time the 10 swings to the nearest 0.1 second. Record your results in the table. After collecting your data, divide each of the times by 10 to estimate the time for one complete swing. Record your answers in the table, rounded to the nearest 0.1 second. heavy object 5 cm sec sec 10 cm sec sec 20 cm sec sec 30 cm sec sec 50 cm sec sec 75 cm sec sec 100 cm sec sec 200 cm sec sec Length of PendulumTime for: One Complete Swing (to nearest 0.1 sec) Ten Complete Swings (to nearest 0.1 sec)

PROJECT 8 The Swing Time of Pendulums cont. 408 Name Date Time Wait for instructions from your teacher before drawing the graph in Problem 5. 5. Construct a graph to show the amount of time it took for each length of the pendulum to complete one swing. Time for One Swing (sec) Length of String (cm) 020 40 60 80 100 120 140 160 180 200 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6

PROJECT 8 Name Date Time The Swing Time of Pendulums cont. 409 6. Experiment with different arc sizes. The largest arc is formed when the string of the pendulum is in a horizontal position. Does the size of the arc make much difference in the amount of time it takes for 10 complete swings? 7. Does the weight of the object at the end of a pendulum affect the time for a complete swing? Using a pendulum with a string 50 cm long, try different numbers of objects to find out if weight makes a difference in the time of the swing. My conclusion: It seems that at rest largerarc at rest smallerarc Length Number of Time for Time for One Swing of PendulumWeights (washers 10 Swings (to (to nearest 0.1 or other objects) nearest 0.1 sec) or 0.01 sec) 50 cm 1 sec sec 50 cm 3 sec sec 50 cm 5 sec sec 50 cm 10 sec sec