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## Chapter 5

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**Chapter 5**Look for a pattern Patterns are everywhere, but not everyone can see them. In this chapter you will learn where to find them and how to use them.**A pattern**is the predictable repetition of certain things, such as shapes, numbers, sound, phenomenon, or behavior etc. In mathematics, we are looking for formulas that can predict the repetitions in patterns.**Simple Patterns in Arithmetic**Do you notice that even # × even # = even # odd # × odd # = odd # even # × odd # = even #**The remainder when a number is divided by 5**11 ÷ 5 = 2 r 1 21 ÷ 5 = 4 r 1 31 ÷ 5 = 6 r 1 12 ÷ 5 = 2 r 2 22 ÷ 5 = 4 r 2 32 ÷ 5 = 6 r 2 13 ÷ 5 = 2 r 3 23 ÷ 5 = 4 r 3 33 ÷ 5 = 6 r 3 14 ÷ 5 = 2 r 4 24 ÷ 5 = 4 r 4 34 ÷ 5 = 6 r 4 16 ÷ 5 = 3 r 1 26 ÷ 5 = 5 r 1 36 ÷ 5 = 7 r 1 17 ÷ 5 = 3 r 2 27 ÷ 5 = 5 r 2 37 ÷ 5 = 7 r 2 18 ÷ 5 = 3 r 3 28 ÷ 5 = 5 r 3 38 ÷ 5 = 7 r 3 19 ÷ 5 = 3 r 4 29 ÷ 5 = 5 r 4 39 ÷ 5 = 7 r 4 Patterns: (4) If a whole number ends in a 4 or 9, then it leaves a remainder of 4 when divided by 5.**Pattern usually means something that repeats. It may be a**design, a behavior, a number, or a formula and so on. Do you notice that red light is always on top of green?**Traffic lights in Key West, Florida.**Remember: Red is never right!**Can you see in the following wallpaper that the same pattern**is repeated many times to generate a large picture?**Look for a pattern.**Fill in the hole with appropriate tiles so that the pattern is preserved.**Look for a pattern.**Here is the answer.**Find a minimum group of tiles that can generate the**following pattern.**IQ Test**1. Study the diagram below and decide what logically should be the missing section from the choices given.**2. Look for a pattern and find the next figure.**Choose from**Look for a pattern**and guess the next 3 terms in the sequence 1, 2, 4, 8, 16, 4, 7, 10, 13, 16, 1, 4, 9, 16, 25, 3, 4, 6, 9, 13, 18, 5, 10, 9, 18, 16, 32, 29, 77, 49, 36, 18, ____ 1, 1, 2, 3, 5, 8, 13, 3, 4, 6, 10, 17, 25, 35, 48,**The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, …**1 pair 1 pair 2 pairs 3 pairs 5 pairs Starting with a male-female pair of baby bunnies, they become matured and productive in one month. By the end of the second month, they can produce another male-female pair of baby bunnies.**Fibonacci was the professional name of an Italian Leonardo**Pisano. Born in 1170, Pisa, Italy. He published the book Liber Abacithat talks about the Fibonacci sequence in 1202. Died in 1250, Pisa, Italy.**Fibonacci sequence in nature.**1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … Sunflower always have 21 counterclockwise spirals and 34 clockwise spirals.**One set of 5 spirals ascends at a shallow angle to the**right, ... a second set of 8 spirals ascends more steeply to the left, ... and the third set of 13 spirals ascends very steeply to the right.**2**3 5 8**Formula for the Fibonacci sequence**for n = 0, 1, 2, 3, 4, … (First discovered by Abraham de Moivre in around 1700.)**“Petals around the Rose” puzzle**Five dice are used to encode a whole number.a) The number is always even.b) The number is between 0 and 20.c) The name of the game is important.d) The order of the dice is not important.For example, in the following roll, the answer is 2. Your mission is to find a way to decode.**Petals around the Rose**Dr. Richard Duke at the University of Michigan used to begin each of his gaming/simulation courses with this exercise. While some students would solve the problem right away, others would struggle all semester. It had taken Dr. Duke well over a year himself, and he would always explain that the smarter you were, the longer it took to figure it out. play this game on computer**Answer: 2**Answer: 4 Answer: 8 Answer: 2 Answer: 14 Answer: 10 Answer: 0 Answer: 6 Answer: 12 Answer: 4**What is the last digit of 758 ?**71 = 7 72 = 49 73 = 343 74 = 2401 75 = 16807 76 = 117649 77 = 823543 78 = 5764801**Change a repeating decimal to a fraction**1/9 = 0.1111… 2/9 = 0.2222… 3/9 = 0.3333… See a pattern? What is 0.555… equal to? 13/99 = 0.131313… 25/99 = 0.252525… 83/99 = 0.838383… What is 0.646464… equal to? Answer: 64/99 Answer: 5/9 More patterns 0.325325325… = 325/999 0.718718718… = ? Answer: 718/999 0.463946394639… = ? Answer: 4639/9999**Change a repeating decimal to a fraction**More patterns 0.027272727… = 27/990 0.0038383838… = 38/9900 0.000545454… = ? Answer: 54/99000 Here is the hardest part, 0.2777777… = 25/90 0.5888888… = 53/90 0.345454545… = 342/990 0.401010101… = 397/990 What is 0.834343434… = ?**Look for a pattern and predict the answer.**512 = 2601 522 = 2704 532 = 2809 542 = 2916 552 = ? 562 = ? 572 = ? 582 = ? 592 = ?**Look for a pattern and predict the answer.**352 = 1225 152 = 225 552 = 3025 252 = 625 452 = 2025 652 = ____ 752 = ____ 852 = ____ 952 = ____ 42 × 48 = 2016 36 × 34 = 1224 59 × 51 = 3009 78 × 72 = 5616 27 × 23 = ____ 54 × 56 = ____ 43 × 47 = ____ 61 × 69 = ____ 621 4225 3024 2021 5625 7225 4209 9025**More patterns in squares**Find a pattern in these squares so that you can easily remember them. 172 = 289 182 = 324 192 = 361 How about these? 272 = 729 282 = 784 292 = 841**Milk Lovers**Alysia and Melissa and Dante and Melody loved milk. They convinced their older brother, Mark, who did all the shopping, to buy each of them their own gallon of milk because they each liked it so much. They all put their names on their gallons.**One day, they were really thirsty and each took 10 drinks**according to a different pattern: Alysia started by drinking ½ of the milk in the container. Then she drank ⅓ of what was left. Then she drank ¼ of what was left, then 1/5 and so on. Melissa started by drinking 1/11 of her milk, then 1/10 of what was left, then 1/9 of what was left, and so on. ……. After each had taken 10 drinks, how much milk remained in each container?**1. SEQUENCE PATTERNS**Write the next three numbers in each sequence and briefly indicate the pattern. a. 2, 5, 10, 17, . . . b. 64, 32, 16, 8, 4, ... c. 5, 10, 9, 18, 17, 34, 33, . . . d. 1, 3, 7, 13, 21, ... e. 2, 3, 5, 9, 17, ... f. 1, 5, 13, 26, 45, 71, ... g. 1, 2, 6, 24, 120, 720, . . .**2. AIR SHOW**In order to keep the spectators out of the line of flight, the Air Force arranged the seats for an air show in a "V" shape. Kevin, who loves airplanes, arrived very early and was given the front seat. There were three seats in the second row, and those were filled very quickly. The third row had five seats, which were given to the next five people who came. The following row had seven seats; in fact, this pattern continued all the way back, each row having two more seats than the previous row. The first twenty rows were filled. How many people attended the air show?**3. RECTANGULAR DOTS**Continue this pattern (rectangular dots). How many dots are there in the thirty-fourth figure?**5. LAST DIGIT**What is the last digit of 2 to the 57th power?**8. BEES**A male bee is born from an unfertilized egg, a female bee from a fertilized one. So, in other words, a male bee only has a mother, while a female bee has a mother and a father. How many ancestors does a male bee have in total, going back ten generations? (Try drawing a diagram to help organize this.)**9. PASCAL'S TRIANGLE**The triangle below is called Pascal's Triangle. Find a pattern that will produce the next row. Then copy the triangle and fill in the next four rows. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1**GOLF MATCH**Clark, Chris, Doug, and Diana are standing on the first tee of their favorite golf course about to begin a best-ball-of-partners match. (A best-ball match pits two golfers against the other two golfers.) They are standing in a square, with two partners standing next to each other on the cart path facing the other two partners standing next to each other on the grass. This standing arrangement is typical of the beginning of a golf match. They shake hands and then they throw a tee in the air and let it hit the ground. Whoever it points to will tee off first. Clark is standing diagonally opposite Diana. Chris is facing the person whose name begins with the same letter as the person who will tee off first. Partners tee off one after the other. Who will tee off second?**2. MACARONI AND CHEESE**Orie was too excited about his new toy to sit still. He came to the lunch table, ate 1/6 of his macaroni and cheese, then left to play with something, came back, ate 1/7 of what was left, ran and played, came back, ate 1/8 of what was left, and so on. After 10 such episodes of eating and then playing, how much of his original macaroni and cheese was left? (Assume that he started with one pound of macaroni and cheese.)**3. COMIC OF THE MONTH**I subscribe to the “Comic of the Month” club. Each month I can buy any number of the 48 titles offered by the club. The first month I bought five comics for $3.07. The second month I bought two comics for $1.72. The next month I bought six of the club offerings for $3.52. In May I bought three more for a charge of $2.17. The club charges a handling fee and then a fee for each comic. How much would it have cost to buy all 48 titles at the same time?**5. Roo and Tigger are having a 200 feet race. They have to**run 100 feet forward and then another 100 feet back. Roo can make 3 two-foot jumps in the same amount of time as Tigger makes 2 three-foot jumps. Who will win the race?