Combinatorics is all about

Transcription

1 SHOW 109 PROGRAM SYNOPSIS Segment 1 (1:21) COMBINATORICS: MORE THAN JUST A WORD In a parody of a TV commercial, Dweezil Zappa discovers the meaning of combinatorics when he determines how many possible outfits he can make from certain numbers of pants, shirts, and sweaters. Segment 2 (7:07) SUPERGUY: NEW CAPE CAPER, PARTS 1 AND 2 While desperate townspeople await his help, Superguy wonders how many outfits he can make from three belts and three capes. Using combinatorics, he develops a systematic way of counting and combining results. Segment 3 (1:42) SQUARE ONE CHALLENGE: SHIRTS AND SHORTS A combinatorics question is asked in this excerpt from the game show: If you have six t-shirts, how many shorts do you need to make at least 20 different combinations of t-shirts and shorts? Buster and Maria work through another combinatorics problem, using a chart to illustrate the steps. HALF-AND-HALF PIZZAS ONE HALF: ONE HALF: mushrooms green peppers anchovies onions pepperoni plain plain Let Me Count the Ways: Counting with Combinatorics INTRODUCTION Combinatorics is all about analyzing situations to plan efficient ways of counting things. Instead of counting things one by one, combinatorics gives systematic shortcuts, but you have to think them through carefully. And then you have to communicate your reasoning so that someone else can see what you ve done. This tape introduces one of the basic principles of counting: If, for example, you have three different capes and three different belts, then there are 3x3, or 9, ways to pair a cape with a belt, because with each of the three capes you can wear any of the three belts. If there were four capes, then there would be 4x3, or 12, ways to pair a cape with a belt. This idea can be extended to more than just two kinds of objects, as the first segment in the tape illustrates. Even simple combinatorial situations often involve large numbers, so they provide a good opportunity to develop and sharpen students sense of the comparative sizes of numbers. Note: Combinatorics provides an ideal context in which to engage in the kinds of reasoning and communication that are emphasized in the NCTM Standards. BEFORE VIEWING Start the discussion with a story about the Half-and-Half Pizza Parlor, which specializes in half-and-half pizzas. On one half of a pizza you can get mushrooms, anchovies, or pepperoni, and on the other half you can get green peppers or onions. (If you want to make this easier, drop the option of having either half or both halves plain.) How many different choices of pizza does the store offer? Begin a list of choices on the board. Then return to this problem after students have watched the tape. 57

2 MathTalk DURING AFTER VIEWING SUPERGUY STOP THE TAPE right after Superguy and the salesman start to put capes and belts on the display rack. Ask students how many different cape-and-belt combinations they think the characters will be able to make, and why they think so. (Some students become so engrossed in the story line here that they lose sight of the math. You can assure them that Superguy will save the town, even though he seems to be more concerned with his wardrobe.) VIEWING What if you could order a pizza in small, medium, or large? SQUARE ONE CHALLENGE STOP THE TAPE after the two cast members give their answers in the Square One Challenge game to give students time to think about and discuss their own ideas. Return to the pizza problem posed earlier. At the Half-and-Half Pizza Parlor, there were four choices for one half of the pizza (including plain), and three for the other. So there are a total of 4x3, or 12, possibilities. Then there would be 36 possibilities. Now we re up to 72! What if you could ask for extra cheese? activity COMBINATORICS STRATEGY: WIN THE JACKPOT Most states now have some kind of state lottery. When you buy a lottery ticket you choose some numbers according to the lottery rules. Lottery tickets are sold for a low price, and the prizes you can win are sometimes huge. But there s a catch: There are so many possible combinations of numbers that your chances of winning are close to zero. Using combinatorics, you can figure out how many possibilities there really are. The activity, which can be done individually or in small groups, shows that even with a small number of numbers to choose for a lottery, the number of possible choices is enormous. The rules are that you pay $1 to play the game. Then you must select one number in each row. If you choose all the correct numbers (ones that are drawn randomly by the state lottery commission), then you win a $50 prize. MATERIALS copies of reproducible page 62 WIN-BIG LOTTERY #

3 NUMBER SENSE 1. Pass out copies of the reproducible page, and explain that it describes three different lottery games. Each game has different rules. To find out your chances of winning each game, you must find the number of ways there are to choose the numbers. 2. Discuss the first lottery game as a class. Give one or two examples of numbers a player might choose for instance, 21, 32, 43, and 54. Those numbers would be printed on the lottery ticket. Finding the total number of ways to select the four numbers, one from each row, is a straightforward application of combinatorics. Ask: How many numbers are there to choose from in the first row? (10) And how many in the second row? (10) So how many ways are there to choose the first two numbers? (10x10, or 100. That is, for each of the 10 ways to choose the first number, there are 10 ways to choose the second, so there are 100 ways in all.) Once you have the first two numbers chosen, how many ways are there to choose a number in the third row? (10 again) So now there are 10x10x10, or 1000, ways to choose the first three numbers. Choosing the fourth number gives 10x10x10x10 ways in all. That s 10,000 ways! What is the probability of winning? (1/10,000) Does $50 seem like a reasonable prize? (no) Why not? (The probability of winning is only 1/10,000. You can think of it like this: If you bought one ticket with each of the 10,000 possible combinations, you d spend $10,000. To be fair, the prize should be what you spent $10,000.) 3. Go on to the second game. You can let a student give an example of what a ticket would look like for this one something like 40, 31, 22, 23, 24, 35, 46, 27, 58, 39. Ask if there will be more or fewer possibilities or just as many with the second game than there were with the first. Some students might expect 10,000 possibilities for the second game card too, since the card looks the same as before. But here there are four ways to choose the first number, then four ways to choose the second, and so on. So there are a total of 4x4x4x4x4x4x4x4x4x4 ways. (If you re used to exponents, that can be written as 4 10.) Use a calculator! That s a total of 1,048,576 ways! (Note that the number of ways in the first lottery game card was 10 4.) 4. The third game card has three hexagons, one inside the other. Each hexagon has a different number of possible numbers to choose. The rule is that you have to pick one number from each hexagon, three numbers in all. How many possible three-number combinations are there with this game card? (6x12x18,or 1296) It s always a good strategy to simplify a tough problem. You could start with a simpler lottery game like this one If you choose one number from each row, there are 4x4x4, or 64 possibilities. If you choose one number from each column, there are 3x3x3x3, or 81 ways. 59

4 MathTalk keep thinking WIN-BIG LOTTERY # PLAYING THE ODDS Students who have completed the third problem on the reproducible page can be given an extra challenge: Suppose the rules were changed so that, at most, one of the three numbers you choose can be at a vertex of a hexagon. In other words, at least two of the numbers must be somewhere other than at a vertex. Then how many possible combinations of three numbers are there? (Note that every number in the smallest hexagon is a vertex number, so choosing any number from that hexagon uses up the allowance of vertex numbers. The other two numbers must be chosen from the sides (not on a vertex). There are six side numbers in the middle hexagon, and 12 in the largest. So there are a total of 6x6x12, or 432 possibilities for choosing three numbers.) Here s another challenge. Suppose the hexagons were drawn as shown below so that two of the numbers, 4 and 10, each appear on two hexagons. Now how many ways are there to choose three different numbers, one from each hexagon? (Hint: One way to approach this is to look at four separate ways of choosing numbers: Neither 4 nor 10 is chosen; 4 is chosen, but not 10; 10 is chosen, but not 4; both 4 and 10 are chosen.) As an example, take the first case: If neither 4 nor 10 are chosen, then there are five ways to choose a number from the first hexagon, 10 from the second, and 17 from the third, so there are 5x10x17, or 850 possible choices of three numbers that use neither 4 nor 10. Now go on to the other cases and add up the results. (There are 1262 possibilities.) 60

5 NUMBER SENSE PORTFOLIO ASSESSMENT Ask students to be on the lookout for situations in which combinatorics might play a role, to choose such a situation, and to write up an analysis of it. It might be the menu in the cafeteria or in a restaurant (how many different meals are there?), or their own state s lottery (how many ways of choosing numbers), or some card game, or the number of different ways the students in the class could line up for recess. If your class has seen the tape BOTH SIDES OF ZERO: PLAYING WITH POSITIVE AND NEGATIVE NUMBERS,a student might want to analyze one of the questions suggested there about the number of paths there are (or might be) in Pauline s pyramids of various sizes. CURRICULUM CONNECTIONS The counting principle that is the focus of this module is one of the basic ways in which multiplication of whole numbers occurs in the real world. Many situations involving multiplication can be recast as problems in combinatorics. For example, suppose there are seven buses and each bus holds 30 people. One would typically ask, How many people will fit? But you can also think of this problem in terms of combinatorics and ask, How many ways are there of assigning a person to a seat? For each of seven buses there are 30 choices, so there are 7x30, or 210, choices in all. Math Talk CONNECTIONS NUMBER SENSE Factor Em In: Exploring Factors and Multiples Both Sides of Zero: Playing with Positive and Negative Numbers Working with common multiples Combinatorial reasoning 61

6 NAME LET ME COUNT THE WAYS: COUNTING WITH COMBINATORICS MathTalk GAME #1 Rules: Pick four numbers, one number from each row. How many ways are there of doing this? WIN-BIG LOTTERY # GAME #2 Same numbers, different rules: This time, pick ten numbers one number from each column. How many ways are there to make the choices? WIN-BIG LOTTERY # GAME #3 Different game card, different rules: Pick three numbers one from each of the three different hexagons. How many possible three-number combinations are there? WIN-BIG LOTTERY # Which game do you have the greatest probability of winning? #1 #2 # Children s Television Workshop