SHOW 107 PROGRAM SYNOPSIS Segment 1 (5:59) DIRK NIBLICK: TOO MANY COOKOUTS, PARTS 1 AND 2 Dirk Niblick, fearless leader of the Math Brigade, helps his neighbor Mr. Beazley plan a barbecue. Together they work out how to save money on the party food by finding the smallest number of packages of hotdogs 12 to a package and buns eight to a package needed to feed his party guests. Segment 2 (4:24) COMMON MULTIPLE MAN When a couple has to figure out how many hors d oeuvres to buy to serve either 12, 16, or 24 guests equally, they call on Common Multiple Man, a superhero with a very strange (but useful!) super power. You know, 0 is a multiple of every number, because 0 times any number is 0. But when we talk about a least common multiple of two numbers, we mean the smallest positive multiple that the two numbers have in common. Factor Em In: Exploring Factors and Multiples INTRODUCTION Factors and multiples have traditionally been an important part of the elementary school mathematics curriculum, and for good reason: Whenever multiplication crops up, so do multiples. Students who know how to recognize and work with multiples have a head start on developing number sense. This module concentrates on finding multiples that two or more numbers have in common that is, common multiples. Multiples and factors appear in many parts of mathematics beyond arithmetic (see CURRICULUM CONNECTIONS, page 49). BEFORE VIEWING You can help your students become familiar with the idea of the factors of a number, the multiples of a number, and the concept of prime numbers (whole numbers that have exactly two factors) by having regular practice with the concepts. Try offering some informal challenges such as, Name a multiple of 4 that s bigger than 20. What could it be? (Any number such as 24, 28, 32, and so on that is a number multiplied by 4.) What s the smallest prime number greater than 20? (23) I m thinking of a number that is a multiple of 5. It s also 3 less than a multiple of 9. What could my number be? (15 is one possibility. 60 and 105 in fact, any number that is 15 more than a multiple of 45 will also work.) Stretch your students awareness by including impossible ones, too but be sure to alert them that there may be no solution for example, I m thinking of a number that is a multiple of 3 and is 1 less than a multiple of 12. (That s impossible, because every multiple of 12 is also a multiple of 3. If a number is 1 less than a multiple of 12, it can t be a multiple of 3.) 45
MathTalk AFTER VIEWING At the end of the last segment it seemed pretty clear that Common Multiple Man wanted to be invited to the party. Then instead of 12, 16, or 24 guests, how many guests would there have been? (13, 17, or 25) Explore how many hors d oeuvres would have been needed then. One way to approach finding a common multiple of 13, 17, and 25 would be to write lists of the multiples of 13, of 17, and of 25, looking for numbers that appear on all three lists. Note: 13, 26, 39, 52,... A calculator is helpful here. 17, 34, 51, 68,... Press 1 3 = = = =... 25, 50, 75, 100,... to get successive multiples of 13. Just looking at these numbers, it isn t immediately clear that there will be any number that appears on all three lists. Some students may not have fully internalized the reasoning here, and they will need to continue the lists for a while to get a feeling for what is happening. Ask students to work in small groups to continue the lists. Can they think of any shortcuts? Some may suggest multiplying the three factors (getting 5525). That s a lot of hors d oeuvres! How do you know whether 5525 is the lowest common multiple or not? You can simplify the problem by first trying to find a number that is a multiple of 13 and of 17. Is 221 a multiple of 25? (No. Keep multiplying to find a multiple of both 221 and 25.) Be on the lookout for shortcuts. It turns out that, by adding one more guest, the number of hors d oeuvres needed has jumped from 48 (which was the least common multiple of 12, 16, and 24) all the way up to 5525. (Maybe that s why the host and hostess didn t want to invite Common Multiple Man!) 221 is the first one! 46
NUMBER SENSE activity GAME: MULTIPLE MANIA This game helps players become more familiar with the idea of multiples and common multiples of numbers in a context in which strategy is important. 1. Pass out a number cube and a copy of the reproducible page to each pair of students. Give each student 10 chips or other small objects of the same color, with each partner in a pair having a different color. 2. Review the rules of the game on the reproducible page with the class. You may want to play one game as a demonstration: Two students can take turns rolling the number cube, while the whole class discusses moves that could be made. 3. Now let pairs of students play the game. 4. Encourage your students to take the game board home and play the game with their parents or other family members. SAMPLE GAME: Number Rolled MATERIALS for each pair of students: one number cube (numbered 1 6) copies of reproducible page 50 20 chips 10 each of two different colors Number Covered on Game Board Red rolls 5 15 (15 already contains an even number [0] of Blue s chips, so it can be covered. 20 is another possible play for Red.) Blue rolls 4 12 Red rolls 4 16 (12 is blocked by Blue s single chip.) Blue rolls 1 13 (Rolling 1 is the only way to cover 13 or any other prime number.) Red rolls 2 18 Blue rolls 6 12 (18 is blocked by Red, so this is the only play. It unblocks 12, because 12 now has an even number of chips. Until Blue gets a third chip on 12, Red can capture both of Blue s chips by rolling 1, 2, 3, 4, or 6.) Red rolls 3 12 (Red captures Blue s two chips on 12, removing them from play. Red s chip remains on 12.) Blue rolls 3 (Each multiple of 3 is covered by one Red chip, so Blue cannot play. Therefore Red wins.) In the course of discussing the rules of the game you can ask questions like these: If it s the first roll of the game and you roll a 3, where could you put your chip? (On any multiple of 3 12, 15, or 18.) What must you roll to be able to put a chip on 18? (1, 2, 3, or 6; note that the other factors of 18 namely 9 and 18 aren t on the number cube.) What must you roll to be able to put a chip on 19? (Only 1.) 47
MathTalk keep thinking VARIANTS OF MULTIPLE MANIA You can discuss variants of the game as a group, and then assign teams of two students to explore how the changes affect the game. For example, the game board could have the numbers 21 through 30, or 100 through 109. The numbers don t even have to be consecutive whole numbers. Or you might change the numbers on the number cube. PARTY PLANNING WITH MULTIPLES Explore how the least common multiple changes as guests are added or subtracted. The information can be displayed in a table like this: Number of guests Least common who might come multiple Some students may need to start with simpler examples. Suppose that the number of guests could have been 4, 8, or 12. The least common multiple of 4, 8, and 12 is 24. What if each number was increased by 1, so that 5, 9, or 13 guests might show up? Start to build a chart like the one below. 12 16 24 48 13 17 25 5525 14 18 26 1638 R. S. V. P. This can be done in small groups with a class discussion at the end. Students should be asking themselves and each other two questions: Is 210 (for example) a multiple of 6, 10, and 14? Is it the smallest multiple of 6, 10, and 14? Number of guests Least common who might come multiple 4 8 12 24 5 9 13 585 6 10 14 210 7 11 15 1155 8 12 16 48 R. S. V. P. 48
NUMBER SENSE FOR THE PORTFOLIO 1. Suggest that students make a more complete list of least common multiples for the chart on page 48, subtracting or adding guests. Examine the list for patterns to try to determine why sometimes the least common multiple is fairly small, like 48, and sometimes much larger, like 5525. (12, 16, and 24 have several factors in common, while 13, 17, and 25 have only one common factor 1.) Encourage students to try lots of examples, including smaller numbers, and only two at a time instead of three. (It turns out that the product of two numbers is the same as their least common multiple times their greatest common factor. For example, 8 x12 is the same as 24 x 4 [and note that 24 is the least common multiple of 8 and 12; 4 is their greatest common factor.]) Have students explain in writing their reasoning. 2. Students may explore and write up strategies for the MULTIPLE MANIA game. What is a good first move for each of the possible rolls of the number cube? Why? CURRICULUM CONNECTIONS Multiples are important when calculating with fractions, since multiplying the numerator and denominator of a fraction by the same positive whole number gives a fraction with the same value. The concept of multiples underlies proportional reasoning. For instance, the ratio of 6 to 5 is the same as the ratio of 12 to 10 because 12 and 10 are the same multiple of 6 and 5, respectively. Each side is two times as long, This idea of proportion permeates geometry and so the shapes are measurement, too. Two geometric figures are similar if all similar. the measurements of one are the same multiple of the corresponding measurements of the other. Math Talk CONNECTIONS NUMBER SENSE Let Me Count the Ways: Counting with Combinatorics GEOMETRY What Shape Is Your Number? Finding Number Patterns in Squares and Triangles Finding multiples Shape by Number: Building Rectangles 49
NAME MathTalk Family Page MULTIPLE MANIA NUMBER OF PLAYERS: Two WHAT YOU NEED: a number cube 10 chips each (a different color for each player) RULES OF THE GAME 1. On each turn a player rolls the number cube and places a chip on any number that is a multiple of the number rolled, as long as the space already contains an even number of the opponent s chips. The opponent s Remember that zero is an even number! chips on that space (if any) are removed. Any of your own chips already on that space stay there. The only case in which a player may not place a chip on a multiple of the number rolled is if that space is blocked by an odd number of the opponent s chips. 2. The game ends when: a. each player has played all 10 chips. The player with more chips on the board is the winner. It s a tie if both players have the same number of chips. or b. one player has no legal move. The other player is the winner. 50 1995 Children s Television Workshop