You can help your students become familiar with the idea of thefactors of a number, the multiples of a number, and the concept ofprime numbers (whole numbers that have exactly two factors) byhaving regular practice with the concepts. Try offering someinformal challenges such as,“Name a multiple of 4 that’s bigger than20. What could it be?” (Any number—such as 24, 28, 32, and soon—that is a number multiplied by 4.) “What’s the smallest primenumber greater than 20?” (23) “I’m thinking of a number that is amultiple 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 includingimpossible ones, too—but be sure to alert themthat there may be no solution—for example,“I’m

thinking of a number that is a multiple of 3 and is 1less than a multiple of 12.” (That’s impossible,

because every multiple of 12 is also a multiple of3. If a number is 1 less than a multiple of 12, it

can’t be a multiple of 3.)

Factor ‘Em In: ExploringFactors and Multiples

INTRODUCTION

Factors and multiples havetraditionally been an importantpart of the elementary schoolmathematics curriculum, and forgood reason: Whenevermultiplication crops up, so domultiples. Students who knowhow to recognize and workwith multiples have a headstart on developing numbersense.

This module concentrates onfinding multiples that two ormore numbers have in common—that is, common multiples.Multiples and factors appear in many parts of mathematicsbeyond arithmetic (see CURRICULUM CONNECTIONS, page 49).

VIEWINGBEFORE

PROGRAM SYNOPSIS

Segment 1 (5:59)DIRK NIBLICK: TOOMANY COOKOUTS,PARTS 1 AND 2

Dirk Niblick, fearless leaderof the Math Brigade, helpshis neighbor Mr. Beazley plana barbecue. Together theywork out how to save moneyon the party food by findingthe smallest number ofpackages of hotdogs—12 toa package—and buns—eightto a package—needed tofeed his party guests.

Segment 2 (4:24)COMMON MULTIPLE MAN

When a couple has to figureout how many hors d’oeuvresto buy to serve either 12, 16,or 24 guests equally, they callon Common Multiple Man, asuperhero with a verystrange (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 positivemultiple that the two numbers

have in common.

SHOW

107

45

46

VIEWINGAFTER

At the end of the last segment it seemed pretty clear thatCommon Multiple Man wanted to be invited to the party.Then instead of 12, 16, or 24 guests, how many guestswould there have been? (13, 17, or 25) Explore how manyhors 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 onall three lists.

13, 26, 39, 52, . . . .17, 34, 51, 68, . . .25, 50, 75, 100, . . . .

Just looking at these numbers, it isn’t immediatelyclear that there will be any number that appears on allthree lists.

Some students may not have fully internalized the reasoninghere, and they will need to continue the lists fora while to get a feeling for what ishappening.

Ask students to work insmall groups tocontinue the lists. Canthey think of anyshortcuts? Some maysuggest multiplying thethree factors (getting 5525).That’s a lot of horsd’oeuvres! How do youknow whether 5525 isthe lowest commonmultiple or not?

Is 221 a multiple of 25?(No. Keep multiplying to find a multiple of both 221 and25.) Be on the lookout for shortcuts.

It turns out that, by adding one more guest, the number ofhors d’oeuvres needed has jumped from 48 (which was theleast common multiple of 12, 16, and 24) all the way up to5525. (Maybe that’s why the host and hostess didn’t wantto invite Common Multiple Man!)

221 is the first one!

You can simplify the problem by first trying to find a number that is a

multiple of 13 and of 17.

MathTalk

Note:A calculator is helpful here.

to get successive multiples of 13.Press 1 3 – = = = = . . .

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 theirparents or other family members.

N UMBE R S E N S E

activityGAME: MULTIPLE MANIA

MATERIALSfor each pair of students: ■ one number cube

(numbered 1–6)■ copies of reproducible page 50■ 20 chips—10 each of two

different colors

Number Rolled 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 getsa 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.)

SAMPLE GAME:SAMPLE GAME:

47

48

Explore how the least common multiple changesas guests are added or subtracted. Theinformation can be displayed in a table like this:

Some students may need to start with simplerexamples. Suppose that the number of guestscould have been 4, 8, or 12. The least commonmultiple of 4, 8, and 12 is 24. What if each numberwas increased by 1, so that 5, 9, or 13 guests mightshow up? Start to build a chart like the one below.

This can be done in small groups with a classdiscussion at the end. Students should be askingthemselves and each other two questions: Is 210 (forexample) a multiple of 6, 10, and 14? Is it the smallestmultiple of 6, 10, and 14?

keepthinking

VARIANTS OF “MULTIPLE MANIA”

You can discuss variants of the game as a group, and then assign teams oftwo students to explore how the changes affect the game. For example,the game board could have the numbers 21 through 30, or 100 through109. The numbers don’t even have to be consecutive whole numbers.Or you might change the numbers on the number cube.

Number of guestsLeast common

who might comemultiple12 16 24

4813 17 25

552514 18 26

1638

R. S. V. P.

R. S. V. P.

Number of guestsLeast common

who might come

multiple

48

12

24

59

13

585

610

14

210

711

15

1155

812

16

48

PARTY PLANNING WITH MULTIPLES

MathTalk

N UMBE R S E N S E

49

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?

NUMBER SENSE

GEOMETRY

Let Me Count the Ways:

Counting withCombinatorics

What Shape Is Your Number?

Finding NumberPatterns in Squares

and Triangles

Shape–by–Number:Building Rectangles

Finding multiples

MathTalk CONNECTIONS

CURRICULUM CONNECTIONS

Multiples are important when calculating with fractions, since multiplyingthe numerator and denominator of a fraction by the same positive wholenumber 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.

This idea of proportion permeates geometry andmeasurement, too. Two geometric figures are similar if allthe measurements of one are the same multiple of the

corresponding measurements of the other.

Each side is two times as long, so the shapes are

similar.

50 ©1995 Children’s Television Workshop

MULTIPLE MANIA

NUMBER OF PLAYERS: Two

WHAT YOU NEED:

■ a number cube ■ 10 chips each

(a different color for each player)

R U L E S O F T H E G A M E

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’schips 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.

orb. one player has no legal move.

The other player is the winner.

Remember that zero is an even number!

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