## The "M.O.M." Game

One of the basic topics of elementary number theory is that of finding the Least Common Multiple of two or more numbers. Testifying to its basic simplicity is the fact that is presented as early as the 4th grade. But too often it is a concept that is only cursorily taught, as evidenced by the confusion that lingers long into one's educational experience. How many times have you heard a student mix up the ideas of LCM with the Greatest Common Factor (GCF) --- in an advanced math class?

Perhaps this confusion is caused by insufficient initial concept development. After an example or two are presented, the learner is expected to find LCMs through a cumbersome process of writing a list of several multiples for each number involved. (Later on an algorithmic procedure, like prime factorization, is introduced.) As long as the numbers are small and/or the LCM is found relatively soon, making a list is not a difficult matter. But let the numbers become a bit larger, and the task becomes something that most students would find tedious, unnecessarily time consuming, and unproductive as a learning activity.

Enter the calculator. Here is a way to make finding LCMs of bigger numbers an adventure. Use two calculators and two students, and it becomes an activity in cooperative learning.

First introduce the whole class how to use the constant add feature of the calculator. Show how this aspect of the calculator produces as many multiples as one needs or desires. (NOTE: some models start producing the next multiple of the number on the first press of the [=] key, others require a second press.)

Next, form pairs of students to work in a cooperative format. Present the two numbers for which the LCM is to be found, say for example, 12 and 14. Each member of the team enters one of the numbers into his/her calculator. Then each presses the [+] key. (Note: for Casio models, the [+] key must be pressed twice, causing a "K" to appear in the display.) From this point on, the [=] key will be pressed.

Team members now "take turns" pressing their own [=] keys according to whomever has the smaller number showing in the display window of his/her own calculator. When the numbers make a match -- and they always will match, eventually -- the goal has been reached, the LCM has been found. For the example cited above, this occurs with 84.

```Turn:	 1    2	   3    4    5    6    7    8    9    10    11
----------------------------------------------------------------
"12":	24        36        48        60        72          84

"14":	     28        42        56        70         84
```

Of course, sometimes depending on the numbers, one person will press [=] two or more times in a row before the second person has his/her turn. Observe: let's use the numbers 6 and 16.

```Turn:	 1    2    3    4    5    6    7    8    9
-----------------------------------------------------
"6":	12   18        24    30  36        42	48

"16":            32                   48
```

So the LCM of 6 and 16 is 48.

It should now be clear that the process is basically a simple one, made more enjoyable by the use of a simple technology: the calculator. And students enjoy it more for that reason.

As a class-group activity, both factors of cooperation and gentle competition are involved. Present a pair of numbers for which the whole class is to find the LCM. Each member of each team takes one of the numbers. At the signal "Begin!", all students start working on the problem. The first team to find the LCM is the winner.

The factor of cooperation is very important in this activity. If one member of a team presses [=] too hastily or incorrectly, his/her team might "overshoot" the LCM. So while speed is important, caution and careful observation of the values in the display window is advisable as well.

A natural follow-up question, after a student has had a certain degree of experience, is: if one continues the basic process after finding the match, when will the next match occur? The answer is, of course, not until twice the first match has been attained. If possible, this should not be revealed too soon to the learners; rather it is hoped that the result would be discovered by them.

One of the nice features of this activity is that there is no need to limit the problem to small numbers. The calculator adds large numbers just as easily and quickly as small ones. It can also be used as a short, 5-minute class-ending activity before the dismissal bell rings. Just have a few exercises prepared in advance. And it doesn't cause any â€˜mess", as no paper or pencils are needed.

Try it! Your students will like it.

P.S. Why is this activity called the MOM Game? Well, it comes from the phrase of "Matching Our Multiples". Get it?

Below are charts of prepared values for classroom use.

 number pair LCM No. of multiples 14, 18 126 9, 7 20, 35 140 7, 4 16, 56 112 7, 2 26, 65 130 5, 2 28, 63 252 9, 4 33, 39 429 13, 11 45, 51 765 17, 15 56, 72 504 9, 7 55, 88 440 8, 5 45, 57 855 19, 15

This chart has numbers that are relatively prime.

 15, 16 240 16, 15 15, 17 255 17, 15

This chart has values that were chosen by students. Note the interesting LCM's that resulted. 1040 is the famous IRS tax form. And 714 is Jack Webb's police badge number on his famous TV series "Dragnet". They were chosen on 8/24/93 and 9/8/94, respectively, by students in the fourth grade of the Escuela Americana, San Salvador, E. S.

 65, 80 1040 16, 13 51, 42 714 14, 17

Notice that the MOM Game can be extended to a three-person activity. The basic rules still apply. Here are some prepared values to use.

 Number trios LCM No. of multiples 2, 3, 5 30 15, 10, 6 2, 3, 7 42 21, 14, 6 6, 8, 10 120 20, 15, 12 6, 7, 8 168 28, 24, 21

Notice the patterns in this chart.

 3, 5, 7 105 35, 21, 15 6, 10, 14 210 same 9, 15, 21 315 same

This chart has a different sort of pattern.

 3, 4, 5 60 20, 15, 12 4, 5, 6 60 15, 12, 10

And finally, when three students have a lot of "time on their hands", or should I say "fingers"?

 9, 13, 15 1755 195, 135, 117