# Least Common Multiple (LCM)

## by Ken Marushige

Ken Marushige is a graduate student in the Educational Technology Masters program at San Diego State University. In his spare time he a 6th grade math and computer teacher, parent of two, assistant soccer coach and a juggler with the Fern Street Circus.

Instructional Objective The learners will be able to determine the least common multiple (LCM) of any two numbers between two and fifteen.

Learners/Context The target audience is 4th-6th grade students who are learning how to add and subtract fractions with different denominators. The learners are already familiar with the multiplication facts 1-10 and have already learned to add and subtract fractions with the same denominator.

The game is designed to be played after a lesson on LCM's/common denominators and prior to the lesson on adding fractions with different denominators. This game lends itself well to a learning center for four to six students (2-3 decks) in which the the students could play a number of times.

Rationale One of the prerequisite skills for adding and subtracting fractions with different denominators is finding the LCM for the two denominators (a.k.a the common denominator). Finding the LCM is a critical yet mundane skill. A game format can reinforce this skill, keep the students motivated as well as encourage them to make the determinations quickly.

Rules

• Number of players: 2
• Materials: (besides the cards) Pencil and paper to keep score
• Basic Game
1. Remove cards numbered eleven through fifteen.
2. Deal half of the deck to each player.
3. Each player simultaneously throws down a card.
4. Each player tries to be the first to state the least common multiple of the two cards on the table. If the first player to respond is correct they get two points. If that player gives an answer that is a common multiple but not the least, they are given one point. If the first player to respond is completely wrong, the other player may get one point by answering correctly. Otherwise no points are given.
• The players can check their answer by looking at both lists of multiples and finding the lowest number that appears on both lists.
5. After points are awarded and the score is updated the two cards go into a discard pile.
6. The game ends when both players run out of cards.
7. The winner is the player with the most points

• Advanced Game: The rules are identical to the basic game except that the entire deck is used.

Card Design The cards are 2.5" x 3.25". In the upper left corner of each card is a number from two to fifteen. Directly below that number is a list of the first fifteen multiples of that number. The same information appears upside-down starting in the lower right corner so that the card is readable from both directions. During the game the players may choose to refer to the list on each card to find the least common multiple. In the center of the card is the "LCM" logo.

Sample Cards

Deck Design The deck consists of cards numbered two to fifteen. There are six of each kind of card for a total of eighty-four cards. The cards numbered eleven through fifteen may be removed for the basic game leaving a deck of 54. In this way the deck size is manageable for both games but also large enough to provide a meaningful amout of practice.

Design Process I want my students to be familiar enough with the numbers two through ten so that they can quickly find, if not memorize, the LCM's for any two numbers in that set. At the same time I want them to know how to how to determine the LCM of two numbers by looking at the multiples of each number and then finding the smallest number on each list. I had considered giving each player a copy of the multiplication table for the numbers two through fifteen but I felt that it would be very difficult for the player to quickly isolate two particular columns (or rows) to find the LCM and would thus interfere with the the reinforcement of the skill objective. For this reason, I decided to put a list of the first fifteen multiples on each card directly under the card's number. I chose to make the list of multiples big enough to read easily but small enough so that a more savy player could choose to ignore them. It is my hope that if students play the game a number of times, they will wean themselves from the lists as they become more familiar with the numbers. In the prototype shown above, the list of multiples is in a larger font size to illustrate this feature. I would use a slightly smaller font size in the production model.

Last updated by Ken Marushige on September 30 , 1996.