1.6 The Factor Game
Teacher's page
Resources needed: 
Microworlds 2.03 (factor.mws)
A set of fifteen 3 by 5 cards (with tape on the back) numbered from 1 to 15.

Curriculum: 
This game fits in nicely in a unit on number theory.

GEPA connections: 
Number sense, factors, composites, primes

Preliminaries
Explain the rules for your students. (See Rules of the Game below)
 
Rules of the Game

Two players (or teams) compete for the highest score by picking numbers from 1 to 15. Let's say that Player A picks 15. This means A gets 15 points. B receives the sum of the factors of that number and they are added to B's total. 1, 3, and 5 are the available factors of 15 so B has 9 (1+3+5) points. The number chosen and its

 factors are then removed from the board or screen. Next its B's turn to pick a number that is still on the board. Play continues until all the cards have been selected.

How to play with your students
As a large group activity. Split the class into two groups and assign a captain to each group.

Tape fifteen 3 by 5 cards (numbered from 1 to 15) on the blackboard or wall.

After explaining the rules, tell the students that your role would be strictly to move the cards. For example, if the first team (A) chooses 15 the teacher moves the 15 card to Team A's total. Team B then has to tell the teacher what cards they are entitled to (factors of 15 still on the board.) In this case 1 and 5 should move to Team Bís hopper. If a team makes a mistake it is the obligation of the other team to catch it. This keeps the students attentive and engaged. If some errors are not picked up by the students, the teacher should make sure they are aware of the problem.

Here is an example of a possible game (Initial number chosen is in blue; chosen factors are in red.)
 
TEAM A
TEAM B
Difference between A & B's score
TEAM A's score
TEAM B's score
Comment
15
1+5+3=9
6
15
9
A leads by 9
 
13
13
15
22
B leads by 7
11
  
11
26
22
A leads by 4
 
9
9
26
31
B leads by 5
7
  
7
33
31
A leads by 2
2
14
12
35
45
B leads by 10!
10
  
10
45
45
Tie game
 
6
6
45
51
B leads by 6
12
4
8
57
55
A leads by 2
 
8
8
57
63
B wins by 6
           

Variations of the Game - Using the Computer

One computer classroom. Use the computerized version of this game (Factor.mws is written in Microworlds 2.03 or Microworlds Pro.) Even though the group game is still my favorite way to play game, the computer does offer some advantages. It allows for students to play with each other in a lab situation. You can also easily change the range of numbers. Watch out for time though. A game of 40 can last a full period!

Extensions & Additional Activities

I discovered an interesting variation because of the range feature. I asked the players to play a game of 10 several times. Player A always goes first. Assuming that both players are using an optimal strategy, does player A always win? Try it for other ranges? Is there a pattern?

The terms prime, composite, deficient, abundant, and perfect can be introduced to explain the how the numbers work. 18 is an example of an abundant number because the sum of its factors (not including 18) is greater than 18. (1+2+3+6+9=21). A variation of this factor game is used in the Connected Mathematics Project in Prime Time - Factors & Multiples published by Dale Seymour Publications.

Books