### Background Information

The Factor Game is a two-person game in which players find factors of numbers on a game board. To play, one person selects a number and colors it (as shown below in blue). The second person colors all the proper factors (as shown below in red) of the first person's number. The roles are switched and play continues until there are no numbers remaining with uncolored factors. Each person then adds up the numbers they've colored, and the winner is the person with the largest total.

### Introductory Problem

Pose the following problem to the class:

Today Jamie is 12 years old. Jamie has three younger cousins: Cam, Emilio, and Ester. They are 2, 3, and 8 years old respectively. The following mathematical sentences show that Jamie is

6 times as old as Cam, | 4 times as old as Emilio, | 1 and 1/2 times as old as Ester |

12 = 6 × 2 | 12 = 4 × 3 | 12 = 1½ × 8 |

Ask students what they notice about the first two examples as opposed to the third example. Students should notice that each of the whole numbers 2, 3, 4, and 6 can be multiplied by another whole number to get 12. We call 2, 3, 4, and 6 *whole number factors* or *whole number divisors* of 12. Although 8 is a whole number, it is not a whole number factor of 12, since we cannot multiply it by another whole number to get 12. To save time, we will simply use the word *factor* to refer to whole number factors.

### Playing the Factor Game

For the next portion of the lesson, students will be working in pairs on the computers.

Students should open the Factor Game. In the "Instructions" section, students can review the rules for playing the game.

Factor Game

Allow students to play several rounds of the game with their partner. During those rounds, ask students to think about the strategies they use to win the game.

After allowing students to play the game for a sufficient period of time, return to a class discussion.

State the following to the class:

Did you find that some numbers are better than others to pick for the first move in the Factor Game? For example, if you pick 22, you get 22 points and your opponent gets only

1 + 2 + 11 = 14 points.

However, if you pick 18, you get 18 points, and your opponent gets

1 + 2 + 3 + 6 + 9 = 21 points!

Allow time for class discussion on this matter, as it will lead naturally into the next portion of the lesson.

Record the following table on the chalkboard or overhead:

**First**
Move | **Proper**
Factors | **My**
Score | **Opponent's**
Score |

1 | none | lose a turn | 0 |

2 | 1 | 2 | 1 |

3 | 1 | 3 | 1 |

4 | 1, 2 | 4 | 3 |

Tell students that they will be creating a table of all the possible moves (numbers from 1 to 30) they could make when playing the game. For each move, list the proper factors, and record the scores the student and the opponent would receive.

Students may work with their partner to create their tables. Once they have completed all of the numbers (1 to 30), they should use their lists to figure out the best and worst first moves.

In pairs, and then as a class, discuss the following:

- What is the best first move? Why?
- What is the worst first move? Why?
- Look for other patterns in your list. Describe an interesting pattern that you find.

The following are some discussion points related to the above questions for students:

*The chart indicates that prime numbers are good first moves, especially large primes like 29. (Note that prime numbers are only legal when they are first moves. Once a first move has been made, all primes are illegal because their only proper factor, 1, will have already been circled.) This chart is also a good display of abundant, deficient, and perfect numbers (explained in Part 3). The number 24, for example, is abundant because the sum of its proper factors is more than 24. The number 16 is deficient because the sum of its proper factors is less than 16. The number 6 is perfect because the sum of its proper factors equals 6. Note that 6 and 28 are the only perfect numbers between 1 and 30.*

Allow time for students to state their definitions of prime and composite numbers as they discovered them through this activity.

Distribute the Factor Game Activity Sheet to students. Allow time for students to work with their partners (or complete as homework, if you so choose).

Factor Game Activity Sheet