**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. The second person **colors**
all the proper factors of the first person's number. The roles are
switched and the play continues till there are no numbers remaining
with uncolored factors. Each person adds up the numbers they've
colored. The winner is the person with the largest total.

The purpose of this investigation is twofold: to help students
determine whether a given number has many or only a few factors and to
show how this property of numbers is useful for problem solving. Part
I, Playing the Factor Game, engages students in a friendly contest in
which winning strategies involve recognizing the difference between
prime numbers and composite numbers. Part II, Playing to Win the Factor
Game, guides students through an analysis of Factor Game strategies and
introduces the definitions of prime and composite numbers. Part III
provides questions that are rich in connections to situations in which
factors, multiples, divisors, products, and prime numbers are
significant.

This investigation is based on the Factor Game from the *Prime Time* unit of the Connected
Mathematics Project, G. Lappan, J. Fey, W Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications, (1996) pp.1-16.

Activity Sheets Packet

### Conducting the Investigation

**Day 1**

**Launch**

This problem gives students an opportunity to learn about factors by playing
a two-person board game. On each turn, one player chooses a number, and the
other player finds the factors of that number. While playing the game, students
become familiar with the factors of the numbers from 2 to 30 and review
multiplication and division of small whole numbers.

Discuss the material at the top of page one and elaborate on what a factor of
a number is. Remind students that there are two ways to think of a factor: as
one of the numbers that is multiplied to get a product, and as a divisor of a
number. You could ask students to give examples:

What factors can you multiply to get a
product of 10?

When you feel your students understand what a factor is, introduce the Factor
Game. The rules for the game and a sample game are given in the investigation,
but the best way to get students started is to play a game against the class
using the game board, in the Activity Sheets Packet. Rather than reading
all the rules at the start, explain the rules as the need arises during the
game. When you play against the class, we suggest that you take your turn first
and that you choose a *non-prime* number, such as 26, for your first move.
This way, students can discover the power of a prime first move. Students can
either use printed copies of the game board or use the provided interactive applet.

The Factor Game

A Note on Calculators

In the Connected Mathematics curriculum, we assume that students have access
to calculators at all times. However, we hope that students will develop good
estimation and mental arithmetic skills. This means that you need to give your
students guidelines about the appropriate uses of calculators. In some classes,
students may be ready to do all of the arithmetic in the Factor Game without the
help of calculators. In other classes, students may need to use calculators to
check their mental computations. You need to make a judgment call about whether
to use the game as an opportunity for practice in mental arithmetic or to
encourage your students to use calculators. After students have a sense of the
Factor Game, you may find it appropriate to encourage them to use calculators to
keep running totals of their scores.

**Explore**

Have students play the game two or three times with a partner.

As you play the game,
think about the questions I am writing on the board.

Write the following questions on the board:

Is it better to go first or second? Why?

What is the best first move? Why?

How do you know when the game is over?

**Summarize**

You may want to have a few students share some ideas they discovered while
playing the game. However, since the next part is an analysis of the game, you
can delay an extensive summary until then.

**Day 2**

**Launch**

This problem engages students in systematically analyzing the Factor Game.
The questions you wrote on the board in the Explore section of the last part
help to launch this part.

Thinking about the best
first move in the Factor Game makes me wonder what the results would be for each
number if I chose it as my first move. What if I chose 1 or 2 or 3? How many
points would I get? How many points would my opponent get?

Let's find the results for
every possible first move. Can you think of a way that we can organize our work
so that we can see patterns and determine which moves are good and which moves
are bad?

Give your students a chance to suggest ways to approach the analysis and
organization of their work. If no good ideas surface, have them consider tabular
organization. Once the students and you have agreed on a scheme for organizing
the data, remind them of what they are trying to determine. (If you want to
provide your students with a chart for recording, see the handouts.)

Remember, you are
exploring what your score and your opponent's score would be if you chose each
of the numbers from 1 through 30 as your first move. When your chart is
complete, write the answers for Part 1and Part 2 in your journal.

**Explore**

Give students 5 to 10 minutes to work on their charts individually. Then
allow time for them to work with a partner to compare, correct, and complete
their charts. You may want to make and display a class chart so you and your
students can refer to it during the rest of the unit. A possible chart is given
in the handouts.

A Note on Factor Game First Moves

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.

**Summarize**

A good way to summarize is to have students share their answers to Part 1 and
Part 2 with the class. Students should record the results of the discussion in
their journals, either in class or as a part of their homework. You may wish to
discuss the fact that the number 1 is neither prime nor composite because it has
no proper factors at all.