In my next few blog posts, I will share you some math games that I use with students. Games are used in many classrooms by many teachers, but very often these games are designed primarily to make practice more palatable. There is certainly a place for games of this type, but I am especially interested in another type of game that involves not just a practice of some skill, but requires an analysis of strategy. For this reason they serve as a natural extension of the investigation and problem-solving that should be at the heart of most math lessons,
One of best-designed and most interesting games I’ve come across is called The Factor Game. It is perfect for 4th grade through 6th grade students who know the multiplication facts well. Here are the rules:
How to Play The Factor Game
Materials:
Factor Game sheet enclosed in a clear plastic protector sheet
dry erase pens
paper or dry erase board for keeping score
This is a game for 2 players. Player 1 selects a number between 1 and 30 inclusive and circles it. This number represents the score for Player 1 on her turn. Player 2 then circles all of the factors of Player 1’s number that have not been circled. The sum of all these factors is Player 2’s score.
Player 2 then circles a number and Player 1 circles all of the uncircled factors. Play continues with players alternating circling an initial number for the round and the factors of the initial number.
If a player who is setting an initial number for the round circles a number for which there are no remaining uncircled factors, that player receives a score of 0 for that round and the number she chose remains uncircled.
Play continues until there are no remaining legal moves. The player with the highest score wins.
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Teachers and tutors should play the game by themselves at first and figure out the best first move, and then the best second move, if the first player makes the best first move. Then use what you have learned to figure out how to determine the best move at any given time during the game.
When you understand the winning strategy, you can then play the game with your student. Read the rules with the student and play a few games together, but on your turn, make more or less random moves, without using the optimal strategy. The student will need a little bit of time to experience the game in order to begin to understand the consequences of certain moves.
After the student has played the game 2 or 3 times, ask her to figure out what the best first move is and to prove it. The best first move, of course, is 29, because that is the greatest prime number on the scoreboard. If a player chooses 29 on her first move, then her opponent gets only 1 point, because 1 is the only remaining factor of 29.
If your student claims that a non-prime number as the best first move, tell her first that there is a better first score than the one she proposed, and then ask her, “What is the lowest score possible for Player 2 as a result of Player 1’s first move?”
If she selects a prime number greater number other than 29 as the best first move, ask her why she selected that number and why it is a good move. Then tell her that there is a prime number on the board that is greater than the one she picked.
I offer these teacher responses because It is important that the student figure out the best moves by virtue of her own analysis. Helping students become fluent at identifying the factors of a number is a useful by-product of the game, but the real purpose of it is to give students work at critical analysis and problem-solving. Therefore, your job as a teacher is to point out any flaws in her thinking (i.e., that there is a better move than the one she proposed) and to ask questions in order to help her clarify the important relationships and patterns she needs to understand in order to be able to come up with the optimal strategy. You should never just tell her the best strategy, or even the best move at a particular point in the game. That is always her job.
As you continue to analyze the game together, help your student put the optimal strategy in words. Over the last 30 years or so there has been a lot more emphasis in math instruction on having students explain their thinking. In general, this has been an important and welcome emphasis. However, in many lessons I’ve seen, teachers have students write down their thinking on their own with little guidance or support. Teachers always need to remember that expressing one’s ideas about mathematical patterns and relationships is quite difficult, and expressing oneself in writing is much more difficult than expressing oneself by speaking alone. Both are important, but they require years of practice, with significant support from teachers. So as you have the student put the strategy for the Factor Game into words, I suggest you do it collaboratively, with you writing down and rephrasing as necessary what the student says. You should also do it iteratively, that is, writing something down, evaluating its adequacy, revising, and then repeating the process until the strategy is both complete and precise.
I also recommend that at first, in order to keep the student’s attention focused on the game, you, the adult, should keep score. But once the student begins to understand the key strategic elements of the game, you should have her keep score. It’s a wonderful opportunity to encourage the use of and to provide excellent practice of mental math skills for doing multi-digit addition.
If you are interested in playing a non-competitive version of the game, you can simply ask the student to figure out the optimum play for each player for a whole game!
As I mentioned, in the upcoming blog posts I will share some other games I use with students. I hope you will share with me games that you have found to help students develop their analytical skills.
The Factor Game was originally published in "Prime Time: Factors and Multiples," Connected Mathematics Project, G. Lappan, J. Fey, W. Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications, (1996), pp. 1‑16.