Divisibility War

by Diane Reed

Diane taught junior high math for three years and Algebra II and Pre Calculus topics for the past two Summers. She will have her MA. in Educational Technology from SDSU Spring 1995. For fun, she skis, works out and sails.

Instructional Objective The learners shall be able to use the divisibility tests to state that a number is divisible by 2, 3,4,5,6,9, and/or 10.

Learners/Context This game is designed for secondary level students studying basic mathematics. Use this card game with students who have been introduced to the divisibility tests for 2, 3,4,5,6,9, and 10.

Rationale Skill in using the divisibility tests makes it possible to recognize that a number can be evenly (no remainder) divided by 2,3,4,5,6,9, and 10, without performing long or short division. Once mastered, the divisibility tests can ease the math mind work involved in long and short division, fractions, percents and consumer skills, etc..

Transfer of this unobservable skill is dependent on the learner having sufficient practice applying the rules with plenty of feedback in circumstances similar to those that make using the divisibility rules useful. Teachers usually introduce students to the divisibility tests, then orally drill the class, assign a worksheet for individual practice and follow up with an oral class test for understanding. Transfer of this skill is promoted by prompting students to use the test at appropriate times in subsequent lessons. This is a haphazard approach that dooms all but the brighter students from learning and applying this skill. Use of worksheets does not provide enough practice or immediate individual feedback. Many students will complete the worksheet by doing the necessary division instead of using the divisibility tests to complete the worksheets; thus defeating the individual practice purpose of the worksheets. Without immediate individual feedback a student may never realize that he is not applying the rules or is applying them incorrectly. In subsequent lessons it is easier for students to use their division skills to learn the new lesson rather than apply the divisibility tests- a skill they have not mastered. Divisibility War could augment, shorten or replace the practice worksheets. This game will provide students practice applying the divisibility tests on large numbers. Divisibility War can fill in gaps in class time and is designed to provide immediate feedback to students as they practice using the divisibility rules. Thus, this game will ensure that students get plenty of practice and feedback using the divisibility rules in circumstances that are similar to those that make the divisibility tests useful; thus promoting transfer for all students.

Rules The game is played with 2-6 players in the following manner:

1. The dealer Shuffles the deck of cards.

2. The dealer deals the pack equally and face down: setting any extra cards aside.

3. Each player stacks his cards into a pile face down in front of him.

4. The player to the left of the dealer takes his turn first.

5. During a player's turn, he should:

(a) place the top card of his stack face up for all to see,

(b) make a statement of divisibility (state that the number on the card he has placed is evenly divisible by 2,3,4,5,6,8,9 and/or 10 or one or none of these numbers), and

(c) place zero to eight count chips, depending on the number of divisibility tests that hold for the number on the card, beside the placed card. For example, if a player states that the number on his card is divisible by 2, 4 ,5 and 10 he should place four count chips beside his card because 4 divisibility tests (2,4,5, and 10)hold for the number on the card.

6. The player to the left of a player takes his turn next.

7. The next player may challenge a player's statement of divisibility after the player places his count chips and before the next player's turn begins. A challenge entails the following:

(a) A challenger must make a statement of divisibility. A challenger may not repeat the player's statement of divisibility.

(b) The player to the left of a challenger may challenge the challenger(s) and the original player as long as the latest challenger does not repeat the statement of divisibility of the original player or any other challenger.

(c) Every player may challenge the previous challenger(s) and original player once before the next player takes his turn.

(d) To end the challenge, players should reference the Divisibility War Key.

(e) If the Divisibility War Key proves any player completely correct, the player or challenger with the correct statement of divisibility gets as many count chips as tests that hold for the number on the card in question.

(f)If the Divisibility War Key proves no player completely correct, no player gets any chips.

8. Each player takes his turn to complete a round.

9. At the end of each round, the player with the most count chips wins the round.

10. The winner of the round collects all cards placed during the round and places these cards at the bottom of his pack. If at the end of a round, two or more players have the same number of count chips there is a war. The following must be done during a war:

(a) The players with the same number of count chips each play three cards face down and a fourth face up ( a player with insufficient cards remaining puts down as many cards as he has, placing the last card face up. His opponents match this number).

(b). The player with the largest number on his face-up card wins all of the cards in the war.

11. Rounds continue until time planned for the game runs out or a player runs out of cards. The player with the most cards wins.

Card Design

Re-Production Reproduce Divisibility War by copying the numbers listed in the Divisibility War Key onto blank 3x5 index cards as pictured above. Complete production of each card set by providing the following:

Design Process My first idea was to create a rummy class game with numbers divisible by a certain number belonging to the same suit. Once I had made a set of cards, I realized that categorizing several three and four digit numbers according to the divisibility rules at one time would require too much of a players memory. This complexity would make a game that is too difficult to be fun. Furthermore, one does not generally use the divisibility rules to categorize large groups of numbers. So, I experimented with a War class game. The war class game could provide opportunities for each student to determine if relatively large numbers are divisible by 2,3,4,5,6,8,9,10 without actually dividing out. These circumstances are similar to the actual use of the divisibility tests.

The count chips , the challenger rule (#7) and the number of players as well as the length of play were created to make the game practical for classroom use, educational and fun. Use of the count chips is meant to ensure speed and accuracy in determining who won a round of the war. The challenger rule is meant to encourage student participation throughout the game, provide immediate feedback to all players, and make winning the game more dependent on a player's skill than the luck of the draw. To make the game flexible for classroom use, I created enough cards for 2-6 players to play and allowed for the game to end as time permits.

I beta tested Divisibility War with six lower level, ESL eighth grade math students. Every student was engaged in the activity. Many students later told the teacher, that they thought the game was fun. During this test, I discovered that the challenger rule without stipulation 7 b and 7c,(in other words, during this test, any player could challenge another player at any time before the next player started his turn) made the game more fun, but could create classroom mayhem and was also unfair to the timid student who was unwilling to shout over the voices of his/her peers. To make the game more fair and compatible to the classroom environment, I added rules 7b and 7c which stipulated that only the next player could challenge the previous player and the next player after that could challenge the previous challenger and so on.

I also realized that the level of difficulty of Divisibility War can be reduced by omitting the challenger rule(#7) at first and having students use three or four of the divisibility tests instead of all seven to determine the winner of a round. Once students have mastered the first four divisibility tests, the difficulty level could be increased by incorporating another divisibility test and finally the challenger rule.. For example, during my beta test, I introduced the students to the game without the challenger rule and limited the students to using the divisibility tests for 2,3,6, and 10. I gradually allowed the students to use the divisibility rules for 4,5, and 9. Once the students mastered using all of the divisibility rules, I introduced the challenger rule (#7).

                      Divisibility War  Helper                                  
      A number is divisible by:         IF                                      
                  2                     the last digit is 0,2,4,6, or 8         
                  3                     the sum of the digits is divisible by   
                  4                     if the last two digits are divisible    
                                        by 4                                    
                  5                     the last digit is 0 or 5                
                  6                     the number is divisible by both 2 and   
                  8                     if the last three digits are            
                                        divisible by 8                          
                  9                     the sum of the digits is divisible by   
                  10                    the last digit is 0                     

                           The Divisibility War Key                             

   #     2        3        4        5        6        8        9        10       
  24        u     u        u                    u     u                          
  34        u                                                                    
  35                                   u                                         
  36        u        u     u                    u                 u              
  44        u                 u                                                  
  46        u                                                                    
  48        u        u        u                 u                                
  55                                   u                                         
  56        u                                                                    
  57                 u                                                           
  60        u        u        u        u        u                          u     
  62        u                                                                    
  65                                   u                                         
  72        u        u        u                 u        u        u              
  74        u                                                                    
  75                 u                 u                                         
  80        u                 u        u                 u                 u     
  84        u        u        u                 u                                
  98        u                                                                    
  115                                  u                                         
  117                u                                            u              
  128       u                 u                          u                       
  130       u                          u                                   u     
  140       u                 u        u                                   u     
  150       u        u                 u        u                          u     
  160       u                 u        u                 u                 u     
  171                u                                            u              
  175                                  u                                         
  190       u                          u                                   u     
  196       u                 u                                                  
  200       u                 u        u                 u                 u     
  216       u        u        u                 u        u        u              
  240       u        u        u        u        u        u                 u     
  256       u                 u                          u                       
  260       u                 u        u                                   u     
  285                u                 u                                         
  308       u                 u                                                  
  309                u                                                           
  338       u                                                                    
  335                                  u                                         
  385                                  u                                         
  408       u        u        u                 u        u                       
  429                u                                                           
   #        2        3        4        5        6        8        9       10     
  438       u        u                          u                                
  447                u                                                           
  495                u                 u                          u              
  524       u                 u                                                  
  567                u                                            u              
  625                                  u                                         
  657                u                                            u              
  666       u        u                          u                 u              
  669                u                                                           
  700       u                 u        u                                   u     
  711                u                                            u              
  715                                  u                                         
  728       u                 u                          u                       
  735                u                 u                                         
  741                u                                                           
  770       u                          u                                   u     
  771                u                                                           
  849                u                                                           
  888       u        u        u                 u        u                       
  915                u                 u                                         
  960       u        u        u        u        u        u                 u     
 1115                                  u                                         
 1135                                  u                                         
 1280       u                 u        u                 u                 u     
 1324       u                 u                                                  
 2204       u                 u                                                  
 2220       u        u        u        u        u                          u     
 2225                                  u                                         
 2318       u                                                                    
 2416       u                 u                          u                       
 2997                u                                            u              
 3000       u        u        u        u        u        u                 u     
 3335                                  u                                         
 5256       u        u        u                 u        u        u              
 5952       u        u        u                 u        u                       
 6000       u        u        u        u        u        u                 u     
 6525                u                 u                          u              
 6720       u        u        u        u        u        u                 u     
 7552       u                 u                          u                       
 8135                                  u                                         
 8228       u                 u                                                  
 8872       u                 u                          u                       
 8886       u        u                          u                                
 9297                u                                            u              
 9927                u                                            u              
 9954       u     u        u                    u                 u