Remainder Patches!

by Natural Math

This fun activity, suggested for Grades 2 – 6,  has students visually discover why some division problems end up with remainders and some do not by filling paper “quilts” with paper “patches.”  Win the game with the most leftover “remainder patches”!

Quilt Patches

Materials (for each team of 2 players):

  • Six, 3”x 3” grid paper game “quilts” 
  • 15 small paper “patch” squares (1”x 1”) in a variety of colors (template for quilt and patches)
  • A recording sheet (template)
  • One 6-sided die

How to Play (for teams of 2 players):

Place the die, all 15 patches, and all 6 quilts before both players. Each player rolls the die.  The player with the highest roll goes first.

Player 1 rolls the die. Except for the number 1, any other number on the die represents the number of quilts to be used for that player during this turn. If a 1 is rolled, roll again for a different number. 

Player 1 takes the 15 patches and tries to divide them equally among the number of quilts they have during this turn by placing the same number of patches onto each of the quilts.  If there are any leftover patches, Player 1 keeps the remaining leftover(s). Player 1 records this result on the Recording Sheet, where “R” stands for the word “remainder.” After this, all quilts and quilt patches (except any patches kept as remainders from the last player) are returned for the next player to use. 

Example:   Player 1 rolls a 2 on the die.  They take 2 quilts and 15 patches. They try to put the same number of patches on both quilts.  The result is that they can place 7 patches on each quilt but end up with one remainder. This is recorded as 15 / 2 = 7 R 1.  Player 1 keeps 1 patch, and then returns the quilts and other patches for the next player’s turn.

 If a player keeps any “remainder patch(es)” from their turn, the next player can only use patches that were not kept by the previous player.  In the example above, this means Player 2 now has 14 patches left to use on their turn.   

Player 2 follows the same process — roll the die, pick the appropriate number of quilts, take all the patches left and place the same number of patches on each quilt.  Record findings. Keep any patches that remain.

Keep playing in turn. On a turn, if the number of quilts is greater than the number of available patches, roll the die again to obtain a new number of quilts. If the remainder is zero for 3 turns in a row, each player receives one patch to keep. 

Players continue the game, alternating turns, until either there are less than 3 patches left to begin a game, or after 10 turns have been played.

At the end of the game, the player with the most “remainder patches” wins!

Other ideas for playing this game:

  • The winner is the player with the least number of remainder patches
  • Only keep patches for numbers that are divisible by a certain number (for example:  keep patches only for numbers divisible by 3)
  • Add more/fewer patches and/or quilts.  What do you notice?

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