Counting Mats Help Build a Foundation of Understanding for Division

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Placing Counters Helps Students Understand Division.

A division mat, a simple template for distributing counters into equal groups. Jerry Webster

Understanding Division

Counting mats for division are incredible tools to help students with disabilities understand division.

Addition and subtraction are in many ways easier to understand than multiplication and division, since once a sum exceeds ten, multi-digit numbers are manipulated using regrouping and place value. Not so with multiplication and division.  Students most easily understand the additive function, especially right after counting, but really struggle with the reductive operations, subtraction and division.  Multiplication, as repetitive addition is not as hard to grasp.  Still, understanding operations is key to being able to apply them appropriately.  Too often students with disabilities begin to 

Arrays are powerful ways to illustrate both multiplication and division, but even these may not help students with disabilities understand division. They may require more physical and multi-sensory approaches to "get it into their fingers."

Using the Templates

  • Use the pdf templates or create your own to make division mats. Each mat has a number by which you are dividing in the upper left corner. On the Mat are the number of boxes.
  • Give each student a number of counter (in small groups, give each child the same number, or have one child help you by counting out the counters.)
  • Use number you know will have multiple factors, i.e. 18, 16, 20, 24, 32.
  • Group Instruction:  Write the number sentence on the board: 32 / 4 = , and have students divide their numbers into equal amounts in the box by counting them out, one at a time into each box. You will see some ineffective techniques: let your students fail, because the struggle to figure it out will help to really cement the understanding of the operation.  
  • Individual Practice:  Give your students a worksheet with simple division problems with either one or two divisors.  Give them multiple counting mats so they can divide them over and over again -- eventually you will be able to withdraw the counting mats, when they understand the operation.

I am only providing counting mats from 2 to 6.  Start with the twos, and after they have done several (say 2's, 3's and 4's) go back and have them apply the strategy to division by one. For ones, just draw a big square in the middle of a whiteboard slate. By the time a students has divided numbers up through 48 by 6, your students should have a strong understanding of the operation: if not, repetition works as well with divisors from six and under as 7 and over.

Introduce Remainders

After your students understand the even division of larger numbers, you can then introduce the itea of "remainders" which is basically math talk for "leftovers."  Divide numbers that are evenly divisible by the number of choice (i.e. 24 divided by 6) and then introduce one close in magnitude so they can compare the difference, i.e. 26 divided by 6.  

2 Division Mat pdf

3 Division Mat pdf

4 Division Mat pdf

5 Division Mat pdf

6 Division Mat pdf