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Division
by Gail Englert

A response to the question:

Please explain the partition method and the subtraction method in teaching division.

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There is a partition method and a subtraction method in teaching division to late elementary students. How are these methods defined. Also, I would appreciate any examples of these two methods. Thanking you in advance.

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I think you might be referring to the different "meanings" for division, rather than the methods. When we divide, we are doing so to find the answer to a problem. There are two types of situations that lead to division. One type, the measurement situation, happens when we have a certain amount, and want to know how many small groups of a specific size we can get from that amount. For example, how many servings of cookies can be served from eight cookies, if two cookies are given for each serving? This is like measuring off groups of two from a line of eight cookies. The size of the original set (how many cookies in all - the dividend, or "the number in the box") is known and the size of each serving, or group, is known. The problem is to determine the number of groups. It is like repeated subtraction.

The other situation is called partitive division. This happens when we know the size of the original amount, and we know how many groups we need from that amount, but we aren't sure how large each group will be. Suppose we had those same four cookies, but now we know there are 4 people to share them. This time the size of the original set (the number of cookies in all) and the number of subsets (how many people get cookies) are known. The problem is to determine the size of each serving. This is like sharing things with friends, or finding equal parts of the group.

One of the most important skills that will create success in division is knowing the ballpark answer, estimating. If your child can tell you about what the answer should be, that shows that he/she understands how the process works, even though the quotient he/she gets might not always be perfectly correct.

Before a child can really complete the division algorithm (process) as we learned it, he/she needs to be able to use many math skills; place value, multiplication facts, multiplication, subtraction facts, and subtraction. Some students continue to have trouble with these skills, but can still move into division with assistance. The traditional algorithm is especially difficult because if you make one mistake, it isn't always clear how far back you must go to correct the error. There is also an alternate method.

I have found that some elementary students do just fine with the traditional method, but some need a little help going from the concrete step to that abstract algorithm.

I ask my students to think about what they think the answer should be. For example, if the problem is 478 divided by 4, about how many groups of 4 do you think there are in 400? How many in 500?

The answer to the problem should be somewhere between those two amounts. Using base ten blocks, and making groups of four (which is tedious), or four equal groups, using the blocks, is one way to make this hands on. Students can arrange the blocks to find that the quotient is somewhere between the number of groups in 400 and the number of groups in 500.

Then, I like to start with an alternate algorithm for division, rather than the traditional method. I have found that the transfer to the traditional is very smooth, and this alternate method gives students who are still struggling to learn the multiplication facts, a chance to be successful, even if they only know the 1's, 2's, 5's and 10's. You might consider an alternate algorithm.

It is easier for them because they do not have to have the perfect factor each time, just one that will work. They can continue finding factors and dividing until they have a remainder that is smaller than the divisor (in this case, 7). When they have found all the factors, they just add them up (not the 7, but the other one each time). That is their quotient, and what is left is the remainder, which can be put in fraction form (in this case, 2 out of 7, or 2/7), or they can place a decimal, and add zeroes to continue dividing.

Besides the help this gives a student, it treats the dividend (in this case, 478) as a whole amount, instead of looking at 4 hundreds, then 48 tens, as the traditional algorithm does.

               ____
           7 /478     7 X 10
             - 70
             _____
              408     7 X 10
              -70
             ____
              338     7 X 10
              -70
             ____
              268     7 X 10
              -70
             ____
              198     7 X 10
              -70
              ____
              128     7 X 10
              -70
              ___
               58     7 X 2
              -14
              ___
              44      7 X 2
             -14
             ___
             30       7 X 2
            -14
             ____
             16       7 X 2
            -14
             ___
              2

           10 + 10 + 10 + 10 + 10 + 10 + 2 + 2 + 2 + 2= 68

See, this student was able to finish the problem without using any of the facts except 7 X 10 and 7 X 2. This could be of great assistance to someone who understood why we needed to divide, but did not have the facts mastered yet.

I try to move my students to try larger "chunks" once we have that first step mastered.

              ____
           7 /478     7 X 30
             -210
             _____
              268     7 X 30
             -210
             ____
               58     7 X 5
              -35
             ____
               23     7 X 3
              -21
             ____
                2

           30 + 30 + 5 + 3 = 68

Then we have a challenge to try to find the one "best chunk" for each place value. In the next example, there is one "chunk" for the tens, and one for the ones. (There is no hundreds chunk, because 7 X 100 would be more than 478).

               _____
           7 /478     7 X 60
             -420
             _____
              58     7 X 8
             -56
             ____

                2

           60 + 8 = 68
           

You will probably notice that that is very similar to the traditional algorithm. It is just a small nudge away for most students, at that point.

This gives the student who is still having some trouble with multiplication facts a way to rely on those facts he/she knows. It also treats the "amount in the box", the dividend, as a whole unit through the entire problem, so place value is easier to see and use. I guess it could be considered "the subtraction method".

-Gail, for the T2T service

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