NCTM: Divide and Conquer: Making Sense of Long Division

Presentation by Dr. Juli Dixon, 

University of Central Florida

Well, this session was just what I needed. For two years, I have had discussion after discussion with fellow mentors and colleagues on how to conceptually teach the long division algorithm and, today, I got the answers that we have been looking for! I will offer an overview of what I learned here,  but I really can not wait to get back to school to engage in some collegial conversation over what I learned today with fellow math geeks Rick, Suzanne, Tom,  Patti,  Sherrie, Tammi, Bridget,  Heather, Katie, and Bobbi! It’s sure to be a lively conversation. Are you ready for some enlightenment? Well, here it is! 

I learned why it has been so difficult for us to try to bridge our students’ understanding from the partial quotients method (or Russian Peasant as we like to call it J), to the procedural method of long division and now it all makes perfect sense. 

After reviewing the differences between measurement and sharing/partitive types of division problems, Dr. Dixon taught us that we need to represent long division with a SHARING (not MEASUREMENT) approach. (This means that there is not a natural or easy transition from Russian Peasant Division to Long Division.) Instead, the use of base ten blocks (hundreds, tens, ones) and/or quick draws of base ten blocks, along with “sharing language”, needs to be reinforced.  

SHARING/PARTITIVE APPROACH:

As an example, for the problem 536 divided by 4, you would need 5 hundreds flats, 3 tens rods, and 6 cubes.  Starting with hundreds, you would equally share “1” hundred flat among four piles (notice the recording method in the example shown below). The one “left over” would then be decomposed into ten tens rods to make 13. Sharing the 13 tens rods equally in the 4 groups would put 3 in each group with one left over, which would again be decomposed (exchanged) into 10 cubes for a total of 16. The 16 cubes shared equally into the four piles would place 4 in each pile with none left over. 

The end result would show 4 EQUAL PILES OF 134.

                    134

            4  )  536

                   -4          (hundreds)

                    13        (tens)

                  -12        (tens)

                    16       (ones)

                  -16       (ones)

                    0

Once students make sense of this, the efficiency of the long division 

algorithm can be introduced.

MEASUREMENT/PARTIAL QUOTIENTS APPROACH

Dr. Dixon also shared a new division model with us that I have not used before which connects to Partial Quotients Division. This involves the decomposition of the dividend (ex. 8,652 = 8,000 + 600 + 50 +2) and left-to-right division (starting with the largest place value first).  See the example below:

                   1,000        200        30        6

          7  )  8,000     1,600     250       42

                 -7,000    -1,400   -210     -42

                  1,000        200        40         0   

Dr. Dixon also reinforced several ideas that we certainly understand at Chets in our own mathematical communities:

1)   1) “Does McDonalds Sell Cheese Burgers?” is A PROBLEM! (Divide, Multiply…) Where's the math in that?

2)   2) Students need a context when working with division so they can model and connect with it.

3)   3) Use language that supports the meaning of division.

4)   4) Students need to see many division strategies, which happen with the Measurement/Partial Quotients/Russian Peasant Method (not with the Sharing/Partitioning or Long Division Methods) so they can see the efficiency in various approaches.

5)   5) Students need lots of experience with both measurement and sharing division types. They need to understand BOTH!

6)   6)Procedurally based algorithms don’t make sense of the math- they make the math quick.

7)   7) Florida’s New Generation Sunshine State Standards have 5^th grade as teaching long division. It’s not a bad thing if we don’t “teach it badly.

Now, that was time well spent! Thank you, Dr. Dixon, for all of this enlightenment! This is exactly what we have been searching for! J