Mistakes in Division

  Over the past few years, I've noticed a trend in my incoming students.  The majority of the students seem to make the same two mistakes when dividing.  I do not know where it comes from.  I have no idea where or when the misconception developed.

It is frustrating because I am not sure how or when to reteach the topic so students start doing it correctly.

First is when students divide, they do not place a zero in as a place holder.  An example would be dividing 5020/10.  So 10 goes into 50 five times.  The student writes 5 above and brings down the 2 but does not put a zero above it.  They bring down the 0 for 20 and put 2 above it because 10 x 2 = 20. 

Instead of 502, they come up with 52.  These students do not recognize they need a zero in there as a place holder.

I've been thinking of having students do their division either on graph paper so they can place one digit per column or use lined paper sideways.  If there is supposed to be a number in each column for the answer, it might provide an automatic reminder to put the zero in for a place holder.

The other situation is thinking the remainder is the number you place next to the decimal.  An example would be 13/5.  The student knows 5 goes into 13, twice.  They write 10 and subtract so 13 - 10 is 3.  They put 5.3 rather than 5.6 because 3/5 = .6.  It is something that occurs with great regularity. 

This one is a bit more challenging.   I am not sure how to have them think about converting the remainder into a decimal.  The only idea I have is to have them create a picture showing the remainder as a fraction of the original. Once they have a fraction, they can convert it from the illustration into a decimal. 

If anyone has any suggestions on ways to help students overcome these misconceptions, I would love to hear.  I realize I could just let them do the math on the calculator and not worry about these misconceptions but the first issue could translate into dividing rational expressions.  They might not put a zero when needed if dividing x^2 + x -3/x+1. In addition, they might not use the remainder properly.