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.

Division

Today, I had a 5th or 6th grader spend some time in my room today. One of his problems required him to use a place holder and he did not recognize it was needed.  He totally omitted it.  I recreated the problem the way he solved it.

 I looked at his work and realized that he did not see the  way the digits were lined up.  this meant he didn't realize he had to look at dividing the 12 into the 3.  He got frustrated at this point because he knew some thing was wrong.  I have noticed that many students, even those in high school, do the same thing.  After some thought, I came up with a solution. It can be done on paper or on an ipad.

If a student uses paper, they can use graph paper to line up columns or they can draw dotted lines in as on the drawing on the right.  The lines help students keep track of the digits.  If its on an ipad, they can use an app such as gloss or take a screen shot of a problem and use an annotating program to draw in lines and do the work.

If anyone else has suggestions that would help students remember to use a zero as a place holders, please feel free to share.  I find this is a big problem with the students who struggle in math.  So please share.