Conceptual Understanding of Decimals - Part 1

This school year I have been trying to focus more on students' conceptual understanding.  When looking at my lessons through this lens, I am realizing there are topics that challenge my conceptually understanding.  I can explain the rules mathematically, but I do not have anything concrete for student to explore.  One of these topics is decimals.  I understand what a decimal is, I can use base ten blocks to show place value, they are even great for adding and subtracting.  Where it all becomes fuzzy for me is multiplying and dividing, and don't even get me started on repeating decimals.
Our textbook has lesson intended to help students understand why some fractions convert to terminating decimals and other fractions become repeating decimals.  The lesson takes the students through the process of dividing cheese sticks to see what is really happening.  While it is a good visual representation, it took me 2 years of teaching it to figure it out.  It is, at best, a good demonstration because in the end I have to explain all the steps to the students.  The model is quickly forgotten after the lesson, and the students go back to dividing on their calculator happy as clams that the answer just shows up.
This year I needed to write sub plans for this lesson.  If it took me 2 years to figure it out, there is no way I could leave it for a guest teacher, so I decided to shake it up a bit.  I left a very open ended task for my students.  I wanted them to create a visual representation to show why fractions convert to different types of decimals. I wanted it to be a stand alone poster (something they would not stand next to and explain) so all of their math thinking would need to be clear in their visual.  I gave them the fractions 3/4 and 2/3 to work with.  Here is a link to the instructions that were shared with the students.  Some students used markers and paper, while others chose digital tools.  There were slight variations, but all the visuals looked something like this:

                   

          

   

Every student gave me some variation on the pictures above.  Now remember, the question is why do some decimal numbers terminate while others repeat.  I want you to look back at the pictures again and see if you notice what I noticed....

While these are all lovely visuals of 3/4 and 2/3, not one of them actually shows decimals!  My students are masters at showing why 3 divided by 4 is 3/4, but then they just said that 3/4 is equal to 0.75.  There was a little more information in some of the explanations, but most of them referenced long division or the fact that 3 doesn't go into 100 like 4 does.  This became the starting point for a great discussion the next day when I returned.  I showed them the drawings and pointed out what I noticed.  We worked together to figure out how to modify our visual representations.  We talked about decimal place values and what that looks like when cutting up a shape into sections (10 sections - the remainder cut into 10 again which becomes hundredths, etc). The pictures below are an attempt at simple visuals to explain the discoveries we made in our class discussion.

If you are looking to better understand your students understanding of decimals give them this task and see what they do with it.  It really was eye opening for me.

If you are looking for a more structured lesson to help build this conceptual understanding you can read part 2 of my post (after I write it).  I will be teaching this lesson with a different class next week and have a new plan for exploring decimal place values and converting fractions to decimals.  I also have a plan for part 3 when I figure out the best way to extend our conceptual understanding to decimal multiplication and division.

So stay tuned...