What IS a fraction, anyway?

As we move forward into our next unit in our new math series, I had real concerns that my fourth graders didn’t have enough background knowledge about fractions to be able to successfully navigate the rigor of the upcoming lessons.  I decided to “frontload” a little with some tried and true lessons that have been successful for me in the past and that would help me not only see what my students already knew, but also any misconceptions that they might have.  I was right!  They were definitely NOT ready and had a number of misunderstandings that we needed to clear up.

The first thing I did was to ask the students to make the following chart in their math notebooks and to take a few minutes to add samples to each side.  They had questions, of course…should they be pictures?  Do I have to use numbers?  I shrugged and told them to do what made sense.  Strangely, that seemed to satisfy them.  (They must be getting used to me!)

I was noticing all sorts of interesting things…poorly drawn “equal parts”, improper fractions (on the “not fraction” side!), and whole numbers.   Hmmmmm.
As I walked around, I made note of the types of things that students were recording and starred samples in a few students’ notebooks and asked them to record them on the smartboard when we were ready.

So I have a few kids put some samples from their page up and we looked them over.

We came to some good agreement about the examples and counterexamples and then I added some of my own–things I had seen on my trip around the room.

The things I drew are in blue and led us to some great discussions!  The arrows are showing where the students felt the drawings ACTUALLY belonged after our chat.  Many students were VERY surprised that we could have a 1 in the denominator and others were quick to erase drawings in their own notebooks when we looked at that rectangle in the upper left corner and agreed that the precision of that drawing made it impossible to call it “equal parts”.

The most fun we had was with the drawing in the bottom right hand corner.  We spent a good 15 minutes debating back and forth about whether or not the diagram showed fractions…and if it DID, what fractional part would the piece in the upper right “corner” be worth?  Some kids were convinced it did not show fractions.  Some felt that piece was 1/3.  Still more felt it was 1/4.  We worked and worked to try to refine our understanding of what a fraction of a region is…and I’m still not sure that all my students are really convinced that the piece is 1/4 of that pie!  It definitely showed my that these students are used to counting pieces out of a whole and are not really fully understanding the concept of equal parts–and that not all “lines” are necessarily drawn!

Want to try an experiment with your class?  Draw it up on the board and ask–is this fractions?  If it is–what fraction is that piece (I had it shaded in to start…)?  I’d love to know if your students did a better job than mine!  I’ll try a similar activity in a week or two and see how their understanding has evolved!  Hope you are all surviving these last few days..