Comparing Fractions - The Fourth Grade Discovery

I have been collaborating with teachers as we plan to make learning accessible to all.  When it comes to fractions, this is a real challenge.  I've really been grappling with why fractions are so challenging for our students and what we can do to support this learning.

The other day a teacher I was working with asked her class, "Are fractions numbers?"  To our complete surprise about a third of the students did not think they were numbers, but that there were numbers in fractions.  So with that misconception in mind, our fraction lessons have centered on concrete visual models and discovery.

In the world of nine and ten year olds, fraction exposure is so limited.  Yes, they hear about a 1/2 price sale or 3/4 of a mile, but their world thus far has been filled with whole numbers.  Their fraction understanding is in its infantile stages.

Hence - concrete models.  The students' tools include fraction bars, compliments of Math Coach's Corner and fraction number lines.

Our plan was to start by introducing the comparison of fractions with the same denominator.  We thought this would be a quick review.  Not!  The students actively used their tools to determine the size of 2/6 as compared to 5/6.  Yes, some quickly grasped the concept, but the majority of the children were relying heavily on their tools.

So we decided to take the approach of coming up with a theory and then testing this theory out.  The students noticed that the denominators were the same.  Then after several trials, they were ready to write a rule:

Next we posed comparisons with the same numerator:  3/4 compared to 3/10.  Again we took the same approach.  We asked the students what they noticed.  They developed a theory, tested it out several times using their tools, and eventually created a rule:

Yes, this took some time, but time so well spent.  The children really persevered and made the discoveries on their own.  As they realized their theories were actually strategies for comparing fractions, an air of confidence filled the room.

At the climax of this two day lesson, a student came up to the teacher and said that she use to not like math, but now she likes it.  Priceless!!!

I am learning so much from the kids - they need visual models, time to explore, discuss, and discover, and lots of repeated practice.  What do you do to foster fraction fundamentals?  Please share your ideas.  We all learn from each other.