Using Our Fractional Reasoning!

fractions
We had a great concept sort today...and I just wanted to share a little about what we did!

We are working on the concept of equivalent fractions...we have drawn pictures, told stories (If I had half a pizza but cut the half into two pieces, what fraction would I have?), and generated lists of equivalent fractions.  What we DIDN'T do is what most math programs do right away--teach students to multiply the numerator and denominator by the same number.  We'll get there-but first I really want students to use their reasoning to really show their understanding of some key fraction concepts.

One of the Standards for Mathematical Practice involves the ability to "reason"--to create strong understanding of key concepts without merely computing.  It states:

"Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects."

By helping students learn to reason about fractions, they become better at understanding without relying on tricks and computation--which helps them with estimating and checking for reasonableness as the math gets more challenging.  I love trying to help students VISUALIZE math and make sense of it before teaching them--so that's what today was all about!


Most students have a pretty decent understanding of the concept of "one half", so I wanted to experiment with a sort and see what my students could do.  We've already talked about the concept of "unit fractions"--and how they can by used to "count" fractions...1/4, 2/4, 3/4 and so on.  We also have used our reasoning to picture the relative size of these unit fractions...that even though "seven" is a bigger number than "three", sevenths are smaller than thirds because more parts must mean smaller parts!


This was really enough information for us to begin this sort--where students used what they know about fractions to sort them into three categories--greater than 1/2, exactly 1/2, and less than 1/2.

equivalent fractions
 One of our rules about concept sorts is that students work in small groups (usually trios) and must go one card at a time where they discuss together and make a decision about which category the cards fall in.  If they have any debate, they set it aside for later.
equivalent fractions
 While students are working, I'm circulating, asking questions, listening--and looking for misconceptions.  Anything "interesting" gets thrown under the document camera at the end!
equivalent fractions
For groups finishing early, I ask them to write their OWN examples for each category...
equivalent fractions
 I LOVED hearing the discussion this group had--they write the example and then couldn't come to an agreement about which category!  One of the students was trying SO hard to explain that HALF of 310 would be 155...so 149/310 HAD to be less than one half.  The other two were NOT understanding her reasoning!
equivalent fractions
After we worked for a while, I picked THIS fraction to discuss...and with NO computation about finding fractions equivalent to 1/2, we had two very justifiable explanations for why 7/15 is less than one half.  One student came up and explained how it HAD to be less than one half because 7/14 would be one half...and fifteens are smaller than fourteenths--so 7/15 had to be smaller than 7/14.  Pretty slick!

The other argument explained that the "halfway" point of fifteenths would have to be "seven and a half" of them...so 7 of them ha to be less than one half.  Such GREAT math discussions...with no computation. This is a perfect example of why I love concept sorts...so much discourse.  So much math.  So  much engagement.  Tomorrow--we learn the algorithm for generating equivalent fractions...and I think they are more than ready for it!
equivalent fractions
This sort is one of the five sorts in this resource.  Check it out if you are curious!
 Looking for just a single sort to address equivalent fractions?  Check out this one!

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