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.
For groups finishing early, I ask them to write their OWN examples for each category...
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!
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!