Class Groups, Fractions, Confusion

This is a very long rambling post. Unfortunately I talk my way around an issue or problem and seldom straight through. My wife and I spent the early part of the evening collecting the materials I'll need for tomorrow's science lesson on making predictions and density. I'll also be teaching or presenting it on June 15 to the same entire group that was present in Dr. Olsen's hijacked classroom today.

Real Life Division or Classroom Groups
The human brain loves to identify patterns. If a 5th grade classroom teacher has divided her students into Orcas, Jellyfish, Dolphins, and Sea Urchins and there are 4 Orcas, 3 Sea Urchins, and the rest of the class is split between Dolphins and Jellyfish, what does that mean? As a sub in that class, I tried to piece together this grouping, came to my own conclusions, and explicitly asked the students. The Dolphins and Jellyfish said the 4 Orcas were troublemakers, but I knew this couldn't be it, because one of the most responsible students in the class was an Orca. The names also connoted something. Were the Dolphins all helpful and the Orcas predatory in the classroom? Are the Sea Urchins prickly students? Do the Jellies just float along?

In my Practicum 2 class, the teacher hands out poker chips to divide her class up to differentiate her math lessons and work with small groups. I became overly-focused on the fact that red chips seemed to connote the bright students who "get it", while "white chips" would spend time with the teacher being retaught. Preparing for my own lesson, I tried to clarify: what would happen if I gave a red chip student a blue chip or a white chip? It seemed the white chips held a stigma (to me at least). This puzzled my teacher. None of the kids, she said, ever complained about getting a white chip. Instead they are excited; they got to spend time with the teacher and oftentimes played math games with her. She happened to bring my chip-focusedness up with our site facilitator in a meeting last week, but the site facilitator too started reading into the chip colors, just like I did. When I told my wife about it, she too started making assumptions about what the color of the chips meant.

Unequal Fractions
Now that my third graders are onto fractions - and now that I'm paying closer attention to the specific vocabulary and terms used in the classroom - I keyed into the whole idea (bad pun) of fractions being equal parts. A pizza cut into 8 slices has 8 near-identical slices. Each slice is a congruent region to every other slice.

Ok, going off on a tangent here (better math pun): how confusing is it to use congruent to describe two shapes that are the same size and shape, yet we also can talk about lines being congruent? I've just had to do a little research to make sure I'm not even making up the idea of congruent lines vs. congruent shapes. Ah, they both can be congruent because it means "coinciding exactly when superimposed"

Anyways, getting back to fractions. Fractions are a way of dividing a whole into parts. A way? A representation? Yes, this is going to get long. Unequal fractions is apparently an actual term for something. I just mean fractions with unequal parts.

For example:
Dividing a 20 person class into 4 groups, normally would have 5 people in each group, but what if:
Red Group: 7
Blue Group: 3
Yellow Group:5
Green Group: 5

So we could talk about one fourth of the groups being confused about a concept, if all of Red Group does not get the idea of what I'm writing about here, because their group is ONE QUARTER of the class. But what if 2 members of Red Group do get it! They are 2/7ths of Red Group. Essentially I'm doing division of fractions, I guess. 2 sevenths of 1 fourth. But that's really convoluted. I'm talking about 2/20 of the class. I thought I was DIVIDING the fractions, but I was just multiplying them. I picked easier numbers. I know if one student out of the 20 does something, he is 1/20th the class. And if I say 1/5 of Green Group did x, I really am saying 1/5 of 1/4, so 1/5*1/4... Hmmn.

That doesn't make much sense, but it does highlight my confusion about the nature of fractions.

In my mind at least, fractions and decimals serve the same purpose. 0.25 is 1/4. But 0.25 makes no assumptions over how the other 0.75 is divided up. But 3/4 does seem to! What's going on here???