This problem sparked quite the discussion about how to go about solving these problems. Of course, as college students we knew how to turn each set into fractions with common denominators, but when trying to reason through it without using fractions, we all struggled a bit.

Some of the methods we attempted was comparing how blue they were in terms of how many they would need to be all blue. So in problem B1, A would need 1 more blue and B would need 3 more blue, so A is more blue. We also attempted to "cancel out" similar containers in each set. So for B4, two blue of each set would cancel out as well as two clear, so the top would be left with 1 blue and the bottom would be left with nothing. So we argued that set A with the one blue would be more blue than set B which has nothing. This method, although seeming intuitive to some of us, caused lots of discussion amongst the class. Especially in the following problem,

We canceled out three blues and two clears from each set, but then were left with nothing in A and a blue and a clear in B. From our logic, B would probably be more blue since a blue and a clear could mix to be a half blue which is more than set A which has nothing. However, when you compute the fractions, A is actually .03 more blue. Our intuitive method of canceling-out failed. But why?

Let's take problem B5 and see why canceling out doesn't fit with fractions. A= 3/5, B=4/7. Cancel out two blues so now A=1/3, B=2/5. If we just stop here we can already see that we've violated the rules of fractions since 3/5 does not equal 1/3 and 4/7 does not equal 2/5! Really, the only way we can solve this problem is by introducing fractions and being able to compare them with common denominators or by turning them into decimals. So once we reveal problems that cannot be solved without fractions, hopefully students will be more motivated to understand fractions. Without the context of fractions representing parts of wholes, it is difficult for students to comprehend the use of fractions. And without the frustration of failing to consistently solve a real world problem, students will have little motivation to learn the sometimes baffling, but important, concept of fractions.

given what we've done with fractions since, do the blue & clear fit into the same idea? Or is the proportion of blue and clear different? Should we be thinkng of blue/clear or blue/total, clear/total?

ReplyDeleteclear, coherent, content, complete +

consolidated: what did this exercise ultimately do for your fraction understanding? Or what's the implication/importance of this problem in clarifying what fraction understanding is? Or give this experience, what comes next for students?

Reading through this solidified my understanding of the problem that we worked on through class. A question that comes to surface for me is whether this holds true for all contexts relating to fractions or is there a way to use the cancelling in specific problems? I don't even know if I know the answer to this question but I was just thinking about it and wanted to put it on paper. Overall, I think this post was very good except I want to know what the exercise in class did for you. (Also, you should consider changing your background.)

ReplyDeleteNo need to change the background haha! I actually came to the opposite conclusion in class when we brought up the cancelling three blue beakers and two white beakers off of each. I think this method actually does work. When B is left with one Blue beaker and one white beaker that is equivalent to .5 or a 1:1 ratio between blue and clear beakers. When you consider that A is a 3:2 ratio of blue to clear, the 1:1 ratio you are left with is a smaller ratio than 3:2, therefore A has a greater concentration of Blue dye than B. The additional one blue and one clear beaker in B decrease the concentration of blue and make it closer to .5 than .6. Though it is a more difficult concept to understand than even a common denominator, your method works when you cancel out A from B!

ReplyDelete