## Friday, February 6, 2015

### Fractions

To introduce the dreaded concept of fractions, our class decided to do exercises that would push the subject and necessitate the learning of fractions.  One such problem that isn't easily explained without the use of fractions was the Mixture Blues problems, some of which can be found in the image above.  Basically, in each container there is either blue dye or water and you have to find out which combination is more blue, A or B.
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.