Mathematical Literacy: Proportional Reasonng and Dimensional Analysis

Fractions … friend or foe?

Our mathematical literacy course started with a variety of topics, some centered around concepts of proportional reasoning.  That work included some unit conversions, just based on a decision to multiply or divide.  Knowing that we would come back to the topic at a more sophisticated level, I did not go beyond the simpler approach in the book … except to mention that we would have a different method later.

Now, we have the different method.  The basic idea of dimensional analysis makes sense to my students — placing units in fractions to produce the result needed.  Of course, doing problems is not easy for all students all the time.  However, it’s clear that students see these fractions as a good thing (for at least this day).

In this work, we’ve been talking about a ‘path’ — the road from the starting unit(s) to the ending unit(s).  I found it interesting that making this explicit seemed helpful to students; this is the more ‘analysis’ part of the method, and I was not expecting students to like it that much.  For the curious, our work began with simple problems involving just one or two conversion factors; the most complicated involved 5 conversion factors.  [This last problem involved converting a rate from mi/hr to cm/sec.]

We included non-linear units (ie, area and volume) which led to “a foot is not 12 inches”.  This, most likely, did not get understood that well — and I’ll see on the quiz at the start of the next class.

It’s possible that students really do ‘get’ the idea of dimensional analysis; that would be a good outcome!  I also hope that success with this type of fraction work does NOT lead to false generalizations to other fraction work (non-product patterns).  Within our math lit course, dimensional analysis is a step in our proportional reasoning topics — and this seems to be working.

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1 Comment

• 

  By schremmer, April 12, 2013 @ 1:37 pm

  Dimensional Analysis is a big word and one should first realize a few things:

  A. That there are two very different situations:
  1. When we are dealing with collections of items which we represent absolutely by number-phrases, that is by (numerator,denominator) pairs e.g. 3 apples.
  2. When we are dealing with amounts of stuff which we can only represent relatively, that is only after we have decided on a unit amount of stuff e.g. 3 quarts of milk.

  B. But that we can use (decimal numerator,metric denominator) in both cases, e.g. 2.3 KiloBytes and 2.3 KiloMeters. However, 2.3456 KiloBytes does not make sense while 2.3456 KiloMeters does.

  C. That, sooner or later and in both cases, we need to involve the denominator dual to the original denominator e.g. cents/apple. Then, co-multiplication is rather nice: 3 apples x 5 cents/apple = 15 cents. (And that is where dimensional analysis really starts.)

  D. That there is no real reason not to deal immediately with spaces, e.g. the space of “baskets” together with the dual space of “unit prices”.

  Regards
  –schremmer

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