From: Pat Ballew (for Teacher2Teacher Service)
Date: Dec 08, 2001 at 10:55:18
Subject: Re: System of equations

Linda,
 I can not speak about your students, but I found that many students who had
difficulty with this type of problem expected to look at math problems
and "know" what to do directly to get a solution.  I think a power of good
mathematical problem solvers is that they take problems that seem different
from their previous experience and "play around" with the information. I
always encouraged my students to do the same thing.

 I don't teach students how to solve any particular TYPE of problem, but I try 
to teach them how to approach problems they don't know how to do with an 
exploratory attitude.

 In this problem, I would ask students to make a table showing something
about the amount of paper produced.  This will immediately produce MORE
questions at first. "by whom?"  "in how long?" to which I usually shrug and
say, whichever is important. Eventually I hope students may stumble across
something like
  DAYS              1        2        3      .........  and I would add N
Plant 1  Med gr    800      1600   2400                            800N
         High gr   300       600    900                            600N
Plant 2  Med gr    200       400    600                            400N
         High gr   700      1400   2100                            700N
both     Med gr    1000     2000   3000
         High gr   1000     2000   3000

We need 1700 med and 2200 high  Obviously we have more than enough of both
after three days, so our answer should require something less than six
working days for the factories.

  At this point we might ask if the factories have to work a full day?  If
both factories have to work the same amount of time?  all of these are
factors in a students understanding of the problem. the problem PROBABLY
assumes that the answer to both of these questions is NO, and so we ask...

    What shall we call the number of days factory X worked   (and pray
someone says X or A or a variable of some sort)

    How much of each product will they make in X days... (and hopefully they
can replace N with X and get 800 x and 600 x for the two types...

   What shall we call the number of days factory Y worked  (now answers will
flow more freely... let's use Y)

    How much of each product will THEY make in Y days and we proceed to
400Y and 700Y

    Now for the hardest part, seperating the variables in their minds.... 
"And how much of the medium grade did both plants make together?"  At this 
point many students will see that it is 800X + 400Y; Others will leap to 1700, 
the given from the problem.  With luck, at least one of each answer will come 
from the class, and you respond, "Both answers are right, so what does that 
mean?" and hopefully with a little prodding they will see that if both answers 
are right, then 800X + 400Y must equal 1700.

  From here it should be downhill, and instead of walking them through it, ask 
them to suggest what questions we might ask next.  I think this point where 
they reflect on what they know, generate information about what they know, and 
constantly keep an eye open for what they want to know, leads to strong 
problem solvers.

  I always try to get students to focus on two things when they encounter hard 
problems
1)  What can I calculate with this information
2)  What would I like to know to be able to answer the question

If there is stuff in 1) that will help in two, find it.  Each part helps to
develop the other.

This is sort of a home-grown approach, based on my own problem solving
method, and it may be of no use to you, but if you have nothing else to work
with, give it a try.


 -Pat Ballew, for the T2T service

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