Q&A #7470

System of equations

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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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