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I am struggling with teaching my 9th grade kid to solve math problems that are just outside of routine.

For e.g.,

An example problem given by math teacher at school.

x, y, z are in geometric progression

x + y + z = C1

x * x + y * y + z * z = C2

Find x, y, z.

My solution (after a couple of failed attempts) for this involves recognizing some patterns

  1. (x + y + z)^2 = ...

  2. Expressing x, y, z as a/r, a, ar

  3. A bit of manipulation of [1] so that I can reuse x + y + z = C1 and eliminate r from the equation and find a.

How does one teach students to independently solve these problems? Are students "wired" to solve these v/s not?

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    $\begingroup$ We need a lot more information to be able to say anything useful about this. Is your kid 6 weeks into a first course in algebra? If so, then there is almost no chance that most kids at this level could do much on this problem. Is this problem meant as a challenging extra credit problem for top students? If so, then why are you worrying? The title is "How do I teach my kid," but your kid has a teacher. Why do you perceive it as your job to teach your kid? Is the instruction inadequate because of covid? $\endgroup$ – Ben Crowell Oct 6 at 20:36
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    $\begingroup$ Are you doing this because your kid expressed an interest in learning more problem solving outside the classroom or do you have a different motivation (and what is it?) $\endgroup$ – Amy B Oct 8 at 12:33
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I should disclaim I haven't managed to solve this equation yet, even with your description. What were your failed attempts like? What was your own approach to this problem, and can't this approach help your son?

Overall my approach for a problem like this would be:

  1. in general, to solve for N variables you need N independent equations. Here the problem gives 3 variables and 2 equations, this means it is either unsolvable or there is extra information in there that can be made into an equation. In this case, "x,y,z are in a geometric progression" which can be converted into your expression of x, y, z according to r and a.

  2. You can kind of generalize that, many twisty logic problems have fun features like knowing that there is an answer, or that somebody found the answer this or that way, are vital missing clues to finding the actual answer. The general principle there is to extract information from the problem to find the answer, and so when stuck you might ask yourself "is there information given in this problem that I haven't used in finding my answer? If so, is there any way I can use this information alongside what I've already done?"

  3. to get back to equations, if you have N equations with N variables, you want to end up with 1 equation with 1 variable, which you can then solve easily and use that variable to find the other N ones. There are many specific techniques for this which your child might have learned and they can pick which one to choose. But in general it means transforming one equation until it looks like part of another one of the equations, at which point you can plug it in and you'll often have gotten rid of some complexity. In your case, I suppose squaring x + y + z brings it a bit closer to the equation with xx+yy+z*z.

  4. This is similar to the larger information issue, but a teacher will usually present a problem in the context of the larger lesson. Was this presented as one example of problems that can be solved through the techniques being learned right now? In that case it's worth trying those techniques even if it doesn't look like they fit. Did your own resolution of the problem end up using techniques your son did learn (or should have learned) recently? You say it's outside of routine, but maybe that's the "trap" or "hook" of the problem, it's actually in the routine but it's presented differently. Or was it presented as something more out-there or advanced, a fun optional challenge? If so maybe the teacher doesn't really expect the student to solve it in a systematic way, maybe it requires grinding effort (like many failed attempts) or a leap of lateral thinking that they expect some students to manage but not all. If so it might be worth not worrying about too much.

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  • $\begingroup$ The key observation I used in this problem is recognizing the relationship between (x + y + z)^2 and x^2 + y^2 + z^2 given that the teacher had given x + y + z and x^2 + y^2 + z^2. The teacher did solve a problem related to (x + y + z)^2 at one point in a different context, so he may expect children to know how to expand (x + y + z)^2 and solve this problem. I am sure some of the "sharper" kids in the class would have seen the pattern and followed that path. there is some expectation of independent problem solving from the teacher that my kid has a hard time meeting at this time. $\endgroup$ – bhakta Oct 6 at 12:58
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My (serious, practical) advice is to teach your kid by

(1) giving more progressive problems to solve (by progressive, I mean easier than this but not bone simple drill). If you were lifting weights, you wouldn't throw 300# on the bar to start. Try for problems with half the complexity of this exercise, for instance. As sort of 150# on the bar.

(2) Use problems like this as an opportunity to explore, deepen knowledge, more than just the getting it without help. IOW, don't simply look at this as a riddle that he gets or not. But if necessary, share the answer (IOW, he doesn't get the aha of solving it without assistance...sure that would be nice...but nobody can figure out every riddle) and then explain the how to solve it and THEN repeat the exercise with a few similar examples (so he will get these problems in the future). Dick Feynman was great at puzzles and riddles...but he mentioned that a key way to being good at riddles (if you can't just intuit it and make the leap without help) is just to have seen the problem before. DON'T devalue the value in this.

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Take a look at Polya's classic "How to solve it". What you are asking is "how do I teach creative problem solving", and for that, sorry to say, there is no royal road. In any case, it isn't fair to expect your kid to solve a problem you struggle with...

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