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I would like questions that demonstrate why a mathematical tool or technique is useful, and which can be used to introduce that idea. Ideally, this would be a compilation of problems organized by the idea they motivate. Does anyone know of such a list? I am thinking of middle school and high school math, but I think such a list would be useful at any level.

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community wiki please. – Dave Penneys Nov 14 '09 at 19:03
I've converted this question to wiki. Please make questions like this community wiki in the future (see for guidelines about what kinds of questions should be wiki). – Anton Geraschenko Nov 14 '09 at 20:48

This is similar to fundamental examples, I recommend you read that post as it was very interesting and at least partially answers your question.

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I suggest you take a look at the Tricki:

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If you're interested in middle school/high school math education, I'm sure you already know about Dan Meyer's excellent blog dy/dan. In case you haven't, he shares (classroom-tested) lesson ideas and media under the label "What Can You Do With This?" (the individual posts are also tagged by subject area).

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In fact, I did not know of this blog. Thank you for pointing it out. There are some very nice lesson plans there! – Peter Tingley Nov 18 '09 at 1:28

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