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Last year, I taught GCD and LCM and then gave my students word problem relating to these concepts ("Two runners with given different speeds; when will they meet again?", "Having three kinds of flowers with different numbers, we want to make similar bunches of flowers. What is the size of the largest one?", etc.)

Some of the students didn't "see" GCD and LCM in these problems and they solved them without reference to these concepts. How can I help them?

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    $\begingroup$ If you want to test GCD and LCM, rather than general problem solving ability, then the problems should be designed so that they would be extremely difficult to solve without making use of GCD or LCM notions. Also, I suspect most any reasonably strong student can probably "reinvent the wheel" on problems such as these. I did this on standardized tests (during the 1970s) before I learned specifically about how GCD and LCM can be used, which I believe I first saw in an undergraduate level number theory book while browsing math books in the library while I was an undergraduate. $\endgroup$ – Dave L Renfro Nov 4 '17 at 15:08
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The premise of your question seems to indicate an undesirable precedent. When the students are able to solve a problem correctly it's immaterial what approach they followed. In fact, giving positive feedback to those who conjure creative methods (instead of following conventional approach taught in the class) is imperative.

But, to be fair, your question might be pertaining to just the pedagogy of teaching these 2 concepts. I suggest the following in that case:

  1. Appreciate the student for solving the question without using LCM/GCD
  2. Ask the student to solve the same question by applying LCM/GCD to let him see by himself the application of LCM/GCD
  3. Whatever approach the student came up with, must be the same as LCM/GCD. Facilitate the student to realise this.
  4. Unless the student's approach is more efficient (which I doubt it will be) let the student appreciate the conventional approach.
  5. As a last resort frame the word problem to force the student to use LCM. For example, in the question "Two runners with given different speeds; when will they meet again?", I figure some students must be writing down the sequences and looking for the next match. In this case you can simply increase the number of runners and the speed as well.

To acheive your goal of letting the students see LCM/GCD, ultimately, they need a lot of practice on problems on these same concepts (preferably varying question types). This is how anyone gains intuition of concepts in maths.

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