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$$ \binom{12}6 = \frac{12\cdot11\cdot10\cdot9\cdot8\cdot7}{6\cdot5\cdot4\cdot3\cdot2\cdot1} = 924. $$ Sometimes it's hard to talk students out of computing both the numerator and the denominator in this expression and then dividing.

I can think of three reasons for this:

  • It takes at least some effort to learn that one can simplify things like this by canceling.
  • Elementary school pupils are told $3\times4$ means "multiply $3$ by $4$", i.e. "$\times$" is a verb in the imperative mood. Even in elementary school, I reminded myself every time I heard this that $3\times4$ is a noun, but I've never seen any evidence of anyone else doing that.
  • Calculators are anesthetics. Students approach math problems with great anxiety, and, confronted with an expression like the one above, frantically reach for their calculators the way a drowning person reaches for anything that floats. A calculator WILL get the student out of the terrible predicament that is a math problem and calculators are INFALLIBLE. If $8/3=2.666666\ldots$ and a calculator says $2.666\times51=135.966$ then that is EXACT and INFALLIBLE. Whoever says the answer is $136$ is rounding it. If $\pi=3.14$ and you find that $3.14^3\cong30.959$, and you want to be more accurate, just add more of the digits that the calculator shows you when you find $3.14^3=30.959144$, and if the exam question says you are to give an EXACT answer, that means give all the digits that your calculator will show you. Calculators COMPLETELY OBVIATE ALL NEED TO THINK ABOUT ANY QUESTIONS THAT THIS PRESENT PARAGRAPH MIGHT SUGGEST. Anyone who doubts that is a lunatic.

In simplifying rational functions, plainly one should cancel first, but with rational numbers, it's not always clear to student what the advantage is, if any. Are there examples not involving algebra, but only arithmetic, that are as cogent to newbies as are examples of simplifying rational functions?

  • 0
    Why don't you tell us how you really feel about calculators?2012-10-01
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    @KevinCarlson : I use calculators and I like calculators, but I don't bow down in idolatrous prayer to them.2012-10-01
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    Personally, I believe calculators should stay out of math class for the most part. Maybe for trigonometry and such as it's hard to appreciate something like $5\cos 55^o$. But if they're using a calculator anyway, how does cancelling from a numerator and denominator as above simplify it for them?2012-10-01
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    I've found another good rule of thumb is, whenever student laziness is suggested as a cause (like here: it takes effort to learn cancellation and they haven't put any effort in) then one should also consider if fear of making a mistake is at fault. No doubt, many such students have been duly punished for some cancelling mistakes.2012-10-01
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    If you linebreak in list mode, and add a double-space in the new paragraph, it will be aligned into the list. You may want to use this information to make the second bullet readable.2012-10-01
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    It's often better NOT to cancel first. Canceling is typically ad hoc and introduces a chance for errors. Also, if I'm doing such a calculation as part of a search for a general solution to something, I want to see the denominator in its natural form. Easier to look up in OEIS and so forth.2012-10-01
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    Once I asked on a test for the students to write the prime factorization of $2^{14}3^55^37^{12}(13)^2(23)^3-2^73^55^67^{12}(13)^2(23)^3$. And every one of them tried multiplying the two terms out!2012-10-02
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    @I.J.Kennedy I'm interested in seeing an example of a case you are thinking of.2012-10-04
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    I would play on the idea that "Mathematicians are lazy" and as such look into making problems simpler with this as only a part of the whole thing.2012-11-18
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    Don't cancel; divide.2013-11-14

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