Example of a student learning*

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yeyui
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Example of a student learning*

Postby yeyui » Thu Jan 06, 2011 8:46 pm UTC

On a recent exam, I asked a question with the answer [math]\frac{\sqrt{3}+1}{\sqrt{3}-1}.[/math] Several students, including some with otherwise correct answers, made the mistake of either writing [imath]\sqrt{3+1}[/imath] and [imath]\sqrt{3-1}[/imath] or, very ambiguously, [imath]\sqrt{3+}1[/imath] and [imath]\sqrt{3-}1[/imath].

On the papers that exhibited this error, I wrote comments such as
Careful! [imath]\sqrt{3+1} \neq \sqrt{3}+1[/imath]. They are different numbers.
I followed up by addressing this error in class. I wanted to address the problem to everyone, since I believe that notational issues like this often actually reflect serious misunderstandings and not just sloppiness or laziness. I also just wanted to point out which version was correct, and which was incorrect for this particular question, since the entire class was about to retake the test and I had given credit for both versions.

One student tried very hard to take this warning to heart when she encountered a problem that was almost identical. On her retest, where the correct answer was [math]\frac{\sqrt{3}-1}{\sqrt{3}+1},[/math] she very carefully wrote [math]{\sqrt{3}-1} \neq {\sqrt{3}+1}.[/math]


*learning --- acquiring new knowledge, behaviors, skills, values or preferences. The knowledge need not be true, nor the behaviors useful, the skills efficient, the values internally consistent, nor the preferences socially accepted.

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Re: Example of a student learning*

Postby Yakk » Thu Jan 06, 2011 9:22 pm UTC

It is true! -1 and 1 are not equal!

You have passed on valuable knowledge to your student. Knowledge you didn't even know you where transmitting.

Feel proud.
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Re: Example of a student learning*

Postby Mindworm » Thu Jan 06, 2011 10:16 pm UTC

Although in some problems I had to explicitly adress 1=-1 and 1 [imath]1 \neq -1[/imath] and do a different proof for both (there are also some statements that fail in this case). Stupid fields with characteristic 2, always being special cases.
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Re: Example of a student learning*

Postby Eastwinn » Thu Jan 06, 2011 10:36 pm UTC

You didn't require them to rationalize the denominator?
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Re: Example of a student learning*

Postby skeptical scientist » Thu Jan 06, 2011 10:58 pm UTC

Eastwinn wrote:You didn't require them to rationalize the denominator?

I never understood why \(\frac{(\sqrt{3}+1)^2}{2}\) was preferable to \(\frac{\sqrt{3}+1}{\sqrt{3}-1}\). Sure, it can sometimes be easier to deal with (like when performing a mental estimation) but often it is just wasted effort to convert from one to the other.
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Re: Example of a student learning*

Postby achan1058 » Thu Jan 06, 2011 11:02 pm UTC

skeptical scientist wrote:
Eastwinn wrote:You didn't require them to rationalize the denominator?

I never understood why \(\frac{(\sqrt{3}+1)^2}{2}\) was preferable to \(\frac{\sqrt{3}+1}{\sqrt{3}-1}\). Sure, it can sometimes be easier to deal with (like when performing a mental estimation) but often it is just wasted effort to convert from one to the other.
That's because [imath]2+\sqrt{3}[/imath] is preferable compared to both of them.

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Re: Example of a student learning*

Postby Yakk » Thu Jan 06, 2011 11:18 pm UTC

The question already implied the answer. If you let people get away with answers that are equivalent to the actual answer, they could just rewrite the question, which is madness.

So clearly we should require exactly one answer, and we should make it as far away from the question as possible.

Wait
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Re: Example of a student learning*

Postby skeptical scientist » Fri Jan 07, 2011 12:19 am UTC

achan1058 wrote:
skeptical scientist wrote:
Eastwinn wrote:You didn't require them to rationalize the denominator?

I never understood why \(\frac{(\sqrt{3}+1)^2}{2}\) was preferable to \(\frac{\sqrt{3}+1}{\sqrt{3}-1}\). Sure, it can sometimes be easier to deal with (like when performing a mental estimation) but often it is just wasted effort to convert from one to the other.
That's because [imath]2+\sqrt{3}[/imath] is preferable compared to both of them.

Yes, but the former form is all that is required to rationalize the denominator, and if the numerator had √5 instead of √3, you wouldn't have been able to do better anyways.
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Re: Example of a student learning*

Postby Eastwinn » Fri Jan 07, 2011 12:40 am UTC

skeptical scientist wrote:I never understood why \(\frac{(\sqrt{3}+1)^2}{2}\) was preferable to \(\frac{\sqrt{3}+1}{\sqrt{3}-1}\). Sure, it can sometimes be easier to deal with (like when performing a mental estimation) but often it is just wasted effort to convert from one to the other.


I prefer rationalized denominators because it's rarely immediately obvious to me that forms like the two in your post are actually equivalent. If I make the habit of rationalizing the denominator, I avoid mistaking the two as unequal. This is something I find useful in math class, but not necessarily when I'm just playing around, so I consider it to be formal. I imagine that it makes it harder on a grader.
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Re: Example of a student learning*

Postby Cleverbeans » Fri Jan 07, 2011 5:25 pm UTC

I always assumed rationalizing the denominator was common before calculators because if you want to calculate an estimate of the quantity by hand it's much easier to do the long division after rationalization, and that it is still taught because teachers simply don't know any better. I would think if you took decimal approximations and divided them directly that you may introduce cumulative errors over time, and certainly be prone to more mistakes in the calculation. I certainly never enjoyed taking approximating square roots by hand anyway.

I think one could argue that it's a valuable technique to know for algebraic manipulation, and to motivate the complex conjugate so I can understand why they would still teach it. Beyond that, I no longer rationalize the denominator unless there is some particularly good reason to favor that representation of the number.
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Re: Example of a student learning*

Postby jestingrabbit » Fri Jan 07, 2011 5:49 pm UTC

When I first encountered it it made me think about how if you just worked with rational numbers and a single surd, like root 2, then you can always write any expression involving multiplication and addition as a sum of a rational and a rational times the surd. I say keep it, because its pointless busy work that can reveal something pretty amazing.
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Re: Example of a student learning*

Postby yeyui » Wed Jan 12, 2011 5:27 pm UTC

For the record, I specifically instructed them not to simplify at all. The item was testing student knowledge of the definition of trigonometric ratios in the context of right triangles. In addition to making grading easier, this reduced the number of factors affecting the score for the item.

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Re: Example of a student learning*

Postby MartianInvader » Wed Jan 12, 2011 9:24 pm UTC

skeptical scientist wrote:
achan1058 wrote:
skeptical scientist wrote:I never understood why \(\frac{(\sqrt{3}+1)^2}{2}\) was preferable to \(\frac{\sqrt{3}+1}{\sqrt{3}-1}\). Sure, it can sometimes be easier to deal with (like when performing a mental estimation) but often it is just wasted effort to convert from one to the other.
That's because [imath]2+\sqrt{3}[/imath] is preferable compared to both of them.

Yes, but the former form is all that is required to rationalize the denominator, and if the numerator had √5 instead of √3, you wouldn't have been able to do better anyways.

::wrinkles brow:: [imath]13 + \sqrt{5}[/imath]?
Last edited by MartianInvader on Thu Jan 13, 2011 5:51 pm UTC, edited 1 time in total.
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Re: Example of a student learning*

Postby skeptical scientist » Wed Jan 12, 2011 11:17 pm UTC

MartianInvader wrote:
skeptical scientist wrote:
achan1058 wrote:
skeptical scientist wrote:I never understood why \(\frac{(\sqrt{3}+1)^2}{2}\) was preferable to \(\frac{\sqrt{3}+1}{\sqrt{3}-1}\). Sure, it can sometimes be easier to deal with (like when performing a mental estimation) but often it is just wasted effort to convert from one to the other.
That's because [imath]2+\sqrt{3}[/imath] is preferable compared to both of them.

Yes, but the former form is all that is required to rationalize the denominator, and if the numerator had √5 instead of √3, you wouldn't have been able to do better anyways.

::wrinkles brow:: 13 + \sqrt{5}?

Emphasis added.
If you rationalize the denominator expression I was indicating, you get $$\frac{\sqrt{5}+1}{\sqrt{3}-1} = \frac{(\sqrt{5}+1)(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)} = \frac{1+\sqrt{3}+\sqrt{5}+\sqrt{3}\sqrt{5}}{2},$$ which is hardly better than the original expression, and you can't simplify further (as you can when both roots are the same).
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Re: Example of a student learning*

Postby Zamfir » Thu Jan 13, 2011 11:19 am UTC

skeptical scientist wrote:which is hardly better than the original expression, and you can't simplify further (as you can when both roots are the same).

I would argue that is definitely better, at least in some contexts. It's a close to a standardized form, making it easier to see whether two numbers are the same or not. As long as there are roots in denominators, or roots of non-prime numbers, there might be answers that are identical but need non-trivial work to recognize as such.

That makes life easier for teachers, but also for students if the result is an in-between step.

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Re: Example of a student learning*

Postby MartianInvader » Thu Jan 13, 2011 5:51 pm UTC

Ah, I thought you meant the numerator of the other expression. All's well then!
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!


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