I Why does the answer key sometimes have a different form compared to my solution?

[PeroK]

I was not referring to Statistics instruction or to practices there. I was referring to instruction in Trigonometry.

It's all mathematics.

 

[symbolipoint]

I have not called anyone ”my professor” in decades. As a university professor I’d happily argue with the people making such bad standards.

So what is this? Cultural difference among the various universities' departments?

 

[symbolipoint]

Still the practices as they may be conducted or expected is the truth. Try arguing about this with your professors and see how much progress you make.

And I see your reaction, but as I say that is (or for sure was) the truth, not kidding. Professors were often enough, busy. Their graders were busy. If a certain form of an answer was specified, then that was what we needed to use. If not followed, then less credit for an exercise. If answer in the back of book was in a different form, then we knew what to do. If grader checks students homework and the work did not conform to a specified outline, then either less credit or no credit; if professor gave a test or quiz and said, "Do not do computations! Give answer in symbolic form only!", then students who did not follow received no credit. If prof. gave test or quiz and said, "round all answers to the nearest tenths place", then any answer not so reported received no credit. If computer science professor expected exercise assignments turned into include data table and flow-diagrams, he meant it. When students were given low scores on such assignments and tried to ask the professor about this, students were cut-off from receiving those discussions or explanations. Most of these professors were not lenient.

 

[Orodruin]

Professors were often enough, busy. Their graders were busy.

Generally I have little pity left over for this. Yes, I am generally quite busy, but quality in teaching is one of the university's main pillars and students should expect better.

 

[Office_Shredder]

I think requiring a rationalized denominator is part of setting a standard for students who don't have good taste in what answers can look like.For example if the problem is to evaluate $\cos(\pi/3)$, is $\sin(\pi/6)$ an acceptable answer? It's equivalent to the right answer, so why not?

Or let's say you have solve $x^2-9=0$. Your student carefully applies the quadratic formula and writes

$\frac{0 \pm \sqrt{0+4*9}}{2*1}$That's their final answer. You have to assign this answer a grade that reflects their understanding of the concepts of Algebra. What do you give them? An A? That's what people are kind of arguing for here.

Write all answers as rationalized fractions is strict, but at this level you need a policy to demonstrate if people understand what they've written or if they're just regurgitating symbols back at you.

 

[PeroK]

And I see your reaction, but as I say that is (or for sure was) the truth, not kidding. Professors were often enough, busy. Their graders were busy. If a certain form of an answer was specified, then that was what we needed to use. If not followed, then less credit for an exercise. If answer in the back of book was in a different form, then we knew what to do. If grader checks students homework and the work did not conform to a specified outline, then either less credit or no credit; if professor gave a test or quiz and said, "Do not do computations! Give answer in symbolic form only!", then students who did not follow received no credit. If prof. gave test or quiz and said, "round all answers to the nearest tenths place", then any answer not so reported received no credit. If computer science professor expected exercise assignments turned into include data table and flow-diagrams, he meant it. When students were given low scores on such assignments and tried to ask the professor about this, students were cut-off from receiving those discussions or explanations. Most of these professors were not lenient.

It seems to me that, if what you say is true, some of your professors were suffering from psychological disorders. Which may be a good reason not to be unduly influenced by them and not to promote their somewhat sadistic methods or excessive pedantry as a productive way to teach mathematics.

 

[symbolipoint]

It seems to me that, if what you say is true, some of your professors were suffering from psychological disorders. Which may be a good reason not to be unduly influenced by them and not to promote their somewhat sadistic methods or excessive pedantry as a productive way to teach mathematics.

Refering to the quote of message in post #36,
Most of those professors actually did teach well, but had some very strict instructional standards. They were tough for some of the students. Some of the reasons could be supportable. As you might imagine, some of the students did complain; to the professors during class time, and among themselves in and out of the classroom.

 

[symbolipoint]

Or let's say you have solve x2−9=0. Your student carefully applies the quadratic formula and writes

0±0+4∗92∗1That's their final answer. You have to assign this answer a grade that reflects their understanding of the concepts of Algebra. What do you give them? An A? That's what people are kind of arguing for here.

Write all answers as rationalized fractions is strict, but at this level you need a policy to demonstrate if people understand what they've written or if they're just regurgitating symbols back at you.

The example you gave would often be a factorization fact exercise for some Algebra 1 students who may have not yet been in Algebra 2; and most typically the students would learn about general solution for quadratic equation in Algebra 2. So that example could be a situation in which instructor expects the use of factorization and not use general formula solution.

(note: the formatting for part of the quoting did not work properly, to show that quadratic formula solution of your example.)

 

[Mark44]

(note: the formatting for part of the quoting did not work properly, to show that quadratic formula solution of your example.)

Just copying or quoting a section of text that contains LaTeX doesn't work

 

[Mark44]

Or let's say you have solve
$x^2-9=0$. Your student carefully applies the quadratic formula and writes

$\frac{0 \pm \sqrt{0+4*9}}{2*1}$

This discussion of whether equivalent answers are allowed or not puts me in mind of Randall Munroe's response after a lengthy and fruitless discussion with a Verizon customer service drone over a bill for $.002. In case the name is not familiar, he's the author of the XKCD blog/cartoons.

The amount on the check is $$\left(0.002 + e^{i\pi} + \sum_{i = 1}^\infty \frac 1 {2^n}\right) \text{ dollars}$$

 

[symbolipoint]

Just copying or quoting a section of text that contains LaTeX doesn't work

Yet in post #40, you did make it work. How?

 

[Mark44]

Yet in post #40, you did make it work. How?

I quoted the section of text up to the start of the LaTeX, and then opened the edit window to copy the unrendered LaTeX and paste it onto the end of what I had quoted. As a mentor I'm able to edit posts, but that capability probably isn't available to regular members.

If you use the Quote feature to quote text with LaTeX in it, you'll undoubtedly need to redo any LaTeX that was there. Since you didn't do this, that's why the quadratic equation solution that you quoted came out as it did.

 

[PeroK]

As a mentor I'm able to edit posts, but that capability probably isn't available to regular members.

Perhaps it's an idea for you to go round all the threads on PF and edit the absurd and inexplicable instances of $\dfrac 1 {\sqrt 2}$? Since it is such an abomination to those with heightened mathematical sensibilities.

 

[Mark44]

Perhaps it's an idea for you to go round all the threads on PF and edit the absurd and inexplicable instances of $\dfrac 1 {\sqrt 2}$? Since it is such an abomination to those with heightened mathematical sensibilities.

No, thanks. I don't have a problem with $\frac 1 {\sqrt 2}$ -- I was just offering a plausible explanation for why elementary algebra textbooks spend so much time on simplifying these types of expressions.

 

[WWGD]

high school algebra text are based of Euler's Elements of Algebra. Where Euler does not rationalize the denominator. Who are we to argue wit Euler.

No wonder I'm a fan of the Houston Eulers.

 

[Mondayman]

If you are learning about rationalizing denominators I can understand why. But as long as the answer is correct it shouldn't matter whether the radical is in the denominator or numerator. Or if it's written with exponents instead. I actually preferred doing so in my calculus and DE courses sometimes, I felt like I could read it more clearly. Instructor had no issue.

 

[SammyS]

To the best of my knowledge, no one writes $\dfrac{\sqrt \pi}{\pi}$ instead of $\dfrac 1 {\sqrt \pi}$.

Square roots appear in the denominator all over statistics, quantum mechanics, and the gamma factor in SR!

Either way, neither of these denominators is rationalized.

$\dfrac{\sqrt \pi\,}{\pi}$ , $\dfrac 1 {\sqrt \pi\, }$

 

[PeroK]

Either way, neither of these denominators is rationalized.

$\dfrac{\sqrt \pi\,}{\pi}$ , $\dfrac 1 {\sqrt \pi\, }$

Ah, so they are both wrong!

 

[WWGD]

Sorry to tell the OP that there are infiitely -many equivalent ways of writing a mathematical expression. You're lucky you're dealig with essentially numerical answers. If the answer was more complex, such as , e.g., a Fourier Series solution, there would be many more different (all correct) expressions for it.

 

[Mondayman]

The most recent Numberphile video is about that. You can integrate functions using different methods and end up with different answers. Shows why the constant of integration is important. I would recommend it to anyone learning calculus.

 

[Office_Shredder]

Ah, so they are both wrong!

Right. It should be $\frac{\sqrt{154}}{22}$

 

[symbolipoint]

Here is maybe another answer:
Because pages cost money and ink to print different answers and different parts of answers cost money.

 

[Orodruin]

Either way, neither of these denominators is rationalized.

$\dfrac{\sqrt \pi\,}{\pi}$ , $\dfrac 1 {\sqrt \pi\, }$

$\dfrac{1/\sqrt\pi}{1}$

Fixed

Anyway, I’d expect teachers to be rational…

 

[Mark44]

$\dfrac{1/\sqrt\pi}{1}$

Fixed

But you still have a denominator (in the numerator fraction) that is irrational.

 

[Orodruin]

But you still have a denominator (in the numerator fraction) that is irrational.

You’re welcome to rationalise it

 

[FactChecker]

Clebsch-Gordan not my areaMy statement (post #7) was about Trigonometry instruction; not other topics.

This is the way it was presented in the precalculus books I taught from in the '70s. I liked it because it made the pattern: $sin(0)=\sqrt{0}/2; sin(30)=\sqrt{1}/2; sin(45)=\sqrt{2}/2; sin(60)=\sqrt{3}/2; sin(90)=\sqrt{4}/2$. The students seemed to like that.
But teaching conventions like that may have changed a lot since the '70s.

 

[symbolipoint]

This is the way it was presented in the precalculus books I taught from in the '70s. I liked it because it made the pattern: $sin(0)=\sqrt{0}/2; sin(30)=\sqrt{1}/2; sin(45)=\sqrt{2}/2; sin(60)=\sqrt{3}/2; sin(90)=\sqrt{4}/2$. The students seemed to like that.
But teaching conventions like that may have changed a lot since the '70s.

Interesting representation in keeping with denominator of 2. I really never saw it done like that.

 

[romsofia]

I would always use this following explanation to high schoolers when I tutored for why they "have" to rationalize their denominators. I personally didn't care, but a lot of teachers would take off points if they don't.

In human history, people first started counting things, like how many cows they own: 1,2,3... These are called the "natural numbers" because you naturally start at one. We next realized that hey, "no cow" should also have a number, which we call 0 and we have a new set of numbers called "whole numbers". Naturally, debt came into the number systems because you owe me a cow (-1) or two cows (-2), and now we have a whole new set of numbers called the integers (negative and positive whole numbers). Well, next humans started discussing parts of the whole. If we have 5 cows, and I take 2 of them, I now have 2 of the 5 cows, or to make life simple, I have $\frac{2}{5}$ of the cows. Number wise, we now have numbers between numbers, and we call these the "rational numbers" because they are just ratios, or parts of wholes. Notice how square roots haven't been "invented" yet? So, your teachers are just following history, and because it's their class, we will too. I don't actually know the history of numbers that well, but the story worked for most of them!

Now, the math reason (Which I do for students in pre-calc/algebra 2)... rational numbers are defined to be $\frac{a}{b}$ where $a,b \in \mathbb{Z}, b \neq 0$ so, yes, your teacher is correct to take points off your test because you're technically wrong to keep it as $\frac{2}{\sqrt{5}}$. But then I also tell my students that, technically, you can't even write $\frac{2\sqrt{5}}{5}$ because $2\sqrt{5} \notin \mathbb{Z}$ and you should have to write it as $\sqrt{5} \times \frac{2}{5}$ so if you ever want to be petty, feel free to bring that up in class.

So, to OP, that's why the answer key in a math book won't have square roots in a denominator, because by definition of rational numbers, that number doesn't "exist". Although, as you can see, most of us won't really care (which I also tell the students I use to tutor if I helped them with these concepts), but when in school grades matter, so best not to let points get away!

 

[PeroK]

so, yes, your teacher is correct to take points off your test because you're technically wrong to keep it as $\frac{2}{\sqrt{5}}$.

It's a convention, nothing more. Real numbers have reciprocals too!

What about calculators and computer language syntax? If you program $1/0$ you get a error. Are you seriously saying that if you program $1/\sqrt 2$ a computer should return a computational error?

How many students are turned off mathematics by this sort of pointless pedantry?

 

[martinbn]

How many students are turned off mathematics by this sort of pointless pedantry?

I would guess none. If that is all it takes to turn someone off mathematics, he wasn't into it in the first place.