Incorrect instruction - split from quadratic

Steven G

Steven G

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firemath said:
You don't need to multiply both sides by 14x.
Here is what you need to do:

[MATH]\frac {1}{2x} = \frac {r-x}{7}[/MATH]
Then, multiplying each side by the other's denominator:

[MATH]7=2x(r-x)[/MATH]
[MATH]7=2rx-2x^2[/MATH]
Can you go on from here?
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continued from https://www.freemathhelp.com/forum/threads/quadratic.120361/#post-482028
What you did is multiply both sides by 14x, which is exactly what the OP said he tried! The OP said he did not know what to do from there. Did you read that part? You also clearly stated that you multiply one side by 7 and the other side by 2x!! First of all that is NOT what you did and 2nd of all, if you did do what you said it will be wrong. You must multiply both sides by the same non-zero expression.
Please only respond to problems you understand. Please
 
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After confrontation by Jomo, I checked my process in Symbolab.

Here's the link.

Jomo, I have tried do do well, and have not always succeeded in answering every question perfectlylike yourself.
 
firemath said:
After confrontation by Jomo, I checked my process in Symbolab.

Here's the link.

Jomo, I have tried do do well, and have not always succeeded in answering every question perfectlylike yourself.
Click to expand...
Symbolab said:

\(\displaystyle fraction\:cross\:multiply:\:if\:\frac{a}{b}=\frac{c}{d}\mathrm{\:then\:}a\cdot \:d=b\cdot \:c\).........This is correct.

but you said:

Then, multiplying each side by the other's denominator: .........This is incorrect.

You must edit your post prior to posting - that is what we preach to the students.
 
What you should have said is

multiplying both sides of the equation by the PRODUCT of the denominators

That is actually the logical basis behind the simplification of cross multiplying.

For some reason preview does not work on my IPad, but I try to edit my posts as quickly as possible. It is possible for those of us who know what we are doing to get what was intended by a typo, but students who are confused should not be asked to do that. Please self-edit.
 
I agree that I did not use the correct terminology. Forgive my error. It was intended to be a "lay" way of describing cross-multiplication.

I seem to have miscommunicated a topic terribly. I apologize, @Richard B.
 
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firemath said:
After confrontation by Jomo, I checked my process in Symbolab.

Here's the link.

Jomo, I have tried do do well, and have not always succeeded in answering every question perfectlylike yourself.
Click to expand...
First of all, it is not true that all my responses are correct. I go to the corner more than anyone else.
I am sorry but I take teaching math very seriously and when you answer so many questions incorrectly I take it personally. In your last response you mentioned cross multiplication. This is a terrible thing to do almost 99% of the times. I only use it to verify if two fractions are equal or not. Instead we multiply both sides by the LCD.

May I please ask you what your background in math is? I am sorry that I am so hard on you but it really irritates me when I see a helper showing something wrong that is obvious and happens frequently.
 
Not a fan of "cross multiply". Just another example of "lay" arithmetic. Learn to multiply. Learn how to find a common denominator. There is just no need to memorize a single-use method in algebra. Stick to basic principles.

My views. I welcome others'.
 
tkhunny said:
Not a fan of "cross multiply". Just another example of "lay" arithmetic. Learn to multiply. Learn how to find a common denominator. There is just no need to memorize a single-use method in algebra. Stick to basic principles.

My views. I welcome others'.
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Absolutely correct. I had to scrub that idea out of my students head. After explaining - why "criss-cross" does not wok always - I had 90% of the students attempting do that that again. So I made a dictatorial rule - if you did not use LCM to simplify fractions - 10% off even if you were right.

I had a revolt in my hand - but I am Khan - I can handle all those uprisings.......
 
Subhotosh Khan said:
if you did not use LCM to simplify fractions - 10% off even if you were right.

I had a revolt in my hand - but I am Khan - I can handle all those uprisings.......
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I bet that you had a revolt on your hand!
The worst I ever did was give a student a -5 on an exam. With one exception his paper was completely blank which would have represented a 0 to me. However the one thing he wrote was something like 7+2 = 12. So I concluded that his paper was worse than a blank paper and gave him a -5. He asked me if in fact it was a -5 and I told him yes. He started to talk but I cut him off and sincerely told him that instead of arguing with me he could be in the library studying or be on his way to my office to ask me for help. He just walked away at that point. I forget how he did on other exams.
 
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Here is why I just hate cross multiplication. 1st reason is that when students do not know what to do they cross multiply. 2ndly, it is almost ALWAY the most inefficient process that I have ever seen.

For example, in solving x/5 = 3/7 if we crossed multiplied we would get 7x=15. Then we divide by 7 and the 7 goes back to where it originally was!! So why did we move the 7? I tell my students that if you just have to cross multiply then just half cross multiply. That is in my example just move the 5 and not the 7.

Cross multiplication makes me think of this example. You have a job (for now) and anytime your boss asks you to do some something you first take the garbage pail by your desk and move it to the office next door for a second and then bring it back to your desk and only then do you do the task that your boss asked you to do. It is completely inefficient, just like cross multiplication! What troubles me the most is that the high school teachers did an excellent job teaching cross multiplication as the students remember it!
 
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tkhunny said:
Not a fan of "cross multiply". Just another example of "lay" arithmetic. Learn to multiply. Learn how to find a common denominator. There is just no need to memorize a single-use method in algebra. Stick to basic principles.

My views. I welcome others'.
Click to expand...
tkhunny said:
Not a fan of "cross multiply". Just another example of "lay" arithmetic. Learn to multiply. Learn how to find a common denominator. There is just no need to memorize a single-use method in algebra. Stick to basic principles.

My views. I welcome others'.
Click to expand...
I do, respectfully, disagree.

[MATH]a + b = c \implies (a + b) - b = c - b \implies a + (b - b) = c - b \implies a + 0 = c - b \implies a = b - c.[/MATH]
That is using the basics. But I bet we all "flip signs" and in a lay way go

[MATH]a + b = c \implies a = b - c.[/MATH]
There are many short-cuts that save time in algebra like the quadratic formula. So long as students clearly understand the reasoning behind a short-cut, I see no reason to discourage its use.

Cross multiplication is just another short-cut, though one easily misapplied.

[MATH]\dfrac{a}{b} = \dfrac{c}{d} \implies ad = bc[/MATH] just skips numerous steps in the non-lay way that is formally justified with

[MATH]\dfrac{a}{b} = \dfrac{c}{d} \implies \dfrac{bd}{1} * \dfrac{a}{b} = \dfrac{bd}{1} * \dfrac{c}{d} \implies \dfrac{ad}{1} * \dfrac{b}{b} = \dfrac{bc}{1} * \dfrac{d}{d} \implies[/MATH]
[MATH](ad) * 1 = (bc) * 1 \implies ad = bc.[/MATH]
The only difference between "flipping signs" and "cross multiplication" is that the latter is a reliable way to clear fractions only in a special case. Students make mistakes when clearing fractions because they forget that cross multiplication does not work generally. This is a reason to make sure they understand why cross-multiplication works and when it does not apply. If they understand clearing fractions generally, then they are ready to understand and use cross multiplication.
 
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Never liked the "change the sign when multiplying with an inequality", either. I was an ivory-towerist stick-in-the-mud at a very early age. :rolleyes:
 
Jomo said:
First of all, it is not true that all my responses are correct. I go to the corner more than anyone else.
I am sorry but I take teaching math very seriously and when you answer so many questions incorrectly I take it personally. In your last response you mentioned cross multiplication. This is a terrible thing to do almost 99% of the times. I only use it to verify if two fractions are equal or not. Instead we multiply both sides by the LCD.

May I please ask you what your background in math is? I am sorry that I am so hard on you but it really irritates me when I see a helper showing something wrong that is obvious and happens frequently.
Click to expand...
I am a math tutor, nothing more. I probably shouldn't help. I'm not even worthy of corner punishment.

I'm sorry if I have offended anyone with my uneducated answers.
 
JeffM said:
I do, respectfully, disagree.

[MATH]a + b = c \implies (a + b) - b = c - b \implies a + (b - b) = c - b \implies a + 0 = c - b \implies a = b - c.[/MATH]
That is using the basics. But I bet we all "flip signs" and in a lay way go

[MATH]a + b = c \implies a = b - c.[/MATH]
There are many short-cuts that save time in algebra like the quadratic formula. So long as students clearly understand the reasoning behind a short-cut, I see no reason to discourage its use.

Cross multiplication is just another short-cut, though one easily misapplied.

[MATH]\dfrac{a}{b} = \dfrac{c}{d} \implies ad = bc[/MATH] just skips numerous steps in the non-lay way that is formally justified with

[MATH]\dfrac{a}{b} = \dfrac{c}{d} \implies \dfrac{bd}{1} * \dfrac{a}{b} = \dfrac{bd}{1} * \dfrac{c}{d} \implies \dfrac{ad}{1} * \dfrac{b}{b} = \dfrac{bc}{1} * \dfrac{d}{d} \implies[/MATH]
[MATH](ad) * 1 = (bc) * 1 \implies ad = bc.[/MATH]
The only difference between "flipping signs" and "cross multiplication" is that the latter is a reliable way to clear fractions only in a special case. Students make mistakes when clearing fractions because they forget that cross multiplication does not work generally. This is a reason to make sure they understand why cross-multiplication works and when it does not apply. If they understand clearing fractions generally, then they are ready to understand and use cross multiplication.
Click to expand...
Jeff,
The main reason I dislike cross multiplication is that most times that students use it they do not use it efficiently. Using your terminology it is not a short cut that saves time. It just irritates me when they use it in the following example.

In solving x/a = b/c it is inefficient to use cross multiplication. In doing so you get cx=ab and then you put back the c where it was by division. How inefficient was that?

I do support cross multiplication to check to verify if two fractions are equal and to clear fractions--but not in the case of my example above.
 
JeffM said:
… [MATH]a + b = c \implies … \; a = b - c.[/MATH]
That is using the basics. But I bet we all "flip signs" and in a lay way go

[MATH]a + b = c \implies a = b - c.[/MATH]
Click to expand...
Hi Jeff. Did you intend to "flip" the sign on those final symbols a?

?
 
Jomo said:
Jeff,
The main reason I dislike cross multiplication is that most times that students use it they do not use it efficiently. Using your terminology it is not a short cut that saves time. It just irritates me when they use it in the following example.

In solving x/a = b/c it is inefficient to use cross multiplication. In doing so you get cx=ab and then you put back the c where it was by division. How inefficient was that?

I do support cross multiplication to check to verify if two fractions are equal and to clear fractions--but not in the case of my example above.
Click to expand...
No. I get your example. As I tried to make clear, cross multiplication is a time-saving tool for a limited number of cases that should be taught only after isolating the unknown and clearing fractions are fully understood and ingrained. .

I'd teach

[MATH]u \ne 0 \text { and } ux = v \implies \dfrac{1}{\cancel u} * \cancel ux = \dfrac{1}{u} * v \implies x = \dfrac{v}{u}[/MATH]
on the second or third day of algebra.

The next day, I'd teach

[MATH]u \ne 0 \text { and } \dfrac{x}{u} = v \implies \dfrac{1}{u} * x = v \implies[/MATH]
[MATH]\dfrac{1}{\left (\dfrac{1}{u} \right )} * \dfrac{1}{u} * x = \dfrac{1}{\left ( \dfrac{1}{u} \right )} * v x \implies = \dfrac{1}{\cancel {\left (\dfrac{1}{u} \right )}} * \cancel{\dfrac{1}{u}} * x = \dfrac{u}{1} * v \implies x = uv.[/MATH]
And then I might teach them the following short-cut.

[MATH]\dfrac{x}{u} = v \implies x = uv.[/MATH]
But at a later time, I would certainly teach them the short cut:

[MATH]x \ne 2 \text { and } \dfrac{x}{3} = \dfrac{5}{x - 2} \implies x^2 - 2x = 15.[/MATH]
 
JeffM said:
… I might teach [the short-cut]

[MATH]\dfrac{x}{u} = v \implies x = uv.[/MATH]
Click to expand...
I use that short-cut most frequently, and I regularly use the following.
\[v = \frac{x}{u}\]
\[u = \frac{x}{v}\]
That is, when I see an object equal to a ratio, I know the object can switch positions with the denominator in the ratio (remaining mindful of possible zeros).

… [MATH]\dfrac{x}{3} = \dfrac{5}{x - 2}[/MATH]
Click to expand...
\[x - 2 = 5 \cdot \frac{3}{x}\]
\[x^{2} - 2x = 15 …\]

Not many short-cuts were taught to me. Most developed over time, like number-sense, as repetition revealed certain patterns. That is, my lazy self would eventually ask, "Why write out the steps, when I already know the result?"

?
 
Otis said:
I use that short-cut most frequently, and I regularly use the following.
\[v = \frac{x}{u}\]
\[u = \frac{x}{v}\]
That is, when I see an object equal to a ratio, I know the object can switch positions with the denominator in the ratio (remaining mindful of possible zeros).


\[x - 2 = 5 \cdot \frac{3}{x}\]
\[x^{2} - 2x = 15 …\]

Not many short-cuts were taught to me. Most developed over time, like number-sense, as repetition revealed certain patterns. That is, my lazy self would eventually ask, "Why write out the steps, when I already know the result?"

?
Click to expand...
It is not laziness. It is experience. In fact, it is the discovery of theorems through experience.

[MATH]a + b = c \implies a = c - b[/MATH] is a theorem taught as a short cut. You call it laziness; I call it experience that can be justified by rigor.

My argument with jomo (if indeed it is an argument at all) is that he proscribes using a universally valid theorem except in rigidly defined cases whereas I recommend using that theorem when both applicable and efficient. The diference that he and I have is that he assumes no student can determine what is applicable and efficient on their own. I prefer to assume that people will be sensible if properly taught. Of course, I have never taught a class so my intuition of human capability is far less informed than his.
 
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