Simplifying a binomial

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chromechris

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I have the following binomial fraction:

[MATH] \frac{6x+12}{6x-5} [/MATH]
Why can I not simpliify this equation by canceling out the [MATH]6x[/MATH] on the top and bottom, then only be left with [MATH]\frac{-12}{5}[/MATH]? Whatever the value of [MATH]6x[/MATH] is, shouldn't I be able to cancel it out? Why am I not allowed to cancel it out? I was answering a question in an online Math platform, and it did not accept [MATH]\frac{-12}{5}[/MATH], so I guess that's wrong. The furthest I was able to simplify was [MATH]\frac{6x+12}{6x-5}[/MATH].
 
chromechris said:
I have the following binomial fraction:

[MATH] \frac{6x+12}{6x-5} [/MATH]
Why can I not simpliify this equation by canceling out the [MATH]6x[/MATH] on the top and bottom, then only be left with [MATH]\frac{-12}{5}[/MATH]? Whatever the value of [MATH]6x[/MATH] is, shouldn't I be able to cancel it out? Why am I not allowed to cancel it out? I was answering a question in an online Math platform, and it did not accept [MATH]\frac{-12}{5}[/MATH], so I guess that's wrong. The furthest I was able to simplify was [MATH]\frac{6x+12}{6x-5}[/MATH].
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You cannot "cancel out" [6x] because [6x] is NOT a factor of the numerator nor the denominator.

When you "cancel out" [6x] - you are DIVIDING the numerator and the denominator by [6x]. In this case that will make the problem more complicated.

On a personal rant, that is why I hate (no wait - abhor) the term "cancel out".
 
chromechris said:
I have the following binomial fraction:

[MATH] \frac{6x+12}{6x-5} [/MATH]
Why can I not simpliify this equation by canceling out the [MATH]6x[/MATH] on the top and bottom, then only be left with [MATH]\frac{-12}{5}[/MATH]? Whatever the value of [MATH]6x[/MATH] is, shouldn't I be able to cancel it out? Why am I not allowed to cancel it out? I was answering a question in an online Math platform, and it did not accept [MATH]\frac{-12}{5}[/MATH], so I guess that's wrong. The furthest I was able to simplify was [MATH]\frac{6x+12}{6x-5}[/MATH].
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If you had the expression [MATH]\frac{6+12}{6-5}[/MATH], would you simplify the fraction by subtracting 6 from the numerator and denominator to get [MATH]\frac{12}{-5}[/MATH]? Or would you be aware enough to realize that subtracting like that doesn't result in an equivalent fraction?[MATH]\frac{18}{1}\ne -\frac{12}{5}[/MATH].

I let my students use the word "cancel", because they will have learned it somewhere; but they have to be able to say what they are really doing, and why it's allowed.

In a fraction, "cancel" means removing a common factor of the numerator and the denominator, because it amounts to removing a factor of 1: [MATH]\frac{6\times 5}{6\times 2} = \frac{6}{6}\times\frac{5}{2} = \frac{5}{2}[/MATH]
"Cancel" (to be legal) can't just mean crossing out two things that are the same anywhere you like.
 
Consider this:

\(\displaystyle \frac{6+2}{6+3} =\frac{8}{9}\) Agree?

But if you "cancel out" the 6s, you'd get \(\displaystyle \frac{2}{3}\) which does not equal \(\displaystyle \frac{8}{9}\)

However if you had \(\displaystyle \frac{6*2}{6*3}=\frac{12}{18} =\frac{2}{3}\) so you can cancel out the 6s.

The difference is that in the second case 6 is a factor of the top and bottom, but in the first case 6 is NOT a factor of the top and bottom.
 
Thanks guys! I can now see how "cancelling" out in my example is a problem, and how it would only work in multiplication and division.
 
If you want to use the phrase cancel out that is fine but you better be able to answer a simple question---cancel out to what?

7-7 cancels out to 0 while 7/7 cancels out to 1 and 14/7 cancels out to 2. Cancel out is not unique.

I agree that 6x/6x = 1 = 1/1 = 7/7= 13/13 =... Can you replace any of those numbers in your work and it not chance the answer. 6x/6x does NOT disappear as it is not 0, it quals 1.
 
This is a very interesting thread. I'm not a teacher and I therefore I have not been exposed to the pitfalls of cancelling out for a very long time (way back when I was a young student myself).

I can see the danger that the word "cancel" covers several mathematical methods which have different results.

Perhaps the bigger danger is that this family of methods can be VERY easy to apply, and usually result in a quick and dramatic simplification. This makes the methods extremely compelling to use. However there's a big but (see * below):- the conditions for when it's OK to use the methods are much harder to grasp than learning how to apply them.

I'm not 100% convinced that students will learn these methods quicker if we don't name them - but I'll bow to experience and try not to use the word "cancel" on the forums! I think my brain works a bit differently anyway :alien:

*- Not the Sir Mix-a-Lot kind of "big butts" ?
 
When I was a grade school sometimes I would not study and try these pitfall methods but not being sure if they were valid I checked common logic. Here is a perfect example: \(\displaystyle \frac{7+4}{7+3}\) Can I cancel out the 7s? Here is why I knew that I could not!
\(\displaystyle \dfrac{7+4}{7+3} = \frac{1+7}{3} = \dfrac{1+4}{1+3}\). I quickly saw that although 7/7 = both 1 and 1/1 that something was wrong.[math]\dfrac{1+4}{1+3} = 5/4 \ while\ \dfrac{1+4}{3}=5/3[/math]. This told me that I could not cancel out the 7s!
 
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Cubist said:
This is a very interesting thread. I'm not a teacher and I therefore I have not been exposed to the pitfalls of cancelling out for a very long time (way back when I was a young student myself).

I can see the danger that the word "cancel" covers several mathematical methods which have different results.

Perhaps the bigger danger is that this family of methods can be VERY easy to apply, and usually result in a quick and dramatic simplification. This makes the methods extremely compelling to use. However there's a big but (see * below):- the conditions for when it's OK to use the methods are much harder to grasp than learning how to apply them.

I'm not 100% convinced that students will learn these methods quicker if we don't name them - but I'll bow to experience and try not to use the word "cancel" on the forums! I think my brain works a bit differently anyway :alien:

*- Not the Sir Mix-a-Lot kind of "big butts" ?
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It is a requirement for your brain to work differently in order to help out on the forum. I know that it is a bit late but welcome to the forum! We need people like you to help out. Thanks. BTW, what is your background?
 
A comment about what cubist said. Cubist suggested that maybe it is not a good idea to use the phrase cancels out in this forum. I myself probably thought that as well. Now after reading what cubist said I think that not using the phrase is wrong. The student knows the phrase cancels out, that is a given. If we instead use that phrase regularly but always point out that the result is not always the same, then the student (after hearing us say it time after time) will (hopefully) learn that they must be careful when concluding that something cancels out. They must learn (from us!!) that when something cancels out it does not always means that it disappears!
 
Jomo said:
It is a requirement for your brain to work differently in order to help out on the forum. I know that it is a bit late but welcome to the forum! We need people like you to help out. Thanks. BTW, what is your background?
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Thanks for the welcome! I studied electronic engineering at uni. Since then I went into computer programming but always with a strong numerical element. I used to struggle a LOT with language (English) when I was at school. But this has much improved over the years, having to write documentation for the code I've worked on and email to co-workers. The importance of words/ communication has become clearer to me as I've aged!

A few years ago I really enjoyed helping my kids with their maths homework. They don't need my help so much now (they are at uni, or at least they were until they returned home due to the virus). So I thought I'd try to continue this hobby (sometimes obsession!) by trying to help others online.
 
Jomo said:
I agree that 6x/6x = 1 = ...
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6x/6x = x^2 because of the Order of Operations. \(\displaystyle \ \ \ \) (edit)

while

6x/(6x) = 1

And 6/6x = 1x = x, while

6/(6x) = 1/x.

So, grouping symbols should be written down for clarity to show what is intended:

6x/(6x) = 1

(6x/6)x = (x)x = \(\displaystyle \ x^2 \)
 
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JeffM said:
Not at all

6x/6x = x^2 because of the order of operations. Of course, if x = 1, then 6x/6x does equal 1.
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I made an edit before you last posted to indicate the correction.
 
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