• Quick note - the problem with Youtube videos not embedding on the forum appears to have been fixed, thanks to ZiprHead. If you do still see problems let me know.

Cross-Multiplication

Madalch

The Jester
Madalch
Joined
Nov 17, 2006
Messages
9,763
Okay, I'm teaching math for elementary school teachers.

The text is explaining how to solve for :) in a ratio such as:
:)/24 = 8/16. (I'm not using x, because it looks just like a multiplication sign.)

According to the text, one must cross-multiply; that is, multiply the left numerator by the right denominator and set that product equal to the product of the right numerator and the left denominator.

So, 16:) = 8x24 =192

Then divide both sides by 16, to give :) = 192/16 = 12.

Or, continues the textbook, we could multiply both sides of the equation by 48, the least common multiple of 16 and 24.

Why the rule 666 would anyone not simply multiply both sides of the equation by 24 to get :) = 8x24/16 = 12? Why waste time throwing in extra steps for no reason?
 
Madalch said:
Okay, I'm teaching math for elementary school teachers.

The text is explaining how to solve for :) in a ratio such as:
:)/24 = 8/16. (I'm not using x, because it looks just like a multiplication sign.)

According to the text, one must cross-multiply; that is, multiply the left numerator by the right denominator and set that product equal to the product of the right numerator and the left denominator.

So, 16:) = 8x24 =192

Then divide both sides by 16, to give :) = 192/16 = 12.

Or, continues the textbook, we could multiply both sides of the equation by 48, the least common multiple of 16 and 24.

Why the rule 666 would anyone not simply multiply both sides of the equation by 24 to get :) = 8x24/16 = 12? Why waste time throwing in extra steps for no reason?
Click to expand...

i quite like the smilie aproach to algebra - :D

are you sure they don't suggest multiplying the left side by 2 and the right side by 3 in order to get two denominators of the same value (ie 48) which gives you 2:) = 24

this would be a common sense approach, multiplying by 48 would be a bit silly....
 
Last edited:
Madalch said:
Why the rule 666 would anyone not simply multiply both sides of the equation by 24 to get :) = 8x24/16 = 12? Why waste time throwing in extra steps for no reason?
Click to expand...
Cross-multiplying is one of my least favorite things. Um... something something something bee stings?

It's a very good example of "here's an algorithm for solving a very specific type of problem, learn it and apply it to all sorts of random things because you don't actually understand what's happening, okay? You'd figure it out yourself if you learned the basics but you haven't so we'll give this mental shortcut that many people who do know what's going on use a name and teach it to you because it's a shortcut and hey, short is better than not short, right?"

You know, just like FOIL. "What?! (3x^2 + 2x + 1)*(x + 2)? My teacher taught me FOIL so I have absolutely no idea how to approach this! *CRY*"

Edit: Here I use "x" as a variable and "*" as the multiplication operator and action delimiter. :)
 
Last edited:
I also want to know why we aren't simplifying our fractions before doing anything else. Life becomes alot easier with 1 and 2 instead of 8 and 16.
 
This is obvious to us, but possibly not to the teachers who are unfamiliar with algebra (well, pre-algebra, really). I suspect what the text is getting at is a broad systematic methodology that it would like to ingrain for other problems that might not be so trivial, and making conceptual connections for later.

'Course, maybe I'm giving it too much credit. It could just be a badly written text. That's not unheard of, after all...

*ETA looks like GreedyAlgorithm beat me to it...sort of! :)
 
GreedyAlgorithm said:
"...it's a shortcut and hey, short is better than not short, right?"
Click to expand...

Except that it's longer....

What really gets me is that my students can't figure out why I would prefer to just multiply both sides of the equation by the appropriate number when (according to them) it's easier to cross-multiply!!!!
 
GreedyAlgorithm said:
Cross-multiplying is one of my least favorite things. Um... something something something bee stings?

It's a very good example of "here's an algorithm for solving a very specific type of problem, learn it and apply it to all sorts of random things because you don't actually understand what's happening, okay? You'd figure it out yourself if you learned the basics but you haven't so we'll give this mental shortcut that many people who do know what's going on use a name and teach it to you because it's a shortcut and hey, short is better than not short, right?"

You know, just like FOIL. "What?! (3x^2 + 2x + 1)*(x + 2)? My teacher taught me FOIL so I have absolutely no idea how to approach this! *CRY*"

Edit: Here I use "x" as a variable and "*" as the multiplication operator and action delimiter. :)
Click to expand...
One of the things maths people do (I don't know if they are taught to /forced to/comes naturally) is have rules for each type of problem. This simplifies things for people (especially small children who do not conceptualize yet -and won't until they are 7th or 8th grade normally) who just want a rule for all situations so they can have a process that works - even if it is not always more efficient or faster. I personally tend to derive equations from the nature of the data when I need them - but I have to learn whatever the textbook of the year is doing for student calculation purposes. Leads to interesting happenings when the book expects the students to come up with a step or two of the lab calculations (doesn't give the specific math) - I can help them do it but it's not in line with the book procedures so it confuses them - and I am not a maths person so I don't know how to muck/slow it up to fit with what the book is doing the rest of the time - cross multiplying to get rid of units is one of those time wasters (YES, as noted previously, I vaguely understand why it's done but such a waste - I spend way too much time trying to set it up while the answer is already in my head' cause you can see it in the numbers!):jaw-dropp :jaw-dropp
 
fuelair said:
One of the things maths people do (I don't know if they are taught to /forced to/comes naturally) is have rules for each type of problem.
Click to expand...
An early rule of fractions is to cancel-out. The first thing to learn about fractions is what they are, FCOL! Whoever set this example clearly didn't. They probably have a deep-seated fear of numbers such as "7" or "5", because multiplication tables remain, to them, a mysterious incantation.
 
Does anyone disagree with using cross-multiplication to solve '24/x = 8/3' ?
 
Last edited:
As a high school math teacher who's taught algebra I a few times, I find that most, if not all, of my students want to know what to do and that's it. I know it's cliche but it's like pulling teeth to get them to think of the numbers and what is really going on beyond "cross multiply."

What I'd prefer the thought to be for something like solving a ratio would be for them to ask themselves what relationship is being given. In the beginning most of the relationships are one half, one third, two to one, etc. While these are fairly straightforward the non-standard relationships (the ones with decimals or other fractions) tend to muddy up the process without some kind of algorithm for solving. If they, however, would focus on the relationship first I think they'd do better later on.

As an alternative--and a another way to relate the ideas to other subjects like English--is to change the form from fraction to analogy. This is by no means revolutionary or uncommon but given the tendency for my students to not know how to evaluate relationships between words OR numbers it can help them build a better understanding. I hope, at least.
 
You must log in or register to reply here.

Back
Top Bottom