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Comparing Fractions with Cross Multiplication (originally posted at parentalcation)
Today my 6th grader asked for with her math homework, specifically how to "cross multiply fractions with whole numbers". I wasn't quite sure what she was talking about, so I took a look at her homework. I saw that she had 10 problems to compare fractions with different denominators, some with whole numbers. I started to explain how to find a common denominator, etc... but she got really upset with me.
"Thats not how my teacher showed us", she said. "My teacher told us to cross multiply."
I still had no idea what she was talking about, so I went to check with my girlfriend, Shannon, to see if she knew what our daughter was talking about. But Shannon was as confused as I was.
Both of us being confused, we did a quick google and came across this explanation of the process from mathleague.com.
Well, we figured it out and were able to help her finish her homework... her way, but we are rather conflicted about it.Comparing Fractions
1. To compare fractions with the same denominator, look at their numerators. The larger fraction is the one with the larger numerator.
2. To compare fractions with different denominators, take the cross product. The first cross-product is the product of the first numerator and the second denominator. The second cross-product is the product of the second numerator and the first denominator. Compare the cross products using the following rules:
a. If the cross-products are equal, the fractions are equivalent.
b. If the first cross product is larger, the first fraction is larger.
c. If the second cross product is larger, the second fraction is larger.Example:
Compare the fractions 3/7 and 1/2.The first cross-product is the product of the first numerator and the second denominator: 3 × 2 = 6.
The second cross-product is the product of the second numerator and the first denominator: 7 × 1 = 7.
Since the second cross-product is larger, the second fraction is larger.Example:
Compare the fractions 13/20 and 3/5.
The first cross-product is the product of the first numerator and the second denominator: 5 × 13 = 65.
The second cross-product is the product of the second numerator and the first denominator: 20 × 3 = 60.
Since the first cross-product is larger, the first fraction is larger.
Though the system works, we aren't quite sure what the purpose of it is. It almost seemed to us to be cheating. Though the system works, neither one of us could give a mathematical explanation of why. Finding a common denominator is relatively easy to explain, and is also an essential skill when it comes to adding unlike fractions. Is this new math, really really old math, or something in between?
rory @ parentalcation
p.s. does anyone else want a sidebar for the blog?
12 comments:
I completely agree that this is a ridiculous method.
Here's why it works. The key is that if you multiply two positive numbers (including fractions) by the same positive number, then the larger one will still be larger. (More specifically, the two numbers will still have the same ratio.)
Let's take the example of 3/7 versus 1/2.
If you multiply both fractions by the number 14, which is the product of the two denominators, then both fractions will be converted into easy-to-compare whole numbers.
3/7 times 14 is 6.
1/2 times 14 is 7.
Therefore the second fraction is larger.
This is superficially similar to cross-multiplying, but that is NOT what you are doing.
SIDEBARS ARE AT THE BOTTOM!
(Apparently you can't put them on the side - I tried. Of course, the fact that I tried & failed does not mean it can't be done...)
That's the method I was taught.
It may be the only method I was taught; it's entirely possible that I figured out finding the same denominator and then comparing on my own.
Math ed in this country really is breathtakingly bad.
Here in Irvington we've got Math TRAILBLAZERS K-5 & then suddenly we've got pure procedural training come 6th grade.
Christopher has approximately zero conceptual knowledge.
Ed has now taken to lamenting this fact on a daily basis.
I was VERY proud of myself back when I first started relearning math because after hours of deep though I figured out why cross multiplication works.
When I finally realized that cross-multiplication means you're multiplying by the reciprocal I was....proud and mortified.
Which was a bit confusing, since usually a person is one or the other.
Well, it's not quite so ridiculous a method if they explain what it is that you're doing when you cross multiply.
Comparing 1/2 to 3/7 requires that you express the two fractions with the same denominator. Since the denominators are 2 and 7, we "rename" the fractions (I hate that term, but that's probably what they call it in her class) the same way we do when we add fractions with unlike denominators. So multiplying 1/2 by 7/7 you get 7/14, and multiplying 3/7 by 2/2 gets you 6/14. Now you can compare 7/14 with 6/14 and see which is larger.
The cross multiplication method is simply a short cut that omits including the denominators, since we know they are going to be the same.
It would be good to have students do it the "long way" first, and after they see what's going on, then introduce the short cut of cross multiplication.
Barry, I meant it was ridiculous as taught.
I'd prefer explaining that the goal is to divide 3/7 by 1/2 and see whether the quotient (or ratio) is more or less than 1. You can just "invert and multiply" and not have to deal with finding a common denominator.
Now that it is explained to us, we understand.
Our issue is that Brianna had no idea of why it worked, just that it did. We agree with Barry that she should have learned it the long way, and then learned the shortcut.
Up is down, new math is old math, cats and dogs living together, the world is a crazy place.
As I mentioned on Rory's site, I usually saw cross-multiplication applied for equations like this:
X/3 = 5/7
I used to tell my students that there is no math rule or definition of cross-multiplication. They tried to apply it to something like this:
X/3 = 5/7 + 5
I told them to NEVER use cross-multiplication.
The thing I don't like about it is that if you don't understand the principle it's easy to mess up the procedure.
For example, if you know vaguely that you are supposed to multiply numerators and denominators, it is easy to see how you might start by taking 7x1, 3x2, and come up with 7 and 6 and guess that means the 1st fraction is bigger (wrong). I find EM does a lot of this sort of thing -- teach an interesting shortcut without explanation, move on to other topics, and when kids cycle back through, they sort of know what to do, but can easily screw up the procedure.
Answer: direct instruction, distributed practice, cumulative sequence.
Susan,
Wow, I hate to admit it but in all these years it never occurred to me to divide the two fractions! (Face red).
Just to mention, in the "cross-multiplication" method, the "finding" of the common denominator is done. What you're left with is (1 x 7)/14 and (3 x 2)/14.
Generally, this type of problem is given starting when students learn how to add and subtract fractions, so it reinforces what they know about equivalent fractions, etc. I think it would be good when teaching fractions, to come back to this type of problem once the students have had fraction division and then show them your way. Good continuity and extension of the concepts they've learned.
"I used to tell my students that there is no math rule or definition of cross-multiplication."
There is actually a rule although it isn't called cross multiplication. It is called the means and extremes property. If you have 2 equal ratios a:b = c:d Then a and d are the extremes and b and c are the means. The rule is that the product of the means is equal to the product of the extremes. Now of course this only is a reasonable property to show and use if your students understand that fractions are ratios.
the word "cross-multiply"
is shunned by careful writers
(and speakers, of course).
different people think it means
different things, so it ends up
being meaningless. this happens
a lot: "ballad" is an example
from outside mathematics.
it shouldn't go without saying
that the denominators must have
the *same sign* for the inequality
to maintain its direction
when invoking this procedure
(by any name ...).
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