Tuesday, January 09, 2007

Comparing Fractions with Cross Multiplication

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.

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.

Well, we figured it out and were able to help her finish her homework... her way, but we are rather conflicted about it.

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?

10 comments:

oregoncelticlady said...

WOW!!!! We love you! "We" being my procrastinating 13 year old daughter and myself! She had to know TONIGHT how to explain crossproducts in multiplication. Soooo, we googled and YOU were first on the list! Who knew it would be a fellow blogger! If you visit my very neglected site..you can see the girl child in the archive fairy pics! Yeah!!! A big tray of cyber brownies for you!!!! Thanks!

Parentalcation said...

Well thank you. For even more info read my post at kitchen table math and especially read the comments.

http://kitchentablemath.blogspot.com/2007/01/da-bomb.html

JulieB said...

Thanks! I knew how to cross multiply, but I always had a problem knowing which would be considered the first product and which the second, once you explained it I thought "duh"--I should've known that. Guess I've been out of school too long! You saved me in getting my 5th grader ready for his test!

Anonymous said...

I was in the same boat trying to figure out exactly what this was to help my daughter with her homework. THANKS!!!

Anonymous said...

It works because you basically are making their denominators the same and then comparing the numerators. Just easier for students. I don't like it because they don't really understand what they are doing.

Tatiana said...

I read the first paragraph of your blog out loud to my fifth grade daughter and said "Hey! That is exactly what happened to us!" Of course she responded "No it's not." But no matter. Your blog saved the day over here. I figured out that she was multiplying her denominators and numerators in the wrong direction and we got it all sorted out. She got her homework done, and her star on the homework chart. Woo Hoo!

Anonymous said...

Yes - Anon at 9:31 pm is almost right. The easy way to find a common denom is to multiply the 2 denoms. In the example, 20 x 5.

So the common denom is 100.

To convert the fractions, need to multiply the first by 5/5, and the second by 20/20. Giving you 65/100, and 60/100.

Numerators look familiar?

I think its a terrible method, because at first glance one has no idea why it works -- a lousy way to teach math!

Anonymous said...

Cross multiplying is such a terrible way to teach kids how to compare fractions. Instead teachers should develop number sense in students by explaining the concept behind what fractions represent and making sure students understand comparing fractions conceptually, not because some rule works somehow.

Ruby (6th grade) said...

I'm kinda proud of my self, because I had no Idea what it was called, but I "discovered" this independently in 5th grade. I know how it works, too. I'm too tired to explain right now, though.

Anonymous said...

i just want to share with u about comparing fraction on this site,,is this same with yours ?

http://www.math-worksheets.co.uk/133-tmd-how-to-compare-fractions/