Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.

An important part of being a flexible mathematician is knowing that one size does not fit all. In other words, mathematicians use different strategies depending on the situation. A good example is comparing fractions. I will go on the record and emphatically state that kiddos should not be cross-multiplying to compare fractions. Yes, I know it’s fast. Yes, I know it works. But it’s critical that our kiddos understand fractions, and cross-multiplying is not a means to that end. So how should we compare fractions?

First, students need to have good fraction sense. That is, they need to deeply understand what a fraction like 1/8 means. That takes lots of concrete experiences with fractions. Check out this blog post for more on fraction number sense.

Next, it totally depends on the fractions being compared. Students need to have multiple strategies for comparing fractions. I wrote a whole series of blog posts outlining the different strategies. You can check out the first post in that series and find links to the whole series here. I made a poster showing all the strategies in the order they should be considered.

Click here to grab a copy of the poster. And if you want some fractions cards to practice, check out this blog post.

I absolutely agree about the cross multiply. When I taught 5th grade math, I showed them the “butterfly method” but only if they absolutely could not get a grip on understanding fractions (and they had a standardized test to pass –TAKS. I think many kids have that “aha” moment with fractions when they really have to use it in a real situation. I always found working with fractions to be easy, but they finally made sense when I was a professional pattern maker in the garment industry (pre-teacher days!)

I really like using the number line and number sense to compare fractions. I used to never teach cross multiplication because I found that it confused them when they would have a multiplication of fractions problem. They tried to cross multiply. I said “NO CROSS MULTIPLYING!!!”….until this year. My 5th graders struggle with number sense in all sorts of the word. I have shown them cross multiplication (Butterfly) as a way to compare fractions for testing purposes until they *hopefully* get their number sense by the end of the year.

I don’t even know the butterfly method. I didn’t learn it in school or in my methods class. I like to have them use fraction bar pieces until they’re ready for common denominators.

The butterfly method is the same thing as cross-multiplying. It’s a short-cut to finding a common denominator. If you Google ‘cross multiply to compare fractions’ you’ll find more information. You are right on, though, with using fractions bars. Fractions are such an abstract concept that without lots of concrete practice, students will never truly understand them.

Excellent post on a crucial concept. Your poster is perfect for journals- I use the same methods but it doesn’t look as nice as yours. As for cross-multiplying, I agree about using it in a standardized test situation and as a way to validate an answer generated in a more concrete model. BUT… I do allow students to use it on formative and summative assessments only AFTER they can explain to me how it works. This helps ensure a deep understanding on a crucial concept: fraction theory

this method is extremely amazing and easy to understand as well…. yet i decided to use this method as my action research. but then, from the web, i found many authors for this method… so can i know who is the first creator for this method? thx for reply~

Hi Donna! I also have a few additional ways of comparing fractions that I love.

First, if the numerators are the same, you can think about the size of the pieces and determine which is more of the whole. So, 3/5 is more than 3/6 because fifths are larger than sixths and in each situation you have three pieces.

Then, if the fractions are missing the same number of pieces from the whole you can compare the relative sizes of the pieces to determine the larger one. For example, 3/4 is less than 5/6. In both cases, we are missing one piece. Since sixths are smaller than fourths, we are closer to the whole.

Finally, with odd numbers in the denominators, I hesitate to compare directly to a whole because you end up having a half in the numerator. I don’t want kids mistakenly thinking that you could ever have a half in the numerator. They instead can think of the size pieces of what would be a half with the same numerator. For example, to see if 3/5 is more or less than 1/2, we can think of a fraction with a 3 in the numerator that equals one half – 3/6. Since fifths are larger than sixths, having three fifths must mean we are over a half. Whereas if it were 2/5, we would realize that 2/4 is the equivalent equal to a whole with the numerator of a 2. Since fifths are tinier than fourths, 2/5 must be smaller than a half.

Granted, these aren’t lower elementary thinking strategies, but I did use them in my 5th grade classroom. They get to the heart of conceptual understanding of fractions.

As a former middle school math teacher cross multiplying should not be used in elementary school to compare fractions because in middle school it is used for solving proportions. Proportions are not fractions they are comparisons in fraction form. If an elementary teacher allow students to use cross multiplication to compare 2 fractions they are creating a major misconception before the student has entered his/her first middle school math classroom.

I absolutely agree about the cross multiply. When I taught 5th grade math, I showed them the “butterfly method” but only if they absolutely could not get a grip on understanding fractions (and they had a standardized test to pass –TAKS. I think many kids have that “aha” moment with fractions when they really have to use it in a real situation. I always found working with fractions to be easy, but they finally made sense when I was a professional pattern maker in the garment industry (pre-teacher days!)

Take Care,

Karen

LittleSecond-GradeSomebodies

Thanks for sounding off on cross-multiplying, Karen!

I really like using the number line and number sense to compare fractions. I used to never teach cross multiplication because I found that it confused them when they would have a multiplication of fractions problem. They tried to cross multiply. I said “NO CROSS MULTIPLYING!!!”….until this year. My 5th graders struggle with number sense in all sorts of the word. I have shown them cross multiplication (Butterfly) as a way to compare fractions for testing purposes until they *hopefully* get their number sense by the end of the year.

I don’t even know the butterfly method. I didn’t learn it in school or in my methods class. I like to have them use fraction bar pieces until they’re ready for common denominators.

Jessica

http://www.learningmetamorphosis.blogspot.com

The butterfly method is the same thing as cross-multiplying. It’s a short-cut to finding a common denominator. If you Google ‘cross multiply to compare fractions’ you’ll find more information. You are right on, though, with using fractions bars. Fractions are such an abstract concept that without lots of concrete practice, students will never truly understand them.

@donna boucher, sorry for disturbing, can i know who is the first author for butterflr method?

I have no idea, Jenny. It is a short-cut trick that does not develop conceptual understanding, so I doubt you’d find it in any research.

Excellent post on a crucial concept. Your poster is perfect for journals- I use the same methods but it doesn’t look as nice as yours. As for cross-multiplying, I agree about using it in a standardized test situation and as a way to validate an answer generated in a more concrete model. BUT… I do allow students to use it on formative and summative assessments only AFTER they can explain to me how it works. This helps ensure a deep understanding on a crucial concept: fraction theory

I agree that being able to explain cross-multiplying should earn a student the right to use it. Great idea!

this method is extremely amazing and easy to understand as well…. yet i decided to use this method as my action research. but then, from the web, i found many authors for this method… so can i know who is the first creator for this method? thx for reply~

Hi Donna!

I also have a few additional ways of comparing fractions that I love.

First, if the numerators are the same, you can think about the size of the pieces and determine which is more of the whole. So, 3/5 is more than 3/6 because fifths are larger than sixths and in each situation you have three pieces.

Then, if the fractions are missing the same number of pieces from the whole you can compare the relative sizes of the pieces to determine the larger one. For example, 3/4 is less than 5/6. In both cases, we are missing one piece. Since sixths are smaller than fourths, we are closer to the whole.

Finally, with odd numbers in the denominators, I hesitate to compare directly to a whole because you end up having a half in the numerator. I don’t want kids mistakenly thinking that you could ever have a half in the numerator. They instead can think of the size pieces of what would be a half with the same numerator. For example, to see if 3/5 is more or less than 1/2, we can think of a fraction with a 3 in the numerator that equals one half – 3/6. Since fifths are larger than sixths, having three fifths must mean we are over a half. Whereas if it were 2/5, we would realize that 2/4 is the equivalent equal to a whole with the numerator of a 2. Since fifths are tinier than fourths, 2/5 must be smaller than a half.

Granted, these aren’t lower elementary thinking strategies, but I did use them in my 5th grade classroom. They get to the heart of conceptual understanding of fractions.

Ann Elise

As a former middle school math teacher cross multiplying should not be used in elementary school to compare fractions because in middle school it is used for solving proportions. Proportions are not fractions they are comparisons in fraction form. If an elementary teacher allow students to use cross multiplication to compare 2 fractions they are creating a major misconception before the student has entered his/her first middle school math classroom.