Person 1 mentioned on Twitter tonight that there aren’t enough blog posts out there about fraction division.

Person 2 recommended using rectangles to model fraction division.

I decided to help Person 1 using Person 2’s suggestion. Though the meat of this post is in this PDF I made and not in the blog post itself:

Fraction Division (1/29/2015 Stacked all fractions and made a cover page.)

Good enough, I say. I made a lot of examples fairly quickly, so I apologize if there are some errors here and there. Let me know and I can easily fix them and re-post the PDF.

And now there are *n* + 1 posts on fraction division on the internet. Woot!

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howardat58Have you tried the “algebra” method?

(2/3)/(1/4) = answer, call it A

Then multiply both sides by 1/4 to get 2/3 = 1/4 of A

How do I get to A ? multiply the quarter of A by 4 (this is so obvious)

and so 2/3 x 4 = A

A = 8/3

No bricks, rectangles, area, just simple common sense.

And it is a short step to the traditional (and now meaningful) rule.

bstockusPost authorHi, thanks for sharing this approach. My purpose with this post was to share some examples of how rectangles (an area model) could be used to model fraction division. Whether using them is “common sense” is up to anyone trying to use and/or teach with this model.

Thank you for sharing an alternative method. There is definitely more than one way to solve fraction division problems. One reason I do appreciate using an area model, at least at first, is because it helps students make some connection between fraction division and whole number division:

“How many ___s are in ___?”

“How many 6s are in 24?”

“How many 1/2s are in 2?”

The modeling can get cumbersome, but I do like the idea of using it to make that connection and to make generalizations about what is going on.

Thanks again for sharing!

xiousgeonzI find it extremely useful to spend time with the idea that taking 1/2 of something is exactly the same as dividing by two … and demonstrating the math language for that to show that ‘dividing by 2/1 is the same as multiplying by 1/2’ … this is extremely useful in real life but I’ve stopped being surprised at how many people simply don’t know it.

Visuals can help with demonstrating why and how dividing by a half is the same as multiplying by 2….

I work at getting across the big idea that multiplying by less than one shrinks your answer; dividing by less than one grows it.

bstockusPost authorI agree with you about making those connections to help students develop better understandings of fraction operations. I’m sure some students who get confused about halving wonder if they are multiplying by 1/2 or dividing by 1/2.

I also agree with looking for patterns in quotients and products to help students understand how the numbers used in the problem affect each other.

For example in 5 x 1/2, I know the product is less than 5 because I’m multiplying 5 by a number less than itself. I’m not even getting one whole group of 5, but something less than a whole group of 5.

On the other hand, I know the product is greater than 1/2 because I’m multiplying 1/2 by something greater than itself. I have 5 groups of 1/2 which is greater than 1 group of 1/2.

Good stuff and a gateway to some interesting conversation.

howardat58Regarding 5 x 1/2, if the students believed that the commutative law of multiplication applied to all numbers then they would see 1/2 x 5, and if they had their feet anywhere near the ground they would see 1/2 x 5 as half of 5. I am sure that 80% of the problems with understanding fractions are due to the willful disregard of real world meaning.

xiousgeonzMy students have been taught that willful disregard; generally, they’re assumed to already have them. Since they don’t, they’ve learned through the years all sorts of insightful other strategies that have gotten them through the courses.

THeir “real world meaning” would indicate that multiplying should make things bigger. Math teachers can throw multiplying by 1/2 out there without addressing that, in which case the students learn “okay, math doesn’t follow its own rules.” The privileged who have the internal and/or external resources to make the bridge on their own do fine and their teachers brand them “smart.” Lots of other students **could** learn it, given the chance.