# Fractions of Fractions

This is day 1 of multiplication of fraction by a fraction and I can already see this will dramatically increase my blogging! So much to write about (for reflection, excitement and possibly confusion). With the implementation of CCSS this year, this is new in the Investigations curriculum and I am finding some things I love about it already and some things I am struggling with just a bit.

Before this lesson, students have worked in the context of a bike race of “x” number of miles and found a fraction of the race various bikers have completed.  Looked like this:

This lesson went very smoothly and I found it was more of a struggle to have them model what was happening on the fraction bar since finding the fraction of the whole number was an action they could do mentally.  To some, it seemed like an unnecessary step and to be honest, I wavered between unnecessary and yet completely necessary to make their thinking visual. I knew how important it would be in fraction x fraction, so I made them construct the model of what was happening in the story.

Today we started fraction of a fraction. It incorporates the same visual image of the fraction bar, so I love that continuation from previous lessons. It did lack a context, which at first bothered me but as we continued working, and heard the discussions, I moved past that.  Tomorrow, I am actually going to have them come up with a story to go along with a few problems to see if they can contextualize the math they are doing.  We started with a fraction of a half and then a fraction of a third, writing the expressions (some equations) as we went:

Of course, you always have the students who fly through the work and finish early as I am walking around and having discussions with the students who need some extra help, so I asked those who finished early to think about the denominator each time. Why is the product’s denominator changing from the denominators of the factors? Did you have an idea what the denominator would be before you used the fraction bar?  There thoughts were so interesting:

Absolutely LOVE all of this scratching out, changing her reasoning!

This one brings up the issue of vocabulary….fours instead of fourths, eights instead of eighths. Something I have to bring out in our discussions.

This one I struggle with because of the words double and triple. I know the number itself is doubling and tripling, but I would like to have them expand that is it happening because there is another half to split or two other thirds to split.

I love that this makes the fractions factors and products are just like whole number factors and product.

Again with the “double” word. Is it just me that struggles with this one??

I am thinking this will be one of MANY multiplication and division of fraction posts! I am just amazed at the ease the students work with the fraction bars and I like what Investigations has done thus far with these lessons. One tweak I would like made would be the directions…students are asked to “stripe 1/2 of the shaded portion” and it is becoming a tongue-twister for me 🙂 I keep saying shaded when I mean striped, minor detail but they keep correcting me!

These conversations are so rich and valuable for this understanding that it blows my mind that a teacher could just say “multiply the numerators. multiply the denominators. That is multiplication of fractions.” If I had learned fractions this way, it would have all made SO much more sense!

To be continued…

## 7 thoughts on “Fractions of Fractions”

1. mrdardy

How much insight do you have about their fluency between these fraction representations and decimal representations of the numbers? I am surprised by the lack of fluency of my high level students, for example very few recognize something like 0.125 = 1/8

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2. mathmindsblog Post author

We do a lot of work with the shading of the 100s and 1,000s grid to establish equivalencies between fractions, decimals and percents. I find that many students who know 2/8 = 1/4, then they can reason that 1/8 is half of 0.25 to get to the .125. It takes a lot of concrete work with the grids for many of my students. Hope that helps?!? Thanks for commenting!

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3. Christopher

I have a super-technical question. It has to do with those equations. I have been an advocate for quite a while now of more, more, more fraction of a fraction work as kids explore fractions, operations and proportionality. So I am delighted to see that work taking place here—the contexts, the abstract number discussions, the visual models. My question is this: What is the reason (from the naive student perspective) for writing 1/2 of 1/3 as 1/2 x 1/3? Why is this a multiplication problem?

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1. mathmindsblog Post author

This is why I think you are great! Thanks for the great question Chris because it is something I struggled with this entire investigation. I don’t feel like that notation is extremely important in this work, but the authors make that translation very explicit. I, maybe naively, trust they have reasons beyond my understanding so I tried to make it make sense for my students.
Prior to this we did multiplication as groups of…2groups of 3=2×3, so 1/2 a group of 6 =1/2 x6, so 1/2 of a group of 1/3 =1/2 x1/3. Some also wrote 1/3 divided by 2. Perfect. So then inverses came out.
I really feel like I tried everything to be sure the “of means multiply” never came out as a rule. It was a tough investigation that I hope to improve this year. Harder still is, while I think leaving it 1/2 of 1/3 is perfect for 5th grade, I have middle school teachers that feel otherwise and I dont want to put students at a disadvantage moving forward:( even if it really is the teachers lack of understanding)
Any help in this explanation would be greatly appreciated bc you are completely right that for the naive child, it makes little sense!

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