From what I know, there are several ways of understanding fractions. See the first one here.
I will probably try to compile all of these methods together in an organized fashion once they're all done. For now I will just try to put them all down.
Without further ado, here's another way of thinking about fractions. I will divide this into 2 parts with 1 example each, since it is a bit long, and I want to be a bit more thorough than last time.
Example 1:
We can extend our thinking with whole numbers, and now think about "how many groups of 1/3 are in 6/7?
First we have to make sure we’re talking
about the same whole. (I will probably
elaborate on the definition of fractions later on and talk about the
differences of different “wholes” and different fraction relationships)
Note
that both shapes have the same sized “wholes.”
In other words, their equivalent of “1” is exactly the same. This is extremely important for establishing
what we will do next.
We can see that it’s difficult to answer
the question “how many groups of 1/3 are in 6/7“ right now since the
sections are not arranged in a way that is easy to see! So why don’t we separate them so that they
contain “subsections” that are the
same?
Let’s split each seventh in three equal
sections
And each third into seven equal sections
This is what basically happened to our
two fractions algebraically
And now we count how many groups of 1/3 are in 6/7
There are 2 complete groups of 1/3, and some leftovers
that can be compared as “part of 1/3”
The leftover part has 4 equal subsections,
and one group of 1/3 has 7 equal subsections. So the leftover part that we have is actually 4/7.
Along with the 2 complete groups, we add
on the 4/7, and we end up with
So we have:
Coming up in part 2:
- a different example where the second fraction is bigger than the first fraction in the division.
- generalization
Great post. Love the graphics!
ReplyDeleteKids always seem to have a hard time with the 4/7. They want to say 4/21 since each "piece" is 1/21 of a whole. Hard for them to grasp that you are diving by groups of seven, so there is 4/7 "leftover."
Looking forward to the next post. Small divided by big is always challenging - unless the "big" is a whole number.
-Cindy
I agree, great visual models to help with the understanding. When using this model I think it's helpful to connect the concept of common denominators so that you have compatible numbers (denominators) to work with. In the past I've been guilty (in hindsight) of telling the students that multiplying and dividing with fractions are "easier" than adding and subtracting, because you don't need to find a common denominator. But the thing is, when you DO find a common denominator, then dividing fractions makes more sense because it can be more easily linked to division with whole numbers! So now you can make the connection to 18 ÷ 7. Of course, this means you first need to be sure your students understand what those remainders really mean!
ReplyDeleteThanks for your response! There is so much that can be generated from the concept of fractions, which is why I planned on making this a series instead of a single post. From division to fractions, to rational expressions, to remainder theorem, to group and number theory, there is just so much to touch on. In hindsight, I probably should have organized these posts better and start with some exploration of the concept of "fractions" first before launching into division of fractions. Oh well, I will leave that for when I summarize everything in a post. This idea and the next one (part 2) does use the idea of common denominators (the next one more explicitly so). I loved the idea of explaining fraction with common denominators all the way through -- although I did find that some students had trouble understanding the explanation. I guess I am attempting to put all the different methods I explain fraction division down somewhere.
DeleteUm... my response above may be a bit erratic... hopefully it made sense!
I always enjoy seeing teachers (and students, of course) dig into this topic. Two tips for you: Gary Davis has a great ebook on this topic at http://republicofmath.files.wordpress.com/2009/12/division_of_fractions_davis__pearn3.pdf, and there are fancy (and for me, easily confused) names for these two types of division. The kind you demonstrated in your first post is called partitive division, and the one you showed above is called quotative. I'd never heard those terms until I went to grad school, but knowing them sure helps me find what I'm looking for when I search on this topic!
ReplyDeleteThanks for the link to the ebook! I skimmed through it just now and I will definitely be able to get something useful out of it! Up until a few months ago, I was largely unaware just how huge this field is. There are a few researchers I know who are trying to nail down understanding of fractions (saw them present at a conference earlier this summer), and there are certainly a lot of terms out there for specific ways of understanding. I have a lot of those notes at home that I still need to read through. It's really invigorating to know that there's still much to learn about what I thought to be a fundamental concept. I managed to navigate through your profile and subscribe to your blog (still trying to learn about all these buttons in the blogging world). Thanks again for the comment!
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