Dividing Fractions by Fractions using Visual Models - 3 examples

Fraction division is one of those topics I find myself needing to refresh on every so often. For me it's definitely one of those "don't use, lose" situations. So this post came from me refreshing. But why now? This tweet:

My students' minds were blown when they found out we CAN divide across. pic.twitter.com/csXfXoL48p

— Howie Hua (@howie_hua) October 4, 2019

After sharing this tweet in our Visual Math Facebook group, it got me reminiscing on fraction division. And full disclosure, I had never considered dividing across as a thing to do. I'm 42 with a Masters in Math for Teaching and this never crossed my mind. I'm not usually too worried about how I come across, but I did appreciate other teachers in the group admitting they never thought of this, either. It allowed us all to learn something new together, and I love when this happens.

In one graduate class we had to write papers on the various algorithms taught in math and why they work. Fraction division was one of those algorithms, and after more than a few "redo"s written at the top of my paper, I finally handed in a paper that could actually be graded. Needless to say that was a taxing class.

Keep, change, flip. Why? I hope to answer that with the 3 examples in this post.


Example 1: (2/3)÷(1/2)

Division asks, "How many of these fit into that?" For 10 divided by 2, for example, we're asking, "How many 2's fit into 10?" We ask the same question when we divide fractions, it's just a little harder to see.

In our first example (2/3)÷(1/2), we're asking, "How many 1/2s fit into 2/3?" It would be easier to answer this question if our fractions were broken into the same number of parts. By creating a common denominator, it will be easier to see how many will fit.

By creating the common denominator 6, we can then see that all 3 of our green bars will fit into the space taken up by our blue bars. There is even room for one more! So all of the green bars (1 whole) can fit into the blue space and 1 more bar out of the 3 (1/3). So (2/3)÷(1/2) = 1 and 1/3. 

Here is a video explaining:



The connection to "Keep-Change-Flip":

When I posted this video on my Instagram feed, one teacher pressed me for the connection to the "keep-change-flip" (ahem, multiplying by the reciprocal) standard algorithm. Like anything in math, there is more than one good way to make a connection. I personally like bringing it back to whole numbers:

If 10 divided by 2 is 5,

Then half of 10 is 5.

So 10 divided by 2 is 1/2 of 10.

We can multiply by a reciprocal for division.

If you'd like to link the algorithm straight to this fraction example, here is how that can be done:

We can see that the numerator of the second fraction becomes the denominator of our answer. This is because we are asking, "How many of the second fraction's numerator (bars) can we fit into the first fraction's numerator [after we create a common denominator]?" 

Example 2: (1/2)÷(2/3)

This example is just like example 1, we just switched the fractions' placements, which is fun because it reinforces that division is not commutative.

In our second example (1/2)÷(2/3), we're asking, "How many 2/3s fit into 1/2?" By creating a common denominator, we can easily see that 3 of the 4 blue bars will fit into the space occupied by the green bars. So (1/2)÷(2/3)=(3/4).

Here is a video:



Example 3: (4/5)÷(2/3)  

Sometimes creating a common denominator with just columns would get too messy to be helpful, so we can create a grid to show it instead. We can always do this, I just prefer columns when there is a chance to use them because I feel they are easier to see.

With (4/5)÷(2/3), our common denominator is 15, so we can create a grid of 15 spaces. 4/5 takes up 12 of these spaces and 2/3 takes up 10 of these spaces. So all of our 2/3 can fit into 4/5, plus an additional 2. We can then see that (4/5)÷(2/3) = 1 and 2/10. 

Here is a video explaining this example:



Summary:

All three fraction by fraction division examples from this post are in this video:


These fraction multiplication and division references are included in my 6th Grade Math Word Wall.

I also just created this set of fraction division task cards to go along with this post. The cards can be laminated and used with a dry erase marker so that they can be reused.

Dividing Fractions Using Models Task Cards

I hope this post has been helpful! 

-Shana

See more fraction activities.