## Pages

### 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 my memory of fraction division. But why now? This tweet:

After sharing Howie Hua's 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.
And for a fun review, this fraction review digital math escape room. In puzzle #5, students are asked to multiply and divide fractions. Students figure out the 4 answers then type their 4-letter code into the answer-validated Google Form to unlock the puzzle.
I hope this post has been helpful!

-Shana McKay. Scaffolded Math and Science

1. Hi, I like to use the example that is you have 4 pizzas and divide them in half, how many pieces do you have?

1. That is a great example. I think that if the kids get comfortable with easier or real-life examples like yours then the concept sticks better when things get more abstract. Thanks so much for sharing.

2. Very informative post! I love the visuals. I've taught math for many years and I don't think I could produce these visuals. Thank you for teaching something new to me!

3. Is there any way you have these videos not on you tube?

1. I'm assuming this is because YouTube is blocked? If there is a platform you can use, I am open to putting them there. My email is shana@scaffoldedmath.com

4. Honestly these show the best examples and its easier :D

1. 5. Thank you for these examples and visuals. I love the visual of multiplying fractions with grids and overlaying the grids. I have been struggling to make sense of dividing fractions with area models. This makes light bulbs light up! I recently saw your last example done with the grids slid together and two of the blocks moved to fill in the spots. I have examined it from every angle and can find no mathematical justification for "Let's just move these blocks because it makes the model look better." Your explaination makes sense and I now understand what they were trying to show.

1. I'm so happy to hear this, Mrs. Thomas. This was a concept I only learned in graduate school so it's shocking to me that kids are learning it now as early as 5th grade. I love hearing about your lightbulbs:) Makes me really happy. I hope you are having a great year!

6. I am glad I found this. My next unit is rational expressions and my students need the fraction review.

1. I loved teaching rationals! I hope you are having a good year!

7. Great job. My son love it!! Especially fraction multiplication!!

1. Yay! That makes me happy to hear:)

8. Thank you so much for sharing such nice illustrations. Appreciated.

1. It's really my pleasure. I had a lot of fun pulling this post together.

9. Great post. Very well explained. Thanks a lot.

10. I don't like to cancel like numerators and denominators in the multiplication step. By rewriting the numerator as 4*6 and the denominator as 6*3 ==> (4*6)/(6*3) we can show that the factors actually divide out to equal 1. Sometimes students don't understand this.

11. How would you use a visual for an eighth divided by a fourth?
And how do you explain that dividing across can work? Is it just that there has to be a condition where the first denominator is a multiple of the second?

1. For 1/8 divided by 1/4, you could change the problem to 1/8 divided by 2/8, or "How much of 2/8 fit into 1/8?" The answer would be "Half [of 2/8 fits into 1/8." This is pretty much how you could think about dividing across for this problem, too. Dividing across will work for fractions where one denominator isn't a multiple of the second, but it gets messy. Here is an example:

2/5 divided by 1/3

numerator: 2/1 = 2
denominator: 5/3

So then we have 2/(5/3). This is the quotient, but we still have to clean it up. We can "keep change flip" to 2/1 x 3/5 to find 6/5.

12. but when you ask 1/4 of 8/20 what do you do?

1. The "of" in "1/4 of 8/20" means it's a multiplication problem. I have video for fraction multiplication here on my blog: https://www.scaffoldedmath.com/2020/05/multiplying-fractions-visual-models-video.html

13. AnonymousMay 02, 2023

How can we use this model to show a whole number divided by a fraction?

1. A whole number divided by a fraction would also be asking the question, "How many fit?" For example, how many quarters (1/4) fit into \$2?

Comment