Division of Fractions by Whole Numbers and Fractions by Fractions

Before the Christmas break we were learning how to divide fractions by whole numbers (and by extension, divide fractions by fractions). I've always found that is much easier to teach this using models so that the kids understand the concepts before applying any rules or algorithms. Students and even teachers (some anyway) are familiar with using "Keep Change Flip" as a procedural method used to solve division with fractions, but don't truly understand why it works. As follows:

Given the following problem: Mr. E has ½ a bar of hersheys chocolate. He wants to split it equally between his three friends. How much chocolate will each friend get?

Chocolate bar before dividing equally between 3 friends (½)

 

Chocolate bar after dividing it equally between 3 friends (⅙)

Therefore:

From there, students develop the understanding that dividing is the same thing as multiplying by the reciprocal (For example: ½ ÷ 3 = ⅙, and can also be written as ½ × ⅓  which also equals ⅙).

Knowing that division is the same as multiplying by the reciprocal, you can easily divide fractions by fractions.

Example:  ½ ÷ ⅙ can be written as ½ ×  6/1 which = 6/2 which = 3.

Lastly, this also applies to whole numbers. Example:  
12 ÷ 4 = 3 
12 × 1/4 = 12/4 = 3


Wait!
If you have time, watch the video below of a student that has difficulty with the model (mainly due to a lack of conceptual knowledge of division) and my attempt (which I don't think was that great) at helping her solve the problem!

P.S. - Division of fractions by fractions can be modeled using fraction strips. I made a post prior to this where students were using them to solve word problems. It can be found here: Dividing Fractions by Fractions - Fraction Strips

Visual Models vs Algorithms

There has been a substantial increase in the use of visual models in my class compared to last year. As a beginner teacher with no experience, I taught like how I was taught. This year however, I understand how important conceptual knowledge can be, and what a better way than to visualize it. It's the reason why some of the weaker classes can keep up. Visual models are good for everyone. Just because a student is in the algorithmic stage (as in, they can solve a problem algorithmically), doesn't mean that they have the conceptual knowledge.

I have been wondering lately though, when should I take students off of visual models and move them onto the division w/ fraction algorithm? Some higher level students are annoyed with drawing models, but I want them to understand that this is a tool they have in their disposal at any point in time, not something that's disposable just because there is a faster way.

We have been doing a lot of division with fractions lately, and we've gone as far as division with fractions with uncommon denominators. And I must say, it did occur to me that the model for division with fractions is a piece of extra work when the algorithm is so easy to follow.

VS.

 

The downside to the algorithm is that it literally conveys no conceptual knowledge at all. I want to make sure students have a solid foundation of what division is before moving onto the next topics (long division/GCF/LCM). In the long run, I'd rather be safe than sorry. I think when students have shown that they can compute or solve word problems fluently with the visual model, then I can teach the algorithm and we can move on.

Visual Models!

The students and I have been getting tons of mileage out of the visual models lately. We've started Unit 2 this past Monday and we've been doing a great job so far. We have been doing division with fractions. In this unit, I really want to go over any sort of topic or lesson that deals with division. It's imperative, as some of the division with fraction questions (word problems) are known to be extremely troublesome for students without strong conceptual knowledge. However, these visual models can help dig up this conceptual knowledge and help students understand what's going on in the question.

I have been staying far away from algorithms (aside from having them rewrite these division w/ fraction problems as multiplication, as making that relationship explicit is equally important). I think if we can make sense of plenty of many different types of these questions via visual models, we'll do a pretty decent job with this unit. 604 has seriously been stepping up their game, and the models have been a  great tool for them to use when tackling division w/ fraction questions. So far, what a great start to Unit #2!