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

Division with Fraction Word Problems

I've been spending a lot of time lately thinking about how I'm not hitting the skill cap there is for teaching. I am trying to increase my ability in facilitating math discussions.  The students are now working on creating their own division (with fraction) word problems.

Division is a very rough topic, as many students do not understand the concept division and do not know when to apply it when given a word problem. I understand that this is quite challenging, but creating your own division problem means you understand what division is and why it's optimal to use it. Therefore, if a lesson like this is successful, that is a big breakthrough for my class. Some questions that were made:

With a topic like this, there is a lot of discussion that has to surface:

#1 - Does our question make sense?
#2 - Does our question require us to find how many (insert fraction) are in (insert fraction)?
#3 - Is it realistic and does the unit we chose make sense considering the quotient? (e.g. we can't have 1.5 people)
#4 - Working on problems that don't make sense and understanding why, and possibly changing it so that it does make sense.

Some misconceptions some of the kids have had so far when making their own problems:

A) The question does not make sense
B) The question does not require us to divide to solve
C) The question is not clear
D) Divisor and dividend have different units

As a teacher, it's been hard clearing up the misconceptions. You have to get students to understand why their own question doesn't make sense. Lastly, when they are finally successful in creating their own questions (like the ones I posted in the pic above), they tend to stick to the same format or don't change the scenario (I'm still happy though!).

A hands-on activity is far better for something like this, or at least should be done before this lesson (note for the future). While this reminds me much more of a lesson for a science class, I have been thinking about purchasing multiple different cups of different measurements and having students solve these types of division word problems in real life.

A question like "Mr. E has 1/3 (cup, gallon, quart) of milk. He wants to share it equally between 2 students. How much milk will each student get?" or "how many 1/5 cups of milk are in 3 cups of milk? Students will actually take the 1/5 cup and continue to fill the container/measuring cup until they find their quotient. The kids would appreciate this as they did with the last chance lesson.

A lesson like this is also good for teaching conversions i.e. finding how many cups are in a quart, converting quarts to gallons, etc. I'll probably do something like this before this unit is over.