Last week, I had a great time teaching & learning division of fractions with one of my teachers. I have not stopped thinking about how models really provide a visual representation of what dividing fractions really mean. Our students use concrete models to gain understanding of what fractions are, how to put them together, take them apart, and take parts “of” it. It is true that when we come to dividing fractions, it gets tricky. This is why models play such an important part.
Here is a picture of a concrete model that shows how to divide ½ by ¼ . Using a concrete manipulative, such as these fraction tiles, allow students to identify the ½ in a whole, and the ¼ in a whole. Then they will think “How many fourths FIT in a half?” or “How many groups of 1 fourth FIT in a half?”
Using fractions tiles is essential when teaching & learning fractions. The great thing about these manipulatives, is that the whole is already identified, and students can use the little fraction tiles to figure out how to divide fractions. These concrete tool gives students different visual representations so they can picture what a half is, or what a fourth is, etc.
THERE IS NO NEED TO INVERT FRACTIONS!
There is no need to teach students strategies such as “flip” the fraction to solve division of fractions. When we provide opportunities for students to visualize what fractions are, they build conceptual understanding, therefore, they can make sense of what dividing fractions is all about.
In this next problem, students solve 4/5 ÷ 1/10 , they understand that this equation means “how many one tenths fit into four fifths?”
Students used a great app called “fraction manipulatives” created by Braining camp, and they used a pictorial representation of what 4/5 looks like, then they made sense of how many 1/10 fit in it. They came to the conclusion that there are eight 1/10 in there. Isn’t this great? No tricks, no rules, no algorithms! Students can see what the answer is, by trusting the model!
To build conceptual understanding, students should move from CONCRETE manipulatives, to PICTORIAL representations, to finally move into more ABSTRACT strategies.
Here is one more example: 3/4 ÷ 3/12 or how many 3/12 fit into 3/4?
Then we moved into another problem that brought some productive struggle. Just like we planned in our lesson.
Here is what happened:
The fraction tiles work when the fractions are alike, but when working with fifths and eights, it can get messy, my favorite part of a math lesson, when it gets messy!
“There is a remainder” they said. Students need to understand what to do with this part that is left.
It is not easy to manipulate eights and fifths in the same whole, but a pictorial model can do that.
After they use a pictorial representation, they can start making groups of 1/8 in their model. Students will be able to visualize and understand that 1/8 of the rectangle is equivalent to 5/40. How many groups of 5/40 are in 3/5?
They will count 4 groups and some leftover. The left over is 4/5. Because 5 of those will make 1 more group, since they do not have 5, that becomes its own fraction.
Concrete manipulatives and pictorial representations (models) are made to build conceptual understanding. That is why we need to use problems that make sense, such as ½ divided by ¼, or 4/5 divided by 1/10. To provide our students with opportunities to visualize and build understanding of what dividing fractions really mean.
In grade 6, when we get to fractions like 3/5 divided by 1/8, students can start making connections, and with the conceptual understanding they built in grade 5, start using other strategies that are more efficient for that grade level. They will be able to answer questions like: Why? How do you know? How does that work? How does that look like?
Models are used to build conceptual understanding. To understand WHY a strategy works and HOW it works.
This reflection made me remember a student in grade 5 last year, who could multiply and divide fractions by multiplying numerator by numerator, and denominator by denominator, without making any sense, just getting the answer quickly. She could divide by inverting the fraction.
I asked her if she wanted to play with fractions tiles, and she said yes. I gave her a problem with fourths and twelves. She started making a whole with fourths, then she made a whole with twelves. Then she solved 3/4 by 4/12. She moved the pieces, put the twelves inside the fourths, and she gasped! She got very quiet. I asked “what is happening?” She looked at me, and said: “Miss Caty, is this why 3/12 is equivalent to 1/4? Because 3 of these fit in 1 fourth? Is this how dividing fractions was invented?”
I said: “Yes. Someone had so much fun playing and understanding fractions with models, that they created a shortcut. The short cut is to invert fractions.”
Making Sense of Mathematics 