Fraction Division via Rectangles

I appreciate Christopher Danielson's post on common numerator fraction division because it's important to examine how various algorithms work and how we can help our students become more flexible with their thinking. It's not surprising that I teach fraction division using rectangles, and I really believe the kids seem to grasp it better because it's visual.

I'll start with this problem: 3/4 ÷ 2/3. But before we do fraction division, I ask kids about whole number division. What is 8 ÷ 2? What is 15 ÷ 5? Eventually we settle on something like: asking what is 8 divided by 2 is the same as asking how many groups of 2 are in 8. Then we apply the same question to 3/4 ÷ 2/3 as "how many groups of 2/3 are in 3/4?" I guide them through this process:

Me: Let's draw out 3/4 and 2/3 on paper.

Half of them draw circles. Awful, drunk, ill-behaved circles.

M: Let's use grid paper instead to draw our rectangles. I think you can show 3/4 much more accurately on grid paper than on a circle. Please draw 2 rectangles of the same size.

(By doing this, we are really dividing two fractions using the common denominator strategy. Christopher writes about it here.)

Students: Any size?

M: What size do you think? Does it matter? Shade the first one to show 3/4 and the second one to show 2/3.

They mess up. They might draw a 1 x 4 rectangle, shade in 3 to show 3/4. But they don't quite know how to shade in 2/3 of a 1 x 4.

M: So maybe we should think about the size of the rectangle more carefully. Look at the problem again. Three-fourths divided by two-thirds. Hmmm... What dimensions should our rectangles have so it's easy to divide into fourths and thirds.

This prompt is enough for someone to say, Draw a 4 by 3 rectangle!

M: Bingo! I'm drawing these with you. Okay, so two rectangles of 4 by 3 — or 3 by 4 — doesn't matter. I'm shading in 3/4 on the first one and 2/3 on the second one. So our question is: How many groups of 2/3 are in 3/4? Because I colored mine in, can you help me ask the question again using colors instead?

Someone responds, How many pinks are in the greens?

M: Yeah. And how many little squares are pink? Okay, eight. So, I'm going over to the green here and round up 8 pink squares. I'm able to round up one group of 2/3 (pink) in the 3/4 (green).

Someone says, There's one left over.

M: How much is this one little green square left over worth? Right! 1/8 because we called 8 little boxes as one, so 1 little box must be 1/8. Our answer then is 1 and 1/8.

A few students say, I get it.

M: How do we know that our answer of 1 1/8 is correct? Okay, we'll use a calculator.

I purposely use an online calculator where I'm entering the fractions as they appear. I don't need to distract them right now with decimals or talk about parentheses. This is from CalculatorSoup.

M: Let's do this again. Now with a mixed number just for fun. Let's do 1 1/2 ÷ 2/5. How many rectangles are we drawing? What dimensions should they be? Oh, but we have more than 1 whole here, so...? We should have something like this then.

They say, How many groups of orange are in blue?

M: So let me round up the groups of orange that are in the blue. I got three. And the leftover is? Right, three. Three out of...?

More students say, Three-fourths! Three and three-fourths. I get this!

This online calculator from Calcul allows for entries of mixed numbers.

M: Okay, your turn to do one all by yourself. Please do 2/3 ÷ 1/2. (Same one Christopher used.)

I think these kids' papers show understanding.

While these are not there yet. I don't know. But it seems that drawing pictures and doing more visual stuff start to disappear in middle school. 

Below is our textbook's treatment of "dividing fractions and mixed numbers" — Chapter 5, Section 7 — the full 3 pages before the Exercises. 

Notice the two circles at the start of the section — that's pretty much it. And circles are great if you have denominators of 2, 4, and 8. I think if I can get my kids to first see the answer, then I can sell them the other algorithms — like multiply the reciprocal — and not come across as a fraud.

I also want to point out that I normally see this visual below for division of fractions. My way is different than this — I deliberately ask kids to draw 2 rectangles whose dimensions are the denominators.

[Update 01/07/2017]

Thank you to Rachel Emily Tabak for creating this accompanying worksheet, 18 - Frac division rectangles