My Other Math Sites
Lessons From the Classroom
Fractions Operations Using Rectangles
A few days ago, Mary had replied to Nat’s tweet.
@fawnpnguyen is the queen of this. Pretty sure she has a blog post on it.
— Mary Bourassa (@MaryBourassa) May 14, 2021
God, I love Mary. She sent me a decadent chocolate bar from 2,800 miles away. She remembers my birthday when 2/3 of my children did not.
I am queen only in my own head, and no, I did not have a blog post on adding fractions. I thought surely there must be a wide assortment of videos on adding fractions using rectangles. But the very first two that I’d clicked on - this and this - really astonished me. They both used grid papers without using the grids. Like, what the heck.
Say we want to add 2/3 and 4/5, same two fractions that popped in my head when I replied to Mary.
Draw two same-size rectangles using the denominators as dimensions.
Students will ask, “Why 3 by 5?” If they don’t, you ask why. And you answer them by asking them to shade in 2/3 for one rectangle and 4/5 for the other. Give them a few seconds to do this and they’ll understand why the 3 by 5 rectangles work pretty well here.
We’re adding the fractions, so let’s combine them.
Similarly with subtraction of fractions.
For multiplication, the word “of” is useful. Of course, the commutative property of multiplication applies too.
To take 2/3 of 4/5, we’d look along the height in this rectangle as we can see the thirds and just grab two of the sections.
Likewise, taking 4/5 of 2/3 is to look along the width of 5 and grab 4 sections of it.
I’ve written about division of fractions previously:
Common Denominator
I already wrote about dividing fractions here and here.
I use the explanation of "dividing by one" to explain why 5/6 divided by 2/3 is the same as 5/6 times 3/2.
But when I was asked recently about how the "common denominator" strategy worked, my muted response was, "Because it does." I didn't mean to be a jerk, rather I just hoped she'd go along with me.
I grabbed a piece of paper and wrote 5 ÷ 3 = 5/3. She was already bored with me. Then I added 1s under the numbers to show 5/1 ÷ 3/1 = 5/3. Right, right? I then changed the problem to 10/2 ÷ 6/2 = 10/6... = 5/3. Still okay, right?
Before I could give another example, she took the paper and rubbed it on my head. Rude.
***
The real common denominator is we're all in this together to #flattenthecurve. This tweet is like rainbow.
My elderly neighbor is self-isolating. So the neighbor kids are playing her a cello concert from her patio. pic.twitter.com/g6Nr2HNVho
— Jackie Borchardt (@JMBorchardt) March 16, 2020
Dividing Fractions
My 6th graders have been working with dividing fractions for the last two weeks. We explore these four ways, in this order:
Number line
Rectangles — I wrote about this here.
Dividing by one
Common denominator
It's completely intentional that we work with the number line and rectangles first. I want my kids to see the answer and that it should match their intuition and understanding.