Dividing Fractions: Quotative Division
Let’s explore ¾ divided by ¼ in a quotative division scenario.
Thinking about dividing fractions quotatively, we are being asked to determine how many 1 1-fourths there are in 3 1-fourths. Or, in other words, how many 1 fourths there are in 3 fourths.
When we focus on the unit of measure (in this case, 1-fourths), it is easy to think of this problem as how many 1 wholes are there in 3 wholes. In a sense, we aren’t changing anything because we are in fact asking how many “1 whole” fourths there are in “3 whole” fourths.
This focus on the unit of measure can really help us better understand much of the work that most learn procedurally alone – like dividing fractions.
While things get a bit more hairy when we introduce fractions with uncommon denominators (i.e.: different units of measure), finding a common denominator takes care of that!
Let’s explore at ? divided by ½ also in a quotative situation.
? divided by ½
= 2 thirds divided by 1 half
Wait. We’re stuck in this different size parts situation. That’s no fun.
Let’s fix that.
= 2 third divided by 1 half
= 4 sixths divided by 3 sixths
Let’s be clear here. I have 4 parts that are a sixth of the original whole and I want to determine how many groups of 3 sixths there are (quotative division).
= 4 sixths divided by 3 sixths
= 4/3 groups
More specifically, since I’m thinking quotatively, I should say that there are 4/3 groups of 3 sixths in 4 sixths. Alternatively, there are 4/3 groups of ½ in ?.
Dividing Fractions: Partitive Division
I think it would be a complete rip-off if we didn’t also look at partitive division as well.
So, let’s first explore something fairly simple. In this case, ¾ divided by 1. Since it is partitive division, we would say how much of ¾ is in 1 whole group.
Pretty straight forward. All of the ¾ must be in the 1 whole group.
Now, let’s ramp it up a bit and work with ¾ divided by ¼ partitively. In other words, we are saying that there is ¾ in ¼ groups. If I think of this highlighting the unit of measure as 1-fourth, we could say that there are 3 fourths in 1 fourth groups. Since our unit of measure is the same, I can now just focus on the 3 “whole” fourths that are in 1 “whole” fourth groups to determine that there must be 3 in each group.
Alternatively, we can show this visually.
When thinking partitively, the goal is to determine how many are in 1 whole group, given a quantity (or total quota) and how many groups (or parts) the quantity is to be divided into.
Using the visual to help us organize this thinking, we can quickly see that since we only have 1 one-fourth of a group to divide the total quantity into, the total quantity must fit into that fractional part of 1 whole group.
Since the goal of partitive division is to determine the quota of 1 whole group, we must actually iterate (or scale) our 1 one-fourth of a group up to 4 one-fourth groups (or 1 whole group).
Our result is 12 one-fourths in 4 one-fourth groups or 3 wholes in 1 whole group.
Wow, that was a lot!
If you made it this far, then you’re clearly interested in how fractions (and other areas of mathematics) work conceptually with a focus on the unit of measure.
By all means I am not suggesting that thinking this way will be faster for students to come up with answers to problems. That’s what procedures are for.
What I am suggesting is that if you want to help students understand the mathematics they are engaging in with an end goal of building procedural fluency, then this would be a great place to invest some time in. When students (and we as teachers) are able to see the connections that exist in mathematics between big ideas, they are more readily able to see mathematics as an interconnected discipline that always makes sense.
Only then can we truly Make Math Moments That Matter for every student in our classroom for weeks, months, and even years.
