I bought a ham. It was touch-and-go there for awhile. As I was picking up and putting down hams of various sizes, I was calculating baking times. My essential question was, can I have this on the table by six? Simultaneously, I was trying to remember if this was partitive or quotative division.
In partitive division problems, a.k.a division as (fair) sharing, the number of groups is known. This type of problem asks how many are in each group. In quotative division problems, a.k.a. division as measurement, the number in each group is known. This type of problem asks how many groups. For example: 6 ÷ 3 = 2 (partitive) means ♦♦ ♦♦ ♦♦; 6 ÷ 3 = 2 (quotative) means ♦♦♦ ♦♦♦. This distinction isn’t limited to collections of objects. Consider 6 ÷ 3 as cutting a 6 m rope into 3 parts (sharing) vs. cutting lengths of 3 m (measurement). Nor are these meanings limited to whole numbers. Which brings me back to my ham…
The directions read “bake approximately 15 minutes per pound (0.454 kg) or until internal temperature reaches whatever.” But here’s the thing:
Kilograms, not pounds. I could have converted from kilograms to pounds by doubling then adding ten percent of that. Instead, I divided 1.214 by 0.454. I know, I know, this still gives me the weight of my ham in pounds. But at the time, I interpreted 2.67 as the number of repeated additions of 15 minutes in my baking time. Either way, I determined how many 0.454s there are in 1.214. Quotative division. By a decimal.
As a math task, this is clunky. The picture book How Much Does a Ladybug Weigh? by Alison Limentani is a more promising jumping off point for quotative division in the classroom. On each page, the weight of one animal is expressed in terms of a smaller animal.
Using the data at the back of the book, we have 3.2 ÷ 0.53 = 6. We could ask children to make other comparisons (e.g., how many grasshoppers weigh the same as one garden snail?).
[Insert link to Marc‘s First Peoples beaded necklace task here]
In the past, I have struggled with partitive division by decimals (or fractions). But I found the following example at The Fair this summer:
It’s not intuitive–at least to me–to think of 1/3 in 12 ÷ 1/3 as the number of groups. Take a step back and think about 26 ÷ 1 = 26. The cost, $26, is shared between 1 rack of ribs; the quotient represents the unit price, $26/rack, if the unit is a rack. This result should be… underwhelming.
Before we think about dividing by a fraction here, let’s imagine dividing by a whole number (not equal to one). What if I paid $72 for 3 racks? (Don’t look for these numbers in the photo above–I’m making them up.) In 72 ÷ 3 = 24, the cost, $72, is shared between the number of racks, 3; again, the quotient represents the unit price, $24/rack. Partitive division.
So what about 12 ÷ 1/3? The cost is still distributed across the number of racks; once again, the quotient represents the unit price, $36/(full) rack. The underlying relationship between dividend, divisor, and quotient hasn’t changed because of a fraction; the fundamental meaning (partitive division) remains the same.
We could have solved this problem by asking a parallel question, how many 1/3s in 12? And this quotative interpretation makes sense with naked numbers. But it falls apart in this context–how many 1/3 racks in 12 dollars? Units, man! If dollars were racks, a quotative interpretation would make sense–how many 1/3 racks in 12 full racks?
As a math task, this, too, is clunky. My favourite math tasks for partitive division by fractions are still Andrew Stadel’s estimation jams.
(Looking for a quotative division problem that involves whole numbers? See Graham Fletcher’s Seesaw three-act math task. For partitive, there’s Bean Thirteen.)