Readers came through for me when I surveyed on ratios. So I come back to the well seeking your insights.
This time on division of fractions.
The Common Core State Standards reserve division of fractions as the only operation on rational numbers at sixth grade. The other three arithmetic operations come in the earlier grades. The Connected Mathematics response to this is going to be to compactly revisit addition and subtraction, linger a bit longer on multiplication and spend some time and depth on division.
As you surely know, the standard approach is to tell students to invert-and-multiply and then move on.
In previous versions of Connected Math, we have done a single Investigation (4 to 5 days worth of study) on division of fractions, with an emphasis on:
- Quotative (or measurement) models, leading to
- Common denominator algorithm
For those unfamiliar, there are two standard problem types for division: quotative (measurement) and partitive (sharing), exemplified by the following.
Quotative (measurement): I have 35 apples to package in bags. I can fit 7 apples in each bag. How many bags can I fill?
Partitive (sharing): I have 35 apples to share equally among 7 people. How many apples does each person get?
A typical quotative fraction division problem goes like this:
I have 3/4 of a cup of grated cheese. Each omelette requires 1/4 cup of cheese. How many omelettes can I make?
These types of problems lend themselves to wanting the denominators to be common; then we can divide the numerators and voila! This is justified by thinking of the 3/4 cup as 3 units and the 1/4 cup as 1 unit. The units are the same, so it doesn’t matter that they are fractional units. The basic question is How many of this are in that? Just like with the apples and bags-how many sevens are in 35?
But partitive is trickier. A typical problem goes like this:
I have 3/4 of a cup of grated cheese. This is enough to make 1/4 of an omelette. How much cheese do I need to make a whole omelette?
Now the common denominator doesn’t matter so much. Instead, I want to multiply by 4, based on the reasoning that if 3/4 is 1/4 of the whole thing, then 4 times 3/4 must be the whole thing.
For most people, this has quite a different feel from the apples and people problem that exemplifies whole-number partitive division. Having a fractional number of groups complicates the partitive problems.
What we have decided to do in Connected Math is to keep the quotative investigation more or less intact (tweaking based on our experience in the field), and create from scratch an investigation on partitive division.
But the typical examples of partitive fraction division don’t resonate with me. It’s easy enough to divide a fraction by a whole number this way (1/2 pound of peanuts shared among 3 people), but fraction by a fraction is tougher (1/2 pound of peanuts is 2/3 of a share?)
Notice that partitive division problems-whether whole-number or fractional-give unit rate answers. Apples per person. Cups of cheese per omelette. Pounds of peanuts per person.
But in those contexts, it is implausible that I would know the rate for a fraction of a unit but not the whole unit. How do I know that 3/4 cup of cheese is enough for 1/4 of an omelette without knowing how much is in a whole omelette? There is no plausible scenario under which I would have this information without also knowing the unit rate.
So now we get to the question…
What are some unit rates in which it’s plausible to know the fractional rate without knowing the unit rate?
My man Sean pointed out in a related discussion that unit rates involving time fit into this category. I can plausibly know that I walked 1/2 miles in 15 minutes, notice that this is 1/2 mile per 1/4 hour and wonder what this means for my miles-per-hour unit rate.
But in an article I reread today, the authors suggested “2/3 of a cake exactly fills 1/2 of my container”. I don’t like this one nearly as well. Exactly fills 1/2? I like it better than the omelette problem, but I feel we can do better.
How about it?