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.
This is a topic with which I have considerable experience.
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?
Overthinking my teaching 
Liping Ma’s book had lots of examples. I may be remembering wrong, but I thought there were 3 models for division that the Chinese teachers used.
I have a hard time seeing any useful distinction between “Quotative” and “Partitive” in the examples you give. The word problems seem so artificial as to be pseudocontext.
Rate and area problems seem to result in fairly natural fraction problems:
“I have half a gallon of paint left, and a gallon of paint covers 350 square feet of wall. How many feet of 8-foot-high wall can I paint? If I have 70 feet of wall to paint, how many more quart (1/4 gallon) cans of paint do I need to buy?”
Sue, thanks for the reminder to revisit Ma. I’ll have to go to campus to dig it up. Might the third model be area? (I know the area and one dimension of a rectangle, what’s the other dimension?)
gasstationwithoutpumps, thanks for the pushback. May I say that I am unsurprised to get the pushback on this topic from the university math/science guy? I have argued informally that these kinds of distinctions are unimportant to mathematicians because on a formal level, division is the inverse operation of multiplication, which is itself abstractly defined. Indeed, it can be argued that the role of mathematics teaching ought to be to erase distinctions such as quotative/partitive.
And that argument is valid; it is a goal of mathematics education to get students to view the world abstractly enough that they don’t worry about those distinctions. But the distinctions are indeed very real in the minds of students. I would be really interested in your take on CGI.
As for the particular problems in this post-I’m not trying to defend them as particularly good examples of the form. They are intended to illustrate the different structures of the two division concepts. Maybe you’ll like this more abstract explanation better:
Quotative: Divisor is a rate.
Partitive: Quotient is a rate.
Oh…but on the specific topic I asked about, gasstation is thinking along the lines I have been working on this morning. Recall that question at hand is this:
I have two categories.
(1) Measuring contexts in which we know some rate per subunit (or group of subunits) and we want to know the rate per other unit; e.g. if I know the number of sq. feet a quart of paint covers, I want to know how much a whole gallon does, or Sean‘s miles per 15 minutes; how much per hour?
(2) Mixtures in which I know the volume of one of the parts and want to know the final volume. So lemonade is mixed at a rate of 1 can concentrate: 3 and 1/3 cans water. If the can holds 6 oz. of concentrate, how big a container do I need to hold my finished lemonade?
These of course need refining as instructional problems. But I think I’ve got the beginnings of something to work with.
But the discussion is still open, gang. Bring it on.
One more thing…gasstationwithoutpumps‘s problem has one fractional divisor (the 1/4 gallon) and is quotative, not partitive. The 1/4 can be seen as a rate: 1/4 quarts per gallon. That rate is the divisor and the quotient is a number of quarts.
Correction 7/5/2011: It’s 1/4 gallons per quart, isn’t it? Oops…
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Distances work well. I ran 3/4 of a mile, which was only 1/4 of the race. How many miles is the entire race?
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Plausible partitive division example (?): My husband purchase a 1/4-lb box of candy for $3.60. How much would he pay for 1 pound?
A kindred spirit. I do love a niche market! I laughed out loud at your blog name and that comment. Thank you SO much for helping me overthink my teaching and be an expert at partitive division of fractions in 5 minutes instead of 50 or more. I hope you are still writing. I am eager to read more of your blog.