The basic way of thinking of multiplication and division is the grouping model: having several sets each of which have the same number of items, and the product (product means reasult of multiplication) is the total number of items in all of the sets.
All of the other types are variations on this:
In rate problems, you have a rate: N things per whatever. Think of this in terms of a unit rate: for each 1 whatever, you have N things. If you were direct modeling, you'd have some way of showing how many whatevers you had, and next to each whatever, you would put N things. This corresponds to the grouping/set model in that a whatever is like a set, and the N things are the items in each set. So these correspond:
| grouping model | rate model: |
price model | multiplicative comparison |
| # of sets | # of whatevers | ||
| how much in each set | N things that go with each whatever | ||
| how much total (product) | total number of things |
Price problems are special types of rates where you have cost per item. The key thing to making it a price problem isn't just that is has prices and money, it's that you are using both a unit price ( The unit price is the cost of buying just one item) and the price of multiple items
| grouping model | rate model: |
price model | multiplicative comparison |
| # of sets | # of whatevers | # of items | |
| how much in each set | N things that go with each whatever | cost for 1 item | |
| how much total (product) | total number of things | total cost for all items |
Multiplicative comparison problems are the most tricky (they are really ratio problems). In a multiplicative comparison problem you compare a measurable attribute (length, height, price) of one item to the same measurable attribute for another item. This usually involves a phrase of a form similar to: there are N times as many this's as that's. If you understand rates, you can make a nice correllation between : "N times as many this's as that's" and the rate" N this's per that" (those phrases mean the same thing).
| grouping model | rate model: |
price model | multiplicative comparison |
| # of sets | # of whatevers | # of items | # of that's |
| how much in each set | N things that go with each whatever | cost for 1 item | N this's for each that |
| how much total (product) | total number of things | total cost for all items | total number of this's |
Your instincts are probably quite good, in the case of whole number problems, for telling you when to multiply and when to divide, but I want you to also slow down and think about which model you are using for division. If you can draw a diagram well enough to identify the sets, and how much is in each set for a given word problem, you can tell by what information is provided, and what is missing, which kind of division problem it is. If you want to find out how much is in one set, and you know how many sets, then you know the partition, and it is a partition division problem. If you want to know how many sets there are, and you know how much or how many is in each set, then you know the measure of the set, and it is a measurement division problem. Division problems are named by what you know, not by what you are trying to figure out.