The structure of a part-part-whole problem is that it describes a relationship between a larger sets, and the smaller sets that it is composed of. The problem tells how many are in some combination of the sets, and asks for the size of the remaining set. Part-part-whole problems can be either "whole unknown" problems, where the problem tells the sizes of the smaller sets, and asks for the size of the larger set, or they can be "part unknown" problems, where the problem tells the size of the larger set and one of the smaller sets, and asks for the size of the remaining part.
A sample Part-part-whole, whole-unknown (PPW-WU) problem is:
Adam has 4 large teddy bears, and 2 small teddy bears. How many teddy bears does Adam have in all?
A sample Part-part-whole, part-unknown (PPW-PU) problem is:
Jenna has 6 balloons. 4 of the balloons are red, and the others are blue. How many blue balloons does Jenna have?
In the first example, the two smaller sets--the parts--are the large teddy bears and the small teddy bears, and together they make up the larger set of all of Adam's teddy bears. There is no action described in the problem, so it is not a join or a separate problem, the problem relies on a relationship between a larger set and the parts it is made up of, so it is a part-part-whole problem. Because the question asks for how many are in the larger set (how many in all), this is a whole unknown problem (the whole is what is asked for).
In the second example, the two part sets are the red balloons and the blue balloons, and together they make up the larger set of all of Jenna's balloons. Notice that part of the description was devoted to describing the second part ("the others are blue") even though no special information is given about it. With some parts and wholes, this description can be omitted (such as girls and boys being the parts that make up the whole set of children). As with the first example, there is no action in the problem (no balloons are blown up, popped, gotten or given away), and the problem relies on the relationship between the whole and the parts for its solution. Because the question tells how many are in the larger set (Jenna has 6 balloons), and the question asks for the unknown amount of one of the parts (how many blue balloons), this is a part-unknown problem.
Part-part-whole problems are somewhat more difficult for children in grades K-1 because the structure doesn't have an action indicating how to act out the problem to solve it, and it may also be difficult because children don't have enough experience with set-subset relationships and the language that describes it.
To give children the background experience they need to understand part-part-whole problems, the teacher may plan lessons where...
When children understand the part-part-whole relationships being described in these problems, and understand what the problem is asking, they start to associate these situations with either join (JRU) or separate (SRU) problem solving strategies.
The most typical way for a child to solve a part-part-whole, whole unknown problem is to model it as though it were a join result unknown problem. The modeling strategy for the problem with Adam's bears is:
Count out 4 counters to represent Adam's large bears
Count out 2 more counters to represent Adam's small bears
Join the small bear counters to the set of large bear counters
Count all of the counters to find how many in all/how many at the end.
The most typical way for a child to solve a part-part-whole, part unknown problem is to model it as though it were a separate result unknown problem. the modeling strategy for the problem with Jenna's balloons is:
Count out 6 counters to represent all of Jennas balloons
Separate out 4 counters to represent Jennas red balloons
Count the remaining counters to find out how many are in the other part (blue balloons)/how many at the end.
Some children may model a part-part-whole, part unknown problem as though it were a join change unknown problem. Modeling the problem about Jennas balloons would work this way:
Count out 4 counters to represent Jennas red balloons
Continue counting more counters intil you reach 6 (keeping these counters slightly separated from the first 4)
Count the "extra counters" (those added after the first 4) to find out how many more were added to the set.
Making the connection between PPW-WU and JRU happens very early, and the relationship between the two types of problems becomes cemented almost as soon as the child understands the PPW-WU structure. Making the connection between PPW-PU and SRU or JCU is more difficult, and children may not be able to solve these problems immediately, even when they understand the problem. During discussions of different ways to solve problems, the teacher may wish to show the process of separating a whole set into two parts to solve PPW-PU problems. Children should generally, however, be given time to make sense of the problem on their own before solution strategies are discussed.
Summary:
Part-Part-Whole problems don't have a unique direct modeling strategy associated with them. Instead, students who successfully solve PPW problems using direct modeling use one of the action-based direct modeling strategies (JRU, SRU, JCU) to solve the problems. This requires children to have a strong enough understanding of the part-part-whole situation to reinterpret it to themselves as either a join or a separate problem. Experience with solving join and separate problems is helpful for learning to solve part part whole problems.
Some children have trouble with understanding the problem situation and question in part-part-whole problems. Understanding the meaning of the problem should the first goals in helping children learn how to solve PPW problems. The teacher may need to explain and provide experiences with sets that are separated into smaller sets, or built out of smaller sets, and also may need to explain and provide instruction with the language that is used in part part whole problems.
In general, children will be find PPW-WU problems harder than JRU or SRU problems, but easier than PPW-PU problems. PPW-PU problems are generally more difficult than JCU problems, but less difficult than CDU problems.
A part-part-whole problem does not have an action associated with the problem. No sets change size over the course of the problem, and there is no before and after scenario in a PPW problem. A PPW problem does have a "whole" set that is comprised of two or more smaller "part" sets. If the problem tells the size of the smaller sets, and asks for the size of the large "whole" set, it is a whole unknown problem (PPW-WU). If the problem tells the size of the larger set and one of the smaller sets, and asks for the size of the other "part" set, then it is a part unknown problem (PPW-PU)