This lesson is a discussion of the differences between Join, Result Unknown problems and Separate, Result Unknown problems. In the Common Core Math Standards, Join problems are called Add To problems, so a Join, Result Unknown problem can also be called an Add To, Result Unknown problem. Separate problems are called Take From problems in the Common Core. Separate, Result Unknown problems correspond to the Take From, Result Unknown problem type.
Thinking about addition and subtraction in terms of an action taking place is the most elementary and concrete approach. When children are learning about addition and subtraction, they need a simple concrete approach at first. When they solve problems that involve adding or subtracting, children at this stage generally use concrete ways of showing the numbers (usually counters of some kind) and they count to find the solution. The strategy of acting out the problem using counters, and then counting to find the answer is called a direct modeling strategy. The problem types that have the simplest addition and subtraction actions associated with them are the Join and Separate (result unknown) problem types.
A Join problem describes a set, and then there is an action where some amount is added to the set, and at the end there is a new (larger) amount. A typical join problem is:
Mary had 4 pencils. Her teacher gave her 2 more pencils. How many pencils does Mary have now?
The story here indicates how to solve the problem by directly modeling it: count out 4 counters into a pile. Count out 2 more counters and add them (join them) to the pile. Now count the counters in the pile to find out how many. It's a very concrete, action-based solution strategy, and the problem makes it easy to figure out the strategy.
A Separate problem describes a set, and then there is an action where some amount is taken away from the set, and at the end there is a new (smaller) amount in the set. A typical separate problem is:
Kyle had 6 pencils. He gave 2 of his pencils to Mike. How many pencils does Kyle have now?
Again, the story indicates how to solve the problem by direct modeling: count out 6 counters into a pile. Take out 2 counters from the pile (separate them from the pile) and count how many are left. This solution is also very concrete, and follows a clear action described in the problem.
Both of these examples are result-unknown problems. In order to be a result unknown type of problem, it must be a problem that has a result--that is, there is a change over time, and the final state (time-wise) is the result. It also must be a problem where the question asks for the result (the state--number of items in the set--at the end of the action). So, the first is a Join Result Unknown (JRU) and the second is a Separate Result Unknown (SRU) problem.
You can tell if a problem is an action-type problem if it has a change over time. So, if you wanted to visually show what's happening in the problem, you should really make a movie so that you could show the beginning, middle and end. If you could draw a single picture of things at a single moment in time that would show the whole problem, then it's probably not an action type problem. Also, in an action-type problem usually has just one set that the problem is about (in the examples above, the set is Mary's pencils), but the number of objects in the set changes.
Because direct modeling problems of these types involved counting out and acting out in a very clear way, these problems are equally easy to solve by direct modeling. SRU (subtraction) problems are more difficult for a child who is using a counting strategy to try to solve them, but they are not more difficult fot a child to solve by direct modeling.
Summary: The two action type problems are Join and Separate problems.
In a join problem, if you made a movie showing the set in the problem, the number of objects in the set at the end would be larger than the number of objects in the set at the beginning:
Scene 1: oooo
Scene 2: oooo oo
Scene 3:oooooo
In a separate problem if you made a movie showing the set in the problem, the number of objects in the sets at the end would be smaller than the number of objects in the set at the beginning:
Scene 1:oooooo
Scene 2:oooo oo
Scene 3:oooo
If it is a result unknown problem, the problem will tell you what happens in scene 1 and scene 2 and ask how many are in scene 3. There can be other join and separate problems that ask about what happens in different scenes of the movie, but the easiest version (the first one children should learn how to solve) is the result unknown version. In a JRU problem, students are using an action to find a sum (a sum is the result of adding). In an SRU problem, students are using an action to find a difference (a difference is the result of subtracting).