How easy it is to solve a word problem by direct modeling (acting it out) is one measure of how easy the problem is. Besides the Join and Separate Result Unknown types, there are some others that children are often successful at direct modeling. This lesson discusses three such problem types that have direct modeling strategies associated with them that are significantly different from the direct modeling strategies for JRU and SRU problems:
Join problems where the change is the unknown, especially "what if" questions can be efficiently solved by direct modeling (JCU). Join problems correspond to the category of Add To problems in the Common Core, so Join, Change Unknown problems correspond to Add To, Change Unknown problems.
Separate problems where the change is unknown. Separate problems correspond to the category of Take From problems in the Common Core, so Separate, Change Unknown problems correspond to Take From, Change Unknown problems.
Problems where you are comparing two amounts, especially straight forward problems where the unknown is how much larger one amount is than the other can be efficiently solved by direct modeling, though the structure of these problems is difficult for some children (CDU). Compare, Difference Unknown problems have the same name as both CGI and Common Core problem types.
Join Change Unknown (JCU): The next-most accessible problem type (after JRU and SRU) is Join problems where the change is unknown. Two typical examples of join change unknown problems are:
Ted made 4 paper airplanes in the morning. How many more paper airplanes does he need to make to have 6 paper airplanes?
Sarah made 4 paper airplanes in the morning. After lunch she made some more paper airplanes. Now she has 6 paper airplanes. How many paper airplanes did she make after lunch?
Both of these have an action (making more paper airplanes) that increases the number in the set. This makes it a join problem. Both of these tell the amounts at the start and at the end the result of the change), but they don't tell the amount of the change--they ask for that amount. Because the change is the amount asked for, not given, these are Join Change Unknown problems (JCU). If you drew a movie showing these problems they would both look something like this:
scene 1: ^^^^
scene 2: ^^^^ ????
scene 3: ^^^^^^
Unlike the JRU and SRU types, however, the ways that a child would direct model to solve a JCU problem is not fundamentally identical to these movie scenes. The thinking behind the direct modeling strategy is clearest for the first example. The problem about Ted's paper airplanes is a "what if" problem; it's format is: I know how things are now, and I need to figure out what to do next. This prompts the direct modeling steps:
Put out 4 counters (for the 4 paper airplanes Ted already has: oooo). Keep counting as you put out more counters until you get to 6 (you want to have 6 eventually so keep going until you get there oooo oo). Go back and count the extras that you put on after the first 4 (if you didn't keep your piles separate enough, you can go back and recount the first 4 to make sure which counters are which oooo oo). The basic action here is keeping track as you count on.
The direct modeling steps for solving Sarah's paper airplanes problem are the same, but thinking of the unknown amount as occurring in the past rather than as a "what if" action in the future makes it harder for young children to figure out the solution strategy.
JCU problems are quite do-able for a child at the direct modeling stage, but only once they have enough experience to start thinking ahead. You need to be able to keep track while you count on, and to keep track of which counters you added after the start amount. Children need a reasonable amount of experience and practice with JRU and SRU problems, and with counting objects, to be ready to keep track of the parts of the set in this way.
Notice that if you were going to solve a JCU problem using arithmetic, you would subtract (6-4=?) which you might think of as a missing number addition problem (4+?=6). This is an example of a join problem--an amount is joined to the initial set, and the end state is larger than the starting state--but because the problem asks for the change rather than the result, the associated arithemetic solution uses subtraction rather than addition. This is a good example of the inverse relationship of addition and subtraction: a missing number addition problem corresponds to a subtraction problem.
Separate, change unknown (SCU): Separate, change unknown problems can also be solved by direct modeling, but are considerably more difficult. Further, the difficulty lies in doing the direct modeling, rather than in figuring out what to do. If the numbers are small and easily counted, SCU problems are appropriate problems for children who are confident with solving JCU and SRU problems.
A typical SCU problem is:
Mike's mom gave him 8 apple slices. He ate some of the apple slices, and now there are 3 apple slices left. How many apple slices did he eat?
To solve this problem by direct modeling, one first counts out 8 counters to show the start amount. Then one takes out some of the counters, and counts the ones that are left, and continues taking out counters, and counting the remaining ones until there are 3 left. then one counts the amount of counters taken out (separated) from the start amount to find out how many were taken out (eaten).
This direct modeling strategy is very similar to the strategy for solving JCU problems, so if a child has experience with JCU problems, they are likely to be able to figure out how to solve SCU problems. Carrying out the SCU solution strategy is significantly more difficult than the JCU strategy because it either involves counting backwards to find out how much is left when counters are taken out, or it involves repeatedly counting the amount left until you get to the desired amount.
Compare Difference Unknown (CDU): One of the fundamental ways of thinking about subtraction is as the difference between two numbers (given two numbers, how much larger is one than the other?). Two typical compare problems that ask for the difference between the two amounts are:
Rachel has 6 markers and 4 pencils. How many more markers does than pencils does she have?
Jacob has 6 toy trucks. David has 4 toy trucks. How many fewer toy trucks does David have than Jacob?
Compare problems are some of the most difficult of all of the basic addition and subtraction problem types for children to decode and solve. Part of the difficulty lies in this being a static problem-type, and there is no associated action in the problems to indicate what to do to solve it. The main thing, however, that makes compare problems difficult is the vocabulary and sentence structure of the problems; children need specific experience and instruction with the structure and meaning of comparison comparison sentences (how many more/fewer of ____ than ____). Children who are successful at understanding the questions can solve them readily by direct modeling. The usual (and most clear) direct modeling strategy for compare problems is to take out counters for each set (in these examples, the child would count out a group of 6 and a group of 4), and then to line up the counters from the sets so they can see how many are the same in both sets (how many have a partner), and then count the number that are left over/don't have a partner:
set 1: oooo oo (count the last two--the ones that don't have a partner in the other set)
set 2: oooo
Practicing using comparison phrases like: "there are two more forks than spoons", and asking related comparison questions (that ask not just "which is larger", but "how much larger") provide the background children need to be able to understand math problems that use comparison. Being able to think of subtraction and addition in terms of comparisons turns out to be very useful for thinking flexibly about numbers, so spending time on modeling and investigating comparison situations is a valuable investment in children's growing number sense in grades K-2. Additionally, we most often phrase comparison questions as "how many more", so teachers need to be aware that it may be necessary to teach the use of the words "less" and "fewer" separately at some point (this is not particularly important from a number sense standpoint, but it's helpful from a communication and vocabulary standpoint).
Unlike SCU problems, which are difficult to solve by direct modeling, even once you figure out how to do it, CDU problems are tricky to figure out how to solve by direct modeling, but once you have it figured out, the process of solving CDU problems by direct modeling is straightforward and fairly easy. Thinking of subtraction in terms of comparison, as we'll see later, is an important thinking strategy, so time spent on understanding and solving compare problems is a valuable investment in developing children's number sense.
Summary:
Join, change unknown (JCU): Join problems have an action that increases the amount that the set started with. In a join change unknown problem, the problem tells us how many there were to start with, and how many the set ended with, or that we would like the set to end with, and it asks how much was joined or should be joined to the set to get that larger amount.
JCU problems are typically direct modeled using a "join to" strategy, where after counting out the starting amount, the child continues counting out counters (and counting on) until the end amount is reached, at which point the cocunters that were added on after the starting amount are recounted to find how much the increase was.
JCU problems where the question is about a hypothetical future end situation that someone would like to reach are generally easier for children to understand and translate into a direct modeling strategy and solution. JCU and part part whole problems where the whole is unknown are fairly comperable in the amount of sophistication they require from children to solve.
Separate, change unknown (SCU): Separate problem has an action that decreases the amount that the set started with. In a separate change unknown problem, the problem tells how many there were to start with, and how many the set ended with, and asks how much was separated from the set to get to that smaller amount. Unlike JCU problems, SCU problems are rarely presented in a "what if" (future change) context.
SCU problems are typically direct modeled using a "separate to" strategy, where after counting out the starting amount, children separate counters from the group until the amount left in the group is the end amount. Children often recount the objects in the group several times during this process. Finally, the counters that were separated or taken away from the group are counted to find the size of the change.
SCU problems are somewhat more difficult for children to figure out how to direct model than are JCU problems. SCU problems can also be difficult to solve by direct modeling if the numbers involved are not small.
Compare, difference unknown (CDU): Compare problems do not have an action--they are a description of the relationship between the number of things in two sets that don't change size over the course of the problems. Compare problems where the difference is unknown have a question asking how much larger (or smaller) one set is than another.
CDU problems are typically direct modeled using a strategy where two sets of counters are laid out, and then matched up side by side so that the compared amounts are clearly related to each other, and then the amount "left over" (not paired) is counted to find the solution.
CDU problems are the most difficult type of problem (to understand) that can be readily solved by direct modeling. Part of the difficulty comes from the lack of action in the problem, but a larger stumbling block is the complexity of sentences describing and asking about comparisons. Part of teaching and preparing students to solve compare problems involves vocabulary and sentence building practice with comparisons. Despite the difficulty, it's valuable for children to learn how to interpret and solve compare problems, because they provide a useful way of understanding subtraction.