In a Compare problem, there are two sets that are being compared and there is the difference between the two sets. The compare relationship between the sets is expressed differently in different problems. In the Compare, Difference Unknown problems that we have worked with so far, the size of both sets is presented, and we are asked for the difference in size between the sets. In the other two types of compare problems, the size of one of the sets is presented, and the difference is described, and the size of the other set is asked for. The two sets, however, play different roles in the sentence that describes the difference, and one version is significantly easier than the other for children to understand and work with, so these compare problems are put into two groupes based on that criterion.
A basic sentence that describes the difference in quantity (in a word problem about toy cars) is:
Mike has 4 more toy cars than John has.
In the English language, the roles that words play in a sentence are determined largely by word order, and that is true in comparison sentences. The key word that controls this description is than. Mike's cars are being compared to John's cars. The set of John's cars is the reference point--things are described acccording to their relationship with the size of the set of John's cars.
We call that set that is being compared to the referent. The set of Mike's cars is described in terms of how different it is from the referent set. We call that amount (the amount in the set being compared) the compared quantity.
Referent problems about discrete sets usually use the comparison words "more" or "fewer" along with "than" to describe the relationship. In problems about continuous quantities, comparison words like taller, shorter, heavier, lighter, and less may also be used.
Practice identifying the referent and compared quantity in 6 sentences, and then come back here and continue reading.
CQU vs. CRU:
If I choose a comparitive sentence: "Michelle has 2 fewer Barbies than Janet." The referent is the number of Barbies Janet has, and the compared quantity is the number of Barbies that Michelle has. I can write word problems with either of those being unknown and the other being given:
Compared quantity unknown:
Janet has 7 Barbies. Michelle has 2 fewer Barbies than Janet. How many Barbies does Michelle have?
Referent Unknown:
Michelle has 5 Barbies. Michelle has 2 fewer Barbies than Janet. How many Barbies does Janet have?
These problems have the same comparison sentence in them, so in both cases the referent is the number of Barbies Janet has. The referent in a comparison statement is the set we are comparing other sets to--it's the benchmark or yardstick against which we measure other things. In a compared quantity unknown problem, that referent/benchmark is known, and the comparison sentence describes how the compared quantity relates to it. In the referent unknown problem, you don't know how big the benchmark set is: that's a really strange situation to be in. You need to deduce the size of the benchmark set from other things around it, which makes referent unknown problems more difficult.
There are other ways, too, of seeing why CQU problems are so much easier than CRU problems:
Compare, Compared Quantity Unknown (CQU)
It turns out that problems where the compared quantity is unknown are a lot easier than problems where the referent is unknown. Sometimes I think (when watching children figure these problems out) that the compared quantity unknown problems are even easier than the difference unknown problems.
Examples:
Janet has 7 Barbies. Michelle has 2 fewer Barbies than Janet. How many Barbies does Michelle have?
Sam has 8 Beyblades. Gus has 2 more Beyblades than Sam. How many Beyblades does Gus have?
I have here one "fewer...than" and one "more...than" problem, both tell the referent quantity, and ask for the compared quantity. Notice what you do to solve each:
In the "fewer...than" problem, what you do to solve is you take the number Janet has and take 2 away from the set to find out how many Michelle has. In this problem, the word "fewer" corresponds to the action of taking away. That's what you expect it to do, so the sense of the problem, and the words in the problem fit with the action you take to solve.
In the "more...than" problem, what you do to solve is take the number that Sam has and add/join on two more to get the number that Gus has. Again, the word "more" corresponds to the action of joining or adding on. The word and the action you take to solve fit together and reinforce each other.
Compare, Referent Unknown (CRU)
So now we'll look at what makes referent unknown problems different and harder for children to understand and solve.
Examples:
Michelle has 5 Barbies. Michelle has 2 fewer Barbies than Janet. How many Barbies does Janet have?
Gus has 10 Beyblades. Gus has 2 more Beyblades than Sam. How many Beyblades does Sam have?
So, first, if you compare these problems to the previous pair, you'll see that the situation is the same, and I have left the comparison statement the same, but I switched whose value I am telling, and whose value I am asking about.
Let's look at how you solve these:
In the "fewer...than" problem, now you know that 5 is 2 fewer than the amount Janet has, so you have to add 2 to 5 to find out how many Janet has. Here the word 'fewer" is paired with the action of adding.
In the "greater...than" problem, 10 is 2 more than the amount Sam has, so you need to take 2 away from 10 to find out how much Sam has. The word "more" is paired with the action of taking away.
That switch between the descriptive word and the action causes some dissonance--it doesn't fit as neatly into your brain, and you have to stop and think to be sure you're doing the right thing. That makes it a lot harder to solve. You have to be pretty good at holding different aspects of a problem in your head in order to make sense of a Compare Referent Unknown problem, and restate it to make it easier to understand and solve (one way of solving a Compare Referent Unknown is to restate it as a Compare, Compared Quantity Unknown problem).
Example of Changing CRU to CQU (blue indicates compared quantity, red indicates referent):
Michelle has 5 Barbies. Michelle has 2 fewer Barbies than Janet. How many Barbies does Janet have?
Reword the middle--comparison--sentence to switch the roles of the compared quantity and the referent:
Michelle has 5 Barbies. Janet has 2 more Barbies than Michelle. How many Barbies does Janet have?Gus has 10 Beyblades. Gus has 2 more Beyblades than Sam. How many Beyblades does Sam have?
Reword the middle--comparison--sentence to switch the roles of the compared quantity and the referent:
Gus has 10 Beyblades. Sam has 2 fewer Beyblades than Gus. How many Beyblades does Sam have?
Notice that I did something different here than I did when I switched originally from CQU to CRU. When I switched from CQU to CRU the first time, I changed the problem I kept the comparison sentence the same. In this switch, I kept the question the same, but I rephrased the comparison sentence.
Introducing CQU and CRU problems
Compare, Compared Quantity Unknown problems can be introduced once children become confident at interpreting and solving Compare, Difference Unknown problems. It's probably a good idea to wait until after children have experience with CDU problems rather than introducing CQU problems earlier, because it's important that children understand the compare sentences correctly. It's so easy with CQU problems to latch onto a rule like "if it says more, then add, and if it says fewer then subtract", but if students are solving CQU problems using that sort of a rule, it's going to make things a lot harder for them when they get to CRU problems, because there the rule is completely backwards, so... Children can solve CQU problems fairly early on, but emphasize making sense of and understanding the problem so that you're not making CRU problems even harder when you/they get to them.
Compare, Referent Unknown problems are considerably more difficult--they require some fairly sophisticated language use and logic. This means they need to be introduced with considerable thought given to how you might help children understand the problems through modeling and pictures, and you can't expect that they will be ready for these problems as soon as they have mastered CQU problems.
Direct Modeling
Both Compare, Compared Quantity Unknown and Compare Referent Unknown problems can be solved by direct modeling. I think they aren't included in the list of problem types that can be solved by direct modeling in the Childrens Mathematics book because aren't ready to understand, interpret and solve them until they are pretty far along, and most children are moving to using counting or derived facts strategies to solve problems. There are two ways of solving CQU and CRU problems by direct modeling:
The first way is to use a JRU or SRU strategy. If you look at the CQU problems, the problem solving strategies I described was a JRU strategy for the "more...than" problem, and an SRU strategy for the "fewer...than" problem. A CRU problem is a CQU problem, just with the sentence structure turned around so, if you can wrap your head around it, you can solve it in the same way you would solve the related CQU problem. So far as I can tell, this is what children more often do simultaneously.
Additionally, you can use a variation on the Compare Difference Unknown direct modeling strategy to solve CQU and CRU problems. This is sometimes useful as a teacher strategy--it can help clarify the comparison statement as part of the problem.
Example: Michelle has 5 Barbies. Michelle has 2 fewer Barbies than Janet. How many Barbies does Janet have?
Start by putting out 5 counters for Michelle's Barbies:
oooooThen put out counters for Janet's Barbies to match Michelle's Barbies:
ooooo
ooooo...Then add or take away counters from Janet's pile until it matches the comparison sentence: Michelle has 2 fewer than Janet:
ooooo
ooooooo
Summary:
In a comparison statement, the word "than" points to the referent. In other words, the noun that follows the word "than" tells what the set is that we are comparing to, and using as our benchmark in describing the other (compared) quantity.
Compare, compared quantity problems are relatively easy because the comparison sentence tells how to modify the referent set to find the compared quantity.
Compare, referent problems are difficult, because the comparison sentence doesn't tell you directly how to solve--you need to do some detective work to figure out the referent problem.
In Compare, Compared Quantity problems, if the compared quantity is described as some amount more than the referent, then you can add to find the compared quantity; similarly, if the compared quantity is described as some amount less or fewer than the referent, then you can subtract to find the compared quantity: the operation matches the idea triggered by the comparison word.
In Compare, Referent Unknown problems, if the compared quantity is described as some amount more than the referent, then you must subtract from the compared quantity to find the referent; similarly if the compared quantity is described as being some amount fewer or less than the referent, then you must add on to the compared quantity to find the referent. The operation is backwards from what is suggested by the comparison word. Mathematically, we say the addition and subtractions are inverse operations, so the operation is the inverse of what is suggested by the comparison word.
CQU and CRU can be solved by direct modeling in two ways. The first way is to perform the appropriate addition or subtraction operation using counters in the same way as is done for a JRU or SRU problem. Children are reasonably likely to make that connection and solve the problems that way for CQU problems. The second way is to line up counters as you would do for a compare difference unknown, and to add or remove counters from the unknown set until the amounts fit the comparison sentence.
If you need to directly show children how to solve CQU and CRU problems, it's probably wisest to start with a process such as direct modeling with counters lined up side-by-side to show the comparison: this manipulative process and/or diagram shows the comparison and the difference, and lets children compare the diagram to the problem statement. Directly telling children to add or subtract in certain cases is likely to backfire because the differences between CQU and CRU problems are subtle enough that trying to get students to differentiate between the two is likely to cause more confusion than it solves (and you can't just give an add/subtract rule for one, because it's likely to be mis-applied to the other situation).