For each of these problems 1-12, identify whether they are Compare, Compared Quantity Unknown (CQU), Compare Referent Unknown) or one of the other problem types you have previously learned about.
1. Peter has 20 shiny stickers. He has 10 fewer plain stickers than shiny stickers. How many plain stickers does he have?
CQU. First there is no action taking place, and there is a comparison sentence (10 fewer... than...). In the comparison sentence, "shiny stickers" (following the word "than") is the thing being compared to--it is the referent. The thing being compared is "plain stickers"--the compared quantity. The question asks how many plain stickers, so the compared quantity is unknown. Also, if you think through it, he should have 10 plain stickers, which is what you get when you subtract, which matches what you expect from the word "fewer", which also indicates that it is a CQU problem.
2. Shauna has 11 joke books. She has 4 more joke books than Blake. How many joke books does Blake have?
CRU. First there is no action taking place, and there is a comparison sentence (4 more... than...). In the comparison sentence, "Blake['s joke books]" (following the word "than") is the thing being compared to--it is the referent. The thing being compared is "[the joke books] she [Shauna] has"--the compared quantity. The question asks how many joke books Blake has, so the referent is unknown. Also, if you think through it, he should have 7 plain stickers, which is what you get when you subtract, which is opposite of what you expect from the word "more", which also indicates that it is a CRU problem.
3. Anne has 3 thick markers and 8 thin markers. How many markers does she have in all?
PPW-WU. There is no action taking place, and no comparison statement or question. The question asks how many in all, when combining two groups, so it is a part-part-whole, whole unknown.
4. Rita has 8 red flowers. She has 5 fewer red flowers than blue flowers. How many blue flowers does she have?
CRU. First there is no action taking place, and there is a comparison sentence (5 fewer... than...). In the comparison sentence, "blue flowers" (following the word "than") is the thing being compared to--it is the referent. The thing being compared is "red flowers"--the compared quantity. The question asks how blue flowers Rita has, so the referent is unknown. Also, if you think through it, she should have 13 blue flowers, which is what you get when you add, which is opposite of what you expect from the word "fewer", which also indicates that it is a CRU problem.
5. John has 8 balloons. He has 6 more balloons than Diana. How many balloons does Diana have?
CRU. First there is no action taking place, and there is a comparison sentence (6 more... than...). In the comparison sentence, "Diana['s balloons]" (following the word "than") is the thing being compared to--it is the referent. The thing being compared is "[the balloons] he [John] has"--the compared quantity. The question asks how many balloons Diana has, so the referent is unknown. Also, if you think through it, she should have 2 balloons, which is what you get when you subtract, which is opposite of what you expect from the word "more", which also indicates that it is a CRU problem.
6. Blake has 3 glow in the dark bugs. Ralph has 10 more glow in the dark bugs than Blake. How many glow in the dark bugs does Ralph have?
CQU. First there is no action taking place, and there is a comparison sentence (10 more... than...). In the comparison sentence, "Blake['s bugs]" (following the word "than") is the thing being compared to--it is the referent. The thing being compared is "[the bugs] Ralph has"--the compared quantity. The question asks how many bugs Ralph has, so the compared quantity is unknown. Also, if you think through it, he should have 13 plain stickers, which is what you get when you add, which matches what you expect from the word "more", which also indicates that it is a CQU problem.
7. David had 8 dinosaurs. For his birthday, he got 1 more dinosaur. How many dinosaurs does he have now?
JRU: There is a change over time, and it is an increase. The question asks for the result of the increase so it is a join result unknown problem.
8. Ralph has 14 big erasers. He has 4 more big erasers than small erasers. How many small erasers does he have?
CRU. First there is no action taking place, and there is a comparison sentence (4 more... than...). In the comparison sentence, "small erasers" (following the word "than") is the thing being compared to--it is the referent. The thing being compared is "big erasers"--the compared quantity. The question asks how many small erasers Ralph has, so the referent is unknown. Also, if you think through it, he should have 10 small erasers, which is what you get when you subtract, which is opposite of what you expect from the word "more", which also indicates that it is a CRU problem.
9. Nora has 14 white T-shirts. She has 9 fewer blue T-shirts than white T-shirts. How many blue T-shirts does she have?
CQU. First there is no action taking place, and there is a comparison sentence (9 fewer... than...). In the comparison sentence, "white T-shirts" (following the word "than") is the thing being compared to--it is the referent. The thing being compared is "blue T-shirts"--the compared quantity. The question asks how many blue T-shirts Nora has, so the compared quantity is unknown. Also, if you think through it, he should have 5 blue T-shirts, which is what you get when you subtract, which matches what you expect from the word "fewer", which also indicates that it is a CQU problem.
10. Laura has 10 stuffed toy animals and 8 hard plastic toy animals. How many more stuffed toy animals than hard plastic toy animals does Laura have?
CDU. First, there is no action taking place, and there is a comparison (more...than...) in the question. Since the comparison is in the question, this is a compare, difference unknown problem.
11. Anne has 7 Bratz. Gina has 8 more Bratz than Anne. How many Bratz does Gina have?
CQU. First there is no action taking place, and there is a comparison sentence (10 more... than...). In the comparison sentence, "Blake['s bugs]" (following the word "than") is the thing being compared to--it is the referent. The thing being compared is "[the bugs] Ralph has"--the compared quantity. The question asks how many bugs Ralph has, so the compared quantity is unknown. Also, if you think through it, he should have 13 plain stickers, which is what you get when you add, which matches what you expect from the word "more", which also indicates that it is a CQU problem.
12. Gwen made 3 clay animals. How many more does she have to make to have 8 clay animals?
JCU. There is a change over time (an increase) so this makes this a join rather than a compare problem. The unknown is how many more to make, which is the change over time, so this is a change unknown problem.
Other questions:
13. How do you identify the referent in a comparison sentence?
The referent is the set that follows the word "than".
14. What makes CQU problems so much easier than CRU problems?
The referent is known, so the comparison sentence tells what to do to figure out the compared quantity.
Another way to look at this is that the comparison words more/fewer tell what to do to the referent to find the compared quantity: we addi and subtract in the way we indicated by the comparison words (more triggers the idea of getting more--addition; fewer triggers the idea of having less--the result of a subtraction). In a CRU problem these relationships are reversed, so it's harder to keep track of.
15. Is it a good idea to tell children to add when they see the word "more" and subtract when they see the word "fewer" or "less"? Why or why not?
No, it's a bad idea because while that works fine for CQU problems, it's exactly wrong (backwards) for CRU problems. We need to do our best to not create misconceptions along the way if we can reasonably avoid it.
16. Explain how to solve the following problem by direct modeling in a way that shows both sets:
Marcia has 8 pencils. Adam has 3 more pencils than Marcia. How many pencils does Adam have?
Put out 8 counters for Marcia's pencils, and 8 lined up side by side for Adam's pencils
oooooooo
oooooooo...Now, add onto Adam's pencils until he has 3 more than Marcia:
oooooooo
ooooooooooo[Count the counters representing Adam's pencils]
17. Explain how to solve the following problem by direct modeling in a way that shows both sets:
Kim has 7 small toy horses. She has 3 more small toy horses than large toy horses. How many large toy horses does she have?
Put out 7 counters for the small horses, and 7 lined up side by side for the large horses.
ooooooo
ooooooo...Now, take away counters from the large horse side until there are 3 more small horses than large ones:
ooooooo
oooo[Count the counters representing the large horses]