| Main
Type |
sub-type |
sub-type |
sub-type |
| Join: A join problem situation is one in which an amount is had, and then more is added to the original amount. Example: John has some paper airplanes (original amount). Then he makes some more paper airplanes (more is added to John's set of paper airplanes). Afterward he has an increased amount of paper airplanes (the result of addition to the original set) Video explanation |
Result
Unknown: In this problem, the original amount and the amount of increase are known, and the final amount is unknown. Example: John had 6 paper airplanes. Later he made 3 more paper airplanes. How many airplanes does he have now? |
Change
Unknown: In this problem the original and the final amount are known, and the amount of increase is unknown. There are two main sub-sub-types, depending on whether the change is in the past or the future. Example (change in future): John had 6 paper airplanes. How many more does he need to make to have 9 paper airplanes. Example (change in past): John had 6 paper airplanes. Later he made some more paper airplanes. Now he has 9. How many more did he make? |
Start
Unknown: In this problem, the change and the final amount are known, and the original amount is asked for. Example: John had some paper airplanes. Later he made 3 more paper airplanes. Now he has 9 airplanes. How many airplanes did he have at first? |
| Separate: A separate problem situation is one where an original amount is had, and then some is taken away from the original, and some amount remains. Example: Mary made some block towers (the original amount), then some of the towers fell down (removed from the original amount), and some number of towers are left (the remaining amount. Video explanation |
Result
Unknown: In this problem, the original amount and the amount taken away are known, and the resulting amount is asked for. Example: Mary made 8 block towers. 3 of the towers fell down. How many block towers does Mary have now? |
Change
Unknown: In this problem the original and the final amounts are known, and the amount of the change (amount taken away) is asked for. Example: Mary made 8 block towers. Some of the towers fell down. Now she has 5 block towers. How many towers fell down? |
Start
Unknown: In this problem the amount of change (amount of decrease) and the final amount are known, and the original amount is asked for. Example: Mary made some block towers. 3 of them fell down, and now there are 5 left. How many towers did she make to begin with? |
| Part-Part-Whole: A part-part-whole problem situation is one where objects are considered as two smaller sets and as part of a larger set. Part of the cognitive task is to recognize objects or amounts as being simultaneously part of a smaller and a larger set. Examples: There are (some amount of) children playing on the playground. Some (amount) are boys, and some (amount) are girls. or There are (some amount of) dishes in the cupboard. Some (amount) are small and some (amount) are large. Video explanation |
Whole
Unknown: In this problem, the amounts in the two smaller sets are known, and the amount in the larger set is asked for. Example: There are 8 small plates and 6 large plates in the cupboard. How many plates are in the cupboard in all? Note: Adults are extremely likely to mix this one up with Join, result unknown, but in this problem there is no change over time. |
Part
Unknown: In this problem, the amount in the larger set and in one of the smaller sets are known, and the amount in the other smaller set is asked for. Example: There are 23 children on the playground. 9 of them are boys and the rest are girls. How many girls are on the playground? |
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| Compare:
A compare problem situation is one where there are two sets of objects, and the sets are compared and the difference is found. The difference is expressed as the amount that one set is larger or smaller than the other. Example: Jan has some (amount of) blue beads, and (some amount of) red beads. She has (some number) more (or fewer) red beads than blue beads. Video explanation |
Difference
Unknown: In this problem the amounts in the two sets are known, and the difference between them is asked for. Example: Jan has 24 blue beads and 18 red beads. How many more blue beads than red beads does Jan have? |
Compared
Quantity Unknown: In this problem the difference between the two sets is known, and the set that is compared to (the referent) in the comparing set is known. The amount in the other set is asked for Example: Jack has 8 marbles. Sam has 3 fewer marbles than Jack. How many marbles does Sam have? Note: the word "than" in the comparing sentence refers to Jack. The set of Jack's marbles is the referent. Sam is the quantity being compared (the unknown). Note also: the comparing phrase "fewer than" indicates the appropriate operation (subtraction) |
Referent
Unknown: In this problem the difference between the two sets is known, and the set that is compared to (the referent) is unknown. The other set is also known. Example: Jenna has 6 pencils. Jenna has 2 more pencils than Erick. How many pencils does Erick have? Note again: "than" identifies the referent (Erick's amount) Note also: the comparing phrase "more than" indicates the opposite operation than the one that is appropriate. |