Addition and Subtraction Categories
Join
Join problems involve the operation of addition. Within this category there are three different types. The most basic kind is when two quantities are joined and the result is unknown. Here is an example:
Daquan has 3 toy cars. His grandma buys him 2 more for his birthday. How many cars does he have now?
This problem would be solved with addition. Daquan starts with 3 toy cars and then 2 more toy cars that his grandma gives him are added to his collection. The answer is unknown, but by adding 3 and 2 the answer can be determined.
3 + 2 = []
start unknown change
The answer is 5 toy cars.
This same scenario could be written slightly differently so that the change is unknown. In this situation,
Daquan has 3 toy cars. His grandma gives him some for his birthday. Now Daquan has 5 toy cars. How many toy cars did Daquan's grandma give him?
The representation of this problem is:
3 + [] = 5
The [] represents 2 toy cars in this problem.
Even though this is an addition problem, the subtraction equation 5 - 3 = 2 is used to solve it. Larry Sowder (1995) refers to this type of problem as a missing addend problem. Missing addend problems are confusing for children because the real world operation is addition, as the equation shows. However, mathematically it is solved with subtraction.
The third type of join problem is when the start is unknown. This is another missing addend problem.
Daquan has some toy cars. His grandma gives him 2 more for his birthday. Now he has 5 toy cars. How many toy cars did Daquan have before his birthday?
The problem looks like this:
[] +2 = 5
The answer is 3 toy cars. Once again subtraction 5 - 2 = 3 is used to solve for the unknown start.
Separate
This dimension is composed of the same three subcategories: result unknown, change unknown, and start unknown.
Sydney has 7 pieces of gum. She gives 2 pieces to her friends. How many pieces of gum does she have left?
This is a result unknown because you know how many Sydney started with and you know the change. The part that is unknown is the result. Subtraction is used to separate the 2 pieces of gum that Sydney gave away from the 7 pieces that she had in the beginning.
7 - 2 = []
start result unknown change
The [] is 5 pieces of gum in this problem.
In the next problem, Sydney has an unknown change and once again subtraction is used to solve this type of problem.
Sydney had 7 pieces of gum. She gave some to her friends. Now she has 5 pieces left.
The number sentence below is used to solve for the answer, which is 2.
7 ñ [] = 5
She gave her friends 2 pieces of gum.
The last type in this suite of problems is when the start is unknown. This is another type of problem, missing start, that Sowder claims especially causes trouble for children. It seems to be a subtraction problem, but requires addition to solve it.
Sydney had some pieces of gum. She gave 2 pieces to her friends. She has 5 pieces left. How many pieces did she have before she gave some to her friends?
The number sentence for this problem can be written
[] - 2 = 5
The addition equation 2+ 5 = [] is used to solve this problem. She had 7 pieces of gum.
In the join and separate suites, three very similar problems can be created. Two of the problems deal with subtraction and one problem deals with addition. It is by revealing this kind of relationship between the problems and the operations that fosters a deeper understanding for the students.
Part-Part-Whole Problems
This dimension only has two different subcategories. As the name suggests either a part will be unknown or a whole will be unknown. There is no preferred order for a missing part. This is in contrast to the first two categories of problems, in which time provided a basis for ordering the numbers in the problem. An example of an unknown whole is:
There are 12 boys in the class and there are 10 girls in the class. How many students are in the class?
Addition is used to solve this problem. The major difference between the part-part-whole problems and the join problems is that there is no change over time: an action is not being performed and a quantity is not being changed or moved. The boy part and the girl part coexist and together equal the whole part. Students are not being added or removed.
The schema for this problem can be written
12 + 10 = []
part whole
The whole equals 22 students.
An example with an unknown part is:
There are 22 students in the class. There are 12 boys. How many are girls?
The problem is looking at the part of the class that is unknown. Despite the addition equation, subtraction is used to solve this one. A number sentence for this one could be
12 + [] = 22
The missing part is found by subtracting. The answer is 10 girls.
22 - 12 = []
Compare Problems
These problems are similar to the part-part-whole problems because there is no action involved. There is a set of objects or people and the problem is comparing the quantities. Carpenter (1999) classifies these problems based on three parts: the referent, the compared set, and the difference.
Mark has 6 baseball cards. Tom has 2. How many more baseball cards does Mark have?
6 - 2 = []
referent compared difference unknown
In this problem, the difference is what is missing. Mark's baseball cards are the referent and Tom's cards are the compared set. These two variables are known. When difference is unknown subtraction is used to solve the problem.
Now, if
Tom has 2 baseball cards and this is 4 less than Mark has, how many baseball cards does Mark have?
Here the unknown is the referent. The compared set and the difference are known. In order to solve this type of compare problem, addition would be needed. The appropriate number sentence is
4 + 2 = [].
A similar problem with an unknown compared set would look like this:
Mark has 6 baseball cards. Tom has less baseball cards than Mark. If the difference between their baseball card collections is 4, then how many baseball cards does Tom have?
In this problem subtraction is used.
6 - 4 = [].
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