Session 4, Part A:
Meanings and Relationships of the Operations

In This Part: Addition | Subtraction | Multiplication | Division

As in addition, no two quantities can be subtracted unless they are measured or reported in the same units. Thus, you cannot subtract 7 hundreds from 9 tens and expect to get 2 of anything. A quantity can only be subtracted from another quantity if we can first find a common unit between the two.

The operation of subtraction can be thought of as:

 • a separator, when the result, starting point, or change is unknown (also known as "take-away") • a comparison, when the result, starting point, or change is unknown • a missing addend problem, where one of the parts is unknown Note 4

This table gives an example of each kind of subtraction problem:

Problem Type

Starting Point Unknown

Change Unknown

Result Unknown

Separator
(Take-Away)

 Billy gave 7 toy cars to Sam, so he now has 16 cars left. How many cars did he have before? ? - 7 = 16
 Billy had 23 toy cars. He gave some to Sam. Billy has 16 cars left. How many cars did he give Sam? 23 - ? = 16
 Billy had 23 toy cars. He gave 7 to Sam. How many cars does he have now? 23 - 7 = ?

Comparison

 Jennie has 15 more stickers than Sara. Sara has 9 stickers. How many stickers does Jennie have? ? - 9 = 15
 Jennie has 15 more stickers than Sara. Jennie has 24 stickers. How many stickers does Sara have? 24 - ? = 15
 Sara has 9 stickers. Jennie has 24 stickers. How many more stickers does Jennie have than Sara? 24 - 9 = ?

 Sara has some red chips and 4 black chips. She has 9 chips altogether. How many red chips does she have? ? + 4 = 9
 Sara has 5 red chips and some black chips. She has 9 chips altogether. How many black chips does she have? 5 + ? = 9
 N/A

The missing addend problems are written as addition problems, but the procedure to solve these problems requires the use of some subtraction strategy. The separating concept always requires some sort of separating action, whereas the comparison concept is static.

When negative numbers are introduced, we can more clearly understand the concept of subtraction as the addition of the inverse. Thus, we can write 13 - 6 as the equivalent of 13 + (-6 ), because -6 is the additive inverse of 6; i.e., 6 + (-6) = 0. Similarly, we can represent 13 - (-6) as 13 + 6, since 6 is the additive inverse of -6. So again, subtracting a number (13 - (-6)) is the same as adding its inverse (13 + 6).

Problem A2

Label each of the subtraction problems with the correct situation label, and identify the units involved:

SR: Separator, result unknown
SS: Separator, starting point unknown
SC: Separator, change unknown
CR: Comparison, result unknown
CS: Comparison, starting point unknown
CC: Comparison, change unknown
MS: Missing addend, starting point unknown

 a. Moisha has 7 cars. She gave 3 away. How many cars does she have left? b. Jake read 5 mystery books. He read 3 more adventure books than mysteries. How many adventure books did he read? c. Bret has \$5. Bret has \$2 more than Wendy. How much money does Wendy have? d. Natasha has 7 rabbits. She gave some rabbits to Joshua, and then she had 3. How many rabbits did she give away? e. Reed's parents gave him some money for his birthday. He now has \$12. He had \$7 before his birthday. How much money did they give him? f. Elyse had some candy bars. She ate 3, and she has 5 left. How many did she have to start?

 Session 4: Index | Notes | Solutions | Video