Addition and Subtraction Problems

We have mentioned that children solve addition and subtraction problems by focusing on the action in the problem. We illustrated joining problems where the action is a joining of two or more quantities (like 6 toy cars and 18 more toy cars).

The opposite of joining is separating. In many word problems the action involves separation of quantities. For example, Louise has 36 cookies and gives 12 to Henry (12 cookies are separated from the original quantity of 36). As with Join problems, there are three distinct quantities in Separate problems. There is a starting quantity, a change quantity (the amount removed), and the result. Any one of these quantities can be the unknown.

Types of Addition and Subtraction Problems

Join: 7 + 9 = ? 7 + ? = 16 ? + 9 = 16
Result Unknown Change Unknown Start Unknown
Byron has 7 shells.

Then Melissa gave
him 9 more shells.

How many shells
does Byron have
now?
Byron has 7 shells.

Melissa give Byron
some shells.

Now Byron has 16 shells.
How many shells
did Mellisa give him?
Byron has some shells.

Melissa gives him
9 more.

Now Byron has 16 shells.
How many shells did
Byron start with?

Separate: 8 - 3 = ? 8 - ? = 5 ? - 3 = 5
Result Unknown Change Unknown Start Unknown
Colleen has 8 guppies.

She gave 3
guppies to Roger.

How many
guppies does
Colleen have left?
Colleen has 8 guppies.

She gave some guppies
to Roger. Then she had
5 guppies left.

How many guppies
did Colleen give Roger?
Colleen has
some guppies.

She gave 3 guppies
to Roger. Then she had
5 guppies left.

How many guppies
did Colleen have to
start with?


Strategies for Solving Separating Problems

Generally, result unknown problems are easiest and start unknown problems are hardest. The reason for this is the types of strategies children use to solve these problems. Children solve separate problems in three distinct ways—physical (direct) modeling, counting, or using facts. We shall discuss each of these three strategies for each type of separate problem.

Mom and daughter solve separating
problems as they make a wreath


Separate-Result-Unknown Problems

(8-3=?) There were 8 seals playing. Three seals swam away. How many seals were still playing?

  1. Direct modeling. Using objects like toothpicks, child uses 8 to show the 8 seals. Then removes 3 toothpicks and counts the 5 remaining toothpicks.
  2. Counting. The child counts backwards from 8 . . . 7, 6, 5. The last number in the counting sequence is the answer. Sometimes children will use fingers to keep track of the number of counts, here holding up one finger for 7, two fingers for 6, and 3 fingers for 5.
  3. Using facts. The child may know that 8 - 2 is 6 and then take one more away to make 5 or the child may just remember that 8 - 3 is 5.

Separate-Change-Unknown Problems

(8-?=5) There were 8 seals playing. Some swam away. Then there were 5 playing. How many seals swam away?

  1. Direct modeling. Using objects like toothpicks, child uses 8 to show the 8 seals. Remove toothpicks until 5 are left. Count the toothpicks that were removed.
  2. Counting. The child counts backward from 8 and continues until 5 is reached. The answer is the number of words in the counting sequence. 7 (1), 6 (2), 5(3). Children will often use their fingers to keep track of the number of counts.
  3. Using Facts. 8 - 4 = 4, so 8 - 3 = 5

Separate-Start-Unknown Problems

(?-3 = 5). Some seals were playing. Three swam away. Then there were 5 playing. How many seals were playing at first?

  1. Direct modeling for this problem is very hard because children don't know how many toothpicks to use to represent "some." They may use trial-and-error—using 7 toothpicks, removing 3 toothpicks, and checking to see whether 5 toothpicks remain.
  2. Counting. Most children begin to solve Start Unknown problems when they no longer need to model the action in the problems directly. At this point, they may use counting strategies like counting on from the larger number: 5 . . . 6(1), 7(2), 8(3).
  3. Using Facts. 8 - 3 = 5