Addition and Subtraction Problems

In this newsletter we continue with joining problems. We have covered join-result-unknown. This week we will be covering the second type, join-change-unknown, and the third type, join-start-unknown.

Types of Joining Problems

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?


Join Change Unknown Problems

The unknown in these problems is how many more need to be added to a number to arrive at a given total. This type of problem illustrates a difference between child and adult thinking. Adults would identify this as a subtraction problem but children do not. Children see this as a problem requiring a joining action.


Join-Start-Unknown Problems

These problems are very hard. The unknown in these problems is how many there were to begin with after a number of items have joined them resulting in a total.


Strategies for Solving Joining Problems

Children solve join-change-unknown and join-start-unknown problems using predictable strategies.


Join-Change-Unknown Strategies

Byron has 7 shells. Then Melissa gives Byron some more shells. Now Byron has 16 shells. How many shells did Melissa give Byron?

  1. Physical (direct) modeling Children use fingers (or counters) to stand for the first set (7 shells). Then they add objects until they reach the total of 16 - 8, 9, 10, 11, 12, 13, 14, 15, 16. They find the answer by counting the number that they added to the original group- 9.
  2. Counting Children sometimes use a counting up strategy to solve this type of problem. For example, the child start with, "7 (pauses then counts up) 8, 9, 10, 11, 12, 13, 14, 15, 16 " while extending a finger on each count from 8 to 16. Then they would look at their fingers and say "9"- she gave him 9 shells.
  3. Using facts Children will begin to use their growing knowledge of facts to solve this type of problem too. For example, they may say "7 and 7 is 14, 16 is 2 more than 14, it must be 2 more than 7- it's 9."


Join-Start-Unknown Strategies

Byron has some shells. His friend gives Byron 9 more shells. Now Byron has 16 shells. How many shells did Byron start with?

  1. Direct modeling Some children will try a trial and error strategy of direct modelling to solve these problems. This means that they will try different numbers of shells (or counters), like 4 or 5, then add 9 more shells, and finally count them all (4 + 9 or 5 + 9) to see if the total is 16. Because children do not see this as a subtraction problem, as an adult would, and because they don't know how many to start with, these problems are difficult for them to solve.
  2. Counting When children realize that they can put the nine shells in the starting place and count up they have made a real gain in their understanding of numbers!
  3. Using facts As children learn their facts they will derive or recall facts to solve these problems just as they do with the other problems. (9 +7 = 16)