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.

Result Unknown |
Change Unknown |
Start Unknown |
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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? |

**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.**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.**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."

**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.**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!**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)