Addition and subtraction within 10 while an area we would presume students encountered in first and second grades is part of the structure of skills that students need to know if they are going to progress and be successful in subtraction with renaming. Students need to know that 7 can be made by adding 4+3 or 3+4, 6+1, and 5+2. The commutative property of addition, and associative property of addition should be learned and students should be allowed to work out these findings with manipulatives. I know that saying these rules should be learned might better be couched in words like “be exposed to”, or “be allowed to construct the rule themselves.” In Saxon Math, which I use in my classroom, the rule is presented to the students as well as the terminology. My students have become quite adept to recognizing the commutative property of addition. I find it gives them the vocabulary to name something they do recognize.
In the commutative property of addition children recognize that since addition requires putting things together it doesn’t matter in which order two numbers are added. If we add 6+2 we get 8 and if we add 2+6 we get 8. The associative property comes in when we add three or more numbers. Since we can only add two numbers at a time it is up to the person doing the calculations to choose which numbers to add first. We usually teach adding three or more single digits first. Within that lesson we try to have students realize that looking for tens will make adding these numbers easier. For example in adding
6+ 3 +4 = ____, if we can see that 6+4=10 then we can add 3 and get 13. Many students would begin with the 6+3 and then go on to add 4, which is more difficult for them. Another rule that students usually begin to pick up on is that zero added or subtracted to any number does not change that number.
Subtraction with renaming also predicates itself on the idea that students have become familiar with the relationship between addition and subtraction. Students would be working to build what are termed “fact families”: 5+3=8, 3+5=8, and also 8-3=5 and 8-5=3. We would also stress those combinations of numbers that make 10.