Children were given a variety of strategies for learning their addition facts in Grade 1. These strategies included counting on by 1, 2, or 3, and using the number-line model, fact families, the addition table, and the properties of addition to “reduce” the number of basic facts to be learned.

Children also have learned to use sums of 10 to simplify learning facts involving 7, 8, and 9. For example, 9 + 8 can be written as 9 + 1 + 7. Since 9 + 1 = 10, then 9 + 8 = 10 + 7. Thus, 9 + 8 becomes 10 + 7, which students generally find easier to solve. This process also builds an understanding of our base-ten numeration system and lays a foundation for future work with the standard addition algorithm.

This process makes use of the Associative Property of Addition. The Associative Property of Addition states that changing the grouping of addends does not change the sum. In the language of variables, for all numbers a, b, and c,

a + (b + c) = (a + b) + c.

In the example of 9 + 8, the Associative Property of Addition was applied in the following way.

9 + 8 = 9 + (1 + 7)
= (9 + 1) + 7
= 10 + 7
= 17

The parentheses serve as grouping symbols and indicate which operation should be completed first; in this case we added 9 and 1 before adding 7.

Another strategy for mastering the addition facts is “doubles” and “doubles-plus-1.” The doubles are 1 + 1, 2 + 2, 3 + 3, 4 + 4, 5 + 5, 6 + 6, 7 + 7, 8 + 8, 9 + 9, and 10 + 10. These are “easier” facts for children to learn because they can be learned by counting by 2s: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

Since children are familiar with the “counting-on” strategy from kindergarten and first grade, the “doubles-plus-1” strategy is an effective way to learn additional facts once the doubles are memorized. An example of a doubles-plus-one fact is 8 + 7. The sum can be easily found by thinking of it as a double plus one.

8 + 7 = (7 + 7) + 1
= 14 + 1
= 15

Although this is a relatively simple process for children to understand, its mathematical significance should not be understated. Mathematically, this process is possible by writing 8 as 7 + 1 and applying the properties of addition.

8 + 7 = (7 + 1) + 7
= 7 + (1 + 7) Associative Property of Addition
= 7 + (7 + 1) Commutative Property of Addition
= (7 + 7) + 1 Associative Property of Addition

Teaching Model 2.5: Make 10 to Add