Level 3: Derived fact methods in which known facts are used to find related facts (mastery by some/many at first grade). Doubles are totals of two of the same addend: 1 + 1, 2 + 2, 3 + 3, etc., up to 9 + 9. These are learned by many children in the United States because of the easy pattern in their totals (2, 4, 6, 8, etc.). Doubles ± 1 is a Level 3 more advanced strategy that uses a related double to find the total of two addends in which one addend is one more or less than the other addend (6 + 7 = 6 + 6 + 1 + 12 + 1 = 13). Makeaten methods are general methods for adding or subtracting to find a teen total by changing a problem into an easier problem involving 10. Children first make a 10 from the first addend and then learn to make a 10 from the larger addend. Make a ten to find a total: 8 + 6 becomes 10 + 4 by separating the 6 into the amount that makes 10 with the 8. Then solving 6 = 2 + ? gives the leftover 4 within the 6 to become the ones number in the teen total: 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14. Make a ten to find an unknown addend: 14 − 8 = ? is 8 ? = 14, so 8 + 2 is 10 plus the 4 in 14 makes 14. So 8 + 6 + 14. In this method subtraction requires adding, which is easier than making a ten to find a total. The first step can also be thought of as subtracting the 8 from 10. Three prerequisites for fluency with makeaten methods can be built up before first grade:

facilitated by children’s earlier work with embedded number experiences of finding partners of a total (e.g., Inside seven, I see five and two) and by fluency with the count word sequence, so they can begin counting from any number (most 2, 3, and 4yearolds need to start at 1 when counting and cannot start from just any number). With larger second addends, children also need a method of keeping track of how many they have counted on. These counting on methods are sufficient for all further quantitative work, especially if children are helped to see subtraction as finding an unknown addend, so that they can use counting on to find that addend. Counting down to subtract is difficult, and children make many errors at it (Baroody, 1984; Fuson, 1984). Just counting backward is difficult, and children make various countcardinal errors in counting down. Counting forward to find an unknown addend for subtraction (e.g., solving 9 − 5 = ? as 5 + ? = 9)