Use place value understanding and properties of operations to add and subtract.
5. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
6. Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
As I discussed briefly in the previous post, these standards are based on students' understanding of ten as a unit, which are the major focus of 1.NBT.2. While in Kindergarten, students considered 10 as a "benchmark" or a "mile marker" to understand numbers 11 through 19, in Grade 1, students learn to count ten, that is, ten as a unit. Thus, extending the idea of composing and decomposing number further, instead of thinking 64 as just 60+4, we want students to think of 60 as 10+10+10+10+10+10, or six 10's. Being able to coordinate two units, i.e., ones and tens, is not a trivial task for young children. Many children at this age can recite the sequence of decade number words correctly, "ten, twenty, thirty, forty, ..." However, some children will have to count by ones to answer, "what is 10 more than 36?" For many young children, the decade number word sequence is just a memorized set of words with no numerical significance. They do not yet understand when they go from "thirty" to "forty," the number increased by 10. Thus, these standards, although they are based on 1.NBT.2, are not necessarily something that comes after students master 1.NBT.2. As students think about addition of 2 multiples of 10 or subtract a multiple of 10 from another multiple of 10, we want to encourage students to think in terms of 10's. So, questions like, "how many 10's are in __?" and "how many 10's are we adding (or taking away)?" must be an important part of teachers' questioning repertoire.
These standards also illustrate an important pedagogical idea that seems to come up several times in the CCSS, that is, as students encounter new forms of calculations, they first take advantage of the structures of numbers and model numbers using concrete materials or visual representations before they formalize them into written procedures. Thus, while "adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10" students should think about the base-10 structure of the numbers. For example, if they are adding 38 + 7, they might think, "well, 38 is 3 tens and 8. But 2 more will make another ten, so that will be 4 tens and 5, so 45 is the answer." Or, if they are adding 38 + 40, they might think "we are adding 4 more tens to 3 that we already have. So there will be 7 tens and 8, so 78." They might use base-10 blocks to think along. However, the important reason for using base-10 blocks is not so that students can find the answer using the blocks but so that students can think in terms of unit of ten (i.e., long's). So, asking students to think about how they might model the addition with base-10 blocks but without actually using them might be a useful activity.
Finally, the last part of 1.NBT.4, "Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten." raises a question about the type of addition (and perhaps subtraction) that should be the focus in Grade 1. Specific types of addition mentioned by 1.NBT.4 are "adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10." Thus, addition problems like 38 + 7 or 38 + 40. However, there is this little word, "including," in this standard. So, does that mean addition of two general 2-digit numbers should be taught in Grade 1? Since addition problems like 38 + 7 or 38 + 40 do not require students to do both addition of ones and addition of tens, it might be a bit difficult for children to develop this understanding. It might be technically true that when children add 38 and 40, they are doing 3+4 and 8+0, I doubt many children will see it that way. Perhaps the statement, "adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10," was included to suggest that these types of addition problems should be discussed before addition of two general 2-digit numbers is addressed. 1.NBT.6 seems to be much clearer in determining the type of subtraction problems to be discussed in Grade 1. A similar, more specific indication on this matter will be helpful for teachers and curriculum developers.
