What is Composing and Decomposing? (And why is it important to Computational Fluency)?

One task that I find students struggle with is seeing the trees in the forest--breaking apart numbers to small ones. This skill is the beginning of place value in the kindergarten and first grade but becomes a powerful addition and multiplication strategy later on. Common Core Standard 2.NTB.B.5 moves students to using the break apart strategy on an Open Number Line to add and subtract.

When talking about computational fluency, many of the current articles use the terms composing and decomposing numbers. These are terms that may not be familiar to most parents. They are really not anything new. These terms refer to the idea that numbers can be put together or broken apart to make other numbers.

For instance, the number ten can be broken up (decomposed) in many ways.
10 = 5 + 5
10 = 4 + 6
10 = 3 + 3 + 2 + 2

This may seem like a simple idea, but to a child just learning about numbers it is not simple at all. We want to foster this understanding because it is a critical understanding in terms of becoming computational fluency.

When solving the problem 28 + 45 a student decomposed 28 into 20 + 5 + 3.
Can you see how that would make this problem easier to solve?
28 + 45 = 20 + 5 + 3 + 45 = (45 + 5) + 20 + 3 = 73

This skill can even be helpful when learning basic facts. For instance, when doing 7 + 8, a student might decompose the 7 or the 8 to make ten and extras.
7 + 8 = 5 + (2 + 8) = 5 + 10 = 15

Students have developed a firm understanding of place value of two-digit numbers and to subtract multiples of ten, and are ready to add and subtract within 100 (including the case of adding or subtracting a two-digit number and a one-digit number, and two two-digit numbers). First students are given problems where regrouping is not necessary, and later, problems where regrouping is necessary. Further, students understand that in addition and subtraction, digits in the ones place are added and subtracted; digits in the tens place are added and subtracted; and sometimes regrouping is necessary. In addition, sometimes we must regroup ten ones to form an additional ten, and in subtraction, sometimes we must break a ten into ten ones. The eventual goal of this standard is fluency. This will not happen all at once; students will build gradually towards having procedures and strategies by which they can fluently add an subtract, including standard algorithms and skip-counting up or down.

In a given addition and subtraction problems, ask students to identify which digits are in the one and tens positions. In addition, they should be able to identify the digits in the ones and tens positions.
Provide students with a variety of manipulatives and technologies (such as base-ten blocks or drawings) which can aid in their practice of addition and subtraction through 100.

Use properties of addition to make addition more fluid. For example, 64 + 8 can be thought of as (62 + 2) + 8, reordering addends allows (2 + 62) + 8, regrouping addends yields 2 + (62 + 8) which gives 2 + 70, which is equal to 72. This is what is meant by using a strategy with the properties of the operations.  Another strategy would be using skip-counting for addition (e.g., 58 + 15 can be found by skip-counting up from 58 by ten, and then by five. An additional strategy is to break down place value: 58 + 15 can be thought of as 50 + 10 + 8 + 5, which is 60 + 13, or 73.

Students should have at least one algorithm in place that is robust and works in all cases, but they also should be encouraged to use alternative strategies if they can do so quickly and accurately.


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