In her book How Children Learn Number Concepts*, Kathy Richardson devotes an entire chapter to composing and decomposing numbers. It may surprise you that the title of the chapter is Understanding Addition and Subtraction: Parts of Numbers. This quote sums it up beautifully:

“If basic facts are to be foundational, they must be based on an understanding of the composition and decomposition of numbers.”

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Both the CCSSM and Texas TEKS have a number of standards in Kindergarten and 1st Grade related to composing and decomposing numbers, as shown in this table.

One important feature of the standards that is often overlooked is that they describe the level of concrete or pictorial support a student should receive. Notice that K.OA.3, for example, states that students should use “objects or drawings” and record the decomposition using a “drawing or equation.” In that one standard, you hear each phase in the concrete, representational, abstract (CRA) sequence of instruction. Notice how the concrete and pictorial are tied to the abstract (equation) to help students make that important connection. Rushing students to abstract, or purely symbolic, learning is a recipe for disaster, and that is recognized in the standards.

Composing and decomposing numbers is such a critical component of number sense that it should constitute a major part of the learning that takes place in Kindergarten and 1st Grade. Richardson states, “As children learn the combinations that make up the numbers to 10, they will reach the point where they know the parts so well, they can identify a missing part when they know the total and one part.” In other words, students need lots of practice composing numbers, working with the various combinations of each number, before they will be able to decompose numbers, or find a missing part. In your bag of instructional tricks, you will want to have a wide variety of activities to practice composing and decomposing numbers.  These blog posts will give you ideas for activities as well as freebies you can use this week!

1. Shake and Spill
2. Mathemagician Make Ten
3. Make 5 Go Fish
4. In the Cave

For all of these activities, it’s important to understand that students should master the combinations for one number before moving on to the next.  It does no good for a child to practice composing and decomposing 6 if he does not know the combinations for 5. That’s where differentiation comes in. Richardson describes using the “hiding assessment” to determine a child’s fluency with each number. To determine if a child knows all the combinations for 3, ask the child to count out 3 counters or linking cubes. Hide some counters and show some, asking the student to identify how many are hidden.  For example, hide 1 counter and show 2.  “If I have 3 counters and 2 are showing, how many are hidden?”  Continue this routine for each combination for 3 (hide 3, show 0; hide 2, show 1; hide 0, show 3).  If the student can name all the missing parts for 3, try the combinations for 4.  When the student can no longer easily name the missing parts, that becomes her number. Use a recording sheet to keep track of each student’s number so you can differentiate activities, such as those described above, based on each student’s needs. For example, all students might be using Shake and Spill in a workstation this week, but each child is using his or her own target number. Every few weeks, “test” your students to determine if they are ready to move on to a new number. Missing part flashcards are a helpful tool for periodic assessment.

While it might seem daunting to differentiate based on each student’s number, the good news is that a few engaging activities go a long way.  Constantly rotating the activities keeps engagement high and allows you to meet the needs of each student without a great deal of prep work.  For more great activities, check out Building Number Sense: Games and Activities to Practice Combinations for 10 or these other blog posts.