# Addition and Subtraction

**Strand:** Number

**Outcome:** 3

## Step 1: Identify Outcomes to Address

### Guiding Questions

- What do I want my students to learn?
- What can my students currently understand and do?
- What do I want my students to understand and be able to do, based on the Big Ideas and specific outcomes in the program of studies?

See Sequence of Outcomes from the Program of Studies

### Big Ideas

*Principles and Standards for School Mathematics* states that computational fluency is a balance between conceptual understanding and computational proficiency (NCTM 2000). Conceptual understanding requires flexibility in thinking about the structure of numbers (base ten system), the relationship among numbers and the connections between addition and subtraction. The ability to generate equivalent representations of the same number provides a foundation for using personal strategies to add and subtract, recognizing that for some problems either operation may be used. Computational proficiency includes both efficiency and accuracy. Personal strategies must be compared and evaluated to derive methods that are efficient as well as accurate.

Understanding the operations of addition and subtraction and the connections between them is crucial. John Van de Walle states, "Addition names the whole in terms of the parts, and subtraction names a missing part" (2001, p. 107). Only addition is used in finding the whole when given the parts; however, either addition or subtraction may be used in finding a missing part when given the whole and the other part(s). He goes on to say that addition and subtraction problems include three main types:

- problems involving change—changing a number by adding to it or taking from it
- part–part–whole problems—considering two static quantities either separately or combined
- comparison problems—determining how much two numbers differ in size.

Recognizing which numbers in a problem refer to a part or to a whole helps the students see the inverse relationship between addition and subtraction and understand their properties (Willis et al. 2006). By using a variety of problems, the students will construct their own meaning for the inverse relationship between addition and subtraction and for the following properties:

- commutative property of addition—numbers can be added in any order
- identity element for addition—any number added to zero remains unchanged.

By solving problems in contexts that relate to their own lives, the students use their prior knowledge to make sense out of the problem, estimate the answer and use computational strategies that they are able to explain and justify. The students should always be encouraged to estimate prior to calculating the answer. The students' understanding of addition and subtraction is enhanced as they develop their own methods and share them with one another, explaining why their strategies work and are efficient to use (NCTM 2000, p. 220).

The use of manipulatives or models helps the students understand the structure of the story problem and also connects the meaning of the problem to number sentence (Van de Walle 2001, p. 108). To develop understanding of the meaning of operations, the students connect the story problem to the manipulatives, to the number sentence and then use personal strategies to solve the problem. Van de Walle states:

"It is useful to think of models, word problems, and symbolic equations as three separate languages. Each language can be used to express the relationships involved in one of the operations. Given these three languages, a powerful approach to helping children develop operation meaning is to have them make translations from one language to an other" (2001, p. 109).