Number & Operations for Teachers 

    Copyright David & Cynthia Thomas, 2009

Set Model

Addition and subtraction of whole numbers {0, 1, 2, 3, 4 …} is naturally motivated by everyday experience. For instance, the question “Mary has two candies. If her mother gives her four more, how many candies will she have?” is readily understood by young learners. One of the most convenient ways to model this exercise is with the use of a set model. In the set model, plastic disks are used to represent the addends in the indicated sum. In Figure 2.1, the first addend is shown as a set containing two disks. The second addend is shown as a set containing four disks. The sum of these two sets is a new set containing all of the disks.

+ =

2 + 4 = 6

Figure 2.1: A set model for the indicated sum 2 + 4

A related question may be stated as follows: “Mary has six cookies. If she gives four cookies to her friends, how many does she have left?” Figure 2.2 models this question.

- =

6 - 4 = 2

Figure 2.2: A set model for the indicated difference 6 – 4

Another related question is suggested by Figure 2.3. How would you phrase the question that it models?

- =

6 - 2 = 4

Figure 2.3: A set model for another indicated difference

Figures 2.1 – 2.3 illustrate a fundamental truth about addition and subtraction: The operations are strongly related. For instance, the following sequence of questions may be viewed as first increasing Mary’s cookie count by four, then decreasing it by the same amount. The net result is that Mary’s cookie count is unchanged after the sequence of indicated operations.

· Mary has two candies. If her mother gives her four more, how many candies will she have?

· Mary has six candies. If she gives four candies to her friends, how many does she have left?

Another way to describe this sequence of operations is to say that the second operation “undoes” or “cancels out” the first operation. In mathematics, pairs of operations that behave in this manner are called inverses. It is in this sense that addition and subtraction are described as being inverses of one another.

With respect to the statement 2 + 4 = 6, the statements 2 + 4 = 6, 6 – 4 = 2, 4 + 2 = 6, and 6 – 2 = 4 are said to form a fact family. Listed in this manner, the first two statements are inverses of one another. The same is true of the last two statements. Fact families for subtraction of whole numbers are similar. For instance, a fact family for the statement 5 – 2 = 3 may be written as follows: 5 – 2 = 3; 2 + 3 = 5; 5 – 3 = 2; and 3 + 2 = 5.

Although young learners may not represent arithmetic relationships using variables, all students should develop number sense based on number facts and number relationships. Failure to do so greatly handicaps them when attempting more challenging problems.