In grades K-4, the mathematics curriculum should include the study of patterns and relationships so that student can--

• recognize, describe, extend, and create a wide variety of patterns;
• represent and describe mathematical relationships;
• explore the use of variables and open sentences to express relationships.

Focus

Patterns are everywhere. Children who are encouraged to look for patterns and to express them mathematically begin to understand how mathematics applies to the world in which they live. Identifying and working with a wide variety of patterns help children to develop the ability to classify and organize information. Relating patterns in numbers, geometry, and measurement helps them understand connections among mathematical topics. Such connections foster the kind of mathematical thinking that serves as a foundation for the more abstract ideas studied in later grades.

From the earliest grades, the curriculum should give students opportunities to focus on regularities in events, shapes, designs, and sets of numbers. Children should begin to see that regularity is the essence of mathematics. The idea of a functional relationship can be intuitively developed through observations of regularity and work with generalizable patterns.

Physical materials and pictorial displays should be used to help children recognize and create patterns and relationships. Observing varied representations of the same pattern helps children identify its properties. The use of letters and other symbols in generalizing descriptions of these properties prepares children to use variables in the future. This experience builds readiness for a generalized view of mathematics and the later study of algebra.

Discussion

A classroom can become a rich environment in which to study patterns. A rug, an afghan, or quilt; various wallpaper borders used to identify boxes of materials; designs painted on the window panes; carefully selected pictures; and even the arrangement of furniture are examples of regularity and patterns that children can recognize and describe (see fig. 13.1).

Fig. 13.1

Regularities can be as simple as an explicit recognition that each child has two eyes. As each child in turn stands, the number of eyes of those standing can be recorded (2, 4, 6, ...) and then represented with tiles to emphasize that even numbers come in pairs, whereas each odd number is an even number and one more. See figure 13.2.

Fig. 13.2

Pattern recognition involves many concepts, such as color and shape identification, direction, orientation, size, and number relationships. Children should use all these properties in identifying, extending, and creating patterns. Identifying the "cores" of patterns helps children become aware of the structures. For example, in some patterns the core repeats, whereas in others, the core grows. (See fig. 13.2a)

Fig. 13.2a

Representing a pattern both geometrically and numerically helps children recognize a variety of relationships in the pattern and make connections between arithmetic and geometry. See figure 13.3.

Fig. 13.3

Organizing data on a pattern in a table helps students identify its structure and describe it symbolically. (See Fig. 13.3a)

Fig. 13.3a

As they work on basic facts, children should be encouraged to look for patterns and relationships. The following sequence of numbers is an excellent example of such an activity:

                9, 18, 27, 36, 45, 54, 63, 72, 81, 90

Children should recognize that each number is nine more than the number before it.

Each element should also be recognized as a multiple of 9 and represented as 9 x n. Replacing n with the numbers from 1 to 10 to generate the original set of numbers validates the symbolic representation of the pattern and reinforces multiplication facts. It also illustrates the concept of a variable.

Coloring the elements on a hundred board represents the pattern in yet another way (see fig. 13.4). Such questions as the following help children to relate many mathematical ideas. Why do you think the numbers lie on a diagonal line? If you extended the hundred board to 200 and colored the next ten numbers in the pattern, where would they lie? Explain your thinking. How can you check your answer? What sequence of numbers starting with 9 would give a column of numbers on the hundred board?

Fig. 13.4

Children should be encouraged to explore other less obvious patterns and relationships in the same sequence. For example, the units digits in succeeding elements decrease by 1, whereas the tens digits increase by 1, and the sum of the digits in each element is 9.

Using the constant function on a calculator, children can construct a table of input and output numbers and then express the relationship as an open sentence. See figure 13.5.

Fig. 13.5

Graphing these sets of numbers helps children see number relationships in yet another format and is an informal extension to algebraic and geometric thinking.