In grades 5-8, the mathematics curriculum should include explorations of patterns and functions so that students can--

• describe, extend, analyze, and create a wide variety of patterns;
• describe and represent relationships with tables, graphs, and rules;
• analyze functional relationships to explain how a change in one quantity results in a change in another;
• use patterns and functions to represent and solve problems.

Focus

One of the central themes of mathematics is the study of patterns and functions. This study requires students to recognize, describe, and generalize patterns and build mathematical models to predict the behavior of real-world phenomena that exhibit the observed pattern. The widespread occurrence of regular and chaotic pattern behavior makes the study of patterns and functions important. Exploring patterns helps students develop mathematical power and instills in them an appreciation for the beauty of mathematics.

The study of patterns in grades 5-8 builds on students' experiences in K-4 but shifts emphasis to an exploration of functions. However, work with patterns continues to be informal and relatively unburdened by symbolism. Students have opportunities to generalize and describe patterns and functions in many ways and to explore the relationships among them. When students make graphs, data tables, expressions, equations, or verbal descriptions to represent a single relationship, they discover that different representations yield different interpretations of a situation. In informal ways, students develop an understanding that functions are composed of variables that have a dynamic relationship: Changes in one variable result in change in another. The identification of the special characteristics of a relationship, such as minimum or maximum values or points at which the value of one of the variables is 0 (x- and y-intercepts), lays the foundation for a more formal study of functions in grades 9-12.

The theme of patterns and functions is woven throughout the 5-8 standards. It begins in K-4, is extended and made more central in 5-8, and reaches maturity with a natural extension to symbolic representation and supporting concepts, such as domain and range, in grades 9-12. Examples appropriate for grades 5-8 are incorporated into other standards for this age group.

Discussion

During the middle years, the study of patterns and functions should focus on the analysis, representation, and generalization of functional relationships. These topics should first be explored as informal investigations.

Students should be encouraged to observe and describe all sorts of patterns in the world around them: plowed fields, haystacks, architecture, paintings, leaves on trees, spirals on pineapples, and so on. As the students mature, instructional efforts can move toward building a firm grasp of the interplay among tables of data, graphs, and algebraic expressions as ways of describing functions and solving problems.

Many problems challenge students to find clever ways to find a solution by counting. Looking for patterns in simple situations can lead to a method of counting generalizable to other situations. For example:

Good news travels fast. Iris saved enough money from her paper route to buy a new bicycle. She immediately told two friends, who, ten minutes later, each repeated the news to two other friends. Ten more minutes later, these friends each told two others. If the news continues to spread in this fashion, how many people will know about Iris's new bicycle after eighty minutes?

Students can approach this problem in different ways. One way is to organize specific cases into a table (fig. 8.1).

Fig. 8.1. Good news table

Each row of the table represents a different function. The "Time" row is a linear function involving multiples of 10. The "People told" row is basically an exponential function involving powers of 2. The pattern in the "Total" row can be seen in different ways. One is to observe that a new entry can be found by adding the current entry in "Total" to the next entry in "People told" as illustrated by the arrows in the table.

Students can use their calculators to find the entries for specific times. More mature students can be challenged to find an algebraic rule that will describe the total for any time (t). Students can also be challenged to develop another representation of the data, such as a graph.

The following illustrates a problem situation involving patterns that can be tackled at many levels:

Investigate what happens when different-sized cubes are constructed from unit cubes, the surface area is painted, and the large cube is then disassembled into its original unit cubes. How many of the 1 x 1 x 1 cubes are painted on three faces, two faces, one face, and no faces?

Students can approach this problem by looking at particular instances, organizing the data in a table, looking for patterns, generalizing the patterns, and graphing the four relationships. See figure 8.2.

Fig. 8.2. Cube painting

An interesting feature of this problem is that it involves linear, constant quadratic, and cubic functions. For example, asking whether 3174 will appear in the "painted on one face" column allows students to apply the number-theory concepts of divisibility and square numbers informally to solve quadratic equations in a situation that gives meaning to the concepts and solution.

Much of mathematics in grades 5-8 can be viewed as an exploration of patterns and regularity. Pi is best understood by students when they investigate the ratio between circumference and diameter by measuring round objects and looking for regularities in the data or the graph of circumferences compared to diameters. The number of diagonals of any polygon becomes predictable when an exploration of patterns reveals the underlying function. Integers and operations on integers become a natural extension of whole numbers when viewed in terms of patterns:

                 3  x  2  =  6     3  x  -2  = -6



                 3  x  1  =  3     2  x  -2  = -4



                 3  x  0  =  0     1  x  -2  = -2



                 3  x -1  = -3     0  x  -2  =  0



                 3  x -2  = -6    -1  x  -2  =  2

The following situation encourages students to reason from patterns to solve a real-world problem.

Pat saw a billboard along the roadside (fig. 8.3). If a plain pizza has just cheese and tomato, how many toppings does La Mozzarella's have?

Fig. 8.3. Pizza patterns

Students can explore this problem by taking different numbers of toppings and listing each possible pizza. This can lead to the development of Pascal's triangle and an exploration of the many other situations that relate to it.

In the following problem, students need to generate, organize, and analyze data; look for patterns; and then use the observed patterns to generalize.

In a village are 3 streets. All the streets are straight. Each crossroad has one lamppost. How many lampposts are needed? How many are needed for a village with 20 streets? Generalize to any number of streets.

This problem is not well formed. Questions about how the streets are laid out must be answered before students can pursue their own solutions. As indicated by all the preceding problem situations, the dynamic relationship between variables in a pattern or function can be viewed physically and in tables of data. Here are some ways to explore change represented in graphs:

Mary and her brother John leave home together to walk to school. Mary thinks they are going to be late, so she starts out running, then tires and walks the rest of the way. John starts out walking and starts to run as he nears the school building. They arrive at the same time. The graphs in figure 8.4 show the distance from their home on the vertical axis and time on the horizontal axis. Which graph best represents Mary's trip? John's trip?

Fig. 8.4. Home-to-school graph: distance versus time

Patterns abound in our world. The mathematics curriculum should help sensitize students to the patterns they meet every day and to the mathematical descriptions or models of these patterns and relationships. A mathematical model's ability to predict is a powerful, fundamental mathematical concept that deserves continuing emphasis in the curriculum.