STANDARD 11 - PATTERNS, RELATIONSHIPS, AND FUNCTIONS
Patterns, relationships, and functions constitute a unifying theme of mathematics. From the earliest age, students should be encouraged to investigate the patterns that they find in numbers, shapes, and expressions, and, by doing so, to make mathematical discoveries. They should have opportunities to analyze, extend, and create a variety of patterns and to use pattern-based thinking to understand and represent mathematical and other real-world phenomena. These explorations present unlimited opportunities for problem solving, making and verifying generalizations, and building mathematical understanding and confidence.
Meaning and Importance
Mathematics is often regarded as the science of patterns. When solving a complex problem, we frequently suggest to students that they try to work on simpler versions of the problem, observe what happens in a few specific cases - that is, look for a pattern - and use that pattern to solve the original problem. This pattern-based thinking, using patterns to analyze and solve problems, is an extremely powerful tool for doing mathematics. Students who are comfortable looking for patterns and then analyzing those patterns to solve problems can also develop understanding of new concepts in the same way. Most of the major principles of algebra and geometry emerge as generalizations of patterns in number and shape. For example, one important fact in geometry is that: For a given perimeter, the figure with the largest possible area that can be constructed is a circle. This idea can be discovered informally by students in the middle grades by examining the pattern that comes from a series of constructions and measurements. Students can be given a length, say 24 centimeters, for the perimeter of all figures to be created. Then they can construct and measure or compute the areas of a series of regular polygons: an equilateral triangle, a square, and a regular hexagon, octagon, and dodecagon (12 sides). The pattern that clearly emerges is that as the number of sides of the polygon increases - that is, as the polygon becomes more "circular"- the area increases.
All of the content standards are interconnected, but this standard is one that is particularly closely tied to all of the others. This is because pattern-based thinking is regularly applied to content in numeration, geometry, operations, discrete mathematics, and the fundamentals of calculus. There is a very special relationship, though, between patterns and algebra. Algebra provides the language in which we communicate the patterns in mathematics. Early on in their mathematical careers, students must begin to make generalizations aboutpatterns that they find, and they should learn to express those generalizations in mathematical terms.
K-12 Development and Emphases
Children become aware of patterns very early in their lives - repetitive daily routines and periodic phenomena are all around them. Breakfast is followed by lunch which is followed by dinner which is followed by bedtime and then the whole thing is repeated again the next day. Each one of the three little pigs says to the wolf, at exactly the expected moment, Not by the hair on my chinny-chin-chin! In the primary grades, children need to build on those early experiences by constructing, recognizing, and extending patterns in a variety of contexts. Numbers and shapes certainly offer many opportunities, but so do music, language, and physical activity. Young children love to imitate rhythmic patterns in sound and language and should be encouraged to create their own. In addition, they should construct their own patterns with manipulatives such as pattern blocks, attribute blocks, and multilink cubes and should be challenged to extend patterns begun by others. Identifying attributes of objects, and using them for categorization and classification, are skills that are closely related to the ability to create and discover patterns and need to be developed at the same time.
Young students should frequently play games which ask them to follow a sequence of rules or to discover a rule for a given pattern. Sequences which begin as counting patterns soon develop into rules involving arithmetic operations. Children in the primary grades, for example, will make the transition from 2, 4, 6, 8, ... as a counting by twos pattern to the rule Add 2 or "+2." The calculator is a very useful tool for making this connection since it can be used for counting up or counting down by any constant amount. Students can be challenged to guess the number that will come up next in the calculator's display and then to explain the pattern, or rule, to the class.
At a slightly higher level, input-output activities which require recognition of relationships between one set of numbers (the "IN" values) and a second set (the "OUT" values) provide an early introduction to functions. One of these kinds of activities, the function machine games, is a favorite among first through fifth graders. In these, one student has a rule in mind to transform any number suggested by another student. The first number is inserted into the imaginary function machine and another number comes out the other side. The rule might be plus 7, or, times 4 then minus 3, or even the number times itself. The class's task is to discover the rule by an examination of the input-output pairs. In the intermediate grades, students can simulate the function machine with a computer spreadsheet secretly programmed to take the number typed in the first column and transform it into another number that is placed in the second column.
Slightly older students begin to work with patterns that can be used to solve problems within mathematics and from the real world. There should also be a more deliberate focus on relationships involving two variables. An exploration of the relationship between the number of teams in a round robin tournament and the total number of games that must be played, or between a number of coins to be flipped and the total number of possible outcomes, provides a real-world context for pattern-based thinking and informal work with functions. Graphing software and graphing calculators are extremely valuable at this level to help students visualize the relationships they discover.
At the secondary level, students are able to bring more of the tools of algebra to the task of analyzing and representing patterns and relationships. Thus we expect all students to be able to construct as well as torecognize symbolic representations such as y = f(x) = 4x+1. They should also develop an understanding of the many other representations and applications of functions as well as of a greater variety of functional relationships. Their work should extend to quadratic, polynomial, trigonometric, and exponential functions in addition to the linear functions they worked with in earlier grades. They should be comfortable with the symbols f, representing a rule, and f(x), representing the value which f assigns to x.
The use of functions in modeling real-life and real-time observations also plays a central role in the high school mathematics experience. Line- and curve-fitting as approaches to the explanation of a set of experimental data help make mathematics come alive for students. Technology must play an important role in this process, since students are now able to graphically explore relationships more easily than ever before. Graphing calculators and computers must be made available to all students for use in these types of investigations.
In summary, an important task for every teacher of mathematics is to help students recognize, generalize, and use patterns that exist in numbers, in shapes, and in the world around them. Students who have such skills are better problem solvers, have a better sense of the uses of mathematics, and are better prepared for work with algebraic functions than those who do not.
Note: Although each content standard is discussed in a separate chapter, it is not the intention that each be treated separately in the classroom. Indeed, as noted in the Introduction to this Framework, an effective curriculum is one that successfully integrates these areas to present students with rich and meaningful cross-strand experiences.
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