Question

Explore rules that can be used to represent patterns using symbol.

Answer 1

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 pattern.      

Answer 2

Answer:

When numbers in a pattern get larger as the sequence continues, they are in an ascending pattern. Ascending patterns often involve multiplication or addition.

When numbers in a pattern get smaller as the sequence continues, they are in a descending pattern.