In math, patterns often occur naturally. Once a pattern is discovered within a problem, it can be used over and over again to determine results. Mathematicians use and search for patterns to solve complex and seemingly unsolvable equations. The Ulam Spiral, for example, was discovered accidentally as Stanislaw Ulam wrote down rectangular numbers in a spiral with the number one at the center. He circled all prime numbers and noted that these numbers formed a pattern--a diagonal line. This pattern is still unexplained, but in math, patterns like this are sought in order to solve a variety of problems.
To solve pattern problems, students must first determine whether the problem is predictable and not probable to be sure the pattern will continue. For example, a pattern problem that shows Julie won the first and second game of chess, and John won the third and fourth and then asks who will win the fifth is based on probable patterns. It's likely that Julie will win the fifth, but because there are other important factors such as strategy and skill of the players, it isn't absolute that she will win. Pattern problems that are easiest to solve are based on a pattern of predictable data, such as a sequence of numbers. For example, a problem asks how many blocks are needed to build a six-layer pyramid. As the pyramid grows from one layer and one block to six layers, students see that the pattern requires one block be added to the previous number to build the layer beneath. The pattern identified is predictable, and won't change so it can be used to determine how many blocks go on the sixth layer.
Patterns problems are given in early grades to aid in teaching math concepts. Students might be asked to find a pattern in the numbers 3, 7, and 11, and then to explain the pattern and predict the next sequence of numbers. Finding patterns in mathematical problems aids in learning concepts such as multiplication facts, which in turn will be useful in other types of math. For example, students see that all the numbers multiplied by 10 end in zero. These patterns aid in other mathematical concepts such as addition, subtraction and division.
In early grades, children count by ones and then learn to count by twos based on a pattern. Once this is mastered in later grades, students can determine any number based on the basic function of that mathematical concept. For example, if given the numbers 2, 4, 6, and asked the 50th number in the pattern, students will know to multiply 2 by 50 to find the answer. High school students learn that mathematics is about function, which is the pattern in which one number changes to another. Pattern problems teach students how to identify a pattern and predict a result, which is directly connected to the concept of mathematical functions.