![]() After about 30 seconds, I ask the students to record this initial thought process so that it is preserved for a discussion that will take place later in the lesson. They are asked to keep this result to themselves so that everyone can determine the next term without being influenced by the thoughts of others. My purpose is not to put them under time pressure but rather to have them find a solution in the manner that occurs most naturally to them. I begin the lesson by asking the students to determine individually as quickly as they can how many marbles would be in the fourth term of the sequence shown in Figure 1. ![]() (g) the whole class reflects on their participation in this constructivist lesson. (f) the whole class discusses this approach versus a more traditional lesson to introduce sequences ![]() (e) the whole class discusses the relative advantages and disadvantages of general versus recursive solutions (d) formulas for the nth term solution are considered leading to the construction of the traditional formula for the nth term of an arithmetic sequence (c) the whole class shares and discusses the means by which the initial individual solutions to the 4th term were found (b) in groups, students develop a variety of formulas for the nth term (a) individually, students determine the number of marbles in the 4th term of the sequence (see Figure 1, source unknown) The lesson is comprised of the following components: It is, also, beneficial to this group as it affords them an opportunity to participate in a cooperative learning activity that lends itself to a wide variety of solution methods. I have taught this lesson in a methods course for secondary mathematics teachers to provide them with a model of a constructivist lesson that they will be able to implement in their own classrooms when they begin teaching. This article will discuss a lesson that introduces arithmetic sequences through a simple, yet rich exploration of a pattern. Further, they should be allowed to explore situations in which pattern recognition plays a vital role in the construction of important mathematical knowledge. Students at all levels should be provided with opportunities to investigate and uncover patterns throughout their mathematical careers. Notation will include Σ and a n.Pattern recognition is a critical component of success in mathematics. AII.16 The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems, including writing the first n terms, finding the nth term, and evaluating summation formulas.AII.03b The student will write radical expressions as expressions containing rational exponents and vice versa.AII.03a The student will add, subtract, multiply, divide, and simplify radical expressions containing positive rational numbers and variables and expressions containing rational exponents.AII.02 The student will add, subtract, multiply, divide, and simplify rational expressions, including complex fractions.AII.01 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers and its subsets, complex numbers, and matrices.7.20 The student will write verbal expressions as algebraic expressions and sentences as equations.7.19 The student will represent, analyze, and generalize a variety of patterns, including arithmetic sequences and geometric sequences, with tables, graphs, rules, and words in order to investigate and describe functional relationships.
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