Standards:
6.P.1: Analyze and determine the rules for extending symbolic, arithmetic, and geometric patterns and progressions, e.g., ABBCCC; 1, 5, 9, 13 ...; 3, 9, 27, ....
6.P.2: Replace variables with given values and evaluate/simplify, e.g., 2(x) + 3 when x = 4.
6.P.3: Use the properties of equality to solve problems, e.g., if __ + 7 = 13, then = 13 - 7, therefore __ = 6; if 3 x __ = 15, then 1/3 x 3 x __ = 1/3 x 15, therefore __ = 5.
6.P.4: Represent real situations and mathematical relationships with concrete models, tables, graphs, and rules in words and with symbols, e.g., input-output tables.
6.P.5: Solve linear equations using concrete models, tables, graphs, and paper-pencil methods.
6.P.6: Produce and interpret graphs that represent the relationship between two variables in everyday situations.
6.P.7: Identify and describe relationships between two variables with a constant rate of change. Contrast these with relationships where the rate of change is not constant.
Patterns, Relations, and Algebra
Algebra emerged through the analysis of solutions to equations, while the concept of a function developed as the insights and techniques of calculus began to spread. Patterns, relations, and algebra are integral elements in the study of mathematics. All students should understand how patterns, relations, and functions are interrelated; be able to represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to understand quantitative relationships; and analyze change in various contexts.
The foundation for the study of patterns, relations, and algebra can be constructed in the PreK-K years and expanded gradually throughout the other years. All students should be aware of the mathematics in patterns and use mathematical representations to describe patterns. Young students can identify, translate, and extend repeating rhythmic, verbal, and visual patterns. They can recognize patterns and relationships among objects, and sort and classify them, observing similarities and differences. They can then probe more deeply into the study of patterns as they explore the properties of the operations of addition and multiplication.
Through numerous explorations, elementary grade students deepen their understanding of pattern and work informally with the concept of function. It is important that the concept of a variable is developed for them through practical situations, for example, as they engage in such basic activities as listing the cost of one pencil at 50¢, two pencils at ?, three pencils at ? , ... n pencils at ?.
Investigating patterns helps older students understand the concept of constant growth as they analyze sequences like 1, 3, 5, 7, .... These students should contrast this type of change with other relationships as evidenced in sequences such as 1, 2, 4, 8, ... ; 1, 3, 6, 10, 15, ... (the sum of the first n positive integers); and 1, 1/10, 1/100, 1/1000, .... In middle and high school, students build on prior experiences as they compare sequences and functions represented in recursive and explicit forms.
As students advance through the grades, their work with patterns, functions, and algebra progresses in mathematical sophistication. They learn that change is a central idea in the study of mathematics and that multiple representations are needed to express change. They identify, represent, and analyze numerical relationships in tables, charts, and graphs. They learn about the importance and strength of proportional reasoning as a means of solving a variety of problems. While understanding linear functions and their graphs is a realistic goal for the middle school student, students deepen their study of functions in the secondary years. They engage in problems that feature additional types of polynomial functions, and rational, exponential, logarithmic, trigonometric, and other families of functions.
Graphing calculators and computer software with spreadsheet and graphics capabilities are ideal resources that help students make connections among different representations of the functions. The meaning and importance of domain, range, roots, optimum values, periodicity, and other terms come alive when experienced through technology. With appropriate instruction, students move readily among symbolic, numeric, and graphic representations of functions. Through insightful examples, secondary students learn that functions are a key concept with connections not only to calculus but also to transformational geometry and topics in discrete mathematics.
