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STANDARD 4.3 (PATTERNS AND ALGEBRA) ALL STUDENTS WILL REPRESENT AND ANALYZE RELATIONSHIPS AMONG VARIABLE QUANTITIES AND SOLVE PROBLEMS INVOLVING PATTERNS, FUNCTIONS, AND ALGEBRAIC CONCEPTS AND PROCESSES.
Descriptive Statement: Algebra is a symbolic language used to express mathematical relationships. Students need to understand how quantities are related to one another, and how algebra can be used to concisely express and analyze those relationships. Modern technology provides tools for supplementing the traditional focus on algebraic procedures, such as solving equations, with a more visual perspective, with graphs of equations displayed on a screen. Students can then focus on understanding the relationship between the equation and the graph, and on what the graph represents in a real-life situation.
Patterns. Algebra provides the language through which we communicate the patterns in 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.
Functions and Relationships. The function concept is one of the most fundamental unifying ideas of modern mathematics. Students begin their study of functions in the primary grades, as they observe and study patterns. As students grow and their ability to abstract matures, students form rules, display information in a table or chart, and write equations which express the relationships they have observed. In high school, they use the more formal language of algebra to describe these relationships.
Modeling. Algebra is used to model real situations and answer questions about them. This use of algebra requires the ability to represent data in tables, pictures, graphs, equations or inequalities, and rules. Modeling ranges from writing simple number sentences to help solve story problems in the primary grades to using functions to describe the relationship between two variables, such as the height of a pitched ball over time. Modeling also includes some of the conceptual building blocks of calculus, such as how quantities change over time and what happens in the long run (limits).
Procedures. Techniques for manipulating algebraic expressions – procedures – remain important, especially for students who may continue their study of mathematics in a calculus program. Utilization of algebraic procedures includes understanding and applying properties of numbers and operations, using symbols and variables appropriately, working with expressions, equations, and inequalities, and solving equations and inequalities.
Algebra is a gatekeeper for the future study of mathematics, science, the social sciences, business, and a host of other areas. In the past, algebra has served as a filter, screening people out of these opportunities. For New Jersey to be part of the global society, it is important that algebra play a major role in a mathematics program that opens the gates for all students.
Cumulative Progress Indicators
Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:
A. Patterns 1. Use models and algebraic formulas to represent and analyze sequences and series. · Explicit formulas for nth terms · Sums of finite arithmetic series · Sums of finite and infinite geometric series 2. Develop an informal notion of limit. 3. Use inductive reasoning to form generalizations.
B. Functions and Relationships 1. Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs. · Estimating roots of equations · Intersecting points as solutions of systems of equations 3. Understand and perform transformations on commonly-used functions. · Translations, reflections, dilations · Effects on linear and quadratic graphs of parameter changes in equations · Using graphing calculators or computers for more complex functions · Symmetry · Increasing/decreasing on an interval
C. Modeling 1. Use functions to model real-world phenomena and solve problems that involve varying quantities. · Direct and inverse variation · Expressions, equations and inequalities · Same function can model variety of phenomena · Growth/decay and change in the natural world · Applications in mathematics, biology, and economics (including compound interest) 2. Analyze and describe how a change in an independent variable leads to change in a dependent one. 3. Convert recursive formulas to linear or exponential functions (e.g., Tower of Hanoi and doubling).
D. Procedures 1. Evaluate and simplify expressions. · Add and subtract polynomials · Multiply a polynomial by a monomial or binomial · Divide a polynomial by a monomial 2. Select and use appropriate methods to solve equations and inequalities. · Linear equations – algebraically · Quadratic equations – factoring (when the coefficient of x2 is 1) and using the quadratic formula · All types of equations using graphing, computer, and graphing calculator techniques 3. Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.
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