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 4, students will:

 

 

 A.     Patterns

 1.         Recognize, describe, extend, and create patterns.

·        Descriptions using words, number sentences/expressions, graphs, tables, variables (e.g., shape, blank, or letter)

·        Sequences that stop or that continue infinitely

·        Whole number patterns that grow or shrink as a result of repeatedly adding, subtracting, multiplying by, or dividing by a fixed number (e.g., 5, 8, 11, . . . or 800, 400, 200, . . .)

·        Sequences can often be extended in more than one way (e.g., the next term after 1, 2, 4, . . . could be 8, or 7, or … )

 

B.     Functions and Relationships

 1.         Use concrete and pictorial models to explore the basic concept of a function.

·        Input/output tables, T-charts

·        Combining two function machines

·        Reversing a function machine

 

C.     Modeling

 1.         Recognize and describe change in quantities.

·        Graphs representing change over time (e.g., temperature, height)

·        How change in one physical quantity can produce a corresponding change in another (e.g., pitch of a sound depends on the rate of vibration)

 2.         Construct and solve simple open sentences involving any one operation (e.g., 3 x 6 = __, n = 15 ¸ 3,  3 x __ = 0,  16 – c = 7).

 

D.    Procedures

 1.         Understand, name, and apply the properties of operations and numbers.

·        Commutative (e.g., 3 x 7 = 7 x 3)

·        Identity element for multiplication is 1 (e.g., 1 x 8 = 8)

·        Associative (e.g., 2 x 4 x 25 can be found by first multiplying either 2 x 4 or 4 x 25)

·        Division by zero is undefined 

·        Any number multiplied by zero is zero.

 2.         Understand and use the concepts of equals, less than, and greater than in simple number sentences.

·        Symbols ( = , < , > )

 

 

Link to Standard 4.3 Grade 3

 

Link to Standard 4.3 Grade 5

 

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