STANDARD 4.3: Mathematics

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

By the end of Grade 2, students will:

A. Patterns

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

· Using concrete materials (manipulatives), pictures, rhythms, & whole numbers

· Descriptions using words and symbols (e.g., “add two” or “+ 2”)

· Repeating patterns

· Whole number patterns that grow or shrink as a result of repeatedly adding or subtracting a fixed number (e.g., skip counting forward or backward)

B. Functions and Relationships

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

C. Modeling

1. Recognize and describe changes over time (e.g., temperature, height).

2. Construct and solve simple open sentences involving addition or subtraction.

· Result unknown (e.g., 6 – 2 = __ or n = 3 + 5)

· Part unknown (e.g., 3 + ÿ = 8)

D. Procedures

1. Understand and apply (but don’t name) the following properties of addition:

· Commutative (e.g., 5 + 3 = 3 + 5)

· Zero as the identity element (e.g., 7 + 0 = 7)

· Associative (e.g., 7 + 3 + 2 can be found by first adding either 7 + 3 or 3 + 2)

Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:

A. Patterns

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

· Descriptions using words and number sentences/expressions

· 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, . . .)

B. Functions and Relationships

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

· Input/output tables, T-charts

C. Modeling

1. Recognize and describe change in quantities.

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

2. Construct and solve simple open sentences involving addition or subtraction (e.g., 3 + 6 = __, n = 15 – 3, 3 + __ = 3, 16 – c = 7).

D. Procedures

1. Understand 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)

· Any number multiplied by zero is zero

2. Understand and use the concepts of equals, less than, and greater than to describe relations between numbers.

· Symbols ( = , < , > )

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 ( = , < , > )

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

A. Patterns

1. Recognize, describe, extend, and create patterns involving whole numbers.

· Descriptions using tables, verbal rules, simple equations, and graphs

B. Functions & Relationships

1. Describe arithmetic operations as functions, including combining operations and reversing them.

2. Graph points satisfying a function from T-charts, from verbal rules, and from simple equations.

C. Modeling

1. Use number sentences to model situations.

· Using variables to represent unknown quantities

· Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations

2. Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events.

· Changes over time

· Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)

D. Procedures

1. Solve simple linear equations with manipulatives and informally

· Whole-number coefficients only, answers also whole numbers

· Variables on one side of equation

Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:

A. Patterns

1. Recognize, describe, extend, and create patterns involving whole numbers and rational numbers.

· Descriptions using tables, verbal rules, simple equations, and graphs

· Formal iterative formulas (e.g., NEXT = NOW * 3)

· Recursive patterns, including Pascal’s Triangle (where each entry is the sum of the entries above it) and the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, . . . (where NEXT = NOW + PREVIOUS)

B. Functions and Relationships

1. Describe the general behavior of functions given by formulas or verbal rules (e.g., graph to determine whether increasing or decreasing, linear or not).

C. Modeling

1. Use patterns, relations, and linear functions to model situations.

· Using variables to represent unknown quantities

· Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations/inequalities

2. Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events.

· Changes over time

· Relations between quantities

· Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)

D. Procedures

1. Solve simple linear equations with manipulatives and informally.

· Whole-number coefficients only, answers also whole numbers

· Variables on one or both sides of equation

2. Understand and apply the properties of operations and numbers.

· Distributive property

· The product of a number and its reciprocal is 1

3. Evaluate numerical expressions.

4. Extend understanding and use of inequality.

· Symbols ( ³ , ¹ , £ )

Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:

A. Patterns

1. Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.

· Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions

· Finite and infinite sequences

· Generating sequences by using calculators to repeatedly apply a formula

B. Functions and Relationships

1. Graph functions, and understand and describe their general behavior.

· Equations involving two variables

C. Modeling

1. Analyze functional relationships to explain how a change in one quantity can result in a change in another, using pictures, graphs, charts, and equations.

2. Use patterns, relations, symbolic algebra, and linear functions to model situations.

· Using manipulatives, tables, graphs, verbal rules, algebraic expressions/equations/inequalities

· Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social studies standard 6.6)

D. Procedures

1. Use graphing techniques on a number line.

· Absolute value

· Arithmetic operations represented by vectors (arrows) (e.g., “-3 + 6” is “left 3, right 6”)

2. Solve simple linear equations informally and graphically.

· Multi-step, integer coefficients only (although answers may not be integers)

· Using paper-and-pencil, calculators, graphing calculators, spreadsheets, and other technology

3. Create, evaluate, and simplify algebraic expressions involving variables.

· Order of operations, including appropriate use of parentheses

· Substitution of a number for a variable

4. Understand and apply the properties of operations, numbers, equations, and inequalities.

· Additive inverse

· Multiplicative inverse

Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:

A. Patterns

1. Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.

· Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions

· Finite and infinite sequences

· Arithmetic sequences (i.e., sequences generated by repeated addition of a fixed number, positive or negative)

· Geometric sequences (i.e., sequences generated by repeated multiplication by a fixed positive ratio, greater than 1 or less than 1)

· Generating sequences by using calculators to repeatedly apply a formula

B. Functions and Relationships

1. Graph functions, and understand and describe their general behavior.

· Equations involving two variables

· Rates of change (informal notion of slope)

2. Recognize and describe the difference between linear and exponential growth, using tables, graphs, and equations.

C. Modeling

1. Analyze functional relationships to explain how a change in one quantity can result in a change in another, using pictures, graphs, charts, and equations.

2. Use patterns, relations, symbolic algebra, and linear functions to model situations.

· Using concrete materials (manipulatives), tables, graphs, verbal rules, algebraic expressions/equations/inequalities

· Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social studies standard 6.6)

D. Procedures

1. Use graphing techniques on a number line.

· Absolute value

· Arithmetic operations represented by vectors (arrows) (e.g., “-3 + 6” is “left 3, right 6”)

2. Solve simple linear equations informally, graphically, and using formal algebraic methods.

· Multi-step, integer coefficients only (although answers may not be integers)

· Using paper-and-pencil, calculators, graphing calculators, spreadsheets, and other technology

3. Solve simple linear inequalities.

4. Create, evaluate, and simplify algebraic expressions involving variables.

· Order of operations, including appropriate use of parentheses

· Distributive property

· Substitution of a number for a variable

· Translation of a verbal phrase or sentence into an algebraic expression, equation, or inequality, and vice versa

5. Understand and apply the properties of operations, numbers, equations, and inequalities.

· Additive inverse

· Multiplicative inverse

· Addition and multiplication properties of equality

· Addition and multiplication properties of inequalities

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 n^th 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.

2. Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies.

· Slope of a line or curve

· Domain and range

· Intercepts

· Continuity

· Maximum/minimum

· Estimating roots of equations

· Intersecting points as solutions of systems of equations

· Rates of change

3. Understand and perform transformations on commonly-used functions.

· Translations, reflections, dilations

· Effects on linear and quadratic graphs of parameter changes in equations