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.

Big Idea Algebra provides language through which we communicate the patterns in mathematics.

4.3.8 A. Patterns

Descriptive Statement: 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.

Essential Questions

Enduring Understandings

- How can change be best represented mathematically? (4.5C1; 4.5F1; 4.5F2; 4.5F3; 4.5F4)

- How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations? (4.5C1)

- The symbolic language of algebra is used to communicate and generalize the patterns in mathematics.

- Algebraic representation can be used to generalize patterns and relationships.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

 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

4.3.8 B. Functions and Relationships

Descriptive Statement: The function concept is one of the most fundamental unifying ideas of modern mathematics. Student 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.

Essential Questions

Enduring Understandings

- How are patterns of change related to the behavior of functions? (4.5F1; 4.5F2; 4.5F3; 4.5F4)

- Patterns and relationships can be represented graphically, numerically, symbolically, or verbally. (4.5E1)

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

 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.

4.3.8 C. Modeling

Descriptive Statement: The function concept is one of the most fundamental unifying ideas of modern mathematics. Student 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.

Essential Questions

Enduring Understandings

- How are mathematical models used to describe physical relationships? (4.5E2)

- How are physical models used to clarify mathematical relationships? (4.5E3)

- Mathematical models can be used to describe and quantify physical relationships. (4.5E2)

- Physical models can be used to clarify mathematical relationships. (4.5E3)

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

 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)

4.3.8 D. Procedures

Descriptive Statement: 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.

Essential Questions

Enduring Understandings

- What makes an algebraic algorithm both effective and efficient? (4.5D1)

-  Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole.

- Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra. (4.5D1; 4.5D3; 4.5D4; 4.5D5)

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

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)

·        Simple literal equations (e.g., A = lw)

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

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

Sample SCR Item: The amount A that principal P will be worth after t years at interest rate r, compounded annually, is given by this formula:

A = P(1 + r)t

Suppose $4,000 principal is invested at 6% interest compounded annually for five years. How much money would the investment yield after 5 years? (Answer: $5,532.90)

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

Sample MC Item: If X > 0 and Y < 0, what must be true about the value of the expression – X – Y ?

* a. It is sometimes positive

b. It is always negative

c. It is never negative

d. It is never zero