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.

4.3.6 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 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)

Sample MC Item: Which equation fits this pattern? 2, 6, 18, 24, . . a. NEXT = NOW + 4

b. NEXT = NOW + 3 *

c. NEXT = 3 * NOW

d. NEXT = NOW / 3

4.3.6 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

Sample Classroom Performance Task:
Students are split into groups. Each group receives a large bottle of water and 10 smaller water bottles. Each group also receives a stack of paper or plastic cups—one group receives 2 oz. cups; one group receives 3 oz. cups; one group receives 4 oz. cups; etc.

The students in each group first fill up the cups from the large bottle only. Students record the number of cups filled. The students then fill up additional cups from one small water bottle, recording the total number of cups filled with water. In each group, the process is repeated until the 10 small water bottles have been emptied.

Each group will then prepare a graph of the number of cups filled vs. the number of bottles emptied. Each group then shares the results with the rest of the class. The class compares and contrasts the graphs.
Individual students then attempt, for each size cup, to describe the relationship between the number of bottles and the number of cups that could be filled.

4.3.6 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

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

Sample MC Item: Dorothy has $3.00 on September 1. Each week she earns $5.00. Which number sentence shows how much money she will have in 10 weeks?

a. D = 3 + 5(10)

b. D = 5(10)

* c. D = 3(5) + 10

d. D = 3(10) + 5

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)

4.3.6 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.       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

Sample MC Item: Kathy tells a friend that she can multiply 2-digit numbers in her head using the distributive property. She gives the following example to her friend. 18 x 12 = (18 x 10) + (18 x 2) = 180 + 36 = 216 Using Kathy's method, how could you multiply 32 x 15? a. (30 x 10) + (20 x 5)

* b. (32 x 10) + (32 x 5)

c. (30 x 10) + (2 x 5)

d. (30 x 5) + (2 x 10)

4.       Extend understanding and use of inequality.

·        Symbols  ( ³ , ¹ , £ )