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 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

By the end of Grade 4:

 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 … )

4.3 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

By the end of Grade 4:

 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

4.3 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

By the end of Grade 4:

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).

Sample Assessment Items:
• MC: If 􀂈 x 8 = 96, what is the value of 􀂈?
* a. 12

b. 88

c. 104

d. 768

• MC: If 84 ÷ 􀂈 = 7, then what is the value of 􀂈?
a. 91

b. 77

* c. 12

d. 7

• MC: Some very old books do not have the pages numbered. Mrs. Jensen is a librarian and has developed a rule for estimating the number of pages (P), given the weight of the book. Which of the following is most likely the rule Mrs. Jensen uses for a book weighing two pounds?
* a. P = 100 x 2

b. P = 100 - 2

c. P = 100 + 2

d. P = 100 ÷ 2

4.3 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

By the end of Grade 4:

 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.

Sample Assessment Items:
• MC: Which expression gives the same result as 2 x 4 x 25?
a. 2 x 9 x 5

b. 6 x 25

*c. 2 x 100

d. 4 x 27

• ECR: Sue thinks that 3/0 is 3.
• Is she correct?
• Explain why you believe she is correct or incorrect.

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

·        Symbols ( = , < , > )

Sample Assessment Item:
• MC: Allison and Michele each had $5.00 to spend at the bookstore. Allison bought a book that cost $3.50 and Michele bought a book that cost $4.25. Which of the following correctly compares the amount of money each girl has left?
a. $5.00 - $3.50 < $5.00 - $4.25
b. $5.00 - $4.25 > $5.00 - $3.50
* c. $5.00 - $3.50 > $5.00 - $4.25
d. $5.00 - $4.25 = $5.00 - $3.50