Standard 8: Patterns and Functions

In grades 5-8, the mathematics curriculum should include explorations of patterns and functions so that students can:
• Describe, extend, and create a wide variety of patterns
Benchmarks 11C page 273 (Common Themes: Constancy and Change)
Some features of things may stay the same even when other features change. Some patterns look the same when they are shifted over, or turned, or reflected or seen form different directions.

• Describe and represent relationships with tables, graphs, and rules
• Benchmarks 9B (The Mathematical World: Symbolic Relationships)
Grades 3-5, page 218
Tables and graphs can show how values of one quantity are related to values of another.

Benchmarks 11C (Common Themes: Constancy and Change)
Grades 3-5, page 273
Things change in steady, repetitive, or irregular ways or sometimes in more than one way at the same time. Often the best way to tell which kinds of change are happening is to make a table or graph of measurements.

Benchmarks 12D (Habits of Mind: Communication Skills)
Grades 6-8, page 297
Organize information in simple tables and graphs and identify the relationships they reveal.

• Analyze functional relationships to explain how a change in one quantity results in a change in another
• Benchmarks 9B (The Mathematical World: Symbolic Relationships)
Grades 6-8, page 219
Mathematical statements can be used to describe how one quantity changes when another changes. Rates of change can be computed from magnitudes and vice versa.

Benchmarks 11C (Common Themes: Constancy and Change)
Grades 6-8, page 274
Physical and biological systems tend to change until they become stable and then remain that way unless their surroundings change.

Benchmarks 11C (Common Themes: Constancy and Change)
Grades 6-8, page 274
A system may stay the same because nothing is happening or because things are happening but exactly counterbalance one another.

Benchmarks 11C (Common Themes: Constancy and Change)
Grades 6-8, page 274
Things that change in cycles, such as the seasons or body temperature, can be described by their cycle length or frequency, what the highest and lowest values are, and when they occur. Different cycles range from many thousands of years down to less than a billionth of a second.

• Use patterns and functions to represent and solve problems
• Benchmarks 2A (The Nature of Mathematics: Patterns and Relationships)
Grades 3-5, page 27
Mathematics is the study of many kinds of patterns, including numbers and shapes and operations on them. Sometimes patterns are studied because they help to explain how the world works or how to solve practical problems, sometimes because they are interesting in themselves.

Benchmarks 9C (The Mathematical World: Shapes)
Grades 6-8, page 219
Graphs can show a variety of possible relationships between two variables. As one variable increases uniformly, the other may do one of the following: always keeps the same proportion to the first, increase or decrease steadily, increase or decrease faster and faster, get closer and closer to some limiting value, reach some intermediate maximum or minimum, alternately increase and decrease indefinitely, increase or decrease in steps, or do something different from any of these.