# 3.16 - Patterns

The student will

• identify, describe, create, and extend patterns found in objects, pictures, numbers, and tables.

### BIG IDEAS

• So that I can understand that patterns exhibit relationships that can be extended, described, and generalized
• So that I can look for trends in order to use what I know to help me solve something that is unknown

### UNDERSTANDING THE STANDARD

• Developing fluency and flexibility in identifying, describing, and extending patterns is fundamental to mathematics, particularly algebraic reasoning.
• The use of materials to extend patterns permits experimentation or problem solving approaches that are almost impossible without them.
• The simplest types of patterns are repeating patterns. In each case, students need to identify the core of the pattern and repeat it.
• Growing patterns are more difficult for students to understand than repeating patterns because not only must they identify the core, they must also look for a generalization or relationship that will tell them how the pattern is changing from step to step.  In many growing patterns the change can be described as an increase or decrease by a constant value.  Students need experiences with growing patterns using objects, pictures, numbers, and tables.
• In numeric patterns, students must determine the difference, called the common difference, between each succeeding number in order to determine what is added to each previous number to obtain the next number.  Students do not need to use the term common difference at this level.
• Sample numeric patterns include:
• 6, 9, 12, 15, 18,... (growing pattern)
• 1, 2, 4, 7, 11, 16,... (growing pattern);
• 20, 18, 16, 14,... (growing pattern); and
• 1, 3, 5, 1, 3, 5, 1, 3, 5,... (repeating pattern).
• Numeric patterns, at this level, will be limited to addition and subtraction of whole numbers.
• In geometric figure patterns, students must often recognize transformations or changes of position in the plane of a figure, particularly rotation or reflection. Rotation is the result of turning a figure around a point or a vertex, and reflection is the result of flipping a figure over a line. Students at this level do not need to know the terms related to transformations of figures.
• Sample geometric patterns include

O Δ O O Δ Δ O O O Δ Δ Δ...(growing pattern)

.... (repeating pattern)
• Sample pattern transfers include:
• 2, 5, 8, 11, 14, 17 has the same structure as 4, 7, 10, 13, 16, 19
• Input/output tables with a given rule provide direction for what to do to the input to get the output and can then be used to determine an unknown value.  Applying rules to the input to find the output builds the foundation for functional thinking.  Sample input/output tables that require applying the rule to the input to get the output:

 Rule: Add 4 Input Output 4 8 5 ? 8 ?

 Rule: Subtract 2 Input Output 6 ? 8 6 9 ?

### ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

• Identify and describe repeating and growing patterns using words, objects, pictures, numbers, and tables.
• Identify a missing term in a pattern (e.g., 4, 6, ___ , 10, 12, 14).
• Create repeating and growing patterns using objects, pictures, numbers, and tables.
• Extend or identify missing parts in repeating and growing patterns using objects, pictures, numbers, and tables.
• Solve problems that involve the application of input and output rules limited to addition and subtraction of whole numbers.
• When given the rule, determine the missing values in a list or table.  (Rules will be limited to addition and subtraction of whole numbers.)

### KEY VOCABULARY

Updated: Aug 22, 2018