Print Version Only Chapter 2 Patterns – Informal Math Modeling

“Patterns are everywhere. Children who are encouraged to look for patterns and to express them mathematically begin to understand how mathematics applies to the world in which they Live.”

(National Council of Teachers of Mathematics [NCTM], 1989, p. 260)

See the corresponding standards below from The National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics

In prekindergarten through grade 2 all students should…

• recognize, describe, and extend patterns such as sequences of sounds and shapes or simple numeric patterns and translate from one representation to another
• analyze how both repeating and growing patterns are generated
• describe quantitative change, such as a student’s growing two inches in one year.

In grades 3–5 all students should

• describe, extend, and make generalizations about geometric and numeric patterns
• represent and analyze patterns and functions, using words, tables, and graphs
• investigate how a change in one variable relates to a change in a second variable.

These skills build up to corresponding middle school standards below

In grades 6-8 all students should

• represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;
• relate and compare different forms of representation for a relationship;

In the Massachusetts Mathematics Curriculum Framework…

Grade 3 – 3.OA.D Solve problems involving the four operations, and identify and explain patterns in arithmetic.

9. Identify arithmetic patterns (including patterns in the addition table or multiplication table) and explain them using properties of operation

Grade 4 – 4.OA.C Generate and analyze patterns.

5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.

Grade 7 – 7.EE.B. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
   3. Extend analysis of patterns to include analyzing, extending, and determining an expression for simple arithmetic and geometric sequences (e.g., compounding, increasing area), using tables, graphs, words, and expressions.

Patterns may be observed in real-world situations, geometric figures, numbers, symbols, and relationships among quantities.  Patterns are an excellent way to introduce students to making predictions and conjectures. Recognizing patterns is often considered the foundation for algebraic thinking. In particular, we can use algebra to simplify the pattern and develop an expression for the general case. Using symbols to describe the general rule of a pattern is central to algebraic thinking.

Teaching of patterns begins in the elementary classroom with repeating patterns followed by growing patterns. The study of patterns prepares students to work with functional relationships which are addressed later in this text.