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# Functional Thinking

## Repeating Patterns

To help reveal your students’ thinking about repeating patterns, show them an AAB or ABB pattern such as ??•??•. Rather than ask: “What comes next?” Ask: “Finish the pattern the way I would” and later “Please make the same kind of pattern using different shapes.”

Young children need help thinking about the pattern unit in repeating pattern – the part that repeats over and over. They often just think about repeating the first item or the last item, without looking for the part that repeats over and over. They also are much more familiar with simple AB patterns and need to be pushed to think about more complex patterns.

In preK and Kindergarten, children should be encouraged to extend repeating patterns through at least one pattern unit, to identify the pattern unit, and to create the same kind of pattern, using different materials. This is a NCTM Curriculum Focal Point for Kindergarten. This early pattern work can support greater success with growing patterns in 1st grade.

Go to the Apply page for ideas on how to help your students reflect on repeating patterns.

## Growing Patterns and Functions

To help reveal your students’ thinking about growing patterns, show them a geometric pattern such as the one below. Rather than ask: “Draw the tree that comes next.” Ask: “How many toothpicks would you need to make the 10th tree? The15th tree? “ Later, ask: “In words, describe how you can figure out the number of toothpicks for any tree position.”

Growing patterns are often presented as a sequence of numbers (e.g., “4, 7, 10, 13”; the list of the number of toothpicks in the picture above) and students simply think about how much you need to add to (or subtract from) one number to get to the next number. This approach does not generalize, as it is too cumbersome to use to figure out far off values (e.g., the 15th number) and cannot be used to state a general rule. Students need help thinking about the relationship between two variables – the position number (1st, 2nd, 3rd) and the “outcome” (e.g., the number of toothpicks). Thinking about the relationship between two variables is functional thinking and is directly needed in Algebra.

In 1st grade and above, students should work with growing patterns, and both variables should be explicit so that children can reason about how the two vary together and can predict instances far away (e.g., number of dots to make the 10th triangle in a set of triangles growing in size). By 2nd grade, children can learn to describe a verbal rule for finding any element in a pattern, and this is a NCTM Curriculum Focal Point for 3rd grade.

Go to the Apply page for ideas on how to help your students reflect on growing patterns and functions.