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Number Patterns

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Overview

Number patterns, or sequences, consist of a list of numbers that follow one after another according to some kind of rule.

There are two ways of articulating a number pattern: recursively and closed. In a recursively-defined number pattern, each new number comes from some operation or operations performed on the previous term or terms. For example, if we think of the sequence

<math> 5, 8, 11, 14, 17...</math>

as "adding 3 to the previous number," we are thinking about it recursively. This is perhaps the most natural way of considering number patterns, because here we focus on the question, "What number comes next?"

Instead of this, we could look for a way of discussing not just what characterizes the next number, but also what all of the numbers in the sequence have in common.

Two basic types of number patters are arithmetic and geometric sequences. Arithmetic sequences can be defined recursively by adding or subtracting a fixed amount from the previous term, as in the above example. Geometric sequences can be defined recursively by multiplying or dividing by a fixed number from term to successive term.

Motivating questions and problems

A game that kids really enjoy and that opens up a whole field of topics is a very simple one I call "Guess the number pattern."

You can download a worksheet here.

The concept is an easy one: before I begin to write each number of a number pattern on the board, I ask my students to write down their guess or prediction as to what the next number will be. For the first number they're just taking a shot in the dark, although they quickly catch onto the fact that I usually start with 1. They predict, I write, they cheer or are puzzled, then they guess again, and I write, and so on until there are seven or so numbers on the board. Then I ask who will give their prediction for the eighth, I write the number, and then I ask that student what pattern they saw in the sequence. Sometimes multiple kids share. Then onto a new number pattern. I incorporate a delta column for them to keep track of differences, as I explain once someone uses that idea in explaining their pattern. Soon enough students will often start constructing double and triple delta columns in order to more capably find patterns.


Lesson plans

Under this heading should be posted more specific ways to approach this topic, including:

  • ideas for particular kinds of tasks that could be required, like "find 234 in base-5."
  • links to more elaborate, already-made or customizable worksheets, with solutions
  • grade level these have been used for (remembering that since whether a less is appropriate for a class depends students' background and ability, recommendations should be kept as broad as possible and include any grades that this concept has been taught).
  • time needed to complete lesson
  • sequences of lessons that have been successful
  • examples of student output
  • places and/or ideas that are stumbling blocks

These can be filed according to the following categories.

Exercises

Posted here should be links to lessons and sheets intended to practice and reinforce concepts, ideas, and vocabulary

Problems

Posted here should be links to lessons and sheets intended to explore the topic in a more abstract and deep way

Extensions

Posted here should be ideas for projects or subtopics

Assessments

Posted here should be quizzes, tests, or other assessment tools

Follow up topics, or Where to next?

Topics that could be explored during or immediately after this topic

External links

On-Line Encyclopedia of Integer Sequences

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This page was last modified on 17 March 2007, at 04:46.
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