Uncovering Patterns in Number Sequences

Learning to make generalizations about sets of numbers is a difficult higher-order thinking skill for many children. The following activity fosters this skill while helping kids learn to think logically and look for patterns and functions. When I teach this lesson, I use words like conclusion, characteristic, and generalization as often as possible, so students can hear them in context. Here's how the lesson went when I taught it to a fourth-grade class.

ACTIVITY 1: What do consecutive numbers have in common?

Purpose: To use consecutive numbers to learn to make generalizations.
Materials: pencil and paper
Time Needed: 20 minutes

1. I divided the class into groups of four and wrote the following sets on the board:
• Set 1: 1 2 3 4
• Set 2: 8 9 10 11
• Set 3: 42 43 44 45
• Set 4: 19 20 21 22
• Set 5: 77 78 79 80
I said, "Each of these rows is a set of four consecutive numbers. What do you think I mean by consecutive numbers?" Children described consecutive numbers in several ways.

2. I went on to say, "When I examine these sets, I notice that when I subtract the first number from the last, I always get 3." Then I asked groups to check to see if this was true for all the sequences. While students talked this over, I wrote the generalization on the board: "The difference between the first and last numbers in a sequence of four consecutive numbers is always 3." Students agreed that this statement was correct.

3. I then gave these instructions to groups: "See what else you can say about all sets of four consecutive numbers. Write sentences to describe your generalizations."

4. After students worked for about 15 minutes, I asked each group to choose one person to read one of the group's generalizations. I added: "I'll go around the room. Each group will report just one conclusion on a turn. Listen carefully because I want you to read one that hasn't been reported."

5. After each group read a statement, I asked the other groups to check to see if it matched one they had written and to talk about whether they agreed with it.

EXTENSION ACTIVITIES

Ask students to try these explorations in which they'll practice making generalizations.

1. What can you say about any 2-by-2 array of numbers on a 0-99 chart? What about a 3-by-3 array?

2. What can you say about any three diagonally adjacent numbers on a 0-99 chart?

3. Try problems 1 and 2 using numbers on a calendar instead. Do your generalizations still hold? Why or why not?