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
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
- I divided the
class into groups of four and wrote the following sets on the board:
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.
- Set 1: 1 2
- Set 2: 8 9
- Set 3: 42
43 44 45
- Set 4: 19
20 21 22
- Set 5: 77
78 79 80
- 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.
- 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."
- 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."
- 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.
Ask students to try
these explorations in which they'll practice making generalizations.
- What can you say
about any 2-by-2 array of numbers on a 0-99 chart? What about a 3-by-3
- What can you say
about any three diagonally adjacent numbers on a 0-99 chart?
- Try problems 1
and 2 using numbers on a calendar instead. Do your generalizations still
hold? Why or why not?
A Collection of Math
Lessons from Grades 3 Through 6 by Marilyn Burns, distributed by Cuisenaire.
For more ideas on consecutive sums, try the extended activity "The Consecutive
Sums Problem" that appears in this book.
Burns is the creator of Math Solutions, inservice workshops offered
nationwide, and the author of numerous books and articles.
This activity was
adapted from 50 Problem-Solving Lessons, Grades 1-6 by Marilyn
Burns, distributed by Cuisenaire.