Title: Laying the Foundation for the Concepts of Even and Odd
Author Name(s): Maureen Oberlin
Summary
Before reading this article, think about your understanding of the concepts of even and odd.
How do you explain even and odd numbers to your students? What are your instructional
strategies for teaching these concepts?
Laying the Foundation for the Concept of Odd and Even
In the Common Core State Standards for Mathematics (CCSSM), the concepts of even and odd numbers
are introduced in the second grade. Standard 2.OA.3 states:
Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing
objects or counting by 2s; write an equation to express an even number as a sum of two equal addends.
Many teachers in both elementary and secondary classrooms take a procedural approach when teaching
their students to distinguish even from odd numbers. They explain that an even number is a number
whose ones digit is either 0, 2, 4, 6, or 8 and an odd number is a number whose ones digit is either 1, 3,
5, 7, or 9. Students dutifully memorize these rules and are prompted to apply them when asked to
determine the parity of a number. There are a few problems with this approach. Although it represents
an efficient test of the parity of a number, it is very removed from the underlying concepts of even and
odd. When mathematical ideas are presented in isolation of their conceptual foundations, they are
often “misapplied.” For example, a student may determine that 23 is an even number by mistakenly
considering the tens digit instead of the ones digit.
At higher grades, an even number is defined to be a number of the form 2k where k is an integer (… -3, 2, -1, 0, 1, 2, 3, …). Applying this definition, one can begin to calculate the nonnegative even numbers by
substituting the integers 0, 1, 2, 3, … for k in the expression 2k:
2·0 = 0
2·1 = 2
2·2 = 4
2·3 = 6
2·4 = 8
Etc.
These calculations result in the set of numbers that we call even: 0, 2, 4, 6, 8, … . It is worth noting that
zero is, by definition, considered even.
The odd numbers are similarly defined. A number is odd if it can be written in the form 2k + 1 where k is
an integer. And again, this definition can be used to generate the nonnegative odd numbers:
2·0 + 1 = 0 + 1 = 1
2·1 + 1 = 2 + 1 = 3
2·2 + 1 = 4 + 1 = 5
2·3 + 1 = 6 + 1 = 7
2·4 + 1 = 8 + 1 = 9
Etc.
These calculations result in the set of numbers that you already know of as the odd numbers: 1, 3, 5, 7,
9, …
It would not be mathematically appropriate to develop the concept of even and odd numbers in
elementary school using these more formal definitions. But it is possible to develop these concepts in a
way that addresses standard 2.OA.3 and lays a foundation for later work with the formal definitions. For
example, blocks can be used to model numbers and determine if they are even.
Example: Is 8 even?
1. Count out eight blocks.
2.
Separate the blocks into two towers. This can be done by working with sequences of pairs.
Take two blocks from the pile which can be used to begin the towers.
Take another pair of blocks and add one to each tower.
Take a third pair of blocks and add one to each tower.
Finally, add the last pair of two blocks to the towers.
3. Since the towers contain the same number of blocks (four) conclude that eight is even and write
8 = 4 + 4.
After repeated experiences with this kind of activity, students will begin to develop a concept of even
that mirrors the formal definition. Be sure to emphasize the last step of writing an even number as the
sum of two equal addends. It is the ability to write the number in this way that makes it even.
It is important for students to be given opportunities to test odd numbers as well.
Example: Is 9 even?
When attempting to conduct the test for even numbers with an odd number such as nine, the result will
look something like this:
Emphasize that since there is one block left over, nine is not even and we call this type of number odd
(as opposed to uneven). Writing odd numbers using an equation is beyond standard 2.OA.3, but it may
be developmentally appropriate for some of your students. For these students, nine can be written as:
9 = (4 + 4) + 1
The parentheses are important. They reflect the way in which the number is modeled in the activity.
The towers were constructed in parallel using pairs of blocks. Consequently, the numbers of blocks in
each tower go together in both the activity and the expression. The parentheses emphasize this in the
mathematical representation. In addition, by exclusion, they reinforce the fact that there is one left
over when trying to build towers of the same height.
There are several additional advantages to having students engage in an activity such as this one:
•
It introduces students to the notion of a mathematical test and logical thinking. If some
number, n, passes the test then n is even. If n fails the test, then n is odd.
•
It provides an opportunity to see an equation written with the sum on the left-hand side of the
equal sign. This format reinforces the meaning of the equal sign (“has the same value as”) and
exposes students to an alternative to the more prevalent form of an equation: a + b = c.
•
It introduces and reinforces the use of grouping symbols, i.e. parentheses. The use of the
parentheses is directly tied to the activity so is not as abstract as it might be in an isolated
expression.
•
It provides an opportunity to decompose a number into three (as opposed to two) addends.
•
It provides an opportunity to see the connection between mathematical models and numerical
expressions.
Activities like this encourage students to form a mental model of an even number as being composed of
two equal addends and an odd number as being composed of two equal addends with one left over.
These mental models foreshadow more sophisticated algebraic representations of even (2k) and odd (2k
+ 1) numbers and can form the foundation for later development of these concepts.
Questions for Discussion
Were your concepts of even and odd different from the ones presented in the article? If so, it
what ways?
How have you introduced these definitions in the past? Was your approach process oriented or
conceptually based?
How might a teacher devise other activities that foreshadow the algebraic definitions of even
and odd?