Even a concept as simple as odd and even numbers can be better understood with a model of some kind. Some of the deeper, related number theory ideas can also be represented for better understanding. The sequence of numbers 1, 3, 5, 7, 9, ... can be represented by arrangements in a tile pattern sequence such as the following, in which the first odd number is one, the second odd number is three, the fifth odd number is nine, and so on:
We can use observations about this pattern to answer questions like
If you're having trouble, think about the number of tiles in the bottom row of each arrangement, and how that relates to the total number of tiles it contains!
We can use similar ideas to represent the even numbers (2, 4, 6, 8, 10, ...) with their own sequence of tile pattern arrangements like this:
We can then answer questions similar to the ones for the odd number pattern:
The shape of the individual tile arrangements in each sequence leads to generalized models for odd and even numbers, models that represent any arrangement in their respective pattern sequences:
These models are very useful for thinking about the behavior of odd and even numbers with the basic operations. For some examples, follow the link above to the next page in this lesson, Modeling Operations with Odd and Even Numbers.