Grouping for Generalizing
In this lesson, we explore some more patterns and functions, only we have the added dimension of involvement with both positive and negative integer values in the patterns. We'll follow the same basic procedure as in the Tile Patterns lesson. The main difference is that now we're looking for the net value of the arrangements rather than the number of tiles the arrangements contain.
Let's explore this pattern with a focus on generating equivalent expressions. What observations can you make about the first four arrangements in the sequence?
Again, one strategy for trying to make sense of the pattern is to look for groups of tiles that correspond to the figure numbers, and that are consistent from one arrangement to the next. A tile grouping that works in the second arrangement, for example, should also work in the third, and every other arrangement as well. Here are some observations that students have made about this pattern:
As before, once we begin to see how the pattern 'works', we can make some predictions about larger figures, and write some equations that represent the net value of any arrangement in the pattern.
For example, what would be the net value of the 20th arrangement in the pattern? Maybe the diagrams below will help...
We can see that by using either the area notation, or the dimensions notation, we can determine the net value of the twentieth arrangement.
We can also take advantage of this situation to generalize the algebraic product of (n - 1) x (n - 1):
Another way to explore the nature of patterns and functions is by graphing. Here is a graph of the above pattern's net values. Notice the labels and other identifying notations on the graph. All graphs students produce should contain this information.
By looking at the graph, we can learn more information about the function. It is clear that the dots representing the (n , Vn) points do not lie in a straight line (they are not co-linear), but seem to rise in increasing increments. This is an opportunity to discuss quadratic functions, and the difference between them and linear functions, whose points are, of course, co-linear.
Students enjoy exploring patterns and generating algebraic expressions. This practice in observation and generalization provides a valuable foundation for exploration of mathematics in general, as well as science. We engage in this type of activity often, beginning in grades 4 and 5 (see Visual Reasoning and Pattern Block Trains), and continuing through 8th grade. Try creating some patterns of your own with your child, and see what kinds of algebraic generalizations you can come up with.