Using the idea of grouping presented in the first page of this lesson, we can explore patterns and use our observations to predict and describe. Here are the first three figures in a simple tile pattern:
By looking at grouping patterns, we can predict what the 10th arrangement in the sequence, for example, would look like. One grouping scheme is shown below:
Seeing this grouping helps us to predict that the 10th arrangement will have a row of three on top, and two 'legs' of ten hanging down, for a total of (2 x 10) + 3 = 23 tiles. We have made an observation about the pattern, and translated our thinking into a mathematical statement that can help others see how we are thinking about the pattern.
We can also use the grouping method to answer questions like "Which arrangement contains 69 tiles"?
Knowing how the tiles are grouped in the arrangements allows us to make predictions about the pattern, and answer questions about various other arrangements in the pattern. We will use these skills in later lessons when we explore writing algebraic formulas to communicate our thinking about patterns.
Of course, there are other ways to group the tiles in the arrangements. Here are some other ways of "seeing" the groups. Can you look at the pattern above and see how these students were thinking?
Here's a challenge: Two consecutive arrangements in this pattern contain a total of 156 tiles. Which two arrangements are they? (Hint: Think about how the arrangements change from one figure to the next.)